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--- abstract: 'A new categorical framework is provided for dealing with multiple arguments in a programming language with effects, for example in a language with imperative features. Like related frameworks (Monads, Arrows, Freyd categories), we distinguish two kinds of functions. In addition, we also distinguish two kinds of equations. Then, we are able to define a kind of product, that generalizes the usual categorical product. This yields a powerful tool for deriving many results about languages with effects.' author: - \| Jean-Guillaume Dumas\ LJK, University of Grenoble, France. [[`J`ean-Guillaume.Dumas@imag.fr]{}]{}\ Dominique Duval\ LJK, University of Grenoble, France. [[`D`ominique.Duval@imag.fr]{}]{}\ Jean-Claude Reynaud\ Malhiver, 38640 Claix, France. [[`J`ean-Claude.Reynaud@imag.fr]{}]{} bibliography: - 'prod.bib' date: 'July 4., 2007' title: Sequential products in effect categories --- \[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Assumption]{} \[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Example]{} \[thm\][Examples]{} \[thm\][Definition]{} \[thm\][Convention]{} \[thm\][Conjecture]{} \[thm\][Problem]{} \[thm\][Open Problem]{} \[thm\][Algorithm]{} \[thm\][Observation]{} \[thm\][Question]{} Introduction ============ The aim of this paper is to provide a new categorical framework dealing with multiple arguments in a programming language with effects, for example in a language with imperative features. In our cartesian effect categories, as in other related frameworks (Monads, Arrows, Freyd categories), two kinds of functions are distinguished. The new feature here is that two kinds of equations are also distinguished. Then, we define a kind of product, that is mapped to the usual categorical product when the distinctions (between functions and between equations) are forgotten. In addition, we prove that cartesian effect categories determine Arrows. A well-established framework for dealing with computational effects is the notion of strong monads, that is used in Haskell [@Moggi91; @Wadler93]. Monads have been generalized on the categorical side to Freyd categories [@PowerRobinson97] and on the functional programming side to Arrows [@Hughes00]. The claims that Arrows generalize Monads and that Arrows are Freyd categories are made precise in [@HeunenJacobs06]. In all these frameworks, effect-free functions are distinguished among all functions, generalizing the distinction of values among all computations in [@Moggi91]. In this paper, as in [@BentonHyland03; @HeunenJacobs06], effect-free functions are called pure functions; however, the symbols ${\mathbf{C}}$ and ${\mathbf{V}}$, that are used for the category of all functions and for the subcategory of pure functions, respectively, are reminiscent of Moggi’s terminology. In all these frameworks, one major issue is about the order of evaluation of the arguments of multivariate operations. When there is no effect, the order does not matter, and the notion of product in a cartesian category provides a relevant framework. So, the category ${\mathbf{V}}$ is cartesian, and products of pure funtions are defined by the usual characteristic property of products. But, when effects do occur, the order of evaluation of the arguments becomes fundamental, which cannot be dealt with the categorical product. So, the category ${\mathbf{C}}$ is not cartesian, and products of functions do not make sense, in general. However, some kind of sequential product of computations should make sense, in order to evaluate the arguments in a given order. This is usually defined, by composition, from some kinds of products of a computation with an identity. This is performed by the strength of the monad [@Moggi91], by the symmetric premonoidal category of the Freyd category [@PowerRobinson97], and by the first operator of Arrows [@Hughes00]. In this paper, the framework of cartesian effect categories is introduced. We still distinguish two kinds of functions: pure functions among arbitrary functions, that form two categories ${\mathbf{V}}$ and ${\mathbf{C}}$, with ${\mathbf{V}}$ a subcategory of ${\mathbf{C}}$, and ${\mathbf{V}}$ cartesian. Let us say that the functions are decorated, either as pure or as arbitrary. The new feature that is introduced in this paper is that we also distinguish two kinds of equations: strong equations and semi-equations, respectively denoted ${\equiv}$ and ${\lesssim}$, so that equations also are decorated. Strong equations can be seen, essentially, as equalities between computations, while semi-equations are much weaker, and can be seen as a kind of approximation relation. Moreover, as suggested by the symbols ${\equiv}$ and ${\lesssim}$, the strong equations form an equivalence relation, while the semi-equations form a preorder relation. Then, we define the semi-product of two functions when at least one is pure, by a characteristic property that is a decorated version of the characteristic property of the usual product. Since all identities are values, we get the semi-product of any function with an identity, that is used for building sequential products of functions. Cartesian effect categories give rise to Arrows, in the sense of [@Hughes00], and they provide a deduction system: it is possible to decorate many proofs on cartesian categories in order to get proofs on cartesian effect categories. As for terminology, our graphs are directed multi-graphs, made of points (or vertices, or objects) and functions (or edges, arrows, morphisms). We use weak categories rather than categories, i.e., we use a congruence $\equiv$ rather than the equality, however this “syntactic” choice is not fundamental here. As for notations, we often omit the subscripts in the diagrams and in the proofs. Cartesian weak categories are reminded in section \[sec:weak\], then cartesian effect categories are defined in section \[sec:effect\]; they are compared with Arrows in section \[sec:related\], and examples are presented in section \[sec:exam\]. In appendix \[app:proof\] are given the proofs of some properties of cartesian weak categories, that are well-known, followed by their decorated versions, that yield proofs of properties of cartesian effect categories. Cartesian weak categories {#sec:weak} ========================= Weak categories are reminded in this section, with their notion of product. Except for the minor fact that equality is weakened as a congruence, all this section is very well known. Some detailed proofs are given in appendix \[app:proof\], with their decorated versions. Weak categories {#subsec:weak-cat} --------------- A weak category is like a category, except that the equations (for unitarity and associativity) hold only “up to congruence”. \[defi:weak-cat\] A weak category is a graph where: - for each point $X$ there is a loop ${\mathrm{id}}_X:X\to X$ called the identity of $X$, - for each consecutive functions $f:X\to Y$, $g:Y\to Z$, there is a function $g\circ f:X\to Z$ called the composition of $f$ and $g$, - and there is a relation $\equiv$ between parallel functions (each $f_1\equiv f_2$ is called an equation), such that: - $\equiv$ is a congruence, i.e., it is an equivalence relation and for each $f:X\to Y$, $g_1,g_2:Y\to Z$, $h:Z\to W$, if $g_1\equiv g_2$ then $g_1\circ f\equiv g_2\circ f$ (substitution) and $h\circ g_1 \equiv h\circ g_2$ (replacement), - for each $f:X\to Y$, the unitarity equations hold: $f\circ{\mathrm{id}}_X\equiv f$ and ${\mathrm{id}}_Y\circ f\equiv f$, - and for each $f:X\to Y$, $g:Y\to Z$, $h:Z\to W$, the associativity equation holds: $h\circ (g \circ f) \equiv (h\circ g) \circ f$. So, a weak category is a special kind of a bicategory, and a category is a weak category where the congruence is the equality. Products {#subsec:weak-prod} -------- In a weak category, a weak product, or simply a product, is defined as a product “up to congruence”. We focus on nullary products (i.e., terminal points) and binary products; it is well-know that products of any arity can be recovered from those. \[defi:weak-term\] A (weak) terminal point is a point $U$ (for “Unit”) such that for every point $X$ there is a function ${{\langle \, \rangle}}_X:X\to U$, unique up to congruence. \[defi:weak-prod\] A binary cone is made of two functions with the same source ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$. A binary (weak) product is a binary cone ${Y_1{\stackrel{q_1}{\longleftarrow}}Y_1\times Y_2{\stackrel{q_2}{\longrightarrow}}Y_2}$ such that for every binary cone with the same base ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$ there is a function ${\langle f_1,f_2 \rangle}:X\to Y_1\times Y_2$, called the pair of $f_1$ and $f_2$, unique up to congruence, such that: $$q_1\circ {\langle f_1,f_2 \rangle}\equiv f_1 \;\mbox{ and }\; q_2\circ {\langle f_1,f_2 \rangle}\equiv f_2 \;.$$ As usual, all terminal points are isomorphic, and the fact of using $U$ for denoting a terminal point corresponds to the choice of one terminal point. Similarly, all products on a given base are isomorphic (in a suitable sense), and the notations correspond to the choice of one product for each base. \[defi:weak-ccat\] A cartesian weak category is a weak category with a chosen terminal point and chosen binary products. Products of functions {#subsec:weak-prodfn} --------------------- \[defi:weak-arr-prod\] In a cartesian weak category, the (weak) binary product of two functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ is the function: $$f_1\times f_2 = {\langle f_1\circ p_1,f_2\circ p_2 \rangle}: X_1\times X_2 \to Y_1\times Y_2 \;.$$ So, the binary product of functions is characterized, up to congruence, by the equations: $$q_1 \circ (f_1\times f_2) \equiv f_1 \circ p_1 \;\mbox{ and }\; q_2 \circ (f_1\times f_2) \equiv f_2 \circ p_2 \;.$$ The defining equations of a pair and a product can be illustrated as follows: $$\xymatrix@C=5pc{ & Y_1 \\ X \ar[ru]^{f_1} \ar[rd]_{f_2} \ar[r]^{{\langle f_1,f_2 \rangle}} & Y_1 \times Y_2 \ar[u]_{q_1} \ar[d]^{q_2} \ar@{}[ld]\|(.3){\equiv} \ar@{}[lu]\|(.3){\equiv} \\ & Y_2 \\ } \qquad\qquad \xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \\ X_1\times X_2 \ar[u]^{p_1} \ar[d]_{p_2} \ar[r]^{f_1\times f_2} \ar@{}[rd]\|{\equiv} \ar@{}[ru]\|{\equiv } & Y_1 \times Y_2 \ar[u]_{q_1} \ar[d]^{q_2} \\ X_2 \ar[r]^{f_2} & Y_2 \\ }$$ So, the products are defined from the pairs (note that we use the same symbols $f_1,f_2$ for the general case $f_i:X_i\to Y_i$ and for the special case $f_i:X\to Y_i$). The other way round, the pairs can be recovered from the products and the diagonals, i.e., the pairs ${\langle {\mathrm{id}},{\mathrm{id}}\rangle}$; indeed, it is easy to prove that for each cone ${X_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}X_2}$ $${\langle f_1,f_2 \rangle} \equiv (f_1\times f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$ In the following, we consider products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$, ${Y_1{\stackrel{q_1}{\longleftarrow}}Y_1\times Y_2{\stackrel{q_2}{\longrightarrow}}Y_2}$ and ${Z_1{\stackrel{r_1}{\longleftarrow}}Z_1\times Z_2{\stackrel{r_2}{\longrightarrow}}Z_2}$. \[prop:weak-equiv\] For each $f_1\equiv f'_1:X_1\to Y_1$ and $f_2\equiv f'_2:X_2\to Y_2$ 1. if $X_1=X_2$ $${\langle f_1,f_2 \rangle} \equiv {\langle f'_1,f'_2 \rangle} \;,$$ 2. in all cases $$f_1 \times f_2 \equiv f'_1 \times f'_2 \;.$$ \[prop:weak-comp\] For each $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$, $g_1:Y_1\to Z_1$, $g_2:Y_2\to Z_2$ 1. if $X_1=X_2$ and $Y_1=Y_2$ and $f_1=f_2(=f)$ $${\langle g_1,g_2 \rangle} \circ f \equiv {\langle g_1\circ f,g_2\circ f \rangle} \;,$$ 2. if $X_1=X_2$ $$(g_1 \times g_2) \circ {\langle f_1,f_2 \rangle} \equiv {\langle g_1\circ f_1,g_2\circ f_2 \rangle} \;,$$ 3. in all cases $$(g_1 \times g_2) \circ (f_1 \times f_2) \equiv (g_1\circ f_1) \times (g_2\circ f_2) \;.$$ Let us consider the products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$ and ${X_2{\stackrel{p'_2}{\longleftarrow}}X_2\times X_1{\stackrel{p'_1}{\longrightarrow}}X_1}$. The swap function is the isomorphism: $$\gamma_{(X_1,X_2)} = {\langle p'_1,p'_2 \rangle}_{p_1,p_2} = {\langle p'_1,p'_2 \rangle}: X_2\times X_1\to X_1\times X_2\;,$$ characterized by: $$p_1\circ \gamma_{(X_1,X_2)}\equiv p'_1 \;\mbox{ and }\; p_2\circ \gamma_{(X_1,X_2)}\equiv p'_2 \;.$$ \[prop:weak-swap\] For each $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$, let $\gamma_Y=\gamma_{(Y_1,Y_2)}$ and $\gamma_X=\gamma_{(X_1,X_2)}$, then: 1. if $X_1=X_2$ $$\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv {\langle f_1,f_2 \rangle} \;,$$ 2. in all cases $$\gamma_Y \circ (f_2\times f_1) \circ \gamma_X^{-1} \equiv f_1 \times f_2 \;.$$ Let us consider the products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$, ${X_1\times X_2{\stackrel{p_{1,2}}{\longleftarrow}}(X_1\times X_2)\times X_3{\stackrel{p_3}{\longrightarrow}}X_3}$, ${X_2{\stackrel{p'_2}{\longleftarrow}}X_2\times X_3{\stackrel{p'_3}{\longrightarrow}}X_3}$ and ${X_1{\stackrel{p'_1}{\longleftarrow}}X_1\times (X_2\times X_3){\stackrel{p'_{2,3}}{\longrightarrow}}X_2\times X_3}$. The associativity function is the isomorphism: $$\alpha_{(X_1,X_2,X_3)} = {\langle {\langle p'_1,p'_2\circ p'_{2,3} \rangle}_{p_1,p_2}, p'_3\circ p'_{2,3} \rangle}_{p_{1,2},p_3}: X_1\times (X_2\times X_3) \to (X_1\times X_2)\times X_3\;,$$ characterized by: $$p_1\circ p_{1,2}\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_1 \,,\; p_2\circ p_{1,2}\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_2\circ p'_{2,3} \;\mbox{ and }\; p_3\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_3\circ p'_{2,3} \;.$$ \[prop:weak-assoc\] For each $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and $f_3:X_3\to Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then: 1. if $X_1=X_2=X_3$ $$\alpha_Y \circ {\langle f_1,{\langle f_2,f_3 \rangle} \rangle} \equiv {\langle {\langle f_1,f_2 \rangle},f_3 \rangle} \;,$$ 2. in all cases $$\alpha_Y \circ (f_1\times (f_2\times f_3)) \equiv ((f_1 \times f_2)\times f_3) \circ \alpha_X \;.$$ In the definition of the binary product $f_1\times f_2$, both $f_1$ and $f_2$ play symmetric rôles. This symmetry can be broken: “first $f_1$ then $f_2$” corresponds to $({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})$, using the intermediate product $Y_1\times X_2$, while “first $f_2$ then $f_1$” corresponds to $(f_1\times{\mathrm{id}}_{Y_2}) \circ ({\mathrm{id}}_{X_1}\times f_2)$, using the intermediate product $X_1\times Y_2$. These are called the (left and right) sequential products of $f_1$ and $f_2$. The three versions of the binary product of functions coincide, up to congruence; this is a kind of parallelism property, meaning that both $f_1$ and $f_2$ can be computed either simultaneously, or one after the other, in any order: \[prop:weak-seq\] For each $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ $$f_1\times f_2 \equiv ({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2}) \equiv (f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2) \;.$$ Cartesian effect categories {#sec:effect} =========================== Sections \[subsec:effect-cat\] to \[subsec:effect-semifn\] form a decorated version of section \[sec:weak\]. Roughly speaking, a kind of structure is decorated when there is some classification of its ingredients. Here, the classification involves two kinds of functions and two kinds of equations. Effect categories are defined in section \[subsec:effect-cat\] as decorated weak categories. In section \[subsec:effect-semi\], semi-products are defined as decorated weak products, then cartesian effect category as decorated cartesian weak categories. Decorated propositions are stated here, and the corresponding decorated proofs are given in appendix \[app:proof\]. Then, in sections \[subsec:effect-seq\] and \[subsec:effect-proj\], the sequential product of functions is defined by composing semi-products, and some of its properties are derived. Effect categories {#subsec:effect-cat} ----------------- A (weak) subcategory ${\mathbf{V}}$ of a weak category ${\mathbf{C}}$ is a subcategory of ${\mathbf{C}}$ such that each equation of ${\mathbf{V}}$ is an equation of ${\mathbf{C}}$. It is a wide (weak) subcategory when ${\mathbf{V}}$ and ${\mathbf{C}}$ have the same points, and each equation of ${\mathbf{C}}$ between functions in ${\mathbf{V}}$ is an equation in ${\mathbf{V}}$. Then only one symbol ${\equiv}$ can be used, for both ${\mathbf{V}}$ and ${\mathbf{C}}$. \[defi:effect-cat\] Let ${\mathbf{V}}$ be a weak category. An effect category extending ${\mathbf{V}}$ is a weak category ${\mathbf{C}}$, such that ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$, together with a relation ${\lesssim}$ between parallel functions in ${\mathbf{C}}$ such that: - the relation ${\lesssim}$ is weaker than ${\equiv}$ for $f_1,f_2$ in ${\mathbf{C}}$, $\,f_1{\equiv}f_2 \Rightarrow f_1{\lesssim}f_2$; - ${\lesssim}$ is transitive; - ${\lesssim}$ and ${\equiv}$ coincide on ${\mathbf{V}}$ for $v_1,v_2$ in ${\mathbf{V}}$, $\,v_1{\equiv}v_2 \iff v_1{\lesssim}v_2$; - ${\lesssim}$ satisfies the substitution property:\ if $f:X\to Y$ and $g_1{\lesssim}g_2:Y\to Z$ then $g_1\circ f{\lesssim}g_2\circ f:X\to Z$; - ${\lesssim}$ satisfies the replacement property with respect to ${\mathbf{V}}$:\ if $g_1{\lesssim}g_2:Y\to Z$ and $v:Z\to W$ in ${\mathbf{V}}$ then $v\circ g_1 {\lesssim}v\circ g_2:Y\to W$. The first property implies that ${\lesssim}$ is reflexive, and when ${\equiv}$ is the equality it means precisely that ${\lesssim}$ is reflexive. Since ${\lesssim}$ is transitive and weaker than ${\equiv}$, if either $f_1{\equiv}f_2{\lesssim}f_3$ or $f_1{\lesssim}f_2{\equiv}f_3$, then $f_1{\lesssim}f_3$; this is called the compatibility of ${\lesssim}$ with ${\equiv}$. An effect category is strict when ${\equiv}$ is the equality. In this paper, there is no major difference between effect categories and strict effect categories. A pure function is a function in ${\mathbf{V}}$. The symbol ${\rightsquigarrow}$ is used for pure functions, and $\to$ for all functions. It follows from definition \[defi:effect-cat\] that all the identities of ${\mathbf{C}}$ are pure, the composition of pure functions is pure, and more precisely a composition of functions is pure if and only if all the composing functions are pure. It should be noted that there can be equations $f{\equiv}v$ between a non-pure function and a pure one; then the function $f$ is proved effect-free, without being pure. This “syntactic” choice could be argued; note that this situation disappears when the congruence ${\equiv}$ is the equality. The relation ${\lesssim}$ is called the semi-congruence of the effect category, and each $f_1{\lesssim}f_2$ is called a semi-equation. The semi-congruence generally is not a congruence, for two reasons: it may not be symmetric, and it may not satisfy the replacement property for all functions. Examples of strict effect categories are given in section \[sec:exam\]. For dealing with partiality in section \[subsec:exam-partial\], the semi-congruence ${\lesssim}$ coincides with the usual ordering of partial functions, it is not symmetric but it satisfies the replacement property for all partial functions. On the other hand, in section \[subsec:exam-state\], the semi-congruence ${\lesssim}$ means that two functions in an imperative language have the same result but may act differently on the state, it is an equivalence relation that does not satisfy the replacement property for non-pure functions. Clearly, if the decorations are forgotten, i.e., if both the distinction between pure functions and arbitrary functions and the distinction between the congruence and the semi-congruence are forgotten, then an effect category is just a weak category. A cartesian effect category, as defined below, is an effect category where ${\mathbf{V}}$ is cartesian and where this cartesian structure on ${\mathbf{V}}$ has some kind of generalization to ${\mathbf{C}}$, that does not, in general, turn ${\mathbf{C}}$ into a cartesian weak category. Semi-products {#subsec:effect-semi} ------------- Now, let us assume that ${\mathbf{C}}$ is an effect category extending ${\mathbf{V}}$, and that ${\mathbf{V}}$ is cartesian. We define nullary and binary semi-products in ${\mathbf{C}}$, for building pairs of functions when at least one of them is pure. \[defi:effect-term\] A semi-terminal point in ${\mathbf{C}}$ is a terminal point $U$ in ${\mathbf{V}}$ such that every function $g:X\to U$ satisfies $g{\lesssim}{{\langle \, \rangle}}_X$. \[defi:effect-prod\] A binary semi-product in ${\mathbf{C}}$ is a binary product ${Y_1{\stackrel{q_1}{\leftsquigarrow}}Y_1\times Y_2{\stackrel{q_2}{\rightsquigarrow}}Y_2}$ in ${\mathbf{V}}$ such that: - for every binary cone with the same base ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{v_2}{\rightsquigarrow}}Y_2}$ and with $v_2$ pure, there is a function ${\langle f_1,v_2 \rangle}_{q_1,q_2}={\langle f_1,v_2 \rangle}:X\to Y_1\times Y_2$, unique up to ${\equiv}$, such that $$q_1\circ {\langle f_1,v_2 \rangle} {\equiv}f_1 \;\mbox{ and }\; q_2\circ {\langle f_1,v_2 \rangle} {\lesssim}v_2 \;,$$ - and for every binary cone with the same base ${Y_1{\stackrel{v_1}{\leftsquigarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$ and with $v_1$ pure, there is a function ${\langle v_1,f_2 \rangle}_{q_1,q_2}={\langle v_1,f_2 \rangle}:X\to Y_1\times Y_2$, unique up to ${\equiv}$, such that $$q_1\circ {\langle v_1,f_2 \rangle} {\lesssim}v_1 \;\mbox{ and }\; q_2\circ {\langle v_1,f_2 \rangle} {\equiv}f_2 \;.$$ The defining (semi-)equations of a binary semi-product can be illustrated as follows: $$\xymatrix@C=5pc{ & Y_1 \\ X \ar@{~>}[ru]^{v_1} \ar@{~>}[rd]_{v_2} \ar@{~>}[r]^{{\langle v_1,v_2 \rangle}} & Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \ar@{}[ld]\|(.3){{\equiv}} \ar@{}[lu]\|(.3){{\equiv}}\\ & Y_2 \\ } \qquad \xymatrix@C=5pc{ & Y_1 \\ X \ar[ru]^{f_1} \ar@{~>}[rd]_{v_2} \ar[r]^{{\langle f_1,v_2 \rangle}\;\;} & Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \ar@{}[ld]\|(.3){{\gtrsim}} \ar@{}[lu]\|(.3){{\equiv}}\\ & Y_2 \\ } \qquad \xymatrix@C=5pc{ & Y_1 \\ X \ar@{~>}[ru]^{v_1} \ar[rd]_{f_2} \ar[r]^{{\langle v_1,f_2 \rangle}\;\;} & Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \ar@{}[ld]\|(.3){{\equiv}} \ar@{}[lu]\|(.3){{\gtrsim}}\\ & Y_2 \\ }$$ Clearly, if the decorations are forgotten, then semi-products are just products. The notation is not ambiguous. Indeed, if ${Y_1{\stackrel{v_1}{\leftsquigarrow}}X{\stackrel{v_2}{\rightsquigarrow}}Y_2}$ is a binary cone in ${\mathbf{V}}$, then the three definitions of the pair ${\langle v_1,v_2 \rangle}$ above coincide, up to congruence: let $t$ denote any one of the three pairs, then $t$ is characterized, up to congruence, by $q_1\circ t {\equiv}v_1$ and $q_2\circ t {\equiv}v_2$, because ${\equiv}$ and ${\lesssim}$ coincide on pure functions. \[defi:effect-ccat\] A cartesian effect category extending a cartesian weak category ${\mathbf{V}}$ is an effect category extending ${\mathbf{V}}$ such that each terminal point of ${\mathbf{V}}$ is a semi-terminal point of ${\mathbf{C}}$ and each binary product of ${\mathbf{V}}$ is a binary semi-product of ${\mathbf{C}}$. Semi-products of functions {#subsec:effect-semifn} -------------------------- \[defi:effect-arr-prod\] In a cartesian effect category, the binary semi-product $f_1\times v_2$ of a function $f_1:X_1\to Y_1$ and a pure function $v_2:X_2{\rightsquigarrow}Y_2$ is the function: $$f_1\times v_2={\langle f_1\circ p_1,v_2\circ p_2 \rangle} : X_1\times X_2 \to Y_1\times Y_2$$ It follows that $f_1\times v_2$ is characterized, up to ${\equiv}$, by: $$q_1 \circ (f_1\times v_2) {\equiv}f_1 \circ p_1 \;\mbox{ and }\; q_2 \circ (f_1\times v_2) {\lesssim}v_2 \circ p_2$$ $$\xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \\ X_1\times X_2 \ar@{~>}[u]^{p_1} \ar@{~>}[d]_{p_2} \ar[r]^{f_1\times v_2} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{{\equiv}} & Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \\ X_2 \ar@{~>}[r]^{v_2} & Y_2 \\ }$$ The binary semi-product $v_1\times f_2:X_1\times X_2\to Y_1\times Y_2$ of a pure function $v_1:X_1{\rightsquigarrow}Y_1$ and a function $f_2:X_2\to Y_2$ is defined in the symmetric way, and it is characterized, up to ${\equiv}$, by the symmetric property. The notation is not ambiguous, because so is the notation for pairs; if $v_1$ and $v_2$ are pure functions, then the three definitions of $v_1\times v_2$ coincide, up to congruence. Propositions about products in cartesian weak categories are called basic propositions. It happens that each basic proposition in section \[sec:weak\] has a decorated version, about semi-products of the form $f_1\times v_2$ in cartesian effect categories, that is stated below. The symmetric decorated version also holds, for semi-products of the form $v_1\times f_2$. Each function in the basic proposition is replaced either by a function or by a pure function, and each equation is replaced either by an equation (${\equiv}$) or by a semi-equation (${\lesssim}$ or ${\gtrsim}$). In addition, in appendix \[app:proof\], the proofs of the decorated propositions are decorated versions of the basic proofs. It happens that no semi-equation appears in the decorated propositions below, but they are used in the proofs. Indeed, a major ingredient in the basic proofs is that a function ${\langle f_1,f_2 \rangle}$ or $f_1\times f_2$ is characterized, up to $\equiv$, by its projections, both up to $\equiv$. The decorated version of this property is that a function ${\langle f_1,f_2 \rangle}$ or $f_1\times f_2$, where $f_1$ or $f_2$ is pure, is characterized, up to $\equiv$, by its projections, one up to $\equiv$ and the other one up to ${\lesssim}$. It should be noted that even when some decorated version of a basic proposition is valid, usually not all the basic proofs can be decorated. In addition, when equations are decorated as semi-equations, some care is required when the symmetry and replacement properties are used. \[prop:effect-equiv\] For each congruent functions $f_1{\equiv}f'_1:X\to Y_1$ and pure functions $v_2{\equiv}v'_2:X{\rightsquigarrow}Y_2$ 1. if $X_1=X_2$ $${\langle f_1,v_2 \rangle} {\equiv}{\langle f'_1,v'_2 \rangle} \;.$$ 2. in all cases $$f_1 \times v_2 {\equiv}f'_1 \times v'_2 \;.$$ \[prop:effect-comp\] For each functions $f_1:X_1\to Y_1$, $g_1:Y_1\to Z_1$ and pure functions $v_2:X_2{\rightsquigarrow}Y_2$, $w_2:Y_2{\rightsquigarrow}Z_2$ 1. if $X_1=X_2$ and $Y_1=Y_2$ and $f_1=v_2(=v)$ $${\langle g_1,w_2 \rangle} \circ f {\equiv}{\langle g_1\circ v,w_2\circ v \rangle} \;,$$ 2. if $X_1=X_2$ $$(g_1 \times w_2) \circ {\langle f_1,v_2 \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle} \;,$$ 3. in all cases $$(g_1 \times w_2) \circ (f_1 \times v_2) {\equiv}(g_1\circ f_1) \times (w_2\circ v_2) \;.$$ The swap and associativity functions are defined in the same way as in section \[sec:weak\]; they are products of projections, so that they are pure functions. It follows that the swap and associativity functions are characterized by the same equations as in section \[sec:weak\], and that they are still isomorphisms. \[prop:effect-swap\] For each function $f_1:X\to Y_1$ and pure function $v_2:X{\rightsquigarrow}Y_2$, let $\gamma_Y=\gamma_{(Y_1,Y_2)}$ and $\gamma_X=\gamma_{(X_1,X_2)}$, then: 1. if $X_1=X_2$ $$\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}{\langle f_1,v_2 \rangle} \;,$$ 2. in all cases $$\gamma_Y \circ (v_2\times f_1) \circ \gamma_X^{-1} {\equiv}f_1 \times v_2 \;.$$ \[prop:effect-assoc\] For each function $f_1:X_1\to Y_1$ and pure functions $v_2:X_2{\rightsquigarrow}Y_2$, $v_3:X_3{\rightsquigarrow}Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then: 1. if $X_1=X_2=X_3$ $$\alpha_Y \circ {\langle f_1,{\langle v_2,v_3 \rangle} \rangle} {\equiv}{\langle {\langle f_1,v_2 \rangle},v_3 \rangle} \;,$$ 2. in all cases: $$\alpha_Y \circ (f_1\times (v_2\times v_3)) {\equiv}((f_1 \times v_2)\times v_3) \circ \alpha_X \;.$$ The sequential product of a function $f_1:X_1\to Y_1$ and a pure function $v_2:X_2{\rightsquigarrow}Y_2$ can be defined as in section \[sec:weak\], using the intermediate products ${Y_1{\stackrel{s_1}{\leftsquigarrow}}Y_1\times X_2{\stackrel{s_2}{\rightsquigarrow}}X_2}$ and ${X_1{\stackrel{t_1}{\leftsquigarrow}}X_1\times Y_2{\stackrel{t_2}{\rightsquigarrow}}Y_2}$. It does coincide with the semi-product of $f_1$ and $v_2$, up to congruence: \[prop:effect-seq\] For each function $f_1:X_1\to Y_1$ and pure function $v_2:X_2{\rightsquigarrow}Y_2$ $$f_1\times v_2 {\equiv}({\mathrm{id}}_{Y_1}\times v_2) \circ (f_1\times{\mathrm{id}}_{X_2}) {\equiv}(f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times v_2) \;.$$ Sequential products of functions {#subsec:effect-seq} -------------------------------- It has been stated in proposition \[prop:weak-seq\] that, in a cartesian weak category, the binary product of functions coincide with both sequential products, up to congruence: $$f_1\times f_2 \equiv ({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2}) \equiv (f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2) \;.$$ In a cartesian effect category, when $f_1$ and $f_2$ are any functions, the product $f_1\times f_2$ is not defined. But $({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})$ and $(f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2)$ make sense, thanks to semi-products, because identities are pure. They are called the sequential products of $f_1$ and $f_2$, and they do not coincide up to congruence, in general: parallelism is not satisfied. \[defi:effect-seq-prod\] The left binary sequential product of two functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ is the function: $$f_1\ltimes f_2 = ({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2}) : X_1\times X_2\to Y_1\times Y_2 \;.$$ So, the left binary sequential product is obtained from: $$\xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\ X_1\times X_2 \ar@{~>}[u]^{p_1} \ar@{~>}[d]_{p_2} \ar[r]^{f_1\times {\mathrm{id}}} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{{\equiv}} & Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2} \ar[r]^{{\mathrm{id}}\times f_2} \ar@{}[rd]\|{{\equiv}} \ar@{}[ru]\|{{\gtrsim}} & Y_1 \times Y_2 \ar@{~>}[u]^{q_1} \ar@{~>}[d]_{q_2} \\ X_2 \ar@{~>}[r]^{{\mathrm{id}}} & X_2 \ar[r]^{f_2} & Y_2 \\ }$$ The left sequential product extends the semi-product: \[prop:effect-seq-lproduct\] For each function $f_1$ and pure function $v_2$, $ f_1\ltimes v_2 {\equiv}f_1\times v_2 $. >From proposition \[prop:effect-comp\], $f_1\ltimes v_2 = ({\mathrm{id}}\times v_2) \circ (f_1\times{\mathrm{id}}) {\equiv}({\mathrm{id}}\circ f_1) \times (v_2\circ{\mathrm{id}}) {\equiv}f_1 \times v_2$. Note that the diagonal ${\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}$ is a pair of pure functions. So, by analogy with the property ${\langle f_1,f_2 \rangle} \equiv (f_1\times f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}$ in weak categories: \[defi:effect-seq-pair\] The left sequential pair of two functions $f_1:X\to Y_1$ and $f_2:X\to Y_2$ is: $${\langle f_1,f_2 \rangle}_l = (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$ The left sequential pairs do not satisfy the usual equations for pairs, as in definition \[defi:weak-prod\]. However, they satisfy some weaker properties, as stated in corollary \[cor:seq-pair\]. The right binary sequential product of $f_1$ and $f_2$ is defined in the symmetric way; it is the function: $$f_1\rtimes f_2 = (f_1\times{\mathrm{id}}_{Y_2}) \circ ({\mathrm{id}}_{X_1}\times f_2) : X_1\times X_2\to Y_1\times Y_2 \;.$$ It does also extend the product of a pure function and a function: for each pure function $v_1$, $ v_1\rtimes f_2 {\equiv}v_1\times f_2 $. The right sequential pair of $f_1:X\to Y_1$ and $f_2:X\to Y_2$ is: $${\langle f_1,f_2 \rangle}_r = (f_1\rtimes f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$ Here are some properties of the sequential products that are easily deduced from the properties of semi-products in \[subsec:effect-semi\]. The symmetric properties also hold. \[prop:seq-equiv\] For each congruent functions $f_1{\equiv}f'_1:X_1\to Y_1$ and $f_2{\equiv}f'_2:X_2\to Y_2$ $$f_1 \ltimes f_2 {\equiv}f'_1 \ltimes f'_2\;.$$ Clear, from \[prop:effect-equiv\]. \[prop:seq-comp\] For each functions $f_1:X_1\to Y_1$, $g_1:Y_1\to Z_1$, $g_2:Y_2\to Z_2$ and pure function $v_2:X_2{\rightsquigarrow}Y_2$ $$(g_1 \ltimes g_2) \circ (f_1 \times v_2) {\equiv}(g_1\circ f_1) \ltimes (g_2\circ v_2) \;.$$ $$\xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \ar@{~>}[r]^{{\mathrm{id}}} & Z_1 \\ X_1 \times X_2 \ar@{~>}[u] \ar@{~>}[d] \ar[r]^{f_1\times v_2} \ar@{}[rd]\|{{\equiv}} \ar@{}[ru]\|{{\equiv}} & Y_1\times Y_2 \ar@{~>}[u] \ar@{~>}[d] \ar[r]^{g_1\times {\mathrm{id}}} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{{\equiv}} & Z_1 \times Y_2 \ar@{~>}[u] \ar@{~>}[d] \ar[r]^{{\mathrm{id}}\times g_2} \ar@{}[rd]\|{{\equiv}} \ar@{}[ru]\|{{\gtrsim}} & Z_1 \times Z_2 \ar@{~>}[u] \ar@{~>}[d] \\ X_2 \ar@{~>}[r]^{v_2} & Y_2 \ar@{~>}[r]^{{\mathrm{id}}} & Y_2 \ar[r]^{g_2} & Z_2 \\ }$$ >From several applications of proposition \[prop:effect-comp\] and its symmetric version:\ $({\mathrm{id}}\times g_2) \circ (g_1\times{\mathrm{id}}) \circ (f_1 \times v_2) {\equiv}({\mathrm{id}}\times g_2) \circ ((g_1\circ f_1) \times v_2) {\equiv}({\mathrm{id}}\times g_2) \circ ({\mathrm{id}}\times v_2) \circ ((g_1\circ f_1)\times{\mathrm{id}}) {\equiv}({\mathrm{id}}\times (g_2\circ v_2)) \circ ((g_1\circ f_1)\times{\mathrm{id}})$. \[prop:seq-swap\] For each functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$, the left and right sequential products are related by swaps: $$\gamma_Y \circ (f_2\rtimes f_1) \circ \gamma_X^{-1} {\equiv}f_1 \ltimes f_2 \;.$$ >From proposition \[prop:effect-swap\] and its symmetric version:\ $ \gamma \circ ({\mathrm{id}}\times f_2) \circ (f_1\times{\mathrm{id}}) {\equiv}(f_2\times{\mathrm{id}}) \circ \gamma \circ (f_1\times{\mathrm{id}}) {\equiv}(f_2\times{\mathrm{id}}) \circ ({\mathrm{id}}\times f_1) \circ \gamma$. \[prop:seq-assoc\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and $f_3:X_3\to Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then: : $$\alpha_Y \circ (f_1\ltimes (f_2\ltimes f_3)) {\equiv}((f_1 \ltimes f_2)\ltimes f_3) \circ \alpha_X \;.$$ >From proposition \[prop:effect-assoc\]. Projections of sequential products {#subsec:effect-proj} ---------------------------------- Let us come back to a weak category, as in section \[sec:weak\]. The binary product of functions is characterized, up to congruence, by the equations: $$q_1 \circ (f_1\times f_2) \equiv f_1 \circ p_1 \;\mbox{ and }\; q_2 \circ (f_1\times f_2) \equiv f_2 \circ p_2 \;,$$ so that for all constant functions $x_1:U\to X_1$ and $x_2:U\to X_2$ $$q_1 \circ (f_1\times f_2) \circ {\langle x_1,x_2 \rangle} \equiv f_1 \circ x_1 \;\mbox{ and }\; q_2 \circ (f_1\times f_2) \circ {\langle x_1,x_2 \rangle} \equiv f_2 \circ x_2 \;.$$ In a cartesian effect category, it is proved in theorem \[thm:seq-prod\] that $f_1\ltimes f_2$, when applied to a pair of constant pure functions ${\langle x_1,x_2 \rangle}$, returns on the $Y_1$ side a function that is semi-congruent to $f_1(x_1)$, and on the $Y_2$ side a function that is congruent to $f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1\circ x_1$, which means “first $f_1(x_1)$, then forget the result, then $f_2(x_2)$”. More precise statements are given in propositions \[prop:seq-val\] and \[prop:seq-com\]. Proofs are presented in the same formalized way as in appendix \[app:proof\]. As above, we consider the semi-terminal point $U$ and semi-products\ ${X_1{\stackrel{p_1}{\leftsquigarrow}}X_1\times X_2{\stackrel{p_2}{\rightsquigarrow}}X_2}$, ${Y_1{\stackrel{q_1}{\leftsquigarrow}}Y_1\times Y_2{\stackrel{q_2}{\rightsquigarrow}}Y_2}$ and ${Y_1{\stackrel{s_1}{\leftsquigarrow}}Y_1\times X_2{\stackrel{s_2}{\rightsquigarrow}}X_2}$. \[prop:seq-val\] For each functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ $$q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1 : X_1\times X_2\to Y_1\;.$$ $$\xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\ X_1\times X_2 \ar@{~>}[u]^{p_1} \ar[r]^{f_1\times {\mathrm{id}}} \ar@{}[ru]\|{{\equiv}} \ar@/_4ex/[rr]_{f_1\ltimes f_2}^{=} & Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar[r]^{{\mathrm{id}}\times f_2} \ar@{}[ru]\|{{\gtrsim}} & Y_1 \times Y_2 \ar@{~>}[u]^{q_1} \\ }$$ \ ------- ------------------------------------------------------------------------- ---------------------------------------- $(a)$ $q_1\circ ({\mathrm{id}}\times f_2) {\lesssim}s_1$ $(b)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}s_1\circ (f_1\times{\mathrm{id}})$ $(a)$, ${\mathit{subst}}_{{\lesssim}}$ $(c)$ $s_1\circ (f_1\times{\mathrm{id}}) {\equiv}f_1\circ p_1$ $(d)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1$ $(b)$, $(c)$, ${\mathit{comp}}$ ------- ------------------------------------------------------------------------- ---------------------------------------- \ \[lem:seq-terminal\] For each function $f_1:X_1\to Y_1$ and pure function $x_2:U{\rightsquigarrow}X_2$ $${\langle {\mathrm{id}}_{Y_1},x_2\circ{{\langle \, \rangle}}_{Y_1} \rangle}\circ f_1 {\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}_{X_1} \rangle} : X_1 \to Y_1\times X_2 \;.$$ Both handsides can be illustrated as follows: $$\xymatrix@C=5pc{ & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\ X_1 \ar[r]^{f_1} & Y_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}} \ar@{~>}[r]^{{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}} \ar@{}[rd]\|{{\equiv}} \ar@{}[ru]\|{{\equiv}} & Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2} \\ & U \ar@{~>}[r]^{x_2} & X_2 \\ } \qquad \xymatrix@C=5pc{ X_1 \ar[r]^{f_1} & Y_1 \\ X_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}} \ar[r]^{{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{{\equiv}} & Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2} \\ U \ar@{~>}[r]^{x_2} & X_2 \\ }$$ \ --------- -------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------- $(a_1)$ $s_1\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}{\mathrm{id}}$ $(b_1)$ $s_1\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}f_1$ $(a_1)$, ${\mathit{subst}}_{{\equiv}}$ $(a_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}x_2\circ{{\langle \, \rangle}}$ $(b_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}x_2\circ{{\langle \, \rangle}}\circ f_1$ $(a_2)$, ${\mathit{subst}}_{{\equiv}}$ $(c_2)$ ${{\langle \, \rangle}}\circ f_1 {\lesssim}{{\langle \, \rangle}}$ semi-terminality of $U$ $(d_2)$ $x_2\circ{{\langle \, \rangle}}\circ f_1{\lesssim}x_2\circ{{\langle \, \rangle}}$ $(c_2)$, ${\mathit{repl}}_{{\lesssim}}$ ($x_2$ is pure) $(e_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\lesssim}x_2\circ{{\langle \, \rangle}}$ $(b_2)$, $(d_2)$, ${\mathit{trans}}_{{\lesssim}}$ $(f)$ ${\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}$ $(b_1)$, $(e_2)$ --------- -------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------- \ \[prop:seq-com\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and pure function $x_2:U{\rightsquigarrow}X_2$ $$q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 : X_1\to Y_2 \;.$$ Both handsides can be illustrated as follows: $$\xymatrix@C=3pc{ X_1 \ar@{~>}[r]^{{\mathrm{id}}} & X_1 & & \\ X_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}} \ar@{~>}[r]^{{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\;} & X_1 \times X_2 \ar@/^4ex/[rr]^{f_1\ltimes f_2}_{=} \ar@{~>}[u]_{p_1} \ar@{~>}[d]^{p_2} \ar[r]^{f_1\times{\mathrm{id}}} \ar@{}[ld]\|{{\equiv}} \ar@{}[lu]\|{{\equiv}} \ar@{}[rd]\|{{\gtrsim}} & Y_1 \times X_2 \ar@{~>}[d]^{s_2} \ar[r]^{{\mathrm{id}}\times f_2} \ar@{}[rd]\|{{\equiv}} & Y_1 \times Y_2 \ar@{~>}[d]^{q_2} \\ U \ar@{~>}[r]^{x_2} & X_2 \ar@{~>}[r]^{{\mathrm{id}}} & X_2 \ar[r]^{f_2} & Y_2 \\ } \quad \xymatrix@C=2pc{ \mbox{ } \\ X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[d]_{{{\langle \, \rangle}}} \\ & U \ar@{~>}[r]^{x_2} & X_2 \ar[r]^{f_2} & Y_2 \\ }$$ \ ------- ---------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- $(a)$ $q_2\circ ({\mathrm{id}}\times f_2) {\equiv}f_2\circ s_2$ $(b)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(a)$, ${\mathit{subst}}_{{\equiv}}$ {\equiv}f_2\circ s_2\circ (f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle}$ $(c)$ $(f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} prop. \[prop:effect-comp\] {\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}$ $(d)$ ${\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle} lemma \[lem:seq-terminal\] {\equiv}{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$ $(e)$ $(f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(c)$, $(d)$, ${\mathit{trans}}_{{\equiv}}$ {\equiv}{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$ $(f)$ $f_2\circ s_2\circ (f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(e)$, ${\mathit{repl}}_{{\equiv}}$ {\equiv}f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$ $(g)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(b)$, $(f)$, ${\mathit{trans}}_{{\equiv}}$ {\equiv}f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$ $(h)$ $p_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}x_2\circ {{\langle \, \rangle}}$ $(i)$ $f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 $(h)$, ${\mathit{subst}}_{{\equiv}}$, ${\mathit{repl}}_{{\equiv}}$ {\equiv}f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1 $ $(j)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(g)$, $(i)$, ${\mathit{trans}}_{{\equiv}}$ {\equiv}f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1$ ------- ---------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- \ \[thm:seq-prod\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and pure functions $x_1:U{\rightsquigarrow}X_1$ and $x_2:U{\rightsquigarrow}X_2$, the function $(f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle}$ satisfies: $$q_1 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\lesssim}f_1 \circ x_1 \;\mbox{ and }\; q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 \circ x_1 \;.$$ $$\xymatrix@C=3pc{ U \ar@{=}[d] \ar@{~>}[rr]^{x_1} && X_1 \ar[rrr]^{f_1} &&& Y_1 \\ U \ar@{=}[d] \ar@{~>}[rr]^{{\langle x_1,x_2 \rangle}} && X_1\times X_2 \ar[rrr]^{f_1\ltimes f_2} \ar@{}[ru]\|{{\gtrsim}} \ar@{}[rd]\|{{\equiv}} &&& Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \\ U \ar@{~>}[r]^{x_1} & X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{{\langle \, \rangle}}} & U \ar@{~>}[r]^{x_2} & X_2 \ar[r]^{f_2} & Y_2 \\ }$$ \ --------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------- $(a_1)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1$ prop. \[prop:seq-val\] $(b_1)$ $q_1\circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} $(a_1)$, ${\mathit{subst}}_{{\lesssim}}$ {\lesssim}f_1\circ p_1 \circ {\langle x_1,x_2 \rangle}$ $(c_1)$ $p_1 \circ {\langle x_1,x_2 \rangle}{\equiv}x_1$ (on values) $(d_1)$ $f_1\circ p_1 \circ {\langle x_1,x_2 \rangle}{\equiv}f_1\circ x_1$ $(c_1)$, ${\mathit{repl}}_{{\equiv}}$ $(e_1)$ $q_1 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} $(b_1)$, $(d_1)$, ${\mathit{comp}}$ {\lesssim}f_1 \circ x_1$ $(a_2)$ ${\langle x_1,x_2 \rangle} {\equiv}{\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} \circ x_1$ (on values) $(b_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\equiv}q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} \circ x_1$ $(a_2)$, ${\mathit{repl}}_{{\equiv}}$ $(c_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} prop. \[prop:seq-com\] {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1$ $(d_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} $(c_2)$, ${\mathit{subst}}_{{\equiv}}$ \circ x_1 {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1\circ x_1$ $(e_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1\circ x_1$ $(b_2)$, $(d_2)$, ${\mathit{trans}}_{{\equiv}}$ --------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------- \ The corresponding properties of left sequential pairs easily follow. \[cor:seq-pair\] For each functions $f_1:X\to Y_1$, $f_2:X\to Y_2$ and pure function $x:U{\rightsquigarrow}X$ $$q_1\circ {\langle f_1,f_2 \rangle}_l {\lesssim}f_1 \,\mbox{, hence }\, q_1\circ {\langle f_1,f_2 \rangle}_l\circ x {\lesssim}f_1\circ x \,\mbox{, and }\, q_2\circ {\langle f_1,f_2 \rangle}_l\circ x {\equiv}f_2 \circ x \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 \circ x \;.$$ Effect categories and Arrows {#sec:related} ============================ Starting from [@Moggi91; @Wadler93], monads are used in Haskell for dealing with computational effects. A Monad type in Haskell is a unary type constructor that corresponds to a strong monad, in the categorical sense. Monads have been generalized on the categorical side to Freyd categories [@PowerRobinson97] and on the functional programming side to Arrows [@Hughes00]. A precise statement of the facts that Arrows generalize Monads and that Arrows are Freyd categories can be found in [@HeunenJacobs06], where each of the three notions is seen as a monoid in a relevant category. Now we prove that cartesian effect categories determine Arrows. In section \[sec:exam\] our approach is compared with the Monads approach, for two fundamental examples. In this section, all effect categories are strict: the congruence ${\equiv}$ is the equality. Arrows {#subsec:related-arrows} ------ According to [@Paterson01], Arrows in Haskell are defined as follows. \[defi:related-arr\] An Arrow is a binary type constructor class ${\mathtt{A}}$ of the form: 999 = 999 = 999 `class Arrow {\mathtt{A}} where`\ ${\mathtt{arr}}:: (X\to Y)\to {\mathtt{A}}\;X\;Y$\ $({>\!\!>\!\!>}):: {\mathtt{A}}\;X\;Y \to {\mathtt{A}}\;Y\;Z \to {\mathtt{A}}\;X\;Z$\ ${\mathtt{first}}:: {\mathtt{A}}\;X\;Y \to {\mathtt{A}}\;(X,Z)\;(Y,Z)$\ satisfying the following equations: ----- ------------------------------------------------------------------------------------------------ --- ------------------------------------------------------------------------------- (1) ${\mathtt{arr}}\; {\mathrm{id}}{>\!\!>\!\!>}f $ = $ f$ (2) $f {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathrm{id}}$ = $ f$ (3) $(f {>\!\!>\!\!>}g) {>\!\!>\!\!>}h $ = $ f {>\!\!>\!\!>}(g {>\!\!>\!\!>}h)$ (4) ${\mathtt{arr}}\;(w.v) $ = $ {\mathtt{arr}}\; v {>\!\!>\!\!>}{\mathtt{arr}}\; w$ (5) ${\mathtt{first}}\; ({\mathtt{arr}}\; v) $ = $ {\mathtt{arr}}\; (v\times{\mathrm{id}})$ (6) ${\mathtt{first}}\;(f {>\!\!>\!\!>}g) $ = $ {\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{first}}\; g$ (7) ${\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{arr}}\; ({\mathrm{id}}\times v) $ = $ {\mathtt{arr}}\; ({\mathrm{id}}\times v) {>\!\!>\!\!>}{\mathtt{first}}\; f$ (8) ${\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathtt{fst}}$ = $ {\mathtt{arr}}\; {\mathtt{fst}}{>\!\!>\!\!>}f$ (9) $\;\;{\mathtt{first}}\; ({\mathtt{first}}\; f) {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathtt{assoc}}$ = $ {\mathtt{arr}}\; {\mathtt{assoc}}{>\!\!>\!\!>}{\mathtt{first}}\; f$ ----- ------------------------------------------------------------------------------------------------ --- ------------------------------------------------------------------------------- where the functions $(\times)$, ${\mathtt{fst}}$ and ${\mathtt{assoc}}$ are defined as: $$\begin{array}{lll} (\times) :: & (X\to X')\to(Y\to Y')\to (X,Y)\to (X',Y') & (f\times g)(x,y)=(f\;x,g\;y) \\ {\mathtt{fst}}:: & (X,Y)\to X & {\mathtt{fst}}(x,y)=x \\ ({\mathtt{assoc}}) :: & ((X,Y),Z)\to (X,(Y,Z)) & {\mathtt{assoc}}((x,y),z) = (x,(y,z)) \\ \end{array}$$ Cartesian effect categories determine Arrows {#subsec:related-cec-arr} -------------------------------------------- Let ${\mathbf{V}}_H$ denote the category of Haskell types and ordinary functions, so that the Haskell notation $\mathtt{(X\to Y)}$ represents ${\mathbf{V}}_H(X,Y)$, made of the Haskell ordinary functions from $X$ to $Y$. An arrow ${\mathtt{A}}$ contructs a type ${\mathtt{A}}\;X\;Y$ for all types $X$ and $Y$. We slightly modify the definition of Arrows by allowing $\mathtt{(X\to Y)}$ to represent ${\mathbf{V}}(X,Y)$ for any cartesian category ${\mathbf{V}}$ and by requiring that ${\mathtt{A}}\;X\;Y$ is a set rather than a type. In addition, we use categorical notations instead of Haskell syntax. So, from now on, for any cartesian category ${\mathbf{V}}$, an Arrow $A$ on ${\mathbf{V}}$ associates to each points $X$, $Y$ of ${\mathbf{V}}$ a set $A(X,Y)$, together with three operations: 999 = 999 = 999 ${\mathtt{arr}}: {\mathbf{V}}(X,Y)\to A(X,Y)$\ ${>\!\!>\!\!>}: A(X,Y) \to A(Y,Z) \to A(X,Z)$\ ${\mathtt{first}}: A(X,Y) \to A(X\times Z,Y\times Z)$\ that satisfy the equations (1)-(9). Basically, the correspondence between a cartesian effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$ and an Arrow $A$ on ${\mathbf{V}}$ identifies ${\mathbf{C}}(X,Y)$ with $A(X,Y)$ for all types $X$ and $Y$. More precisely: \[thm:related-arr\] Every cartesian effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$ gives rise to an Arrow $A$ on ${\mathbf{V}}$, according to the following table: The first and second line in the table say that $A(X,Y)$ is made of the functions from $X$ to $Y$ in ${\mathbf{C}}$ and that ${\mathtt{arr}}$ is the convertion from pure functions to arbitrary functions. The third and fourth lines say that ${>\!\!>\!\!>}$ is the (reverse) composition of functions and that ${\mathtt{first}}$ is the semi-product with the identity. Let us check that $A$ is an Arrow; the following table translates each property (1)-(9) in terms of cartesian effect categories (where $\rho_X:X\times U \to X$ is the projection), and gives the argument for its proof. ----- ----------------------------------------------------------------- --- --------------------------------------------------------- -------------------------------------------------- (1) $f\circ {\mathrm{id}}$ = $f$ unitarity in ${\mathbf{C}}$ (2) $ {\mathrm{id}}\circ f $ = $f$ unitarity in ${\mathbf{C}}$ (3) $h\circ (g\circ f) $ = $ (h\circ g)\circ f $ associativity in ${\mathbf{C}}$ (4) $w\circ v$ in ${\mathbf{V}}$ = $w\circ v$ in ${\mathbf{C}}$ ${\mathbf{V}}\subseteq{\mathbf{C}}$ is a functor (5) $v\times{\mathrm{id}}$ in ${\mathbf{V}}$ = $ v\times{\mathrm{id}}$ in ${\mathbf{C}}$ non-ambiguity of “$\times$” (6) $(g\circ f)\times{\mathrm{id}}$ = $ (g\times{\mathrm{id}})\circ (f\times{\mathrm{id}})$ proposition \[prop:effect-comp\] (7) $({\mathrm{id}}\times v) \circ (f\times{\mathrm{id}}) $ = $ (f\times{\mathrm{id}}) \circ ({\mathrm{id}}\times v)$ proposition \[prop:effect-comp\] (8) $\rho\circ (f\times{\mathrm{id}}_U) $ = $ f\circ \rho$ definition \[defi:effect-arr-prod\] (9) $\alpha^{-1}\circ ((f\times{\mathrm{id}})\times{\mathrm{id}}) $ = $ (f\times{\mathrm{id}})\circ\alpha^{-1}$ proposition \[prop:effect-assoc\] ----- ----------------------------------------------------------------- --- --------------------------------------------------------- -------------------------------------------------- The translation of the Arrow combinators follows easily, using ${\langle f,g \rangle}_l=(f\ltimes g) \circ {\langle {\mathrm{id}},{\mathrm{id}}\rangle}$ as in section \[subsec:effect-seq\]: --------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------- $ (id \times f) = \gamma \circ (f \times id) \circ \gamma $ ${\mathtt{second}}\;f = {\mathtt{arr}}\;{\mathtt{swap}}{>\!\!>\!\!>}{\mathtt{first}}\;f {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{swap}}$ $f \ltimes g = ({\mathrm{id}}\times g) \circ (f\times{\mathrm{id}})$ $f {\!\!\!\!}g = {\mathtt{first}}\;f {>\!\!>\!\!>}{\mathtt{second}}\;g $ ${\langle f,g \rangle}_l = (f\ltimes g) \circ {\langle {\mathrm{id}},{\mathrm{id}}\rangle}$ $f {\&\!\!\&\!\!\&}g = {\mathtt{arr}}(\lambda b \rightarrow (b,b)) {>\!\!>\!\!>}(f {\!\!\!\!}g) $ --------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------- For instance, in [@Hughes00], the author states that ${\&\!\!\&\!\!\&}$ is not a categorical product since in general $(f {\&\!\!\&\!\!\&}g) {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{fst}}$ is different from $f$. We can state this more precisely in the effect category, where $(f {\&\!\!\&\!\!\&}g) {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{fst}}$ corresponds to $q_1 \circ {\langle f,g \rangle}_l $. Indeed, according to corollary \[cor:seq-pair\]: $$q_1 \circ {\langle f,g \rangle}_l {\lesssim}f \;.$$ Examples {#sec:exam} ======== Here are presented some examples of strict cartesian effect categories. Several versions are given, some of them rely on monads. Partiality {#subsec:exam-partial} ---------- Let ${\mathbf{V}}={\mathbf{Set}}$ be the category of sets and maps, and ${\mathbf{C}}={\mathbf{Part}}$ the category of sets and partial maps, so that ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. Let ${\lesssim}$ denote the usual ordering on partial maps: $f{\lesssim}g$ if and only if ${\mathcal{D}}(f)\subseteq{\mathcal{D}}(g)$ (where ${\mathcal{D}}$ denotes the domain of definition) and $f(x)=g(x)$ for all $x\in{\mathcal{D}}(f)$. The restriction of ${\lesssim}$ to ${\mathbf{V}}$ is the equality of total maps. Clearly ${\lesssim}$ is not symmetric, but it satisfies all the other properties of a congruence, in particular the replacement property with respect to all maps. So, ${\lesssim}$ is a semi-congruence (which satisfies replacement), that makes ${\mathbf{C}}$ a strict effect category extending ${\mathbf{V}}$. Warning: usually the notations are $v:X\to Y$ for a total map and $f:X{\rightharpoonup}Y$ for a partial map, but here we use respectively $v:X{\rightsquigarrow}Y$ (total) and $f:X\to Y$ (partial). Let us define the pair ${\langle f,v \rangle}$ of a partial map $f:X\to Y_1$ and a total map $v:X{\rightsquigarrow}Y_2$ as the partial map ${\langle f,v \rangle}:X\to Y_1\times Y_2$ with the same domain of definition as $f$ and such that ${\langle f,v \rangle}(x)={\langle f(x),v(x) \rangle}$ for all $x\in{\mathcal{D}}(f)$. It is easy to check that we get a cartesian effect category. For illustrating the semi-product $f\times v$, there are two cases: either $f(x_1)$ is defined, or not, in which case we note $f(x_1)=\bot$. We use the traditional notation $\xymatrix@=1.5pc{ x \ar@{\|->}[r]^{f} & y \\ }$ when $y=f(x)$ and its analog $\xymatrix@=1.5pc{ x \ar@{\|~>}[r]^{v} & y \\ }$ when $y=v(x)$ and $v$ is pure. $$\xymatrix@C=5pc{ x_1 \ar@{\|->}[r]^{f} & y_1 \\ {\langle x_1,x_2 \rangle} \ar@{\|~>}[u] \ar@{\|~>}[d] \ar@{\|->}[r]^{f\times v} \ar@{}[rd]\|{=} \ar@{}[ru]\|{=} & {\langle y_1,y_2 \rangle} \ar@{\|~>}[u] \ar@{\|~>}[d] \\ x_2 \ar@{\|~>}[r]^{v} & y_2 \\ } \qquad \mbox{ or }\qquad \xymatrix@C=5pc{ x_1 \ar@{\|->}[r]^{f} & \bot \\ {\langle x_1,x_2 \rangle} \ar@{\|~>}[u] \ar@{\|~>}[d] \ar@{\|->}[r]^{f\times v} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{=} & \bot \ar@{\|~>}[u] \ar@{\|~>}[d] \\ x_2 \ar@{\|~>}[r]^{v} & y_2\ne \bot \\ }$$ It can be noted that, in the previous example, ${\mathbf{C}}$ is a 2-category, with a 2-cell from $f$ to $g$ if and only if $f{\lesssim}g$. More generally, let ${\mathbf{C}}$ be a 2-category and ${\mathbf{V}}$ a sub-2-category where the unique 2-cells are the identities. Then by defining $f{\lesssim}g$ whenever there is a 2-cell from $f$ to $g$, we get a strict effect category. In such effect categories, the replacement property holds with respect to all functions in ${\mathbf{C}}$, but the semi-congruence is usually not symmetric. Let us come back to the partiality example, from the slightly different point of view of the Maybe monad. First, let us present this point of view in a naive way, without monads. Let $U=\{\bot\}$ be a singleton, let “$+$” denote the disjoint union of sets, and for each set $X$ let $GX=X+U$ and let $\eta_X:X\to GX$ be the inclusion. Each partial map $f$ from $X$ to $Y$ can be extended as a total map $Gf$ from $X$ to $GY$, such that $Gf(x)=f(x)$ for $x\in{\mathcal{D}}(f)$ and $Gf(x)=\bot$ otherwise. This defines a bijection between the partial maps from $X$ to $Y$ and the total maps from $X$ to $GY$. Let ${\mathbf{C}}$ be the category such that its points are the sets, and a function $X\to Y$ in ${\mathbf{C}}$ is a function $X\to GY$ in ${\mathbf{Set}}$; we say that $X\to Y$ in ${\mathbf{C}}$ stands for $X\to GY$ in ${\mathbf{Set}}$. Let $J:{\mathbf{Set}}\to{\mathbf{C}}$ be the functor that is the identity on points and associates to each map $v_0:X\to Y$ the map $\eta_Y\circ v_0$. Let ${\mathbf{V}}=J({\mathbf{Set}})$. Then ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. For all $f,g:X\to Y$ in ${\mathbf{C}}$, that stand for $f,g:X\to GY$ in ${\mathbf{Set}}$, let: $$f{\lesssim}g \iff \forall x\in X\; (f(x)\ne\bot\Rightarrow (g(x)\ne\bot \wedge g(x)=f(x))\;.$$ This yields a strict effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$, with the semi-congruence ${\lesssim}$, and as above the replacement property holds with respect to all functions in ${\mathbf{C}}$ but ${\lesssim}$ is not symmetric. Let $f:X\to Y_1$ in ${\mathbf{C}}$ and $v:X\to Y_2$ in ${\mathbf{V}}$, they stand respectively for $f:X\to GY_1$ and $v=\eta_{Y_2}\circ v_0$ with $v_0:X\to Y_2$. Then, in ${\mathbf{Set}}$, the pair ${\langle f,v_0 \rangle}:X\to GY_1 \times Y_2$ can be composed with: $$t:GY_1\times Y_2=(Y_1+U)\times Y_2\to (Y_1\times Y_2)+U =G(Y_1\times Y_2)\;,$$ that maps ${\langle y_1,y_2 \rangle}$ to itself and ${\langle \bot,y_2 \rangle}$ to $\bot$. Now, let ${\langle f,v \rangle}:X\to Y_1\times Y_2$ in ${\mathbf{C}}$ stand for ${\langle f,v \rangle}=t\circ {\langle f,v_0 \rangle}:X\to G(Y_1\times Y_2)$ in ${\mathbf{Set}}$. Then ${\langle f,v \rangle}$ is a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category. The diagrams for illustrating the semi-product $f\times v$ are the same as above. This point of view can also be presented using the the Maybe monad for managing failures, as follows. We have defined a functor $G:{\mathbf{Part}}\to{\mathbf{Set}}$, that is a right adjoint to the inclusion functor $I:{\mathbf{Set}}\subseteq{\mathbf{Part}}$. The corresponding monad has endofunctor $M=GI$ on ${\mathbf{Set}}$, the category ${\mathbf{C}}$ is the Kleisli category of $M$, and $J:{\mathbf{Set}}\to{\mathbf{C}}$ is the canonical functor associated to the monad. In addition, this monad $M$ is strong, and $t$ is the $(Y_1,Y_2)$ component of the strength of $M$. But the definition of the semi-congruence ${\lesssim}$, as above, is not part of the usual framework of monads. State {#subsec:exam-state} ----- Let ${\mathbf{V}}_0$ be a cartesian category, with a distinguished point $S$ for “the type of states”; for all $X$, let $\pi_X:S\times X\to X$ denotes the projection. Let ${\mathbf{C}}$ be the category with the same points as ${\mathbf{V}}_0$ and with a function $f:X\to Y$ for each function $f:S\times X\to S\times Y$ in ${\mathbf{V}}_0$; we say that $f:X\to Y$ in ${\mathbf{C}}$ stands for $f:S\times X\to S\times Y$ in ${\mathbf{V}}_0$. Let $J:{\mathbf{V}}_0\to{\mathbf{C}}$ be the identity-on-points functor which maps each $v_0:X\to Y$ in ${\mathbf{V}}_0$ to the function $J(v_0):X\to Y$ in ${\mathbf{C}}$ that stands for ${\mathrm{id}}_S\times v_0:S\times X\to S\times Y$ in ${\mathbf{V}}_0$. Let ${\mathbf{V}}=J({\mathbf{V}}_0)$, it is a wide subcategory of ${\mathbf{C}}$. For all $f,g:X\to Y$ in ${\mathbf{C}}$, let: $$f{\lesssim}g \iff \pi_Y\circ g = \pi_Y\circ f \;.$$ We get a strict effect category, where the semi-congruence ${\lesssim}$ is symmetric, but does not satisfy the replacement property with respect to all functions in ${\mathbf{C}}$. The semi-product of $f:X\to Y_1$ and $v:X{\rightsquigarrow}Y_2$ is defined as follows. Since $f:S\times X\to S\times Y_1$ in ${\mathbf{V}}_0$ and $v={\mathrm{id}}_S\times v_0$ for some $v_0:X\to Y$ in ${\mathbf{V}}_0$, the pair ${\langle f,v_0\circ\pi_X \rangle}:S\times X\to (S\times Y_1)\times Y_2$ exists in ${\mathbf{V}}_0$. By composing it with the isomorphism $(S\times Y_1)\times Y_2 \to S\times (Y_1\times Y_2)$ we get ${\langle f,v \rangle}:S\times X\to S\times (Y_1\times Y_2)$ in ${\mathbf{V}}_0$, i.e., ${\langle f,v \rangle}:X\to Y_1\times Y_2$ in ${\mathbf{C}}$. It is easy to check that this defines a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category, where the characteristic property of the semi-product $f\times v$ can be illustrated as follows: $$\xymatrix@C=5pc{ (s,x_1) \ar@{\|->}[r]^{f} & (s',y_1) \\ (s,x_1,x_2) \ar@{\|~>}[u] \ar@{\|~>}[d] \ar@{\|->}[r]^{f\times v} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{=} & (s',y_1,y_2) \ar@{\|~>}[u] \ar@{\|~>}[d] \\ (s,x_2) \ar@{\|~>}[r]^{v} & (s,y_2)\ne(s',y_2) \ar@{\|~>}[r]^{\quad\pi_Y} & y_2 \\ }$$ The example above can be curried, thus recovering the State monad. A motivation for the introduction of Freyd categories in [@PowerRobinson97] is the possibility of dealing with state in a linear way, as above, rather than in the exponential way provided by the State monad. Now ${\mathbf{V}}_0$ is still a cartesian category with a distinguished point $S$, the “type of states”, and in addition ${\mathbf{V}}_0$ has exponentials $(S\times X)^S$ for each $X$. Then the endofunctor $M(X)=(S\times X)^S$ defines the State monad on ${\mathbf{V}}_0$, with composition defined as usual. It is well-known that $M$ is a strong monad, with strength $t_{Y_1,Y_2}=(S\times Y_1)^S\times Y_2 \to (S\times Y_1\times Y_2)^S$ obtained from ${\mathrm{app}}_{S\times Y_1}\times{\mathrm{id}}_{Y_2}: S\times (S\times Y_1)^S\times Y_2 \to S\times Y_1\times Y_2$, where “${\mathrm{app}}$” denotes the application function. Hence, from $f:X\to M(Y_1)$ and $v_0:X\to Y_2$ in ${\mathbf{V}}_0$, we can build ${\langle f,v \rangle}=t_{Y_1,Y_2}\circ{\langle f,v_0 \rangle}:X\to M(Y_1\times Y_2)$. Let ${\mathbf{C}}$ be the Kleisli category of the monad $M$, let $J:{\mathbf{V}}_0\to{\mathbf{C}}$ be the canonical functor associated to the monad, and let ${\mathbf{V}}=J({\mathbf{V}}_0)$, then ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. A function $f:X\to Y$ in ${\mathbf{C}}$ stands for a function $f:X\to (S\times Y)^S$ in ${\mathbf{V}}_0$. Now, in addition to the usual framework of monads, for all $f,g:X\to Y$ in ${\mathbf{C}}$, i.e., $f,g:X\to (S\times Y)^S$ in ${\mathbf{V}}_0$, let: $$f{\lesssim}g \iff {\pi_Y}^S \circ g = {\pi_Y}^S \circ f \;,$$ where ${\pi_Y}^S:(S\times Y)^S\to Y^S$ associates to each map $m:S\to S\times Y$ the map $\pi_Y\times m:S\to Y$. The relation ${\lesssim}$ defines a semi-conguence on ${\mathbf{C}}$, and ${\langle f,v \rangle}$ is a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category. The characteristic property of the semi-product $f\times v$ can be illustrated as follows: $$\xymatrix@C=5pc{ x_1 \ar@{\|->}[r]^{f} & (s\mapsto(s',y_1)) \\ (x_1,x_2) \ar@{\|~>}[u] \ar@{\|~>}[d] \ar@{\|->}[r]^{f\times v} \ar@{}[rd]\|{{\gtrsim}} \ar@{}[ru]\|{=} & (s\mapsto(s',y_1,y_2)) \ar@{\|~>}[u] \ar@{\|~>}[d] \\ x_2 \ar@{\|~>}[r]^{v} & (s\mapsto(s,y_2))\ne(s\mapsto(s',y_2)) \ar@{\|~>}[r]^{\qquad\pi_Y^{S}} & (s\mapsto y_2) \\ }$$ Conclusion ========== We have presented a new categorical framework, called a cartesian effect category, for dealing with the issue of multiple arguments in programming languages with computational effects. The major new feature in cartesian effect categories is the introduction of a semi-congruence, which allows to define semi-products and to prove their properties by decorating the usual definitions, properties and proofs about products in a category. Forthcoming work should study the nesting of several effects. In order to deal with other issues related to effects, we believe that the idea of decorations in logic can be more widely used. This is the case for dealing with exceptions [@B-DuvRey05] (note that a previous attempt to define decorated products can be found in [@F-DuvRey04]). The framework of decorations might be used for generalizing this work in the direction of closed Freyd categories [@PowerThielecke99]. or traced premonoidal categories [@BentonHyland03]. Moreover, with one additional level of abstraction, decorations can be obtained from morphisms between logics, in the context of diagrammatic logics [@F-DuvLai02; @A-Duv03]. Proofs in cartesian effect categories {#app:proof} ===================================== Here are proofs for some results in section \[subsec:weak-prod\], called basic proofs, followed by their decorated versions for the corresponding results in section \[subsec:effect-semi\]. All basic proofs are straightforward. All proofs are presented in a formalized way: each property is preceded by its label and followed by its proof. For the basic proofs, the properties of the congruence are denoted ${\mathit{trans}}$, ${\mathit{sym}}$, ${\mathit{subst}}$, ${\mathit{repl}}$, for respectively transitivity, symmetry, substitution, replacement. For the decorated proofs, the properties of the congruence and the semi-congruence are still denoted ${\mathit{trans}}$, ${\mathit{sym}}$, ${\mathit{subst}}$, ${\mathit{repl}}$, with subscript either ${\equiv}$ or ${\lesssim}$. It should be reminded that ${\mathit{sym}}_{{\lesssim}}$ does not hold, and that ${\mathit{repl}}_{{\lesssim}}$ is allowed only with respect to a pure function: if $g_1{\lesssim}g_2:Y\to Z$ and $v:Z{\rightsquigarrow}W$ then $v\circ g_1 {\lesssim}v\circ g_2:Y\to W$. In addition, ${\mathit{comp}}$ means compatibiblity of ${\lesssim}$ with ${\equiv}$, which means that if either $f_1{\equiv}f_2{\lesssim}f_3$ or $f_1{\lesssim}f_2{\equiv}f_3$ then $f_1{\lesssim}f_3$. In decorated proofs, “like basic” means that this part of the proof is exactly the same as in the basic proof. Proofs of propositions \[prop:weak-assoc\], \[prop:effect-assoc\](associativity) and \[prop:weak-seq\], \[prop:effect-seq\] (parallelism) are left to the reader. \ ----------- ---------------------------------------------------------------- -------------------------------------- 1.$\quad$ When $X_1=X_2$ $(a_1)$ $q_1\circ {\langle f_1,f_2 \rangle} \equiv f_1$ $(b_1)$ $f_1 \equiv f'_1$ $(c_1)$ $q_1\circ {\langle f_1,f_2 \rangle} \equiv f'_1$ $(a_1)$, $(b_1)$, ${\mathit{trans}}$ $(c_2)$ $q_2\circ {\langle f_1,f_2 \rangle} \equiv f'_2$ like $(c_1)$ $(d)$ ${\langle f_1,f_2 \rangle} \equiv {\langle f'_1,f'_2 \rangle}$ $(c_1)$, $(c_2)$ 2.$\quad$ In all cases $(e_1)$ $f_1\equiv f'_1$ $(f_1)$ $f_1\circ p_1\equiv f'_1\circ p_1$ $(e_1)$, ${\mathit{subst}}$ $(f_2)$ $f_2\circ p_2\equiv f'_2\circ p_2$ like $(f_1)$ $(g)$ ${\langle f_1\circ p_1,f_2\circ p_2 \rangle} \equiv $(f_1)$, $(f_2)$, $(1)$ {\langle f'_1\circ p_1,f'_2\circ p_2 \rangle}$ ----------- ---------------------------------------------------------------- -------------------------------------- \ \ ----------- ----------------------------------------------------------------------------------------------------- ------------------------------------- 1.$\quad$ When $X_1=X_2$ $(c_1)$ $q_1\circ {\langle f_1,f_2 \rangle} {\equiv}f'_1$ like basic $(a_2)$ $q_2\circ {\langle f_1,f_2 \rangle} {\lesssim}f_2$ $(b_2)$ $f_2 {\equiv}f'_2$ $(c_2)$ $q_2\circ {\langle f_1,f_2 \rangle} {\lesssim}f'_2$ $(a_2)$, $(b_2)$, ${\mathit{comp}}$ $(d)$ ${\langle f_1,f_2 \rangle} {\equiv}{\langle f'_1,f'_2 \rangle}$ $(c_1)$, $(c_2)$ 2.$\quad$ In all cases $(g)$ ${\langle f_1\circ p_1,f_2\circ p_2 \rangle} {\equiv}{\langle f'_1\circ p_1,f'_2\circ p_2 \rangle}$ like basic ----------- ----------------------------------------------------------------------------------------------------- ------------------------------------- \ \ The three left handsides can be illustrated as follows: $$\xymatrix@C=3pc@R=1.5pc{ & & Z_1 \\ X \ar[r]^{f} & Y \ar[ru]^{g_1} \ar[rd]_{g_2} \ar[r] & \bullet \ar[u] \ar[d] \ar@{}[ld]\|(.3){\equiv} \ar@{}[lu]\|(.3){\equiv} \\ & & Z_2 \\ } \quad \xymatrix@C=3pc@R=1.5pc{ & Y_1 \ar[r]^{g_1} & Z_1 \\ X \ar[ru]^{f_1} \ar[rd]_{f_2} \ar[r] & \bullet \ar[u] \ar[d] \ar[r] \ar@{}[ld]\|(.3){\equiv} \ar@{}[lu]\|(.3){\equiv} & \bullet \ar[u] \ar[d] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} \\ & Y_2 \ar[r]_{g_2} & Z_2 \\ } \quad \xymatrix@C=3pc@R=1.5pc{ X_1\ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \\ \bullet \ar[u] \ar[d] \ar[r] & \bullet \ar[u] \ar[d] \ar[r] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} & \bullet \ar[u] \ar[d] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} \\ X_2 \ar[r]_{f_2} & Y_2 \ar[r]_{g_2} & Z_2 \\ }$$ ----------- ------------------------------------------------------------------------------------ --------------------------------------------- 1.$\quad$ When $f_1=f_2(=f)$ $(a_1)$ $r_1\circ {\langle g_1,g_2 \rangle} \equiv g_1$ $(b_1)$ $r_1\circ{\langle g_1,g_2 \rangle}\circ f\equiv g_1\circ f $ $(a_1)$, ${\mathit{subst}}$ $(b_2)$ $r_2\circ {\langle g_1,g_2 \rangle} \circ f \equiv g_2 \circ f $ like $(b_1)$ $(c)$ ${\langle g_1,g_2 \rangle} \circ f \equiv {\langle g_1\circ f,g_2\circ f \rangle}$ $(b_1)$, $(b_2)$ 2.$\quad$ When $X_1=X_2$ $(d)$ $(g_1\times g_2) \circ {\langle f_1,f_2 \rangle} $(1)$ \equiv {\langle g_1\circ q_1 \circ {\langle f_1,f_2 \rangle}, g_2\circ q_2 \circ {\langle f_1,f_2 \rangle} \rangle}$ $(e_1)$ $q_1 \circ {\langle f_1,f_2 \rangle} \equiv f_1$ $(f_1)$ $g_1\circ q_1\circ{\langle f_1,f_2 \rangle} \equiv g_1\circ f_1$ ${\mathit{repl}}$ $(f_2)$ $g_2\circ q_2 \circ {\langle f_1,f_2 \rangle}\equiv g_2\circ f_2$ like $(f_1)$ $(g)$ ${\langle g_1\circ q_1 \circ {\langle f_1,f_2 \rangle}, $(f_1)$, $(f_2)$, prop. \[prop:weak-equiv\] g_2\circ q_2 \circ {\langle f_1,f_2 \rangle} \rangle} \equiv {\langle g_1\circ f_1,g_2\circ f_2 \rangle}$ $(h)$ $(g_1\times g_2) \circ {\langle f_1,f_2 \rangle} \equiv $(d)$, $(g)$, ${\mathit{trans}}$ {\langle g_1\circ f_1,g_2\circ f_2 \rangle}$ 3.$\quad$ In all cases $(k)$ $ (g_1 \times g_2) \circ {\langle f_1\circ p_1,f_2\circ p_2 \rangle} \equiv $(2)$ {\langle g_1\circ f_1\circ p_1,g_2\circ f_2\circ p_2 \rangle}$ ----------- ------------------------------------------------------------------------------------ --------------------------------------------- \ \ The three left handsides can be illustrated as follows: $$\xymatrix@C=3pc@R=1.5pc{ & & Z_1 \\ X \ar@{~>}[r]^{v} & Y \ar[ru]^{g_1} \ar@{~>}[rd]_{w_2} \ar[r] & \bullet \ar@{~>}[u] \ar@{~>}[d] \ar@{}[ld]\|(.3){{\gtrsim}} \ar@{}[lu]\|(.3){{\equiv}} \\ & & Z_2 \\ } \quad \xymatrix@C=3pc@R=1.5pc{ & Y_1 \ar[r]^{g_1} & Z_1 \\ X \ar[ru]^{f_1} \ar@{~>}[rd]_{v_2} \ar[r] & \bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r] \ar@{}[ld]\|(.3){{\gtrsim}} \ar@{}[lu]\|(.3){{\equiv}} & \bullet \ar@{~>}[u] \ar@{~>}[d] \ar@{}[ld]\|{{\gtrsim}} \ar@{}[lu]\|{{\equiv}} \\ & Y_2 \ar@{~>}[r]_{w_2} & Z_2 \\ } \quad \xymatrix@C=3pc@R=1.5pc{ X_1\ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \\ \bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r] & \bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r] \ar@{}[ld]\|{{\gtrsim}} \ar@{}[lu]\|{{\equiv}} & \bullet \ar@{~>}[u] \ar@{~>}[d] \ar@{}[ld]\|{{\gtrsim}} \ar@{}[lu]\|{{\equiv}} \\ X_2 \ar@{~>}[r]_{v_2} & Y_2 \ar@{~>}[r]_{w_2} & Z_2 \\ }$$ ----------- --------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------ 1.$\quad$ When $f_1=v_2(=v)$ $(b_1)$ $r_1\circ{\langle g_1,w_2 \rangle}\circ v{\equiv}g_1\circ v $ like basic $(a_2)$ $r_2\circ {\langle g_1,w_2 \rangle} {\lesssim}w_2 $ $(b_2)$ $r_2\circ{\langle g_1,w_2 \rangle}\circ v{\lesssim}w_2\circ v $ $(a_1)$, ${\mathit{subst}}_{{\lesssim}}$ $(c)$ ${\langle g_1,w_2 \rangle} \circ v {\equiv}{\langle g_1\circ v,w_2\circ v \rangle}$ $(b_1)$, $(b_2)$ 2.$\quad$ When $X_1=X_2$ $(d)$ $(g_1\times w_2) \circ {\langle f_1,v_2 \rangle} $(1)$ {\equiv}{\langle g_1\circ q_1 \circ {\langle f_1,v_2 \rangle}, w_2\circ q_2 \circ {\langle f_1,v_2 \rangle} \rangle}$ $(f_1)$ $g_1\circ q_1\circ{\langle f_1,v_2 \rangle} {\equiv}g_1\circ f_1$ like basic $(e_2)$ $q_2 \circ {\langle f_1,v_2 \rangle} {\lesssim}v_2$ $(f_2)$ $w_2\circ q_2\circ{\langle f_1,v_2 \rangle} {\lesssim}w_2\circ v_2$ ${\mathit{repl}}_{{\lesssim}}$ ($w_2$ is pure) $(g)$ ${\langle g_1\circ q_1 \circ {\langle f_1,v_2 \rangle}, $(f_1)$, $(f_2)$, prop. \[prop:weak-equiv\] w_2\circ q_2 \circ {\langle f_1,v_2 \rangle} \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle}$ $(h)$ $(g_1\times w_2) \circ {\langle f_1,v_2 \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle}$ $(d)$, $(g)$, ${\mathit{trans}}_{{\equiv}}$ 3.$\quad$ In all cases $(k)$ $ (g_1 \times w_2) \circ {\langle f_1\circ p_1,v_2\circ p_2 \rangle} {\equiv}{\langle g_1\circ f_1\circ p_1,w_2\circ v_2\circ p_2 \rangle}$ $(2)$ ----------- --------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------ \ \ The two left handsides can be illustrated as follows: $$\xymatrix@C=2pc@R=1.5pc{ & Y_1 \ar[r]^{{\mathrm{id}}} & Y_1 \\ X \ar[ur]^{f_1} \ar[dr]_{f_2} \ar[r] & Y_2\times Y_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]\|(.3){\equiv} \ar@{}[lu]\|(.3){\equiv} & Y_1\times Y_2 \ar[u]\ar[d] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} \\ & Y_2 \ar[r]_{{\mathrm{id}}} & Y_2 \\ } \quad \xymatrix@C=2pc@R=1.5pc{ X_1 \ar[r]^{{\mathrm{id}}} & X_1 \ar[r]^{f_1} & Y_1 \ar[r]^{{\mathrm{id}}} & Y_1 \\ X_1\times X_2 \ar[u]\ar[d] \ar[r] & X_2\times X_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} & Y_2\times Y_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} & Y_1\times Y_2 \ar[u]\ar[d] \ar@{}[ld]\|{\equiv} \ar@{}[lu]\|{\equiv} \\ X_2 \ar[r]_{{\mathrm{id}}} & X_2 \ar[r]_{f_2} & Y_2 \ar[r]_{{\mathrm{id}}} & Y_2 \\ }$$ ----------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------- 1.$\quad$ When $X_1=X_2$ $(a_1)$ $q_1\circ\gamma_Y \equiv q'_1$ $(b_1)$ $q_1\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} $(a_1)$, ${\mathit{subst}}$ \equiv q'_1 \circ {\langle f_2,f_1 \rangle} $ $(c_1)$ $q'_1 \circ {\langle f_2,f_1 \rangle} \equiv f_1$ $(d_1)$ $q_1\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv f_1$ $(b_1)$, $(c_1)$, ${\mathit{trans}}$ $(d_2)$ $q_2\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv f_2 $ like $(d_1)$ $(e)$ $\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv {\langle f_1,f_2 \rangle}$ $(d_1)$, $(d_2)$ 2.$\quad$ In all cases $(f)$ ${\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} prop. \[prop:weak-comp\], ${\mathit{sym}}$ \equiv {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $ $(g)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} ${\mathit{repl}}$ \equiv \gamma_Y \circ {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $ $(h)$ $\gamma_Y \circ {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $(1)$ \equiv {\langle f_1\circ p'_1\circ \gamma_X^{-1},f_2\circ p'_2\circ \gamma_X^{-1} \rangle}$ $(i_1)$ $p'_1\circ \gamma_X^{-1} \equiv p_1$ $(j_1)$ $f_1\circ p'_1\circ \gamma_X^{-1} \equiv f_1\circ p_1$ $(i_1)$, ${\mathit{repl}}$ $(j_2)$ $f_2\circ p'_2\circ \gamma_X^{-1} \equiv f_2\circ p_2$ like $(j_1)$ $(k)$ $ {\langle f_1\circ p'_1\circ \gamma_X^{-1},f_2\circ p'_2\circ \gamma_X^{-1} \rangle} $(j_1)$, $(j_2)$, prop. \[prop:weak-equiv\] \equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$ $(l)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} $(g)$, $(h)$, $(k)$, ${\mathit{trans}}$ \equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$ ----------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------- \ \ The two left handsides can be illustrated as follows: $$\xymatrix@C=2pc@R=1.5pc{ & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\ X \ar[ur]^{f_1} \ar@{~>}[dr]_{v_2} \ar[r] & Y_2\times Y_1 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r] \ar@{}[ld]\|(.3){{\gtrsim}} \ar@{}[lu]\|(.3){{\equiv}} & Y_1\times Y_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{}[ld]\|{{\equiv}} \ar@{}[lu]\|{{\equiv}} \\ & Y_2 \ar@{~>}[r]_{{\mathrm{id}}} & Y_2 \\ } \quad \xymatrix@C=2pc@R=1.5pc{ X_1 \ar@{~>}[r]^{{\mathrm{id}}} & X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\ X_1\times X_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r] & X_2\times X_1 \ar@{~>}[u]\ar@{~>}[d] \ar[r] \ar@{}[ld]\|{{\equiv}} \ar@{}[lu]\|{{\equiv}} & Y_2\times Y_1 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r] \ar@{}[ld]\|{{\gtrsim}} \ar@{}[lu]\|{{\equiv}} & Y_1\times Y_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{}[ld]\|{{\equiv}} \ar@{}[lu]\|{{\equiv}} \\ X_2 \ar@{~>}[r]_{{\mathrm{id}}} & X_2 \ar@{~>}[r]_{v_2} & Y_2 \ar@{~>}[r]_{{\mathrm{id}}} & Y_2 \\ }$$ ----------- ------------------------------------------------------------------------------------ ---------------------------------------- 1.$\quad$ When $X_1=X_2$ $(d_1)$ $q_1\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}f_1$ like basic $(a_2)$ $q_2\circ\gamma_Y {\equiv}q'_2$ $(b_2)$ $q_2\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} $(a_2)$, ${\mathit{subst}}_{{\equiv}}$ {\equiv}q'_2 \circ {\langle v_2,f_1 \rangle} $ $(c_2)$ $q'_2 \circ {\langle v_2,f_1 \rangle} {\lesssim}v_2$ $(d_2)$ $q_2\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} {\lesssim}v_2$ $(b_2)$, $(c_2)$, ${\mathit{comp}}$ $(e)$ $\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}{\langle f_1,v_2 \rangle}$ $(d_1)$, $(d_2)$ 2.$\quad$ In all cases $(l)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} like basic \equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$ ----------- ------------------------------------------------------------------------------------ ---------------------------------------- \
--- author: - 'Iskander A. TAIMANOV [^1]' title: Singular spectral curves in finite gap integration --- Introduction {#introduction .unnumbered} ============ This article is an extended version of the talk given by the author at the conference “Geometry, Dynamics, Integrable Systems — GDIS 2010” (Serbia, September 7–13, 2010) dedicated to the 60th birthdays of B.A. Dubrovin, V.V. Kozlov, I.M. Krichever, and A.I. Neishtadt. In the talk as well as in the text we restrict ourselves to a description of a pair of examples studied by us together with A.E. Mironov [@MT1; @MT2] and P.G. Grinevich [@GT1; @GT2] and which demonstrate how in integrable problems from differential geometry there naturally appear singular spectral curves: - [in the construction of finite gap orthogonal curvilinear coordinate systems and solutions of the associativity equations [@Krichever97] it is naturally to consider the degenerate case when the geometrical genus of a singular spectral curve is equal to zero [@MT1; @MT2].]{}\ In the case the construction of the Baker–Akhiezer function and of finite gap solutions is reduced to solving linear systems, solutions are expressed in terms of elementary functions and, in particular, one may obtain solutions, to the associativity equations, satisfying the quasihomogeneity condition which gives a construction of infinitely many unknown before Frobenius manifolds. <!-- --> - [it appears that for soliton equations with self-consistent sources the spectral curve may be deformed and a deformation reduces to creation and annihilation of double points [@GT1; @GT2].]{} We recall the main notions in §1, expose examples 1 and 2 in §2 and §3 respectively and briefly note in §4 more interesting examples of integrable problems in which singular spectral curves appear. Baker–Akhiezer functions on singular spectral curves {#s1} ==================================================== Spectral curves {#subsec1.1} --------------- The definitions of spectral (complex) curves [^2] may be splitted into three types which are related to each other: 1\) for one-dimensional differential operators these curves are defined via Bloch functions and first that was done for Schrödinger operators by Novikov [@Novikov1974]. This procedure includes an explicit construction of Bloch functions; 2\) for two-dimensional differential operators the spectral curves are defined implicitly for a fixed energy level via the “dispersion” relation between quasimo menta. These curves were introduced by Dubrovin, Krichever, and Novikov [@DKN] who did that for the Schrödinger operator in a magnetic field \[magnetic\] L = \| + A \| + U. In [@DKN] the inverse problem of reconstruction of such operators from algebro-geometrical spectral data which include the spectral curve was also solved for operators which are finite gap on a fixed energy level; 3\) for problems which are integrated by using Baker–Akhiezer functions, introduced by Krichever [@Krichever1977], the spectral curves are Riemann surfaces on which these functions are defined. [^3] We briefly expose the main notions referring for details to surveys [@DMN; @Krichever1977; @KN; @Dubrovin1981]. 1\) In the finite gap integration spectral curves first appear in the initial article by Novikov [@Novikov1974] as Riemann surfaces which parameterize Bloch functions of the operator $L = -\frac{d^2}{dx^2} + u(x)$ which comes into the Lax representation $$\frac{\partial L}{\partial t} = [L,A]$$ for the Korteweg–de Vries (KdV) equation. A Bloch function of an operator $L$ with a periodic potential $u(x) = u(x+T)$ is a joint eigenfunction of $L$ and of the translation operator $\widehat{T}$: $\widehat{T}(f)(x) = f(x+T)$, i.e., a solution of the system \[bloch\] L = E, = e\^[ip(E)T]{}. Since $L$ is of order two, for every value of $E$ it has a two-dimensional space $V_E$ formed by all solutions of the equation $L\psi = E\psi$ and this space is invariant under $\widehat{T}$. Therewith [the spectral curve]{} $\Gamma$ appears as a two-sheeted cover $$\pi: \Gamma \to \C,$$ onto which the Bloch function $\psi(x,P)$ is correctly defined as a meromorphic function in $P \in \Gamma$, i.e. $\Gamma$ parameterizes all Bloch functions. Since $\det \widehat{T}\vert_{V_E} =1$, the eigenvalues of $\widehat{T}\vert_{V_E}$ are equal to $\pm 1$ at branch of this covering and $\widehat{T}$ is not diagonalized at these points. If there are finitely many such points then the operator $L$ is called [finite gap]{}, [^4] and the Riemann surface $\Gamma$ is completed up to a (complex) algebraic curve by a branch point at infinity $E=\infty$. Let us recall that the KdV equation is the initial equation of a hierarchy of equations which are represented in the form $$\frac{\partial L}{\partial t_k} = [L,A_k]$$ where $A_k$ is a differential operator of order $2k+1$ and the flows corresponding to these equations do commute. In particular, Novikov proved two fundamental results which lies in the basement of the finite gap integration method [@Novikov1974]: if $u(x)$ meets the equation of the form $$[L,A_N + c_1 A_{N-1} + \dots + c_{N-1}A_1] = 0$$ (Novikov’s equation), then the spectral curve is a Riemann surface of the form $w^2 = Q(E)$ with a polynomial $Q(E)$ of degree $2N+1$ (such solutions of the KdV equation are also called [finite gap]{}); the spectral curve $\Gamma$ and the quasimomentum function $p(E)$ which is correctly defined on $\Gamma$ up to $\frac{2\pi k}{T}, k \in \Z$, are first integrals of the KdV equation. In particular, for the roots of $Q(E)=0$ are first integrals of finite-gap solutions and the classical Kruskal–Miura integrals are described in terms of the branch points. For smooth real-valued solutions of the KdV equation the spectral curve is non-singular and real: the polynomial $Q(E)$ has no multiple roots and all roots are real. Analogously there are defined the spectral curves for all one-dimensional (scalar and matrix) differential and difference operators with periodic coefficients. The construction of the spectral curve reduces to a solution of ordinary differential equations to which the equation $L\psi= E\psi$ (see (\[bloch\])) reduces. 2\) For two-dimensional differential operators $L$ with periodic coefficients we define [Floquet functions]{} as solutions of the following problem: $$L\psi = E\psi, \ \ \psi(x+T_1,y) = e^{ip_1 T_1}\psi(x,y), \ \ \psi(x,y+T_2) e^{ip_2 T_2} \psi(x,y),$$ where $T_1$ and $T_2$ are the periods, and $p_1$ and $p_2$ are the quasimomenta. [The multipliers]{} Floquet functions are defined as $e^{ip_1 T_1}$ and $e^{ip_2 T_2}$. For hypoelliptic operators (for instance, for $\partial\bar{\partial} + \dots$ and $\partial_y - \partial^2_x + \dots$ where we denote by points the terms of lower order) one may show that the energy $E$ and the quasimomenta $p_1$ and $p_2$ meet an analytical “dispersion” relation \[curve\] F(E,p\_1,p\_2) = 0. The complex curve in the $p$-space defined by (\[curve\]) for a fixed value of $E$ is called [the spectral curve on the energy level $E$]{} [@DKN]. The existence of such a curve (possibly of infinite genus) can be established either by perturbation method [@Krichever1989], either by using the Keldysh theorem [@Taimanov2006], In particular, by using the Weierstrass representation, for tori in $\R^3$ (and in $\R^4$), via solutions to the equation $\D \psi = 0$ with $\D$ a two-dimensional Dirac operator with periodic coefficients in [@Taimanov1998] there was introduced the spectral curve of a torus, immersed into $\R^3$, as the spectral curve of the operator $\D$ on the zero energy level. It appeared that this spectral curve reflects the geometrical properties of the surface [@Taimanov2006] (see also [@Taimanov1997]). In §\[subsec4.2\] we discuss an example related to spectral curves of tori. In difference with the one-dimensional situation the spectral curve can not be constructed by solution of the direct problem. However it enters into the inverse problem data for the Baker–Akhiezer function method. 3\) Let us demonstrate a solution of the inverse problem and a construction of solutions of nonlinear equations by using Baker–Akhiezer functions as that was done initially for the Kadomtsev–Petviashvili (KP) equation. By developing methods, by Dubrovin and Its–Matveev, of constructing finite gap solutions of the KdV equations, Krichever defined the [Baker–Akhiezer function]{} (for the KP equation) and by using it solved the inverse problem of constructing solutions to the KP equation from the algebro-geometrical spectral data as follows [@Krichever1976]. Let $\Gamma$ be a smooth Riemann surface of genus $g$, $P$ be a fixed point on it and $k^{-1}$ be a local parameter near this point such that $k(P) = \infty$, and let $D = \gamma_1+\dots+\gamma_g$ be a generic effective divisor of degree $g$ ($\gamma_1,\dots,\gamma_g \in \Gamma$). Then - [there exists a unique function $\psi(x,y,t,\gamma)$, $\gamma \in \Gamma$ which meromorphic in $\gamma$ on $\Gamma \setminus P$, has poles only at points from $D$ (more precise, divisor of poles $\leq D$) and has an asymptotic $$\psi = e^{kx+k^2y+k^3t}\left(1+ \sum_{m>0} \frac{\xi_m(x,y,t)}{k^m}\right) \ \ \mbox{as $\gamma \to P$};$$ The function $u(x,y,t)= -2\partial_x \xi_1$ satisfies the KP equation $\frac{3}{4}\,u_{yy} = \frac{\partial}{\partial x}\left[u_t - \frac{1}{4}\left(6uu_x + u_{xxx}\right)\right]$, for all $\gamma \in \Gamma \setminus P$ we have \[spectral\] = L, , and there is an explicit formula for $u(x,y,t)$ in terms of the theta function of $\Gamma$: \[its\] u(x,y,t) = 2 (Ux+Vy+Wt+z\_0)+c. ]{} The set $(\Gamma,P,k^{-1},D)$ defines Krichever’s “spectral data” for the inverse problem for solutions to the KP equation and for the operator $\frac{\partial}{\partial y}-L$, for which, by (\[spectral\]), $\psi$ is a Floquet function on the zero energy level in the case when the potential $u$ is periodic. If on $\Gamma$ there exists a meromorphic function with a unique pole which is of the second order and at the point $P$, then there exists a two-sheeted covering $\Gamma \to \C P^1$ which is ramified at $P$, the function $u$ does not depend on $y$, $\psi$ reduces to the Bloch function of the Schrödinger operator $L$, the formula (\[its\]) reduces to the Its–Matveev formula for finite gap solutions of the KdV equation, and the set $(\Gamma,P,k^{-1},D)$ becomes the inverse problem data for a finite gap Schrödinger equation. These results for the KdV equation were obtained before in the articles by Dubrovin, Novikov, Its, and Matveev (see the survey [@DMN]). A Riemann surface $\Gamma$ is called [the spectral curve]{}, and solutions obtained by this method are called [finite gap]{}. Therewith quasiperiodic solutions are not excluded (it is easy to notice from (\[its\]) that such solutions may be obtained by using this method). This terminology is used for all problems which are solved by using Baker–Akhiezer functions (by using [the finite gap integration method]{}). Real-valued periodic solutions of the KdV, KP, and sine-Gordon equations are constructed from smooth Riemann surfaces. Theta functional formulas are rather complicated because parameters coming into them are related by complex transcendent equations. At the same time for some equations to which finite gap integration is applied interesting solutions are constructed from singular spectral curves including curves with geometrical genus zero. In the latter case a solution is expressed in terms of elementary functions and admits a simple qualitative investigation (see §§2 and 3). Algebraic curves with singularities {#subsec1.2} ----------------------------------- Let us recall the main notions related to complex algebraic curves with singularities (singular curves; see details in [@Serre Chapter 4]). Let $\Gamma$ be a complex algebraic curve with singularities. Then there exists a morphism of a smooth algebraic curve $\Gamma_\nm$: $$\pi: \Gamma_\nm \to \Gamma,$$ such that 1\) in $\Gamma_\nm$ there is a finite set of points $S$, splitted into subclasses and $\pi$ maps $S$ exactly into the set of singular points $\Sing = \Sing \Gamma$ of $\Gamma$ and moreover the preimage of every point from $\Sing$ consists in a certain subclass of $S$; 2\) the mapping $\pi: \Gamma_\nm \setminus S \to \Gamma \setminus \Sing$ is a smooth one-to-one projection; 3\) every regular mapping $F: X \to \Gamma$ of a nonsingular algebraic variety $X$ with everywhere dense image $F(X) \subset \Gamma$ descends through $\Gamma_\nm$, i.e $F = \pi G$ for some regular mapping $G: X \to \Gamma_\nm$. A mapping $\pi$ satisfying these conditions is called [the normalization]{} of $\Gamma$ and is uniquely defined by the conditions. Sometimes the curve $\Gamma_\nm$ itself is called the normalization. The genus of $\Gamma_\nm$ is called [the geometrical genus]{} of $\Gamma$ and is denoted by $p_g(\Gamma)$. Into the Riemann–Roch formula there enters [the arithmetic genus]{} $p_a(\Gamma)$ which is a sum of the geometric genus and a positive contribution of singularities (the points from $\Sing$). For a smooth curve we have $p_a = p_g$. A meromorphic $1$-form $\omega$ on $\Gamma_\nm$ defines [a regular differential]{} on $\Gamma$ if for every point $P \in \Sing$ we have $$\sum_{\pi^{-1}(P)} \mbox{Res}\ (f\omega) = 0$$ for all meromorphic functions $f$, on $\Gamma_\nm$, which descends to functions on $\Gamma$, i.e have the same value at points from every divisor $D_i$ and have no poles at $\pi^{-1}(P)$. Regular differentials may have poles at the preimages of singular points. The dimension of the space of regular differentials is equal to $p_a(\Gamma)$. Let us take on an irreducible algebraic curve $\Gamma_\nm$ $s$ families $D_1,\dots,D_s$, consisting in $r_1,\dots,r_s$ points all of which are different and let us construct $\Gamma$ by identifying all points from the same family. Then $$p_a(\Gamma) = p_g(\Gamma) + \sum_{i=1}^s (r_i-1).$$ Let us formulate the Riemann–Roch theorem for algebraic curves with singularities: - [Let $L(D)$ be a space of meromorphic functions on $\Gamma$ with from $D = \sum n_P P$ of order $\leq n_P$ and $\Omega(D)$ be a space of regular differentials, on $\Gamma$, which has at every point $P \in \Sing$ a zero of order not less than $n_P$. Then $$\dim L(D) - \dim \Omega(D) = \deg D + 1 - p_a(\Gamma).$$ For generic divisor $D$ with $\deg D \geq p_a$ we have $\dim \Omega(D)=0$ and $$\dim L(D) = \deg D + 1 - p_a(\Gamma).$$ ]{} The standard scheme of proving the uniqueness of the Baker–Akhiezer function is based on the Riemann–Roch theorem ad the genus $g$ of a smooth spectral curve $\Gamma$ comes into all reasonings as the arithmetic genus. For the case of singular spectral curves it is enough to replace $g$ by $p_a$ in all arguments and definitions. Let us expose the simplest examples of one-dimensional finite gap Schrödinger operators with singular spectral curves. [Example 1 (a curve with a double point).]{} Let $\Gamma_\nm = \C P^1 = \C \cup \{\infty\}$ with a parameter $k$. We construct a curve $\Gamma$ by identifying the points $k=\pm \lambda$ on $\Gamma_\nm$. We have $p_a(\Gamma)=1$ and therefore put $D = \{k=0\}$. From the spectral data $(\Gamma,\infty,k,D)$ there are constructed the Schrödinger operator [^5] $$L = -\frac{d^2}{dx^2} + \frac{2\lambda^2}{\sin^2\,(\lambda x)}$$ and the Baker–Akhiezer function $$\psi(k,x) = \left(1 + \frac{i\lambda}{k}\frac{\cos(kx)}{\sin(kx)}\right)e^{ikx}.$$ [Example 2 (a cuspidal curve).]{} Let $\Gamma$ be the same as in the previous example. The curve $\Gamma$ which homeomorphic to $\Gamma_\nm$ but has a cuspidal singularity at $k=0$ is obtained by restricting of the class of locally holomorphic functions on $\Gamma_\nm$: we assume that a function $f$ which is holomorphic near $k=0$ on $\Gamma_\nm$ is holomorphic in a neighborhood of $k=0$ on $\Gamma$ if and only if \[cusp\] f\^\_[k=0]{} = 0. We have $p_a(\Gamma)=1$. The spectral data $(\Gamma,\infty,k,D = \{k=0\})$ defines the potential $$u(x) = \frac{2}{x^2},$$ and the function $$\psi_1(k,x) = \left(-\frac{\partial}{\partial x} + \frac{1}{x}\right) e^{ikx} = \left(-ik+\frac{1}{x}\right)e^{ikx}$$ parameterizes all Bloch function and satisfies (\[cusp\]). After diving it by $-ik$ we obtain the Baker–Akhiezer function normalized by the asymptotic: $$\psi(k,x) = \left(1 +\frac{i}{kx}\right)e^{ikx} \ \ \ \ \mbox{as $k \to \infty$}.$$ This potential is obtained from the potentials described in Example 1 in the limit as $\lambda \to 0$. [Example 3.]{} The potential from Example 2 enters a series of rational soliton potentials: \[hcusp\] u\_l(x) = . The Baker–Akhiezer function for an operator with the potential (\[hcusp\]) is defined on the curve $\Gamma$ which is obtained from $\Gamma_\nm = \C P^1$ by assuming the condition which is analogous to (\[cusp\]): $$f^\prime = f^{\prime\prime} = \dots = f^{(l)} = 0 \ \ \ \mbox{при $k=0$}.$$ We have $p_a(\Gamma) = l$, $D = lQ$, where $Q = \{k=0\}$, and another spectral data are the same as in Example 2. The Baker–Akhiezer function has the form $$\psi_l(k,x) = \frac{1}{(-ik)^l}\left(-\frac{\partial}{\partial x} + \frac{l}{x}\right)\dots \left(-\frac{\partial}{\partial x} + \frac{1}{x}\right) e^{ikx}.$$ The Riemann surface $\Gamma$ is defined by the equation $y^2 = x^{2l+1}$ and the normalization has the form $$\C \to \Gamma \setminus\{\infty\}, \ \ \ t \to (x = t^2, y = t^{2n+1}).$$ Usually the Baker–Akhiezer function is constructed as a function $\psi$ on $\Gamma_\nm$ and multiple points and cuspidal singularities are defined by additional conditions $$\psi(Q_1)=\psi(Q_2), \ \ \ \psi^\prime (Q) = \dots = \psi^{(l)}(Q).$$ At the same time in soliton theory there were considered more general constraints of the form \[sing\] \_[j=1]{}\^M \_[i=1]{}\^[m\_j]{} a\_[ij]{} \^[(j)]{}(k,x,…)\_[k=Q\_j]{} = 0, Q\_1,…,Q\_M \_, or even systems of such conditions (see, for instance, [@DKMM]). In this case $\psi$ is a section of a certain bundle over the curve $\Gamma$ which we may determine as follows. It is evident that generically a constraint of the form (\[sing\]) posed on rational functions on $\Gamma_\nm$ does not distinguish any subfield: a product of two functions meeting such a condition does not satisfy it. At the same time sections of the bundle which we are looking for form a module over the field of rational functions on $\Gamma$. Let $f$ be such a function and $\psi$ be a section of the bundle defined by (\[sing\]), then the minimal conditions to which $f$ has to satisfy for $f\psi$ to satisfy (\[sing\]) for all sections are as follows: $$f(Q_1) = \dots = f(Q_M), \ f^\prime(Q_j) = \dots = f^{(m_j)}(Q_j) = 0, \ \ j=1,\dots,M,$$ i.e. $\Gamma$ is obtained from $\Gamma_\nm$ by gluing points $Q_1,\dots,Q_M$ and therewith every such a point with $m_j \geq 1$ is singular (see Example 1). However some of these functions (apparently all of them) may be obtained from functions on smooth spectral curves by degenerations of curves. Let expose the following [Example 4.]{} (A.E. Mironov) Let $\Gamma_1$ be a connected algebraic curve of genus $g$ on which there is constructed the Baker–Akhiezer function $\psi$. For simplicity we assume that $\psi$ is constructed from the spectral data for the KP equation however it satisfies an additional condition $$\psi(Q_1) = \alpha \psi(Q_2),$$ where $\alpha \neq 1$. By the Riemann–Roch theorem, the fixing of poles divisor of $\psi$ (it consists in $g+1$ generic points $\gamma_1,\dots,\gamma_{g+1}$ taken together with their multiplicities) together with the asymptotic at the essentially singular point determines $\psi$ uniquely. Let us consider a rational curve $\Gamma_2 = \C P^1$ with marked points $a$ and $b$, which differ from $\infty$, and let us identify $Q_1$ with $a$ and $Q_2$ with $b$ to obtain a reducible curve $\Gamma = \Gamma_1 \cup \Gamma_2 / \{Q_1 \sim a, Q_2 \sim b\}$ with the arithmetic genus $p_a(\Gamma)=g+1$. Let us construct on $\Gamma$ the Baker–Akhiezer function $\widetilde{\psi}$ with $g+2$ poles at $\gamma_1,\dots,\gamma_{g+1} \in \Gamma_1$ and at the point $q \in \Gamma_2$ such that $(b-q)/(a-q) = \alpha$. The number of poles is greater by one than the arithmetic genus and we pose an additional normalization condition $\psi(\infty) = 0$ where $\infty \in \Gamma_2$. Then the restriction of $\psi$ onto $\Gamma_2$ is the function $\varphi = \frac{c}{z-q}, c = \mathrm{const} \neq 0$. By the choice of $q$, we have $\varphi(a) = \alpha \varphi(b)$ and therefore the restriction of $\widetilde{\psi}$ onto $\Gamma_1$ is the desired function $\psi$. The Riemann surface $\Gamma$ is obtained by a degeneration of smooth surfaces via pinching a pair of contours into the points $Q_1 \sim a$ and $Q_2 \sim b$ and $\widetilde{\psi}$ is a degeneration of the Baker–Akhiezer functions on these surfaces. Since the inverse problem is solved from the asymptotics of $\psi$ at the essential singularities and there are no such singularities on $\Gamma_2$, then the restriction of $\widetilde{\psi}$ onto $\Gamma_2$ does not play any role. At the same time the real spectral curve on which $\widetilde{\psi}$ is defined as the Baker–Akhiezer function is $\Gamma$ and not $\Gamma_1$. Apparently one may obtain by this procedure the potential Schrödinger operators from [@Taimanov2003], [^6] which are constructed from singular spectral curves with odd arithmetic genus (for smooth spectral curves in this problem the genus is always even [@Krichever1989]). In [@Malanyuk] Malanyuk constructed finite gap solutions of the KPII equation for which the spectral curve is reducible and the essential singularity of the Baker–Akhiezer function lies in the rational component. These solutions are nonlinear superpositions of soliton waves. It is easy to notice that the construction from Example 4 is generalized for all conditions of the form $\alpha_1\psi(Q_1) + \dots + \alpha_M\psi(Q_M)=0$ with generic coefficients $\alpha_1,\dots,\alpha_M$. Therewith $\Gamma$ is obtained from $\Gamma_1$ by adding a rational curve $\C P^1$ which intersects $\Gamma_1$ at $Q_1,\dots,Q_M$. Orthogonal curvilinear coordinate systems and Frobenius manifolds {#s2} ================================================================= Orthogonal curvilinear coordinates and integrable systems of hydrodynamical type {#subsec2.1} -------------------------------------------------------------------------------- Into theory of integrable systems orthogonal curvilinear coordinates came from two sides: they do naturally appeared in the integrability problem for one-dimensional systems of Dubrovin–Novikov hydrodynamical type [@DN83; @N85; @DN89] and they also do appear as a result of a differential reduction as a particular case of metrics with diagonal curvature to whose explicit construction the inverse problem method was applied by Zakharov [@Z]. [Orthogonal curvilinear coordinate system]{} in a Riemannian manifold is a coordinate system $(u^1,\dots,u^N)$ such that in these coordinates the metric tensor takes the diagonal form $$\label{diagonal} ds^2 = \sum_{i=1}^N H_i^2(u) \left(du^i\right)^2.$$ We note that for $N \geq 4$ not every Riemannian metric admits locally orthogonal curvilinear coordinates. In fact, the existence of such coordinates is a very strong condition on a metric. The coefficients $H_i$ are called [the Lame coefficients]{}, and the expressions $$\label{rotation} \beta_{ij} = \frac{1}{H_j}\frac{\partial H_i}{\partial u^j}$$ define [the rotation coefficients]{}. A metric is called of [Egorov]{} type if it admits locally orthogonal curvilinear coordinates with symmetric rotation coefficients: $$\beta_{ij} = \beta_{ji}, \ \ \ \ 1 \leq i,j \leq N.$$ In this case the metric is of [potential]{} type: $$g_{ii} = H_i^2 = \frac{\partial V}{\partial u^i}, \ \ \ i=1,\dots,N,$$ for a certain potential $V(u)$. In theory of hydrodynamical systems the case of Riemannian metrics is not distinguished and there is considered the general case of pseudoriemannian metrics for which in terms of orthogonal coordinates the metric tensor takes the form $$ds^2 = \sum_{i=1}^n \varepsilon_i H_i^2(u) \left(du^i\right)^2, \ \ \varepsilon_i = \pm 1,$$ the rotation coefficients are defined similar to the Riemannian case, and the Egorov condition takes the form $$\beta_{ij} = \varepsilon_i \varepsilon_j \beta_{ji}.$$ A one-dimensional system of hydrodynamical type is an evolution system of the form $$u^i_t = \sum_{j=1}^N v^i_j(u) u^j_x, \ \ \ \ i=1,\dots,N,$$ where $u^1(x,t),\dots,u^N(x,t)$ are functions of one-dimensional spatial variable $x$ and a temporal variable $t$. Such a system is Hamiltonian if it admits a representation of the type $$u^i_t = \{u^i(x), \widehat{H}\},$$ whereе $$\widehat{H}(u) = \int H(u)dx$$ and the Poisson brackets (of hydrodynamical type, or called also now [Poisson–Dubrovin–Novikov brackets]{}) have the form $$\label{poisson} \{u^i(x),u^j(y)\} = g^{ij}(u(x))\delta^\prime(x-y) - g^{is}\Gamma^j_{sk}u^k_x \delta(x-y), \ \ \ g^{ij}=g^{ji}.$$ These notions were introduced by Dubrovin and Novikov [@DN83; @DN89] who proved that 1\) the expression (\[poisson\]) with a nondegenerate pseudoriemannian metric $g^{ij}$ (with upper indices) in the $N$-dimensional $u$-space defines Poisson brackets if and only if the metric is flat (has zero curvature) and $\Gamma^i_{jk}$ is the corresponding Levi-Civita connection; 2\) Hamiltonian systems of hydrodynamical type do appear by averaging such integrable systems as the KdV and sine-Gordon equations, and the nonlinear Schrödinger equation (NS). Novikov stated the conjecture that Hamiltonian systems of hydrodynamical type with a diagonal matrix $\left( v^i_j \right)$ are integrable. For such systems the coordinates $u^1,\dots,u^N$, are called [Riemann invariants]{} and the matrix $\left( g^{ij} \right)$ is also diagonal the $u$-space with orthogonal curvilinear coordinates $u^1$,…,$u^N$. This conjecture was proved by Tsarev who also introduced the integration procedure — the generalized godograph method [@Tsarev]. In theory of integrable systems an interest to Egorov metrics is due to Dubrovin who showed that if a flat metric $g^{ij}$, written in terms of Riemannian invariants, is of Egorov type then the system is superintegrable [@Dubrovin90]. [^7] In [@MF] the following nonlocal generalization of Poisson–Dubrovin–Novikov brackets was proposed: $$\{u^i(x),u^j(y)\} = g^{ij}(u(x))\delta^\prime(x-y) - g^{is}\Gamma^j_{sk}u^k_x \delta(x-y) + \frac{c}{2} \sgn (x-y) u^i_x u^j_y,$$ where $c = \mathrm{const}$, and it was proved that this expression defines Poisson brackets if and only if $\Gamma^i_{jk}$ is the Levi-Civita connection for the metric $g^{ij}$ with constant sectional curvature $c$ and that the generalized godograph method is applicable to Hamiltonian systems with such Poisson brackets. Therewith Riemann invariants define orthogonal curvilinear coordinates in the space of constant curvature $c$. The Lame equations {#subsec2.2} ------------------ In the 19th century the theory of orthogonal curvilinear coordinates was actively studied by leading geometers (Dupin, Gauss, Lame, Bianchi, Darboux). The classification problem for these coordinates was basically solved up to the beginning of the 20th century and the state of the theory for this moment was summarized in Darboux’s book [@Darboux]. All such coordinate systems are constructed from solutions of the Lame equations as follows. Let us split the nontrivial components of the Riemann curvature tensor $R_{ijkl}$ into three groups: 1\) all indices $i,j,k,l$ are pair-wise different; 2\) $R_{ijik}$ with pair-wise different $i,j,k$; 3\) $R_{ijij}$ with $i \neq j$. We recall that the Riemann tensor is skew-symmetric in the first and in the second pairs of indices: $R_{ijkl} = - R_{jikl} = -R_{ijlk}$ and $R_{ijkl} = R_{klij}$ for all $i,j,k,l$. For diagonalized metrics (\[diagonal\]) all components of the first type vanish and the vanishing of the components of the second type is equivalent to the system \[1\] = + , ij k, and the equations \[2\] ( ) + ( ) + \_[ki j]{} = 0, i j, are equivalent to the vanishing of the components of the third type: $R_{ijij} = 0$. The systems (\[1\]) and (\[2\]) consist in $\frac{N(N-1)(N-2)}{2}$ and $\frac{N(N-1)}{2}$ equations, respectively. From counting the numbers of equations and of variables it is clear that the system (\[1\])–(\[2\]) on the Lame coefficients is strongly overdetermined. The order of the system (\[1\])–(\[2\]) is minimized by introducing the rotation coefficients. In this case the equations (\[1\]) take the form \[4\] = \_[ik]{}\_[kj]{}, i j k, and the equation (\[2\]) are written as \[5\] + + \_[ki,j]{} \_[ki]{}\_[kj]{}=0, i j. The systems (\[4\]) and (\[5\]) form the system of [the Lame equations]{} and the equations (\[4\]) are just the compatibility condition for (\[rotation\]). A general solution of these equations depends on $\frac{N(N-1)}{2}$ arbitrary functions of two variables. Given a solution $\beta_{ij}$ to the Lame equations, the Lame coefficients are found from (\[rotation\]) as a solution to the Cauchy problem $$H_i(0,\dots,0,u^i,0,\dots,0)=h_i(u^i).$$ Therewith a solution depends on the initial data which are $N$ functions $h_i$ of one variable. The determination of Euclidean coordinates $x^1,$ $\dots,x^N$ as functions of $u^1,\dots,u^N$ (the immersion problem) is reduced to solving an overdetermined system of linear equations \[6\] = \_[l=1]{}\^N\_[ij]{}\^l, where the Christoffel symbols have the form $$\Gamma_{ij}^k=0,\ i\ne j\ne k; \ \ \Gamma_{kj}^k=\frac{1}{H_k}\frac{\partial H_k}{\partial u^j}; \ \ \Gamma_{ii}^k=-\frac{H_i}{(H_k)^2}\frac{\partial H_i}{\partial u^k},\ i \neq k.$$ By (\[1\]) and (\[2\]), the system (\[6\]) is compatible and determines an orthogonal curvilinear coordinate system up to motions of $\R^N$. Egorov metrics with zero curvature as an integrable system: Dubrovin’s theorem {#subsec2.3} ------------------------------------------------------------------------------ In [@Dubrovin90] Dubrovin did show that the problem of constructing flat Egorov metrics is integrated by methods of soliton theory. We have 1\) A Egorov (Riemannian or pseudoriemannian) metric is flat if and only if its rotation coefficients satisfy the equations $$\label{egorov1} \frac{\partial \beta_{ij}}{\partial u^k} = \beta_{ik}\beta_{kj}, \ \mbox{where $i,j,k$ are pair-wise different},$$ $$\label{egorov2} \sum_{k=1}^N \frac{\partial \beta_{ij}}{\partial u^k} = 0, \ \ \ i \neq j.$$ 2\) The system consisting of the equations (\[egorov1\]) and (\[egorov2\]), is the compatibility condition for the linear system $$\frac{\partial \psi_i}{\partial u^j} = \beta_{ij}\psi_j, \ \ \ i \neq j,$$ $$\sum_{k=1}^N \frac{\partial \beta_{ij}}{\partial u^i} = \lambda \psi_i, \ \ \ i=1,\dots,N,$$ where $\lambda$ is a spectral parameter. 3\) For a flat Egorov metric a restriction of the rotation coefficients $\beta_{ij}(u)$ onto every plane $u^i = a^i x + c^i t$ satisfies the $N$ waves equations $$[A, B_t] - [C, B_x] = [[A,B],[C,B]],$$ где $$A = \diag (a^1,\dots,a^N), \ C = \diag (c^1,\dots,c^N), \ B = (\beta_{ij}),$$ with an additional reduction $$\Im B =0, \ \ B^\top = JBJ, \ \ J = \diag (\varepsilon_1,\dots, \varepsilon_N)$$ (for Riemannian metrics all $\varepsilon_i = 1$). This theorem is proved by straightforward computations. Zakharov’s method of constructing metrics with diagonal curvature and orthogonal curvilinear coordinates {#subsec2.4} -------------------------------------------------------------------------------------------------------- Zakharov applied the inverse scattering method to constructing a wide class of metrics with diagonal curvature [@Z]. Let us expose it. The Riemann curvature tensor is interpreted as the curvature operator on the space of tangent bivectors. Let $M^N$ be a Riemannian (or pseudoriemannian) manifold, $\Lambda^2 TM^N$ be a linear bundle over $M^N$ for which a general fiber over a point is the space of bivectors at this point. The metric on $M^N$ defines a standard metric on fibers: if $e_1,\dots,e_N$ is an orthonormal basis for $T_x M^N$, the tangent space at в точке $x$, then $\{e_i \wedge e_j, i < j\}$ is an orthonormal basis in $\Lambda^2 T_x M^N$. Then the Riemann curvature tensor $R_{ijkl}$ defines the curvature operator $R$ by the formula $$\langle R \xi,\eta \rangle = R_{AB}\xi^A\eta^B, \ \ R_{AB} = R_{ijkl}, \ \ A = [ij], B = [kl],$$ where $\xi = \sum_A e_A \xi^A$ and $\eta = \sum_B e_B \eta^B$ is a decomposition of bivectors in the basis $e_A = e_i \wedge e_j, i<j$. Let us remark that the well-known in the relativity theory Petrov’s classification of four-dimensional solutions to the Einstein equations is bases on the classification of algebraic types of the curvature tensor. [^8] In [@Z] a manifold with diagonal curvature is defined as a manifold such that near every its point the metric is diagonalized, i.e reduces to the form (\[diagonal\]), and in thse coordinates the quadratic form $R_{AB}$ (or the curvature operator $R$) is also diagonal. The equations (\[egorov1\]) exactly describe the rtoation coefficients of metrics with diagonal curvature and are written as follows: \[7\] \_[i,j,k]{}\_[ijk]{}(I\_j I\_k - I\_i B I\_j B I\_k )=0, where $B(u) = (\beta_{ij})$ is an $(N\times N)$-matrix function, formed by the rotation coefficients, and $I_j = \diag (0,\dots,0,1,0,\dots,0)$ (the unity is at the $j$-th entry). Let us consider an auxiliary function $\widetilde{B}=\widetilde{B}(u,s)$, where $s = u^{N+1}$ is an additional variable, which satisfies the equations \[8\] \[I\_i,\] - \[I\_j,\] + I\_i I\_j - I\_j I\_i - \[\[I\_i,\],\[I\_j,\]\]=0, where $i,j=1,\dots,N+1$ and $I_{N+1}$ is the unit matrix. If $\widetilde{B}$ satisfies (\[8\]), then for any fixed value of $s$ the matrix-valued function $\widetilde{B}$ satisfies the $N$ waves equation (\[7\]). The system (\[8\]) admits the Lax representation $$[L_i,L_j]=0, \ \ \ L_j=\frac{\partial}{\partial u^j} + I_j \frac{\partial}{\partial s} + [I_j,\widetilde{B}].$$ Let us apply to it the dressing method by considering an integral equation of the Marchenko type \[9\] K(s,s\^,u)=F(s,s\^,u)+\_s\^K(s,q,u)F(q,s\^,u)dq, where $F(s,s^\prime,u)$ is a matrix-valued function. \[zakharov-theorem\] 1) If $F(s,s^\prime,u)$ meets the following conditions: а) the holds the equation \[10\] + I\_i F + F I\_i = 0, б) the equation (\[9\]) has a unique solution, then the function \[11\] (s,u) = K(s,s,u) satisfies (\[8\]) and therefore for any fixed value of $s$ the function $B(u) = \widetilde{B}(s,u)$ satisfies (\[7\]) and defines the rotation coefficients of a metric with diagonal curvature. The class of metrics with diagonal curvature is rather wide and, for instance, as it was noted by Zakharov it includes many solutions of the Einstein equations including the Schwarzschild metric. The problem posed by him on finding a reduction, of the inverse problem data, corresponding to Ricci-flat metrics stays an important open problem. By using a new and original trick, i.e. [a differential reduction]{}, he distinguishes on the language of inverse problem data, i.e. in terms of the function $F$, the class of flat metrics with diagonal curvature. Let us assume the conditions of Theorem \[zakharov-theorem\] and assume that $F$ satisfies the relation \[12\] (s,s\^,u) + (s\^,s,u)=0. Then $\widetilde{B}$ satisfies the equations (\[5\]) and the rotation coefficients constructed from $\widetilde{B}$ correspond to flat metrics of the form (\[diagonal\]), i.e. to orthogonal curvilinear coordinate systems. The system consisting of (\[10\]) and (\[12\]) has the following solution [@Z]: Let $\Phi_{ij}(x,y), i<j$, be $\frac{N(N-1)}{2}$ arbitrary functions of two variables and let $\Phi_{ii}(x,y)$ be $N$ arbitrary skew-symmetric functions: $$\Phi_{ii}(x,y)=-\Phi_{ii}(y,x).$$ Let us put $$F_{ij}=\frac{\partial \Phi_{ij}(s-u^i,s^\prime-u^j)}{\partial s},\ F_{ji}=\frac{\partial \Phi_{ij}(s^\prime-u^i,s-u^j)}{\partial s}, \ i \neq j,$$ $$F_{ii}=\frac{\partial \Phi_{ii}(s-u^i,s^\prime-u^i)}{\partial s}.$$ Then $F=(F_{ij})$ satisfies (\[10\]) and (\[12\]), and [for any fixed value of $s$ a solution $K$ of the equation (\[9\]) with such a matrix $F$ defines the rotation coefficients of an orthogonal coordinate system: $\beta_{ij}(u) = K_{ij}(s,s,u)$.]{} Notice that we have $\frac{N(N+1)}{2}$ functional parameters $\Phi_{ij}, i \leq j$, and a general solution depends on $\frac{N(N-1)}{2}$ functional parameters which implies that this method gives equivalent classes of dressings. A general theory of reductions, including differential reductions, which may be helpful for distinguishing other classes of metrics is discussed in [@ZM]. In particular, by a variation of the reduction (\[12\]) it is possible to distinguish metrics with diagonal curvature and constant sectional curvature $K \neq 0$, i.e. orthogonal curvilinear coordinates in the spaces of constant curvature $K$ (Zakharov). In this case the system (\[5\]) is replaced by the equations $$\frac{\partial \beta_{ij}}{\partial u^i} + \frac{\partial \beta_{ji}}{\partial u^j} + \sum_{k\ne i,j} \beta_{ki}\beta_{kj} = -KH_iH_j, \ \ \ i \neq j.$$ Krichever’s construction of finite gap orthogonal curvilinear coordinates {#subsec2.5} ------------------------------------------------------------------------- In [@Krichever97] Krichever proposed the finite gap version of a construction of orthogonal curvilinear coordinates in the Euclidean spaces, which in difference with [@Z] does not split into two parts: a construction of metrics with diagonal curvature and then distinguishing flat metrics among them but straightforwardly gives such coordinates. Let $\Gamma$ be a smooth complex algebraic curve, i.e. a compact smooth Riemann surface. Let us choose three effective divisors on $\Gamma$: $$P=P_1+\dots+P_N, \ \ D = \gamma_1+\dots+\gamma_{g+l-1}, \ \ R=R_1+\dots+R_l,$$ where $g$ -is the genus of $\Gamma$, $P_i,\gamma_j,R_k\in\Gamma$. Let us take a local parameter $k_i^{-1}$ near every point $P_i$, $i=1,\dots,N$, such that this parameter vanishes at the corresponding point. The Baker–Akhiezer function which corresponds to this spectral data $S = \{\Gamma,P,D,R\}$ is a function $\psi(u^1,\dots,u^N,z),\ z\in\Gamma$ such that 1\) $\psi\exp(- u^i k_i)$ is analytical near $P_i$, $i=1,\dots,N$; 2\) $\psi$ is meromorphic on $\Gamma\backslash\{\cup P_i\}$ with poles in $\gamma_j$, $j=1,\dots,g+l-1$; 3\) $\psi(u,R_k)=1$, $k =1,\dots,l$. For generic divisor $D$ such a function exists, unique and is expressed in terms of the theta-function of $\Gamma$. If $\Gamma$ is not connected it is assumed that the restriction of $\psi$ on every connected component meets such conditions. Let us take an additional divisor $Q=Q_1+\dots+Q_N$, on $\Gamma$, such that $Q_i \in \Gamma \setminus \{P \cup D \cup R\}, i=1,\dots,N$ and let us put $$x^j(u^1,\dots,u^N)=\psi(u^1,\dots,u^N,Q_j), \ j =1,\dots,N.$$ We have \[krichever-theorem\] 1) Let a holomorphic involution $\sigma:\Gamma\rightarrow\Gamma$ be defined on $\Gamma$ such that a\) $\sigma$ has exactly $2m$ fixed points, $m \leq N \leq 2m$: $P_1,\dots, P_N$ and $2m-N$ points from $Q$; b\) $\sigma(Q)=Q$, i.e. for points from $Q$ the involution either interchanges them, either let them fixed: $$\sigma(Q_k) = Q_{\sigma(k)}, \ \ \ k=1,\dots,N;$$ c\) $\sigma(k_i^{-1}) = -k_i^{-1}$ near $P_i$, $i=1,\dots,N$; d\) there exists a meromorphic differential $\Omega$ on $\Gamma$ such that its divisors of zeros and poles are of the form $$(\Omega)_0= D + \sigma D +P, \ \ \ (\Omega)_{\infty}=R+\sigma R+Q;$$ e\) $\Gamma_0 = \Gamma/\sigma$ is a smooth algebraic curve. Then $\Omega$ is a pullback of some meromorphic differential $\Omega_0$ on $\Gamma_0$ and we have the following equalities: $$\sum_{k,l} \eta_{kl}\frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} = \varepsilon^2_i H^2_i(u) \delta_{ij},$$ where $$H_i = \lim_{P \to P_i} \left(\psi e^{-u^i k_i}\right), \ \ \ \eta_{kl} = \delta_{k,\sigma(l)} \res_{Q_k} \Omega_0,$$ $$\Omega_0 = \frac{1}{2} \left(\varepsilon_i^2 \lambda_i + O(\lambda_i)\right) d\lambda_i, \ \lambda_i = k_i^{-2}, \ \mbox{at $P_i$, $i=1,\dots,N$}.$$ 2\) If moreover there exists such an antiholomorphic involution $\tau: \Gamma \to \Gamma$ such that all fixed points of $\sigma$ are fixed by $\tau$ and $$\tau^\ast(\Omega) = \overline{\Omega},$$ then $H_i(u)$ are real-valued for $u^1,\dots,u^N \in \R$, and $u^1,\dots,u^N$ are orthogonal curvilinear coordinates in the flat $N$-dimensional space with the metric $\eta_{kl} dx^k dx^l$. [^9] If all points from $Q$ are fixed by $\sigma$ and \[diff\] \_[Q\_1]{} \_0 = …= \_[Q\_N]{} \_0 = \^2\_0 > 0, then the functions $x^1(u^1,\dots,u^N),\dots,x^N(u^1,\dots,u^N)$ solve the immersion problem for orthogonal curvilinear coordinates $u^1,\dots,u^N$ and for the metric $ds^2 = H_1^2 (du^1)^2 + \dots + H_N^2 (du^N)^2$ where $$H_i = \frac{\varepsilon_i h_i}{\eta_0}, \ \ \ i=1,\dots,N.$$ [Remark 1.]{} Theorem still holds if instead of a) it is assumed that the functions $\psi\exp(-f^i(u^i)k_i)$ are analytical near $P_i$ where $f^i$ are some functions of one variable, which are invertible near zero, $i=1,\dots,N$. Therewith we do not differ orthogonal coordinates which are obtained by change of variables of the form $$u^i\rightarrow f^i(u^i).$$ It is also possible to change the condition $\psi(u,R_k) = 1, k=1,\dots,l$, by \[baker1\] (u,R\_k) =d\_k, k=1,…,l, where all constants $d_k$ do not vanish. Moreover we even can assume only that $d_k$ do not vanish simultaneously: \[baker2\] \|d\_1\|\^2 + …+ \|d\_l\|\^2 0, and still Theorem holds. For the case of orthognal coordinates in spaces of constant curvature Krichever’s method was recently generalized in [@BR]. Coordinate systems corresponding to singular spectral curves {#subsec2.6} ------------------------------------------------------------ Krichever’s construction (Theorem \[krichever-theorem\]) leads to theta-functional formulas are difficult for qualitative analysis. At the same time after degenerating the spectral curve $\Gamma$ we may obtain simpler formulas satisfying to solutions which lie on the boundary of the moduli space found by Krichever. Also for understand the structure of finite gap orthogonal curvilinear systems is helpful to find among them classical coordinates. The following theorem may serve for that. \[th1\] 1) Theorem \[krichever-theorem\] holds for a singular algebraic curve after replacing $g$ by $p_a(\Gamma)$, the arithmetic genus of $\Gamma$, and replacing the smoothness condition d) for $\Gamma/\sigma$ by the condition that $P_1,\dots,P_N$ and the poles of $\Omega$ are nonsingular points. Moreover we may assume that $\psi$ satisfies (\[baker1\]) and (\[baker2\]) instead of the equalities $\psi(u,R_k)=1, k=1,\dots,l$. In the case when $\Gamma_\nm$ is a union of smooth rational curves, i.e. of copies of $\C P^1$, the constructions of the Baker–Akhiezer function and of orthogonal coordinates reduce to simple computations with elementary functions and goes no further than solving linear systems. We demonstrate that by examples. We remark that the simplest spectral curves correspond to exotic coordinates therewith for classical coordinates the spectral curves are rather complicated. Let us recall that a regular differential $\Omega$ on $\Gamma$ is defined by differentials $\Omega_1,\dots,\Omega_s$ on irreducible components $\Gamma_1,\dots,\Gamma_s$. By its definition, we have $\sum_i \sum_{\pi^{-1}(P) \in \Gamma_i} \ \Omega_i$ $=0$ where $\pi: \Gamma_\nm \to \Gamma$ is the normalization and the summation is taken over all singular points $P$. The arithmetic genus $p_a$ is the dimension of the space of regular differentials. [Example 5.]{} [^10] Let us consider the simplest singular spectral curve: $\Gamma$ falls into two copies of $\C P^1$, i.e. $\Gamma_1$ and $\Gamma_2$, which intersects at two points which implies $p_a(\Gamma)=1$: $a \sim b, (-a) \sim (-b), \{a,-a\} \subset \Gamma_1, \{b,-b\} \subset \Gamma_2$ (see Fig. 1). We consider the case with the essential singularities lying in different irreducible components: at points $P_1=\infty\in\Gamma_1$ and $P_2=\infty\in\Gamma_2$. The Baker–Akhiezer function takes the form $$\psi_1(u^1,u^2,z_1)=e^{u^1z_1}\left(f_0(u^1,u^2)+ \frac{f_1(u^1,u^2)}{z_1-\alpha_1}+\dots +\frac{f_k(u^1,u^2)}{z_1-\alpha_{s_1}}\right),\ z_1\in\Gamma_1,$$ $$\psi_2(u^1,u^2,z_2)=e^{u^2z_2}\left(g_0(u^1,u^2)+ \frac{g_1(u^1,u^2)}{z_2-\beta_1}+\dots +\frac{g_n(u^1,u^2)}{z_2-\beta_{s_2}}\right),\ z_2\in\Gamma_2.$$ $$\psi_1(a)=\psi_2(b),\ \ \psi_1(-a)=\psi_2(-b).$$ (170,100)(-100,-80) (-10,0)(-10,30)(40,30) (40,30)(90,30)(90,0) (90,0)(90,-30)(40,-30) (40,-30)(-10,-30)(-10,0) (35,33) (60,0)(60,30)(110,30) (110,30)(160,30)(160,0) (160,0)(160,-30)(110,-30) (110,-30)(60,-30)(60,0) (105,33) (-10,0) (90,0) (60,0) (160,0) (75,24) (75,-24) (-25,0) (165,0) (95,0) (45,0) (63,20) (54,-26) (85,18) (80,-26) (60,-50) The general normalization condition is as follows \[normal\] \_1(R\_[1,i]{})= d\_[1,i]{}, \_2(R\_[2,j]{})= d\_[2,j]{}, where $R_{1,i}\in\Gamma_1, i=1,\dots,l_1$ и $R_{2,j} \in \Gamma_2, j=1, \dots,l_2$. We also have $l = l_1 + l_2 = s_1 + s_2$. We assume that $$\Omega_1=\frac{(z_1^2-\alpha_1^2)\dots (z_1^2-\alpha_{l_1}^2)} {z_1(z_1^2-a^2)(z_1^2-R_{1,1}^2)\dots(z_1^2-R_{1,l_1}^2)}\, dz_1,$$ $$\Omega_2=\frac{(z_2^2-\beta_1^2)\dots (z_2^2-\beta_{l_2}^2)} {z_2(z_2^2-b^2)(z_2^2-R_{2,1}^2)\dots(z_2^2-R_{2,l_2}^2)}\, dz_2.$$ Let us put $Q_1 = 0\in\Gamma_1, Q_2=0\in\Gamma_2$. If the following inequalities $$\res_a \Omega_1 = - \res_b \Omega_2,\ \ \res_{-a}\Omega_1 = -\res_{-b}\Omega_2,\ \ \res_{Q_1}\Omega_1 = \res_{Q_2}\Omega_2$$ then the differential $\Omega$, defined by the differentials $\Omega_1$ and $\Omega_2$, is regular and the condition (\[diff\]) is satisfied. In this case, by \[th1\], the coordinates $u^1$ and $u^2$ such that $$x^1(u) = \psi_1(u,0), \ \ x^2(u) = \psi_2(u,0),$$ are orthogonal. Let us consider the simplest case $l_1=0$ and $l_2 = 1$. Then we have $$\psi_1=e^{u^1z_1}f_0(u^1,u^2),\ \ \psi_2=e^{u^2z_2}\left(g_0(u^1,u^2)+\frac{g_1(u^1,u^2)}{z_2-c}\right).$$ The gluing conditions at the intersection points and the normalization condition take the form $\psi_1(a)=\psi_2(b), \psi_1(-a)=\psi_2(-b), \psi_2(r)=1, r = R \in \Gamma_2$ which imply $$\psi_1=e^{u^1z_1} \frac{2b(c-r)e^{au^1+(b-r)u^2}}{(b+c)(b-r)e^{2bu^2}-(b+r)(b-c)e^{2au^1}},$$ $$\psi_2=e^{u^2z_2}\left( \frac{e^{-ru^2}((b-c)e^{2au^1}+(b+c)e^{2bu^2})(c-r)} {(b+c)(b-r)e^{2bu^2}-(b-c)(b+r)e^{2au^1}}+\right.$$ $$\left. \frac{1}{z_2-c}\frac{(b^2-c^2)(r-c)e^{-ru^2}(e^{2au^1}-e^{2bu^2})} {(b+c)(r-b)e^{2bu^2}+(b-c)(b+r)e^{2au^1}}\right).$$ On the components the differential $\Omega$ is defined by the differentials $$\Omega_1 = -\frac{dz_1}{z_1(z_1^2-a^2)}, \ \ \Omega_2 = - \frac{(z_2^2-c^2)dz_2} {z_2(z_2^2-b^2)(z_2^2-r^2)},$$ and their residues at singular points are equal to $\res_{a}\Omega_1=\res_{-a}\Omega_1 = -\frac{1}{2a^2}= - \res_{b}\Omega_2= -\res_{-b}\Omega_2= \frac{(b^2-c^2)}{2b^2(b^2-r^2)}$. The regularity condition (\[diff\]) takes the form $\res_{Q_1}\Omega_1= \frac{1}{a^2}= \res_{Q_2}\Omega_2 = \frac{c^2}{r^2 b^2}$, which implies $a=\frac{br}{c},\ r=\frac{b}{\sqrt{2 + \frac{b^2}{c^2}}}$. By straightforward computations we derive $$(x^1)^2+\left(x^2-e^{-ru^2}\frac{b(c-r)}{c(b^2-r^2)}\right)^2= e^{-2ru^2}\frac{b^2(c-r)^2}{c^2(b^2-r^2)^2}.$$ The coordinate lines $u^2 = \const$ are circles centered at the $x^2$ axis. For $b = \pm 1$ these circles touch the $x^1$ axis, and the second family of coordinate lines $u^1 = \const$ consists in circles which are centered at the $x^1$ axis and which touch the $x^2$ axis (see Fig. 2). (170,100)(-100,-80) (60,-30)[(0,4)[130]{}]{} (-15,25)[(3,0)[185]{}]{} (60,35) (60,45) (70,25) (80,25) (160,30) (65,92) (50,-50) The case with the same spectral curve and the essential singularities lying in the same component is considered in [@MT1]. [Example 6. The Euclidean coordinates.]{} Let $\Gamma$ be a union of $n$ copies $\Gamma_1,\dots,\Gamma_n$ of the complex projective line $\C P^1$. We put $$P_j=\infty, \ \ Q_j=0, \ \ R_j=-1\in\Gamma_j, \ \ \psi_j(R_j)=1, \ \ \ j=1,\dots,N.$$ The differential $\Omega$ is given by the differentials $\Omega_j=\frac{dz_j}{z_j (z_j^2-1)}$ on the components, the Baker–Akhiezer function $\psi$ equals $\psi_j=e^{u^jz_j}f_j(u^j)$, $j=1,\dots,N$, and we obtain the Euclidean coordinates in $\R^N$: $x^j = e^{u^j}$. [Example 7. The polar coordinates.]{} The curve $\Gamma$ consists in five irreducible components $\Gamma_1,\dots, \Gamma_5$, which do intersect as it is shown on Fig. 3. We have $p_a(\Gamma)=1$. Let us define an involution $\sigma$ on $\Gamma$ as follows: а) the involution has the form $\sigma(z_j)=-z_j$, on $\Gamma_1, \Gamma_2$, and $\Gamma_3$; б) $\Gamma_3$ and $\Gamma_4$ are interchanged by $\sigma$ and $b_1,c_1,\infty\in\Gamma_3$ are mapped into $b_2,c_2,\infty\in\Gamma_4$. For a special choice of the divisors $D,P,Q$, and $R$ [@MT1] these spectral data correspond to the polar coordinates: $$x^1 = \psi_5(Q_1) = r \cos \varphi, \ \ x^2 = \psi_5(Q_2) = r \sin \varphi, \ \ r = e^{u^1}, \ \ \varphi = u^2.$$ (85,50)(-150,-80) (-100,0)(-100,20)(-70,20) (-70,20)(-40,20)(-40,0) (-40,0)(-40,-20)(-70,-20) (-70,-20)(-100,-20)(-100,0) (-50,0)(-50,40)(-20,40) (-20,40)(10,40)(10,0) (10,0)(10,-40)(-20,-40) (-20,-40)(-41,-37)(-46,-23) (7,39)(14,44)(30,45) (30,45)(47,44)(54,39) (60,25)(60,5)(30,5) (30,5)(0,5)(0,25) (0,-25)(0,-5)(30,-5) (30,-5)(60,-5)(60,-25) (54,-39)(47,-44)(30,-45) (30,-45)(14,-44)(7,-40) (50,0)(50,40)(80,40) (80,40)(110,40)(110,0) (110,0)(110,-40)(80,-40) (80,-40)(50,-40)(50,0) (-100,0) (-49,15) (-43,15) (-57,7) (10,0) (14,-3) (50,0) (37,-3) (110,0) (113,-4) (10,8) (2,3) (14,10) (10,-9) (-6,-9) (14,-17) (50,8) (41,10) (54,1) (50,-9) (41,-17) (53,-10) (-113,0) (-72,25) (-25,45) (25,50) (25,-40) (78,45) (15,-60) [Example 8. The cylindrical coordinates.]{} The curve $\Gamma$ is a disjoint union of the curve $\widehat{\Gamma}$ from the previous example (the polar coordinates) and a copy $\Gamma_6$ of $\C P^1$. All data related to $\widehat{\Gamma}$ are the same as before. On $\Gamma_6$ we put $Q_3 = 0, P_3 = \infty, R_4 = -1$ и $\psi(R_4) = 1$. Then we have $\psi_6(u^3)=e^{u^3(z_6+1)}$ and $$x^1 = \psi_5(Q_1) = r \cos \varphi, \ \ x^2 = \psi_5(Q_2) = r \sin \varphi, \ \ x^3 = \psi_6(Q_3) = z,$$ where $r = e^{u^1}, \varphi= u^2$ and $z = u^3$. [Example 9. The spherical coordinates in $\R^3$.]{} The curve $\Gamma$ consists in $9$ irreducible components which intersect as it is shown on Fig. 4. We have $p_a(\Gamma = 2$. (170,100)(-90,-80) (-85,0)(-85,20)(-55,20) (-55,20)(-25,20)(-25,0) (-25,0)(-25,-20)(-55,-20) (-55,-20)(-85,-20)(-85,0) (-85,0) (-96,-4) (-65,25) (-42,18) (-50,10) (-37,20) (-45,0)(-45,40)(-15,40) (-15,40)(15,40)(15,0) (15,0)(15,-40)(-15,-40) (-15,-40)(-36,-37)(-41,-23) (-15,45) (15,0) (15,7) (15,-7) (18,-3) (7,2) (0,-7) (18,8) (19,-15) (7,39)(14,44)(30,45) (30,45)(47,44)(54,39) (60,25)(60,5)(30,5) (30,5)(0,5)(0,25) (27,47) (45,7) (36,8) (0,-25)(0,-5)(30,-5) (30,-5)(60,-5)(60,-25) (54,-39)(47,-44)(30,-45) (30,-45)(14,-44)(7,-40) (27,-40) (45,-7) (36,-15) (45,0)(45,40)(75,40) (75,40)(105,40)(105,0) (105,0)(103,-29)(97,-31) (75,-40)(45,-40)(45,0) (45,0) (33,-3) (48,1) (47,-8) (72,43) (90,36) (80,31) (97,31) (75,0)(75,40)(105,40) (105,40)(135,40)(135,0) (135,0)(135,-40)(105,-40) (105,-40)(75,-40)(75,0) (102,43) (135,0) (135,7) (135,-7) (139,-3) (126,2) (120,-7) (138,9) (138,-15) (127,39)(134,44)(150,45) (150,45)(167,44)(174,39) (180,25)(180,5)(150,5) (150,5)(120,5)(120,25) (150,47) (165,7) (155,8) (120,-25)(120,-5)(150,-5) (150,-5)(180,-5)(180,-25) (174,-39)(167,-44)(150,-45) (150,-45)(134,-44)(127,-40) (150,-40) (165,-7) (155,-14) (165,0)(165,40)(195,40) (195,40)(225,40)(225,0) (225,0)(225,-40)(195,-40) (195,-40)(165,-40)(165,0) (165,0) (152,-3) (169,1) (168,-8) (225,0) (228,-3) (192,43) (75,-60) For certain choices of divisors $D,P,Q$ and $R$ (see [@MT1]) these spectral data lead to the spherical coordinates: $$x^1 = \psi_5(Q_1) = r \sin \varphi,\ \ x^2 = \psi_9(Q_2) = r \cos \varphi \sin \theta,$$ $$x^3 = \psi_9(Q_3) = r \cos \varphi \cos \theta, \ \ r = e^{u^1}, \ \varphi = u^2, \ \theta = u^3.$$ [Example 10. The spherical coordinates in $\R^N$.]{} Let $\Gamma^{(N-1)}$ be the spectral curve and let $\psi^{(N-1)}$ the Baker–Akhiezer function for $(N-1)$-dimensional spherical coordinates. The spectral curve $\Gamma^{(N)}$ for the $N$-dimensional spherical coordinates is obtained from $\Gamma^{(N-1)}$ and the curve from Fig. 5 via their intersection at $0\in\Gamma_{4N-7} \subset \Gamma^{(N-1)}$ and $0\in\Gamma_{4N-6}$ (the number of irreducible components of $\Gamma^{(k)}$ equals $4k-3$). We have $p_a(\Gamma^{(N)}) = N-1$. (85,50)(-150,-80) (-50,0)(-50,40)(-20,40) (-20,40)(10,40)(10,0) (10,0)(10,-40)(-20,-40) (-20,-40)(-50,-40)(-50,0) (-50,0) (-46,-3) (7,39)(14,44)(30,45) (30,45)(47,44)(54,39) (60,25)(60,5)(30,5) (30,5)(0,5)(0,25) (0,-25)(0,-5)(30,-5) (30,-5)(60,-5)(60,-25) (54,-39)(47,-44)(30,-45) (30,-45)(14,-44)(7,-40) (50,0)(50,40)(80,40) (80,40)(110,40)(110,0) (110,0)(110,-40)(80,-40) (80,-40)(50,-40)(50,0) (10,0) (14,-3) (50,0) (28,-3) (110,0) (113,-3) (10,8) (3,2) (14,10) (10,-9) (-5,-8) (14,-17) (50,8) (41,10) (54,1) (50,-9) (41,-17) (53,-10) (-27,45) (23,50) (23,-40) (76,45) (15,-60) It would be interesting to find the spectral data for other known orthogonal coordinates and, in particular, for the elliptic coordinates $u^1,u^2$ which are related to the Euclidean coordinates $(x^1,x^2)$ as follows $$x^1 = \cosh u^1 \, \cos u^2, \ \ \ x^2 = \sinh u^1\, \sin u^2.$$ A remark of discrete orthogonal coordinates {#subsec2.7} ------------------------------------------- In an actively developing discretization of differential geometry by discrete coordinates in $\R^N$ it is meant a mapping $$x: \Z^N \to \R^N$$ which is an embedding of lattice. The translation operator $T_i$ along the $i$-th coordinate acts on functions $F: \Z^N \to \R^k$ as follows $$T_i F(u^1,\dots,u^{i-1},u^i,u^{i+1},\dots,u^N) = F(u^1,\dots,u^{i-1},u^i+1,u^{i+1},\dots,u^N),$$ and the partial derivation $\Delta_i$ in the $i$-th direction is defined as $$\Delta_i F(u) = T_i F(u) - F(u).$$ By [@CDS], a coordinate system is orthogonal if two conditions holds: 1\) (the planarity condition) the points $\x(u), T_i\x(u), T_j\x(u), T_iT_j \x(u)$ lies in one plane for any given triple $i,j,u$; 2\) (the circular condition) a planar polygon spanned by $\x(u)$, $T_i\x(u)$, $T_j\x(u)$, and $T_i T_j \x(u)$ is inscribed into a circle. In [@AVK] there was found a procedure for construction discrete Darboux–Egorov coordinates (flat Egorov coordinates), based on the formal discretization of the Baker–Akhiezer function corresponding to continuous Darboux–Egorov coordinates from [@Krichever97]. Therewith discrete Darboux–Egorov coordinates satisfy the planarity condition and the circular condition is is strengthened as follows: 3\) (the discrete Egorov condition) the edges $X^+_i(u) = T_i\x(u)-\x(u)$ and $X^-_j(u) = T^{-1}_j\x(u) - \x(u)$ of the lattice are orthogonal for all given triples $i,j,u$ (this implies that in any quadrangle from the circular condition two right angles which are opposite to each other and therefore lean onto a diameter of the circle into which this quadrangle is inscribed). For discrete Darboux–Egorov coordinates there exists a discrete potential — a function $\Phi$ such that $\Delta_i \Phi(u) = \|T_i\x(u) - \x(u)\|^2$. Analogously one may do a formal discretization (on the level of Baker–Akhiezer functions) of coordinates given by Theorem \[th1\]. Since everything is easily computed in this case, it is possible to find explicitly examples of coordinates which, although do satisfy the planarity condition, do not satisfy the circular condition. This demonstrates the difficulties which appear in the study of an interesting problem consisting in finding a geometrically meaningful discretization of the Lame equations. Frobenius manifolds {#sec3} =================== The associativity equations and Frobenius manifolds {#subsec3.1} --------------------------------------------------- Let us consider a finite-dimensional algebra generated by $e_1,\dots,e_n$ and with a commutative multiplication $$e_\alpha \cdot e_\beta = c^\gamma_{\alpha \beta} e_\gamma, \ \ \ c_{\alpha \beta \gamma} = \frac{\partial^3 F(t)}{\partial t^\alpha \partial t^\beta \partial t^\gamma}, \ \ \ c^\gamma_{\alpha \beta}=\eta^{\gamma\delta}c_{\alpha \beta \delta}, \ \ \eta^{\alpha\beta} = \eta^{\beta\alpha},$$ where $F=F(t)$ is a function of $t=(t^1,\dots,t^N)$. The algebra is associative, i.e. $$(e_\alpha \cdot e_\beta)\cdot e_\gamma = e_\alpha \cdot (e_\beta \cdot e_\gamma) \ \ \ \mbox{дл€ всех $\alpha,\beta,\gamma$,}$$ if and only if $F$ satisfies [the associativity equations]{} \[e1\] \^ = \^ . In fact, if these conditions holds we obtain a family of associative algebras depending on an $N$-dimensional parameter $t$. These equations first appeared in the quantum field theory and together with the following conditions: 1\) $c_{1\alpha\beta} = \eta_{\alpha\beta}, \alpha,\beta=1,\dots,N$; 2\) $\det (\eta^{\alpha\beta}) \neq 0$, $\eta^{\alpha\beta}\eta_{\beta\gamma} = \delta^\alpha_\gamma$ and the metric $\eta_{\alpha\beta}$ is constant; 3\) (the quasihomogeneity condition) \[e2\] F(\^[d\_1]{}t\^1,…,\^[d\_N]{}t\^N) = \^[d\_F]{}F(t\^1,…,t\^N) they form the system of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations [@W; @DVV]. There are two generalizations of the quasihomogeneity conditions: it is assumed that there exists a vector field $E = (q^\alpha _\beta t^\beta + r^\alpha) \partial_\alpha$ which meets one of the following conditions: 3$^\prime$) the equality $$E^\alpha \partial_\alpha F = d_F F$$ holds. In (\[e2\]) $E$ has the form $E = d_1 t^1 \partial_1 + \dots + d_N t^N \partial_n$. This generalization covers the case of quantum cohomology; 3$^{\prime\prime}$) since by [@DVV] it is only important that the correlators $c_{ijk}$, i.e. the third derivatives of $F$, are homogeneous in the sense of (\[e2\]) it is sufficient to assume that \[quasi\] E\^\_F = d\_F F + (t\^1,…,t\^N). This generalization is important for us because in our examples some exponents $d_i$ are equal to $-1$. A geometrical counterpart to solutions of the WDVV equations are Frobenius manifolds whose notion was introduced by Dubrovin [@Dubrovin93]: [a Frobenius manifold]{} is a domain $U \subset \R^N$ endowed with a constant nondegenerate metric $\eta_{\alpha\beta}du^\alpha du^\beta$ and with a solution to the associativity equations ([a prepotential]{}) $F$ satisfying the conditions 1), 2) and the most general quasihomogeneity condition 3$^{\prime\prime}$). The quasihomogeneity condition came from physics and in addition to quantum filed theory Frobenius manifolds play an important role in the theory of isomonodromic deformations [@Dubrovin2008]. [^11] However this condtions is the most restrictive and sometimes in the modern Frobenius geometry it is omitted. Before [@MT2] all know Frobenius manifolds were given by Dubrovin’s examples of Frobenius structures on the orbits of Coxeter groups (here the flat metric is the Sato metric; the solutions the WDVV equations corresponding to singularities on type $A_n$ were found in [@DVV]) and on the Hurwitz spaces, by quantum cohomology and the extended moduli spaces of complex structures on Calabi–Yau manifolds [@BK], and by the “doubles” of the Hurwitz spaces found by Shramchenko [@Sh]. In every of these examples a Frobenius manifold has a special geometrical meaning. The examples from [@MT2] are obtained by analytical methods (via finite gap integration), are algebraic in the sense that the correlators $c_{ijk} = \frac{\partial^3 F}{\partial x^i \partial x^j \partial x^k}$ are algebraic functions, and are not semisimple. Finite gap Frobenius manifolds {#subsec3.2} ------------------------------ There is an important relation between solutions of the associativity equations and flat Egorov metrics discovered by Dubrovin [@D0]. It is as follows. Let $$\eta_{\alpha\beta} dx^\alpha dx^\beta = \sum_{i=1}^N H^2_i(u) \left(du^i\right)^2$$ be a flat Egorov metric and $x^1,\dots,x^N$ be coordinates in some domain which the coefficients $\eta_{\alpha\beta}$ are constant. We have $$\eta^{\alpha\beta} = \sum_{i=1}^N H^{-2}_i \frac{\partial x^\alpha}{\partial u^i} \frac{\partial x^\beta}{\partial u^i}$$ and the conditions of symmetry for rotation coefficients and of zero curvature imply the existence of a function $F(t)$ such that \[e4\] c\_ = \_[i=1]{}\^N H\_i\^2 = and the associativity equations $$c^\lambda_{\alpha \beta} c^\mu_{\lambda\gamma} = c^\mu_{\alpha\lambda} c^\lambda_{\beta\gamma} \ \ \ \mbox{for all} \ \alpha,\beta,\gamma=1,\dots,N,$$ holds where $$c^\alpha_{\beta\gamma} = \sum_i \frac{\partial x^\alpha}{\partial u^i} \frac{\partial u^i}{\partial x^\beta}\frac{\partial u^i}{\partial x^\gamma}.$$ Assuming that the associative algebra is semisimple the converse is also true: one may construct Egorov metric satisfying (\[e4\]) from a solution $F(t)$ of the associativity equations. Since $u^1,\dots,u^N$ are orthogonal curvilinear coordinates in flat space, one may apply Krichever’s method for obtaining such coordinates (see Theorem \[krichever-theorem\]). In his article [@Krichever97] there are given additional constraints on the spectral data which correspond to Egorov metrics and therefore to solutions of the associativity equations. However it sounds that from the properties of theta functions one may derive that explicit theta-functional formulas obtained there do not give quasihomogeneous solutions. At the same time one may expect that solutions expressed in terms of elementary functions and corresponding to singular curves may be quasihomogeneous and define Frobenius manifolds. That was shown in [@MT2]. The following theorem distinguishes a special case when the construction of Theorem \[th1\] gives flat Egorov metrics and quasihomogeneous solutions of the associativity equations. \[th2\] 1) Let the assumptions of Theorem \[th1\] hold and $\Gamma$ be a spectral curve such that all its irreducible components are rational curves (complex projective lines $\C P^1$). Let us pick on every component $\Gamma_i$ a pair of points $P_i = \infty,\ Q_i=0$ and a global parameter $k_i^{-1} = z_i$, $i=1,\dots,N$. Let us assume that all singular points are double points of intersections of different components, every such an intersection point $a \in \Gamma_i \cap \Gamma_j$ has the same coordinates on both components: $$z_i(a)=z_j(a),$$ and the involution $\sigma$ on every component has the form $$\sigma(z_i)=-z_i.$$ Then the metric $$ds^2 = \eta_{kl} d x^k d x^l = \sum_{i} H^2_i \left(du^i\right)^2, \ \ H_i=H_i(u^1,\dots,u^N), \ i=1,\dots,N,$$ constructed from this spectral data is a flat Egorov metric (Darboux–Egorov metric). 2\) Let us also assume that the spectral curve is connected and the Baker–Akhiezer function is normalized at one point $r$: $$\psi(u,r)=1, \ \ \ R = r \in \Gamma.$$ Then the functions $$c_{\alpha \beta \gamma}(x)=\sum_{i=1}^n H_i^2\frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i}{\partial x^\beta}\frac{\partial u^i}{\partial x^\gamma},$$ are homogeneous: $$c_{\alpha\beta\gamma}(\lambda x^1,\dots,\lambda x^n)= \frac{1}{\lambda}c_{\alpha\beta\gamma}(x^1,\dots,x^n).$$ To obtain from the derived solution of the associativity equations a Frobenius manifold it needs to add to the generators $e_1,\dots,e_N$ of the associative algebra the unit $e_0$ and a nilpotent element $e_{N+1}$. Such an extension is given by the following algebraic lemma. \[[@MT2]\] Let $F(t^1,\dots,t^N)$ be a solution of the associativity equations with a constant metric $\eta_{\alpha\beta}$. Then the function $$\widetilde{F}(t^0,t^1,\dots,t^n,t^{N+1}) = \frac{1}{2}\left(\eta_{\alpha\beta}t^\alpha t^\beta t^0 + \left(t^0\right)^2 t^{N+1}\right) + F(t^1,\dots,t^N)$$ satisfies the associativity equations (\[e1\]) with the metric $$\widetilde{\eta} = \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & \eta & 0 \\ 1 & 0 & 0 \end{array} \right)$$ and the associative algebra generated by $e_0,e_1,\dots,e_N,e_{N+1}$ with the multiplication rules $$e_i \cdot e_j = c^k_{ij} e_k, \ \ \ c^k_{ij} = \widetilde{\eta}^{kl} \frac{\partial^3 \widetilde{F}}{\partial t^l \partial t^i \partial t^j},$$ has the unit $e_0$: $e_o \cdot e_k =e_k$ for all $k=0,\dots,n+1$, and a nilpotent element $e_{N+1}$: $e_{N+1}^2 = 0$. Moreover if $F$ quasihomogeneous and $d_\alpha + d_\beta =c$ for all $\alpha,\beta$ such that $\eta_{\alpha\beta} \neq 0$, then $\widetilde{F}$ is also quasihomogeneous with $d_0 = d_F-c, d_{N+1}= 2c - d_F$ and the same values of $d_\alpha$, $\alpha=1,\dots,N$ as for $F$. Let us present two simplest examples of Frobenius manifolds given by Theorem \[th2\]. We note that the statement 1) of this theorem gives solutions of the associativity equations, and the statement 2) distinguishes a certain subclass of quasihomogeneous solutions: there are quasihomogeneous solutions which do not satisfy to the sufficient conditions form the statement 2). Example 11 is covered by the statement 2), and Example 12 is not covered by it. [Example 11.]{} Let $\Gamma$ is formed by two complex projective lines $\Gamma_1$ and $\Gamma_2$ which intersect at a pair of points (see Fig. 6): $\{a,-a\in\Gamma_1\}\sim\{a,-a\in\Gamma_2\}$. (170,100)(-100,-80) (-10,0)(-10,30)(40,30) (40,30)(90,30)(90,0) (90,0)(90,-30)(40,-30) (40,-30)(-10,-30)(-10,0) (35,33) (60,0)(60,30)(110,30) (110,30)(160,30)(160,0) (160,0)(160,-30)(110,-30) (110,-30)(60,-30)(60,0) (105,33) (-10,0) (90,0) (60,0) (160,0) (75,24) (75,-24) (-25,0) (165,0) (95,0) (45,0) (63,20) (54,-26) (85,18) (80,-26) (60,-50) Let $N=2$, $l=1$ and the Baker–Akhiezer function $\psi$ be normalized by the condition $\psi_2(r)=1$ at $r \in \Gamma_2$. The prepotential takes the form $$F_{a,c}(x^1,x^2) = \frac{1}{4ac}\left(2x_2\sqrt{(a^2-c^2)x_1^2 +c^2x_2^2}\right.$$ $$+2cx_1^2\log \left(-\frac{cx_2+\sqrt{(a^2-c^2)x_1^2+c^2x_2^2}} {x_1}\right)-\sqrt{2c^2-a^2}(x_1^2+x_2^2)$$ $$\left.\times\log\left(c^2(x_1^2-3x_2^2)+a^2(x_2^2-x_1^2)- 2x_2\sqrt{2c^2-a^2}\sqrt{(a^2-c^2)x_1^2+c^2x_2^2}\right)\right)$$ and satisfies the associativity equations for $\eta_{\alpha\beta}=\delta_{\alpha\beta}$. It depends on two additional parameters $a$ and $c$ and for $a=1, c=\frac{2}{\sqrt{7}}$ the formulas for the coordinates and for the correlators are rather simple: $$x^1=\frac{4(7-\sqrt{7})e^{u^1-u^2}}{(21-6\sqrt{7})e^{2u^1}+(7+2\sqrt{7})e^{2u^2})},$$ $$x^2=\frac{e^{-2u^2}(3(\sqrt{7}-3)e^{2u^1}+(5+\sqrt{7})e^{2u^2})} {3(\sqrt{7}-2)e^{2u^1}+(2+\sqrt{7})e^{2u^2}},$$ $$c_{111}=-\frac{9x_1^8+51x_1^6x_2^2+88x_1^4x_2^4+(2x_1^2x_2^3+4x_2^5)\sqrt{(3x_1^2+4x_2^2)^3}+48x_1^2x_2^6} {2x_1(3x_1^4+7x_1^2x_2^2+4x_2^4)^2},$$ $$c_{112}=\frac{9x_1^6x_2+15x_1^4x_2^3-8x_1^2x_2^5+(2x_1^2x_2^2+4x_2^4)\sqrt{(3x_1^2+4x_2^2)^3}-16x_2^7} {2(3x_1^4+7x_1^2x_2^2+4x_2^4)^2},$$ $$c_{122}=-\frac{9x_1^7+15x_1^5x_2^2-8x_1^3x_2^4+(2x_1^3x_2+4x_1x_2^3)\sqrt{(3x_1^2+4x_2^2)^3}-16x_1x_2^6} {2(3x_1^4+7x_1^2x_2^2+4x_2^4)^2},$$ $$c_{222}=\frac{-27x_1^6x_2-16x_2^7-72x_1^2x_2^5+(4x_1^2x_2^2+2x_1^4)\sqrt{(3x_1^2+4x_2^2)^3}-81x_1^4x_2^3} {2(3x_1^4+7x_1^2x_2^2+4x_2^4)^2}.$$ [Example 12.]{} Let $\Gamma$ be the same as in Example 11. In difference with Example 11 we assume that $$P_1 = \infty \in \Gamma_1, \ \ P_2 = 0\in \Gamma_1, \ \ Q_1 = \infty \in \Gamma_2, \ \ Q_2 = 0 \in \Gamma_2,$$ the normalization point $R=r$ lies in $\Gamma_1$ and the poles divisor $D=c$ lies in $\Gamma_2$ (see Fig. 7). However we do not assume that the intersection points have the same coordinates (the assumptions of the statement 2) of Theorem \[th2\] are not satisfied): $$a \sim b, \ \ -a \sim -b, \ \ \ \pm a \in \Gamma_1, \ \pm b \in \Gamma_2, \ a \neq b.$$ (170,100)(-100,-80) (-10,0)(-10,30)(40,30) (40,30)(90,30)(90,0) (90,0)(90,-30)(40,-30) (40,-30)(-10,-30)(-10,0) (35,33) (60,0)(60,30)(110,30) (110,30)(160,30)(160,0) (160,0)(160,-30)(110,-30) (110,-30)(60,-30)(60,0) (105,33) (-10,0) (90,0) (60,0) (160,0) (75,24) (75,-24) (-25,0) (165,0) (95,0) (45,0) (63,20) (54,-26) (85,18) (80,-26) (60,-50) The prepotential $F(x^1,x^2)$ equals $$F(x^1,x^2) = -\frac{1}{8}\left(\left(x^1\right)^2+\left(x^2\right)^2\right) \log\left(\left(x^1\right)^2+\left(x^2\right)^2\right)$$ and is included in a linear family of quasihomogeneous functions $$F_q(x^1,x^2) = q \left(\left(x^1\right)^2+\left(x^2\right)^2\right)\arctan\left(\frac{x^1}{x^2}\right)$$ $$-\frac{1}{8}\left(\left(x^1\right)^2+\left(x^2\right)^2\right) \log\left(\left(x^1\right)^2+\left(x^2\right)^2\right), \ \ q\in \R,$$ which satisfy the associativity equations for $\eta_{\alpha\beta}=\delta_{\alpha\beta}$. The correlator for $F$ (i.e. $q=0$) has the simplest form: $$c_{111} = -\frac{3}{2}\frac{x^1}{\left(x^1\right)^2+\left(x^2\right)^2} + \frac{\left(x^1\right)^3}{\left(\left(x^1\right)^2+\left(x^2\right)^2\right)^2}, \ \$$ $$c_{112} = -\frac{1}{2}\frac{x^2}{\left(x^1\right)^2+\left(x^2\right)^2} + \frac{\left(x^1\right)^2 x^2 }{\left(\left(x^1\right)^2+\left(x^2\right)^2\right)^2},$$ and the formulas for $c_{122}$ and $c_{222}$ are obtained by interchanging indices $1 \leftrightarrow 2$. Examples 11 and 12 describe quasihomogeneous deformations of the cohomology ring of $\C P^2 \sharp \C P^2$. Indeed, there are the standard generators $e_0,\dots,e_3$ in $H^\ast(\C P^2 \sharp \C P^2;\C)$: $$e_0 \in H^0, \ \ e_1, e_2 \in H^2, \ \ e_3 \in H^4, \ \ e_1^2 = e_2^2 = e_3, e_1e_2=0.$$ We have $d_i = \frac{1}{2}\deg e_i$. The deformations change the multiplication rules for two-dimensional classes by adding two-dimensional terms: $$e_i e_j = e_3 + c_{ij}^k(t) e_k, \ \ \ 1 \leq i,j \leq 2.$$ Soliton equations with self-consistent sources and the corresponding deformations of spectral curves {#s4} ==================================================================================================== Equations with self-consistent sources {#subsec4.1} -------------------------------------- In difference with $(1+1)$-dimensional soliton equations which have the Lax representation $$L_t = [L,A],$$ equations with self-consistent sources are represented in the form $$L_t = [L,A] + C,$$ where $C$ is an expression in terms of solutions $\psi_{k}$ of linear problems $L\psi_k = \lambda_k\psi_k$, $k=1,\dots,l$. To equations with self-consistent sources there is applied the inverse problem method and it looks that first, from the formal algebraical point of view, they were derived by Melnikov [@Melnikov83]. In the physical literature first they did appear in the article by Zakharov and Kuznetsov [@ZK] who, in particular, descried the physical meanings of the KdV (see the formula (\[soliton2\] below) and KP equations $$3u_{yy} - \frac{\partial}{\partial x}(u_t + 6uu_x +u_{xxx}) = 3\frac{\partial^2}{\partial x^2}\|\psi\|^2, \ \ \ i\psi_t = \psi_{xx} + u\psi$$ with self consistent sources. Let us note two important facts: 1\) every such an equation has as its predecessor a soliton equation $L_t=[L,A]$; 2\) for a correct posing the problem it needs to normalize the functions $\psi$ (they have to satisfy to certain fixed spectral characteristics) and have the expression $C$ which fits with the class of solutions in study — for instance, to be fast decaying or periodic. Solutions of such equations do have many interesting qualitative properties: for instance, a creation and an annihilation of solitons do appear [@Melnikov89]. We discuss that in в §\[subsec4.3\]. In the periodic case to $L$ there corresponds a spectral curve which, as it appears, may be deformed by an equation with self-consistent sources and therewith such a deformation consists in creation and annihilation of double points. First this was observed in [@GT1] for the conformal flow which appears in differential geometry of surfaces and later that was studied for the KP equation with self-consistent sources [@GT2]. These two facts are exposed below in §\[subsec4.2\] and §\[subsec4.3\] respectively. Spectral curves of immersed tori and the conformal flow {#subsec4.2} ------------------------------------------------------- In theory of the Weierstrass representation of immersed surfaces in $\R^3$ and $\R^4$ (see the survey [@Taimanov2006]) surfaces are described in terms of solutions of the equation $\D\psi=0$ for surfaces in $\R^3$ and of the equations $\D\psi = \D^\vee \varphi = 0$ for surfaces in $\R^4$ where $$\D = \left(\begin{array}{cc} 0 & \partial \\ -\bar{\partial} & 0 \end{array} \right) + \left(\begin{array}{cc} U & 0 \\ 0 & \bar{U} \end{array} \right), \ \ \ \D^\vee = \left(\begin{array}{cc} 0 & \partial \\ -\bar{\partial} & 0 \end{array} \right) + \left(\begin{array}{cc} \bar{U} & 0 \\ 0 & U \end{array} \right).$$ For surfaces in $\R^3$ the potential $U$ is real-valued and $\D = \D^\vee$. For tori these operators are double-periodic and it is natural to consider their spectral curves on the zero energy level. It appeared that these spectral curves contain an information on the values of the Willmore functional which is defined as follows: $${\mathcal W} (M) = \int_{M} \|{\bf H}\|^2 d\mu,$$ where ${\bf H}$ is the mean curvature vector and $d\mu$ is the volume form, on $M$, induced by the immersion. This functional is conformally invariant in the sense that $${\mathcal W}(M) = {\mathcal W}(f(M)),$$ where $M$ is a closed surface and $f: S^k \to S^k$ is a conformal transformation of $S^k = \R^k \cup \{\infty\}$, $k\geq 3$, such that $M \subset \R^k$ and $f(M) \subset \R^k$. For tori of revolution the potential $U$ depends on one variable and we obtain the reduction of the problem $\D \psi = 0$ to the Zakharov–Shabat spectral problem for the one-dimensional Dirac operator. Therewith the Willmore functional ${\mathcal W}$ becomes the first Kruskal–Miura integral for the modified KdV (mKdV) hierarchy related to this operator. Other Kruskal–Miura integrals of the mKdV hierarchy are generalized to (nonlocal) conservation laws for the hierarchies related to the two-dimensional operator $\D$ (the modified Novikov–Veselov equations and the Davy–Stewartson equations). This leads us to an observation on the existence of relation between the geometry of surface and the spectral properties of the operator $\D$ which comes into the Weierstrass representation of the surface [@Taimanov1997]. The approach to proving the Willmore conjecture which was proposed by us and which is based on this relation until recently has not achieve a success. Another our conjecture that higher two-dimensional generalizations of the Kruskal–Miura integrals and together with them the whole spectral curve are conformally invariant was confirmed in [@GS]. For that there was used the conformal flow introduced by Grinevich. Since the conformal group of $\R^k \cup \{\infty\}$ for $k \geq 3$ is generated by translations, dilations and inversions, and since the potential $U$ is invariant under translations and dilations, it is enough to prove the invariance of spectral curve under inversions. Let us consider the generator of inversions and in the conformal group and let us write down its action of the potential. It is as follows: \[conformal-flow-3\] U\_t = \|\_2\|\^2 - \|\_1\|\^2, where a torus in $\R^3$ is defined by a solution $\psi=(\psi_1,\psi_2)^\top$ of the equation $\D\psi=0$. All Floquet multipliers of $\D$ are preserved by the flow (\[conformal-flow-3\]), which corresponds to an inversion of torus in $\R^3$, there fore they are preserved by inversions. It [@GT1] an analogous problem was considered for tori in $\R^4$ and it was showed that to the following generator of inversions $$\partial_t x^1 = 2 x^1 x^3, \ \ \ \ \partial_t x^2 = 2 x^2 x^3,$$ $$\partial_t x^3 = (x^3)^2 -(x^1)^2 - (x^2)^2 - (x^4)^2, \partial_t x^4 = 2 x^4 x^3$$ there corresponds a deformation of $U$ of the form $$\label{conformal-flow} \partial_t U= \varphi_1\bar\psi_1-\bar\varphi_2\psi_2, \ \ \ \partial_t \bar U= \bar\varphi_1\psi_1-\varphi_2\bar\psi_2, \ \ \ \D \psi = \D^\vee \varphi = 0,$$ where $\psi = (\psi_1,\psi_2)^\top$ and $\varphi=(\varphi_1,\varphi_2)^\top$ define a torus in $\R^4$ via the Weierstrass representation. Analogously to the three-dimensional case it was showed that All Floquet multiplies of $\D$ are preserved by the transformation, of $U$, of the form (\[conformal-flow\]) where $\psi$ and $\varphi$ satisfy $\D\psi = \D^\vee \varphi = 0$ and the quadratic expressions $\psi_i^2,\varphi_j^2, i,j=1,2$, are double-periodic. Let us note that the equations (\[conformal-flow-3\]) and (\[conformal-flow\]) are equations with self-consistent sources which in the absence of these sources reduce to the stationary equation $U_t=0$. However there are known the explicitly computed spectral curves of the Clifford torus $$x_1^2 + x_2^2 = x_3^2 + x_4^2 = \frac{1}{2}$$ in the three-sphere $S^3 \subset \R^4$ and of its stereographic projection into $\R^3$. The Willmore conjecture stays that on the Clifford torus the Willmore functional attains its minimal value (for all immersed tori) which is equal to $2\pi^2$. For the Clifford torus in $S^3 \subset \R^4$ the spectral curve is the complex projective line: $\Gamma = \C P^1$, and for its projection into $\R^3$ it is a rational curve with two double points [@Taimanov2005]. A detailed analysis of the flow (\[conformal-flow\]) led us to the following result. The evolution flow (\[conformal-flow\]), acting on double-periodic potentials, preserve the Floquet multipliers of $\D$ however it may deform its spectral curve (on the zero energy level) and the deformation consists in creations and annihilations of double points. Since the higher conservation laws for the modified Novikov–Veselov hierarchy and for the Davy–Stewartson hierarchy are expressed in terms of the asymptotics of multipliers, the corresponding quantities are also preserved by the flow (\[conformal-flow\]). Finite gap solutions of KdV and KP equations with self-consistent sources {#subsec4.3} ------------------------------------------------------------------------- Let $\Gamma$ be a Riemann surface of genus $g$ (the spectral curve) with a marked point $P$ and with a local parameter $k^{-1}$, $k(P)=\infty$, near this point and let there be defined a generic divisor $D = \gamma_1 + \dots + \gamma_g$, $\gamma_i \in \Gamma, i=1,\dots,N$. Let us also assume that on $\Gamma$ there are marked $2N$ pair-wise different $R_l^{\pm}, l =1,\dots,N$, which also differ from $P$ and from all points from $D$. By the theory of Baker–Akhiezer functions, there exists a unique function $\psi(\gamma,x,y,t,\tau)$, $\tau=(\tau_1,\dots,\tau_N)$, $\gamma \in \Gamma$ such that 1\) $\psi$ is meromorphic in $\gamma$ on $\Gamma \setminus P$ and has $g+N$ simple poles at $\gamma_1,\dots,\gamma_g,R^+_1,\dots,R^+_N$; 2\) $\res \psi(\lambda,x,y,t,\tau)\vert_{\lambda = R^+_l} = \tau_l \psi(R^-_l,x,y,t,\tau),\ l =1,\dots,N$, 3\) $\psi$ has an essential singularity at and $$\psi(\gamma,x,y,t,\tau) = e^{k x + k^2 y + k^3 t} \left(1+ \sum_{m >0} \frac{\xi_m(x,y,t,\tau)}{k^m}\right) \ \ \mbox{при $\gamma \to P$}.$$ There exists the unique adjoint Baker–Akhiezer $1$-form $\psi^\ast(\gamma,x,y,t,\tau)$ [^12] with the following properties: 1$^\ast$) $\psi^\ast$ is meromorphic in $\gamma$ on $\Gamma \setminus P$ and has $g+N$ simple zeros at $\gamma_1,\dots,\gamma_g,R^-_1,\dots,R^-_N$; 2$^\ast$) $\res \psi^\ast(\lambda,x,y,t,\tau)\vert_{\lambda = R^-_l} = -\tau_l \frac{\psi^\ast(\lambda,x,y,t,\tau)}{d\lambda}\vert_{\lambda=R^+_l},\ l =1,\dots,N$, 3$^\ast$) $\psi^\ast$ has an essential singularity at $P$ and $$\psi^\ast(\gamma,x,y,t,\tau) = e^{-k x - k^2 y - k^3 t} (1+ o(1))\,dk \ \ \mbox{при $\gamma \to P$}.$$ Let us put \[kpm\] u(x,y,t) = 2 \_1(x,y,t,), \_l = \_l+\_l t, l=1,…,N. Analogously we construct from $\psi(\gamma,x,y,t,\tau)$ and $\psi^\ast(\gamma,x,y,t,\tau)$ the functions $\psi(\gamma,x,y,t)$ and the forms $\psi^\ast(\gamma,x,y,t)$. We have The function $u(x,y,t)$ of the form (\[kpm\]) satisfies the KP equation with $N$ self-consistent sources: \[kpmf\] u\_t = KP\[u\] + 2\_[l=1]{}\^N \_l \_[= R\^+\_l]{}, where $u_t = KP[u]$ is the KP flow. If there exists a two-sheeted covering $\pi:\Gamma \to \C P^1$, ramified at $P$, $\pi(P)=\infty$, $\pi(k)=\pi(-k)$ and $\pi(R^+_l)=\pi(R^-_l)$, $l=1,\dots,N$, then the formula (\[kpmf\]) defines a solution $u(x,t)$ to the KDV equation with self-consistent sources (in (\[kpmf\]) it needs to replace $KP[u]$ by $KDV[u]$ where $u_t=KDV[u]$ is the Korteweg–de Vries equation). Analogously one may construct finite gap solutions of all equations of the KdV and KP hierarchies with self-consistent sources [@GT2]. The proof of this theorem is essentially based on the theory of Cauchy–Baker–Akhiezer kernels introduced by Grinevich and Orlov [@GO]. [Remark 2.]{} The Baker–Akhiezer function $\psi$ from this theorem is defined on the spectral curve, with double points $R_l$, which is obtained from $\Gamma$ by pair-wise identifications of $R^+_l$ and $R^-_l$, $l=1,\dots,N$. The double point $R_l$ annihilates if and only if $\tau_l=0$. If even for the initial potential $u(x,y,t)$, $t=0$, the spectral curve is regular, then equations with self-consistent sources immediately lead to creation of double points on the spectral curve (for almost all times their number is equal to the number of sources $N$) and these singularities are preserved for almost all times. [Example 13.]{} Let $\Gamma = \C \cup \{\infty\} = \C P^1$ with a parameter $k$, $N=1$ and $R^\pm = \pm \kappa$. After simple computations we obtain the formula $$u(x,t,\tau) = -\frac{16 \tau\kappa^3}{(\tau e^{-(\kappa x + \kappa^3 t)} + 2\kappa e^{\kappa x + \kappa^3 t})^2},$$ which gives a (regular) soliton for $\tau>0$, a zero solution for $\tau=0$ and a singular soliton for $\tau <0$. We derive that the function $u(x,t) = u(x,t,\alpha+\beta t)$ satisfies the KdV equation with a self-consistent source \[soliton2\] u\_t = u\_[xxx]{} - uu\_x + 2\_x \^2(-,x,t), where $\psi(-\kappa,x,t) = \left(1 - \frac{\tau}{\tau +2\kappa e^{2(\kappa x +\kappa^3 t)}}\right)e^{-\kappa x -\kappa^3 t}$, $ \tau = \alpha + \beta t$. There the following qualitative effects first noticed by Melnikov [@Melnikov89]: 1. starting with a small initial value $c=c(0)$ we achieve $c=0$ in finite time (an annihilation of soliton); 2. by inverting the flow (\[soliton2\]) for the initial data $c(0)=0$ we immediately obtain a soliton for $t>0$ (a creation of soliton). For fast decaying potentials these effects are the analogs of an annihilation and of a creation of double point for the spectral curves of double-periodic potentials. Some other examples of integrable problems with singular spectral curves {#s5} ======================================================================== Let us expose a pair of interesting examples of integrable physical problems with singular spectral curves: 1\) Recently Grinevich, Mironov and Novikov distinguished a class of algebro-geometrical spectral data from which by using two-point Baker–Akhiezer functions introduced in [@DKN] there are constructed operators of the form $$L = (\partial +A)\bar{\partial}$$ (a magnetic Pauli operator) [@GMN1]. The spectral curve $\Gamma$ reduces to two smooth components $\Gamma^\prime$ and $\Gamma^{\prime\prime}$ from which is obtained by pair-wise gluing $k+1$ pairs of points $Q_1^\prime \sim Q_1^{\prime\prime}$, $\dots$, $Q_{k+1}^\prime \sim Q_{k+1}^{\prime\prime}$, $Q^\prime_i \in \Gamma^\prime, Q^{\prime\prime}_j \in \Gamma^{\prime\prime}, i,j=1,\dots,k+1$. Moreover there exists an antiholomorphic involution $\sigma: \Gamma \to \Gamma, \sigma^2(\gamma)=\gamma, \gamma \in \Gamma$, which interchanges the components: $$\sigma(\Gamma^\prime) = \Gamma^{\prime\prime}, \ \ \sigma(Q^\prime_k) = Q^{\prime\prime}_{\sigma(k)}.$$ This implies that $\Gamma^\prime$ and $\Gamma^{\prime\prime}$ has the same genus $g$ and $p_a(\Gamma) = 2g+k$. Examples which are interesting from the physical point of view do appear already in the case when $\Gamma^\prime$ and $\Gamma^{\prime\prime}$ are complex projective lines.. With another real reduction of $L$: $$\widetilde{L} = (\partial_x + A)\partial_y,$$ which is constructed from the same spectral curves, there is related an integrable two-dimensional generalization of the Burgers equation [@GMN2]. 2\) In the beginning of 1990s Krichever applied algebro-geometrical methods to a study of solutions of the Yang–Baxter equations for $(4 \times 4)$-matrices [@Krichever1981]. A construction of these solutions does not use Baker–Akhiezer functions however the general ideology is taken from finite gap integration and the role of the spectral curve $\Gamma$ is played by a smooth elliptic curve. Solutions are classified by their rank which takes values $l=1,2$ and, as it is shown in [@Krichever1981], all solutions of rank one are gauge equivalent to the Baxter solution or obtained from it by transformations corresponding to simple symmetries. Dragovich considered degenerated cases of this construction and showed that if $\Gamma$ into two rational components which intersect at two points then we have the well-known Yang solution [@D1] and if $\Gamma$ is a rational curve with a double point then we get Cherednik’s solution [@D2] (in both cases up to gauge equivalence). [MMM]{} Mironov, A.E., and Taimanov, I.A.: Orthogonal curvilinear coordinate systems that correspond to singular spectral curves. Proc. Steklov Inst. Math. 2006, no. 4 (255), 169–184. Mironov, A.E., and Taimanov, I.A.: On some algebraic examples of Frobenius manifolds. Theoret. and Math. Phys. [151]{} (2007), 604–613. Grinevich, P.G., and Taimanov, I.A.: Infinitesimal Darboux transformations of the spectral curves of tori in the four-space. International Mathematics Research Notices 2007 (2007), rnm005, 1–21. Grinevich, P.G., and Taimanov, I.A.: Spectral conservation laws for periodic nonlinear equations of the Melnikov type. Amer. Math. Soc. Transl. 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Grinevich, P.G., Mironov, A.E., and Novikov, S.P.: The two-dimensional Schrödinger operator: evolution $(2+1)$-systems and their new reductions, the two–dimensional Burgers hierarchy, and inverse problem data. Russian Math. Surveys [65]{}:3 (2010), 580–582. Krichever, I.M.: The Baxter equations and algebraic geometry. Functional Anal. Appl. [15]{}:2 (1981), 92–103. Dragovich, V.I.: Solutions of the Yang equation with rational spectral curves. St. Petersburg Math. J. [4]{} (1993), 921–931. Dragovich, V.I.: Solutions of the Yang equation with rational irreducible spectral curves. Russian Acad. Sci. Izv. Math. [42]{}:1 (1994), 51–65. [^1]: Institute of Mathematics, 630090 Novosibirsk, Russia; e-mail: taimanov@math.nsc.ru [^2]: Topologically these are Riemann surfaces however, since the finite gap integration method deals with finite genus curves, it uses the terminology from complex algebraic geometry. [^3]: This includes the case when the spectral curve $$P(z,w)=0, \ \ \ P \in \C[z,w],$$ is defined algebraically via the Burchnall–Chaundy theorem which reads that commuting ordinary differential operators $A_1$ and $A_2$ meet an algebraic relation $P(A_1,A_2)=0$. [^4]: This terminology — finite gap operators and Bloch functions — came from solid state physics. Later in English the term of “finite zone” was replaced by “finite gap”. [^5]: Its from differs from (\[spectral\]) by a sign at $d^2/dx^2$ which is reflected in the change of the asymptotic of $\psi$: $\psi \approx e^{ikx}$ as $k \to \infty$. [^6]: These examples were constructed at the beginning of 1990s and at the same time Malanyuk argued that their spectral curves are obtained via degenerations from smooth curves. [^7]: For the modern state of theory of such systems we refer to [@Pavlov] and references therein. [^8]: Now by Petrov’s classification it is mean the analogous classification based on the algebraic types of the Weyl operator which is analogously constructed from the Weyl tensor (the tensor of conformal curvature). [^9]: It is easy to notice that if there are points $Q_k$ and $Q_l, l =\sigma(k)$ which are interchanged by the involution then the metric $\eta$ is indefinite. [^10]: We expose this simple example in details to demonstrate the construction procedure, for details of other examples we refer to [@MT1]. [^11]: For isomonodromic deformations only [semisimple manifolds]{}, i.e. without nilpotent elements, are interesting. [^12]: The adjoint Baker–Akhiezer $1$-forms were introduced in [@Krichever82]. The proof of its existence and uniqueness is analogous to proofs of the same facts of Baker–Akhiezer functions
--- abstract: 'We present a study of electrical transport properties of $R$Bi$_2$ ($R$ = La, Ce) under hydrostatic pressure up to $\sim$ 2.5 GPa. These measurements are complemented by thermodynamic measurements of the specific heat on CeBi$_2$ at different pressures up to 2.55 GPa. For CeBi$_2$, we find a moderate increase of the antiferromagnetic transition, $T_\text N$, from 3.3 K to 4.4 K by pressures up to 2.55 GPa. Notably, resistance measurements for both CeBi$_2$ and LaBi$_2$ show signatures of superconductivity for pressures above $\sim$ 1.7 GPa. However, the absence of superconducting feature in specific heat measurements for CeBi$_2$ indicates that superconductivity in CeBi$_2$ (and most likely LaBi$_2$ as well) is not bulk and likely originates from traces of Bi flux, either on the surface of the plate-like samples, or trapped inside the sample as laminar inclusions.' author: - Li Xiang - Elena Gati - Kathryn Neilson - 'Sergey L. Bud’ko' - 'Paul C. Canfield' title: 'Physical properties of $R$Bi$_2$ ($R=$La, Ce) under pressure' --- Introduction ============ Bi-rich compounds manifest a rich variety of ground states. For example, Bi-based families such as $A$Bi ($A$ = Li and Na)[@Sambongi1971; @Kushwaha2014], $A$Bi$_2$ ($A$ = K, Rb, Cs and Ca)[@Roberts1976; @Winiarski2016] and $A$Bi$_3$ ($A$ = Sr, Ba, Ca, Ni, Co, and La) are superconducting (SC) at low temperature [@Matthias1952PR; @Shao2016; @Kinjo2016; @Xiang2018; @Gati2018; @Tence2014]. $R$Bi ($R$ = Ce, Nd, Tb and Dy) and $R$Bi$_2$ ($R$ = La-Nd, Sm) families have low-temperature magnetic ground states with complex $H - T$ phase diagrams[@Nereson1971; @Petrovic2002]. Moreover, due to the strong spin-orbit coupling of Bi-6$p$ electrons they can have substantial ferromagnetic anisotropy, like MnBi[@Anna2013; @Taufour2015], or, more recently, they have became candidates for realizing novel topological phases, such as topological insulators or topological superconductors[@Hasan2010; @Qi2011; @Hor2010; @Xia2009; @Chen2009]. Among these, the $R$Bi$_2$ family displays different magnetic ground states depending on the choice of $R$[@Petrovic2002]. Structurally, $R$Bi$_2$ forms in an orthorhombic structure with single layers of Bi separated from each other by $R$Bi bilayers that are stacked along the crystallographic $b$ axis[@Petrovic2002; @Zhou2018PRB]. When $R$ is chosen to be the moment-bearing Ce ion, an antiferromagnetic (AFM) ground state below $T_\text N \sim$ 3.3 K can be stabilized[@Petrovic2002]. A recent study shows that CeBi$_2$ is a Kondo system with a Sommerfeld coefficient $\gamma$ over 200 mJ/mol K$^2$ and Kondo temperature of an order of $\sim $ 2 K[@Zhou2018PRB]. On the other hand, for $R=$ La (non-moment bearing), LaBi$_2$ reveals metallic behavior without indications of magnetic ordering or superconductivity down to 1.8 K[@Petrovic2002]. In this study, we perform a comparative study of the ground-state tunability of these two members by external pressure. We explore the temperature-pressure phase diagram of CeBi$_2$ and LaBi$_2$ by resistance measurements and complement these, in case of CeBi$_2$, with specific heat measurements. Our results show that $T_\text N$ of CeBi$_2$ is moderately increased upon increasing pressure. Surprisingly, resistance measurements of both CeBi$_2$ and LaBi$_2$ show signatures pressure-induced superconductivity at low temperature ($T\lesssim$ 4 K) above very similar threshold pressures ($p\gtrsim$ 1.68 GPa). However, specific heat measurement of CeBi$_2$ does not reveal any anomaly that could be associated with a transition into the superconducting state. We assign these effects to filamentary SC that likely originates from traces of Bi flux, either on the surface of the plate-like samples, or trapped inside the sample as laminar inclusions. Finally, the analysis of pressure-dependent resistance data at fixed temperatures for CeBi$_2$ suggests that there might be a pressure-induced crossover most likely associated with pressure-induced changes in the Kondo temperature and crystal electric field splitting. Experimental details ==================== Single crystals of CeBi$_2$ and LaBi$_2$ were grown by a Bi self-flux technique with the help of a frit-disk alumina Canfield Crucible Set[@Canfield1992; @Canfield2016a]. For CeBi$_2$, Ce and Bi in the molar ratio 9:91 were loaded into a crucible set and sealed into a fused silica ampoule under partial argon atmosphere. The ampoule was heated to 1000 $^\circ$C in 5 h and dwelled at this temperature for another 4 h. It was then slowly cooled to 600 $^\circ$C over 45 h. At this temperature, the ampoule was removed from the furnace and excess liquid was decanted by the help of a centrifuge. For LaBi$_2$, La and Bi in the molar ratio 8:92 were loaded into the crucible set, heated to 1000 $^\circ$C in 5h, dwelled at 1000 $^\circ$C for 2 h, and slowly cooled to 350 $^\circ$C over 80 h. The resulting crystals of CeBi$_2$ and LaBi$_2$ are millimeter-size and plate-shaped. Both CeBi$_2$ and LaBi$_2$ crystals are air-sensitive, the preparation of experiments was therefore performed in a N$_2$ glovebox. The $ac$, in-plane resistance measurements were performed in a Quantum Design Physical Property Measurement System (PPMS) using a 1 mA excitation with frequency of 17 Hz, on cooling using a rate of - 0.25 K/min. The magnetic field was applied perpendicular to the current direction. For CeBi$_2$, two different samples (labeled as S1 and S2) were used in resistance measurements. S1 was measured at ambient condition outside pressure cell and S2 was measured under pressure. The temperature-dependent resistance data for S1 is normalized by extrapolating $p\leq 1.23 GPa$ pressure-dependent resistance data, $R(p)$, at 300 K from S2 back to 0 GPa (see Fig. \[fig1\_RT\]). For LaBi$_2$, only one sample was measured under pressure with the pressures 0.60 GPa $\leq p \leq$ 2.52 GPa. For both compounds, a standard four-contact configuration was used with contacts made by Dupont 4929N silver paint. Specific heat measurements under pressure were performed using an ac calorimetry technique on a third sample (sample S3) in a cryogen-free cryostat from ICEOxford (Lemon-Dry) with base temperature of 1.4 K. Details of the setup used and the measurement protocol are described in Ref. . In this study, a Be-Cu/Ni-Cr-Al hybrid piston-cylinder cell, similar to the one described in Ref. , was used to apply pressure. Good hydrostatic conditions were achieved by using a 4:6 mixture of light mineral oil:n-pentane as pressure medium, which solidifies, at room temperature, in the range $3-4$ GPa, i.e., well above our maximum pressure[@Budko1984; @Kim2011; @Torikachvili2015]. Pressure values were inferred from the $T_{c}(p)$ of lead[@Bireckoven1988], determined via resistance measurements. Results ======= CeBi$_2$ -------- Figure \[fig1\_RT\] shows the temperature-dependent resistance of CeBi$_2$ at ambient pressure (sample S1) and pressure up to 2.44 GPa (sample S2). The temperature-dependent resistance data for S1 is normalized by extrapolating the 300 K pressure-dependent resistance data ($R(p)$ for $p\leq$ 1.23 GPa) measured from S2 back to 0 GPa. As shown in the figure, the resistance decreases upon cooling, showing a metallic behavior. At $T\sim$ 50 K, a broad drop of resistance is observed. In an earlier work, it was suggested that this drop in $R(T)$ is associated with either the coherence in Kondo scattering or crystal electric-field (CEF) splitting of Ce atoms[@Zhou2018PRB]. At $T\sim$ 3.3 K, the resistance shows a kink-like anomaly due to loss of spin-disorder scattering as CeBi$_2$ undergoes an AFM transition at $T_\text N$[@Petrovic2002; @Zhou2018PRB]. Sample S2 was measured under pressure and at lowest pressure (0.12 GPa), resistance of S2 shows very similar feature as S1. Upon increasing pressure, the resistance gradually increases over a large temperature range (essentially everywhere in the paramagnetic state). This is a typical behavior for a Ce-based Kondo lattice under pressure[@Thompson1994; @Hegger2000; @Nicklas2001; @Nicklas2003]. The broad drop of resistance at ambient pressure becomes progressively more pronounced, as pressure is increased, and evolves into a local maximum at highest pressures. The temperature of this broad drop/hump feature is labeled as $T '$ and indicated by arrow in the figure (see below for the description of the criterion used). The evolution of this feature will be analyzed and discussed in more details below. As we move to the low-temperature region (inset to Fig. \[fig1\_RT\]), for $p\le$ 1.23 GPa, the kink-like anomaly, which is associated with the magnetic transition[@Petrovic2002; @Zhou2018PRB], is shifted to higher temperatures upon increasing pressure. Even with this slight increase in $T_\text N$, the loss of spin disorder scattering below $T_\text N$ remains fundamentally the same. As a result, the resistance at 1.8 K, $R$(1.8 K), does not show a significant change. Upon increasing from 1.23 GPa to 1.68 GPa, $R$(1.8 K) shows a sudden decrease. For $p>$ 1.68 GPa, the resistance as a function of temperature, $R(T)$, undergoes a much sharper drop and reaches a zero value, suggesting a pressure-induced superconducting phase at low temperature. The critical temperature of this phase is increased upon increasing pressure. The temperature-derivative of the resistance data is shown in Fig, \[dRdT\_fig3\] to better differentiate between the low $p$ and high $p$ feature at low temperature as well as to trace the broad feature at $T \sim$ 50 K. As shown in Fig.\[dRdT\_fig3\] (a), at low pressures ($p\le$ 1.23 GPa), the magnetic transition shows up as a jump-like feature in the d$R$/d$T$. We therefore define $T_\text N$ as the midpoint of the jump-like feature in d$R$/d$T$ (see dotted lines and arrow in Fig. \[dRdT\_fig3\] (a) as well as Figs. \[fig3\_cT\] (b) and (c) below). As a result, $T_\text N$ increases with increasing $p$ with a slope of $\sim$ 0.48 K/GPa. At higher pressures ($p\ge$ 1.68 GPa), the superconducting transition can be seen as a sharp peak in d$R$/d$T$. Figure \[dRdT\_fig3\] (b) shows d$R$/d$T$ curves over a larger temperature range. As shown in the figure, the broad drop/hump features in $R(T)$ are reflected in minima d$R$/d$T$. We therefore define the crossover temperature $T '$, which marks the change between two different resistance regimes, by the minima in the the d$R$/d$T$ as indicated by the dashed lines in the figure. It is clearly seen that $T '$ increases upon increasing pressure. To trace the magnetic transition to higher pressures, the temperature-dependent resistance under magnetic fields up to 9 T applied along the $b$-axis was studied. The applied field can suppress the superconducting transition which masks the signature of the magnetic transition for $p\ge$ 1.68 GPa. The results for selected pressures are presented in Fig. \[fig2\_RTH\]. As shown in Figs. \[fig2\_RTH\] (a) and (c), at 0.12 GPa the kink-like anomaly in $R(T)$ associated with magnetic transition is broadened in higher fields, yet not much shifted with an applied field of 3 T. In the temperature derivative of the resistance data, the corresponding jump-like feature is suppressed with increasing magnetic fields until it disappears at higher fields. At 2.44 GPa, the sharp drop of the resistance in $R(T)$ associated with superconducting transition at $\sim$ 5 K is suppressed to lower temperatures with magnetic fields and the kink-like anomaly re-emerges at $\sim$ 4 K. Further increasing magnetic fields broadens the kink-like anomaly until it disappears. Similarly, in the temperature derivative d$R$/d$T$, we first observed a sharp peak associated with the superconducting transition at low magnetic fields. Upon increasing the field, the sharp peak is suppressed and shifted to lower temperatures, at the same time, a second jump-like feature emerges. At even higher fields, both features disappear. By analogy we associate this re-emerged kink-like anomaly in $R(T)$ (jump-like feature in d$R$/d$T$) with the same magnetic transition that is observed at low pressures. The resistance does not become zero at 1.8 K for magnetic field $B\geq$ 2 T indicating a critical field of $\sim$ 2 T at 1.8 K. ![Resistance of CeBi$_2$ as a function of temperature at ambient pressure (measured on sample S1) and at different finite pressures up to 2.44 GPa (measured on sample S2). The ambient pressure data for S1 is normalized by extrapolating $p\leq$ 1.23 GPa pressure-dependent resistance data, $R(p)$, at 300 K from S2 back to 0 GPa. A broad hump feature is present in all data sets. The inferred crossover temperature $T '$ is exemplarily marked for the data set at 2.44 GPa (for more details, see text). Inset: Blowup of the resistance data at low temperatures showing the magnetic and superconducting transitions. AFM transition temperature $T_\text N$ is indicated by arrow. Criterion for $T_\textrm{c}^\textrm{offset}$ is indicated by arrow. \[fig1\_RT\]](fig1_RT){width="8.6cm"} ![(a) Temperature derivative of the resistance,d$R$/d$T$, in the low-temperature region ($T\le$ 15 K). The criterion for the determination of the AFM transition temperature $T_\text N$ is illustrated by dashed lines and marked by the arrow (midpoint of the jump-like feature). At high pressures, the magnetic anomaly is masked by a strong drop of resistance, likely due to spurious SC (see main text). The respective temperature is denoted by $T_\text c$ (see arrow). (b) Temperature derivative of the resistance,d$R$/d$T$, showing the evolution of the temperature associated with the broad hump feature in $R(T)$ curves. $T'$ is determined by the minimum in d$R$/d$T$ curves. Data sets in (a) and (b) are offset for clarity. \[dRdT\_fig3\]](dRdT_fig3){width="8.6cm"} ![(a), (b) Temperature-dependent resistance of CeBi$_2$ S2 under magnetic fields up to 9 T for selected pressures. Fields are applied along the $b$-axis. (c), (d) Temperature-derivative of the resistance data, taken in applied magnetic fields, shown in (a) and (b), respectively. Data sets are offset for clarity. Criteria for $T_\text N$ at 0 T and 3 T are indicated by arrows (midpoint of the jump-like feature). \[fig2\_RTH\]](fig2_RTH){width="8.6cm"} ![Pressure dependence of resistance, $R(p)$, at fixed temperatures for CeBi$_2$. A change of slope between 1.68 GPa and 1.97 GPa is indicated by the cross of the dashed line. \[RP\_CeBi2\]](RP_CeBi2){width="8.6cm"} To further investigate the overall increase of resistance with pressure, we present in Fig. \[RP\_CeBi2\] the pressure dependent resistance $R(p)$ at fixed temperatures. As shown in the figure, a change of slope is observed when pressure is increased from 1.68 GPa to 1.97 GPa at 10 K, this feature persists up to 300 K, the highest temperature investigated in this study. The strongest pressure responses are for $T\lesssim T '$, suggesting shifts in the Kondo feature around $T '$. Whereas the $R(p)$ data for 300 K are quite similar to what is found for LaBi$_2$ in Fig. \[RP\_LaBi2\] (see below). ![(a) Evolution of the temperature-dependent specific heat, $C_p(T)$, with pressure up to 2.55 GPa for CeBi$_2$ S3. Criterion for $T_\text N$ is indicated by arrow (midpoint of the specific heat jump). The inset shows the data near 4.6 K for 2.36 GPa, the gray vertical line indicates a 13.2 mJ/mol K specific heat jump at 4.6 K (details are discussed in the main text). (b), (c) Temperature-dependent specific heat data and temperature-derivative of the resistance data at two sets of nearly identical pressures ((b) 0.04 GPa and 0 GPa, (c) 1.28 GPa and 1.23 GPa). Note that the midpoint criterion gives same $T_\text N$ values for both data sets. \[fig3\_cT\]](fig3_cT){width="8.6cm"} The observation of a state with zero resistance in CeBi$_2$ calls for a thermodynamic investigation of the temperature-pressure phase diagram. Thus, we studied the specific heat of CeBi$_2$ (sample S3) under pressure and the results are presented in Fig. \[fig3\_cT\] (a). At lowest pressure (0.04 GPa), very close to ambient pressure, the specific heat, $C_p(T)$, nicely reveals a nearly mean-field-like anomaly at $T\sim$ 3.2 K, which speaks in favor of a second-order phase transition. The shape, position, and size of the feature is consistent with the specific results of a previous study and therefore allows us to assign this feature to the magnetic transition at $T_\text N$. Figures. \[fig3\_cT\] (b) and (c) show the comparison between temperature dependent $C_p$ and d$R$/d$T$ at two sets of nearly identical pressures (0.04 GPa and 0 GPa, 1.28 GPa and 1.23 GPa). As shown in the figure, temperature-dependent $C_p(T)$ and d$R$/d$T$ exhibit similar jump-like feature at the transition temperature which is consistent with the Fisher-Langer relation[@Fisher1968; @Alexander1976]. Thus, to determine the transition temperature, $T_\text N$, from specific heat measurement, same criterion as in the resistance measurement is used (midpoint of jump-like anomaly as indicated by dashed lines and arrow in Fig. \[fig3\_cT\] (a)). As pressure is increased up to 2.55 GPa, $T_\text N$ is monotonically increased. At the same time, the jump size of the anomaly does not significantly change indicating that the amount of entropy released at $T_\text N$ is unchanged. However, we did not observe a second feature at any pressure, thus suggesting that CeBi$_2$ does not undergo any other phase transition than the magnetic one. This includes in particular also a possible superconducting transition for $p>$ 1.68 GPa inferred from our resistance data. One might argue that a possible superconducting feature in specific heat is masked by the huge entropy release at the magnetic transition, as $T_\text N$ and the resistive $T_\text c$ are very close. However, even at high pressure, at which we expect that $T_\text N$ and $T_\text c$ are well separated, no feature in specific heat occurs (see inset of Fig. \[fig3\_cT\] (a)). Another possibility for the apparent absence of a specific heat feature might be that the superconducting jump size is very small and therefore falls below the resolution limit. In the following, we provide estimates for the lower and upper bound of superconducting jump size in CeBi$_2$. For a phonon-mediated BSC superconductor, the specific heat jump at the superconducting transition can be written as, $$\Delta C = 1.43\gamma T_\text c, \label{C jump}$$ where $\gamma$ is the electronic Sommerfeld coefficient and $T_\text c$ is the superconducting transition temperature. To estimate a possible lower limit of $\Delta C$, we first assume that superconductivity is unrelated to the Kondo-lattice-nature of CeBi$_2$. Thus, for the choice of $\gamma$, we refer to the nonmagnetic reference LaBi$_2$ which is isostructural to CeBi$_2$. Since LaBi$_2$ has a $\gamma$ value of 2 mJ/mol K$^2$[@Petrovic2002], with $T_\text c$ $\sim$ 4.6 K from Fig. \[fig1\_RT\], we get $\Delta C = $13.2 mJ/mol K. Compared to the noise level, such value of specific jump (gray vertical line in the inset of Fig. \[fig3\_cT\] (a)) should be resolvable. For an upper limit, we take the $\gamma$ value of the Kondo-lattice CeBi$_2$, 200 mJ/mol K$^2$[@Zhou2018PRB], we get $\Delta C = $1.32 J/mol K, which would be one hundred times larger than the gray vertical line in the inset of Fig. \[fig3\_cT\] (a). The absence of any resolvable specific heat jump feature, which can be associated with superconductivity, suggests that the pressure-induced superconductivity is likely filamentary rather than bulk. This conclusion will be related to again below after presentation of data on LaBi$_2$. ![Temperature-pressure phase diagram of CeBi$_2$ as determined from resistance and specific heat measurements. Black squares and diamonds represent the magnetic transition $T_\text N$ determined from resistance measurement for 0 T and 3 T respectively. Black stars represent $T_\text N$ determined from specific heat measurement. Black open symbols represent the superconducting transition $T_\textrm{c}^\textrm{offset}$ determined from resistance measurement. Blue pentagons represent $T '$ determined from resistance measurement (Note the right axis used here for $T '$). Gray and red areas represent the antiferromagnetically ordered and filamentary-superconducting regions, respectively. \[fig4\_phasediagram\]](fig4_phasediagram){width="8.6cm"} We summarize our $T_\text N$ and $T '$ data for CeBi$_2$ as well as our $T_\text c^\text {offset}$ (filamentary) data in the temperature-pressure ($T-p$) phase diagram shown in Fig. \[fig4\_phasediagram\]. For the magnetic transition, both $T_\text N$ at zero field and 3 T from resistance measurement (Fig. \[fig2\_RTH\]) and $T_\text N$ from zero field specific heat data are included. For superconducting transition, $T_\text c^\text {offset}$ is determined from resistance measurement (Fig. \[fig1\_RT\] (b)). The $T_\text N$ values, inferred from $R(T,p)$ and $C(T,p)$ agree reasonably well within their experimental resolution. As shown in Fig. \[fig4\_phasediagram\], magnetic field suppresses magnetic transition $T_\text N$ slightly ($\sim$ 0.2 K by 3 T), as is often the case for antiferromagnets. $T_\text N$ increases monotonically with pressure with a rate of 0.48 K/GPa up to 2.55 GPa. For superconductivity, it first sets in at $\sim$ 1.68 GPa with a sharp drop in $R(T)$, yet not give rise to zero resistance down to 1.8 K. Upon increasing pressure, the drop in $R(T)$ becomes progressively sharper and zero resistance at low temperature is reached as well. Furthermore, from 1.68 GPa to 2.44 GPa, $T_\text c^\text {offset}$ monotonically increases from 2.1 K to 4.8 K, appearing to saturate at our highest pressure. Finally, the temperature $T '$ associated with Kondo coherence scattering or CEF splitting is suppressed upon increasing pressure, with $T '\simeq$ 98 K at 0 GPa and 74 K at 2.44 GPa. LaBi$_2$ -------- Next, we discuss our resistance data for the non-magnetic, LaBi$_2$, member of the $R$Bi$_2$ family. Figure \[fig5\_RT\] presents the pressure evolution of the temperature-dependent resistance for LaBi$_2$ with pressures 0.60 GPa $\leq p \leq$ 2.52 GPa. For all pressures, resistance decreases upon cooling, showing metallic behavior. For a large temperature range ($T \gtrsim$ 50 K), the resistance shows linear dependence on temperature. In the low-temperature region (upper inset of Fig. \[fig5\_RT\]), for $p \le$ 1.03 GPa, resistance as a function of temperature is relatively flat suggesting that the low-temperature resistance is dominated by impurity scattering. At 1.68 GPa, $R(T)$ shows a faster drop of resistance below $\sim$ 2.5 K. When pressure is further increased, this drop of resistance becomes more pronounced. At 2.52 GPa, resistance actually drops to zero below 2.7 K, suggesting pressure-induced superconductivity. The drop of resistance, visible for 1.68 GPa $\le p \le$ 2.34 GPa, is likely to be associated with traces of superconducting phase. Using the criterion defined in the upper inset of Fig. \[fig5\_RT\], the superconducting transition temperature, $T_\text c^\text {offset}$, can be traced and the results are shown in the bottom inset of Fig. \[fig5\_RT\]. As shown in the figure, $T_\text c^\text {offset}$ increases from 1.2 K to 3 K when pressure is increased from 2.10 GPa to 2.52 GPa. ![Resistance of LaBi$_2$ as a function of temperature at different pressures for 0.60 GPa $\leq p \leq$ 2.52 GPa. Upper inset: blowup of the resistance data at low temperatures showing the superconducting transition. Criterion for $T_\text c^\text {offset}$ is indicated by arrow. Bottom inset: superconducting transition temperature, $T_\text c^\text {offset}$, as a function of pressure. Red area represent the superconducting region as inferred from resistance measurement. \[fig5\_RT\]](fig5_RT){width="8.6cm"} ![Evolution of the field-dependent resistance at 2 K of LaBi$_2$ with pressure 0.60 GPa $\leq p \leq$ 2.52 GPa and fields applied along the $b$-axis. The lowest pressure data (0 GPa) is not included due to excessive noise. Inset shows the blowups of the low-field region. \[fig6\_RH\]](fig6_RH){width="8.6cm"} The field dependence of the resistance at 2 K was studied and is presented in Fig. \[fig6\_RH\]. For $p\le$1.03 GPa, resistance gradually increases with magnetic field with a slightly up-bending curvature. For $p\ge$1.68 GPa, at low fields, the resistance first undergoes a fast increase upon increasing fields, which is likely due to the suppression of superconductivity. At higher fields, $R(H)$ curves behave similarly with the ones at lower pressures. Moreover, at 2.52 GPa the zero resistance at 2 K is lifted for $H\gtrsim$ 500 Oe, indicating a critical field of $\sim$ 500 Oe. Bearing in mind that close to ambient pressure the magnetoresistance clearly deviates from the conventional $H^2$ behavior, we observe that pressures up to $\sim$ 2.5 GPa do not modify this behavior (besides the lower field effects of superconductivity) in any conspicuous way. The data in Figs. \[fig5\_RT\] and \[fig6\_RH\] are consistent with traces of SC phase, with distributions of $T_\text c$ values existing in the LaBi$_2$ sample. The mean $T_\text c$ of these filamentary traces increases with pressure for $p>$ 1.68 GPa. ![Pressure dependence of resistance, $R(p)$, at fixed temperatures for LaBi$_2$. The lowest pressure data (0 GPa) is not included due to excessive noise. \[RP\_LaBi2\]](RP_LaBi2){width="8.6cm"} To better visualize the pressure evolution of the higher temperature resistance for LaBi$_2$, Fig. \[RP\_LaBi2\] presents the pressure-dependent resistance $R(p)$ at fixed temperatures. The resistance of LaBi$_2$ first decreases and then increases with pressure, giving rise to a broad minimum between 1.03 GPa and 1.68 GPa. Compared with the $R(p)$ of CeBi$_2$, $R(p)$ of LaBi$_2$ has a similar higher-pressure, higher-temperature up-turn, but lacks the larger $T\lesssim T '$ pressure dependence seen in CeBi$_2$. Discussion ========== Before discussing the implications of the zero-resistive state, which we observed in CeBi$_2$ and LaBi$_2$ at higher pressures, we first focus on the increase of $T_\text N$ and decrease of $T '$ under pressure in CeBi$_2$, as this is robustly established by our resistance and specific heat study. The properties of a Kondo lattice system are usually dominated by two characteristic energy scales, which are both susceptible to externally applied pressure: Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction energy $T_\text {RKKY} \propto J^2$ and Kondo interaction energy $T_\text K \propto \text e^{-1/J}$ where $J$ is the exchange interaction[@Ruderman1954PhyRev; @Kasuya1956; @Yosida1957PhyRev; @Kondo1964; @Hewson1993]. When $T_\text {RKKY} \gg T_\text K$, the ground state is magnetic and for $T_\text K \gg T_\text {RKKY}$, it is nonmagnetic. The competition between them and the resulting ground state is often described by the Doniach phase diagram[@Doniach1977]. For Ce-based compounds, the ground state is often magnetic. Applying external pressure can suppress magnetic transition temperature to zero and lead to non-magnetic ground state via a quantum critical point[@Steglich1979; @Jaccard1992; @Mathur1998; @Park2006; @Knebel2006; @Jiao2015]. In our study, the AFM transition temperature $T_\text N$ of CeBi$_2$ is moderately increased by pressure up to $\sim$ 2.5 GPa. This suggests that at ambient pressure, CeBi$_2$ is deeply in its magnetic state and higher pressure is needed to suppress $T_\text N$[@Knebel2006a; @Chen2006; @Kimura2007; @Bauer2010]. This is compatible with the Doniach picture, as there is a maximum of $T_\text N$ due to the explicit functional dependences of $T_\text {RKKY}$ and $T_\text K$. Moreover, in the Doniach picture, when pressurizing a Ce-based Kondo lattice, an increase of $T_\text K$ is often observed due to the enhancement of exchange interaction $J$[@Thompson1994; @Goltsev2005]. This, in turn, should give rise to a shift of broad resistive features, associated with $T_\text K$, to higher temperatures with pressure. Therefore, a suppression of $T '$ observed in this study suggests that the broad drop/hump feature in $R(T)$ can not be explained by only the Kondo coherence scattering[@Hegger2000; @Muramatsu2001; @Nicklas2003]. The resistance measurements for both CeBi$_2$ and LaBi$_2$ reveal a zero-resistive state at high pressures, suggesting a pressure-induced SC phase for these compounds. By comparing their $T-p$ phase diagrams (Figs. \[fig4\_phasediagram\] and \[fig5\_RT\] (a) inset), we see that the two phase diagrams exhibit similar SC phase regions, but with slightly different onset pressures and $T_\text c$ values. For CeBi$_2$ $T_\text c$ saturated at $\sim$ 4.8 K by 2.44 GPa whereas $T_\text c$ of LaBi$_2$ reaches $\sim$ 3 K but seems still rising with pressure. Moreover, at the highest pressures in this study (2.44 GPa for CeBi$_2$ and 2.52 GPa for LaBi$_2$), CeBi$_2$ and LaBi$_2$ have very different critical fields at $\sim$ 2 K ($\sim$ 2 T for CeBi$_2$ and $\sim$ 500 Oe for LaBi$_2$). Despite the zero-resistive state and relative sharp resistance drop at high pressures for CeBi$_2$ and LaBi$_2$, we would like to argue that the observed SC feature is extrinsic for the following reasons. First of all, specific heat measurement under pressure for CeBi$_2$ does not reveal any SC feature which strongly speaks in favor of filamentary SC. Second, similar $T_\text c$ values for Ce and La are unlikely in bulk $R$Bi$_2$. On one hand, if the SC in these two compounds is standard BSC SC, then hybridizing rare earths such as Ce or Yb suppresses $T_\text c$ aggressively[@Maple1972; @Canfield1998; @Budko2006]. On the other hand, if CeBi$_2$ at high pressures becomes a heavy fermion superconductor, the specific heat jump anomaly at $T_\text c$ should be even bigger. Then similar SC onset pressure and $T_\text c$ between LaBi$_2$ and CeBi$_2$ are unlikely again as LaBi$_2$ is not a heavy fermion compound. To speculate about the possible origin of the filamentary SC, we refer to literature. First we notice that similar situation has been found in other Bi compounds as well where SC is attributed to Bi flux or thin films of Bi[@Thamizhavel2003; @Mizoguchi2011; @Lin2013]. Moreover, it is know that single-crystalline Bi undergoes sequential structural transitions upon increasing pressure and possesses rich physics under pressure[@Klement1963; @Degtyareva2004; @Li2017a]. Specifically, at low temperature, Bi-II exists between 2.55 GPa and 2.70 GPa with $T_\text c \sim$ 3.9 K and upper critical field $\mu_0 H_{c2}$(2 K)$\sim$ 0.05 T, Bi-III exists between 2.70 GPa and 7.7 GPa with $T_\text c \sim$ 7 K and $\mu_0 H_{c2}$(2 K)$\sim$ 3 T[@Li2017a]. Owing to the very similar $T_\text c$ of Bi-II to our results on CeBi$_2$ in the almost identical pressure range, we suspect that the filamentary SC we observed in the resistance measurement of CeBi$_2$ originates from traces of Bi flux. It is likely that the SC in LaBi$_2$ is non-bulk and origins from Bi flux as well. Slight differences in onset pressure and $\mu_0 H_{c2}$ could arise from details of the unit cell parameters which could give rise to slightly different strain conditions. Conclusion ========== In conclusion, the resistance of $R$Bi$_2$ ($R$ = La and Ce) under pressure up to $\sim$ 2.5 GPa and ac specific heat of CeBi$_2$ under pressure up to 2.55 GPa have been studied. Our studies show that for CeBi$_2$ the antiferromagnetic transition temperature, $T_\text N$, increases upon increasing pressure with the rate of $\sim$ 0.48 K/GPa. This fits into the Doniach phase diagram and suggests that there might be a maximum of $T_\text N$, followed by its decrease and finally a quantum critical point at possibly significantly higher pressures. Resistance and ac specific heat measurements of CeBi$_2$ together suggest that the pressure-induced superconductivity in CeBi$_2$ is likely not bulk. It is likely that the SC phase is filamentary Bi either on the surface or as laminar in the bulk of the sample. We suspect the pressure-induced superconductivity in LaBi$_2$ to arise from a similar extrinsic origin giving that the onset pressure and transition temperature of superconductivity are very similar to that of CeBi$_2$. Further pressure-dependent resistance analyses for CeBi$_2$ and LaBi$_2$ indicate some anomalies in the $R(p)$ curves, a change of slope between 1.68 GPa and 1.97 GPa for CeBi$_2$ and a broad minimum between 1.03 GPa and 1.68 GPa for LaBi$_2$. Taken together, these suggest that the stronger, low-temperature features see near and below $T '$ for CeBi$_2$ are related to the pressure dependent hybridization and crystal electric field splitting of the Ce. Finally, we would like to point out, again, that when studying the properties of Bi-rich compounds under pressure, one needs to be very careful and mindful for the various phases elemental Bi has and the rich physics they display at different pressures[@Klement1963; @Degtyareva2004; @Li2017a]. Work at the Ames Laboratory was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DEAC0207CH11358. L.X. was supported, in part, by the W. M. Keck Foundation and the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4411. [60]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty [**, ()](\doibase 10.1143/jpsj.30.294) [, ()](http://dx.doi.org/10.1088/0953-8984/26/21/212201) [, ()](\doibase 10.1063/1.555540) [, ()](http://dx.doi.org/10.1039/C6CP02856J) [, ()](\doibase 10.1103/PhysRev.87.799) [, ()](http://dx.doi.org/10.1038/srep21484) [, ()](http://stacks.iop.org/0953-2048/29/i=3/a=03LT02) [, ()](https://link.aps.org/doi/10.1103/PhysRevB.98.214509) [, ()](http://dx.doi.org/10.1088/1361-648X/aaf03a) [, ()](http://dx.doi.org/10.1088/0953-8984/26/39/395701) [, ()](\doibase 10.1063/1.1660369) [, ()](http://www.sciencedirect.com/science/article/pii/S0304885302002780) [, ()](\doibase 10.1002/pssr.201206405) [, ()](\doibase 10.1103/PhysRevApplied.4.014021) [, ()](\doibase 10.1103/RevModPhys.82.3045) [, ()](\doibase 10.1103/RevModPhys.83.1057)[, ()](\doibase 10.1103/PhysRevLett.104.057001)[, ()](http://dx.doi.org/10.1038/nphys1274)[, ()](http://science.sciencemag.org/content/325/5937/178.abstract)[, ()](\doibase 10.1103/PhysRevB.97.195120)[, ()](\doibase 10.1080/13642819208215073)[, ()](\doibase 10.1080/14786435.2015.1122248)[, ()](\doibase 10.1063/1.5084730)[ ()](http://www.jetp.ac.ru/cgi-bin/e/index/e/59/2/p454?a=list)[, ()](\doibase 10.1103/PhysRevB.84.134525)[, ()](http://aip.scitation.org/doi/abs/10.1063/1.4937478)[, ()](http://stacks.iop.org/0022-3735/21/i=9/a=004)in [](http://www.sciencedirect.com/science/article/pii/S0168127305800625), Vol. (,) pp. [**, ()](https://link.aps.org/doi/10.1103/PhysRevLett.84.4986)[, ()](http://dx.doi.org/10.1088/0953-8984/13/44/104)[, ()](\doibase 10.1103/PhysRevB.67.020506)[, ()](https://link.aps.org/doi/10.1103/PhysRevLett.20.665)[, ()](https://link.aps.org/doi/10.1103/PhysRevB.13.304)[, ()](\doibase 10.1103/PhysRev.96.99)[, ()](\doibase 10.1143/PTP.16.45)[, ()](\doibase 10.1103/PhysRev.106.893)[, ()](\doibase 10.1143/PTP.32.37)[](\doibase 10.1017/CBO9780511470752), Cambridge Studies in Magnetism (, )[**, ()](\doibase http://dx.doi.org/10.1016/0378-4363(77)90190-5)[, ()](https://link.aps.org/doi/10.1103/PhysRevLett.43.1892)[, ()](http://www.sciencedirect.com/science/article/pii/037596019290860O)[, ()](https://doi.org/10.1038/27838)[, ()](https://doi.org/10.1038/nature04571)[, ()](https://link.aps.org/doi/10.1103/PhysRevB.74.020501)[, ()](http://www.pnas.org/content/112/3/673.abstract)[, ()](https://link.aps.org/doi/10.1103/PhysRevB.74.020501)[, ()](https://link.aps.org/doi/10.1103/PhysRevLett.97.017005)[, ()](\doibase 10.1143/jpsj.76.051010)[, ()](https://link.aps.org/doi/10.1103/PhysRevB.81.180507)[, ()](http://dx.doi.org/10.1088/0953-8984/17/11/011)[, ()](\doibase 10.1143/jpsj.70.3362)[, ()](http://www.sciencedirect.com/science/article/pii/0038109872902815)[, ()](https://physicstoday.scitation.org/doi/10.1063/1.882396)[, ()](http://www.sciencedirect.com/science/article/pii/S1631070505001799)[, ()](\doibase 10.1143/jpsj.72.2632)[, ()](https://link.aps.org/doi/10.1103/PhysRevLett.106.057002)[, ()](http://www.sciencedirect.com/science/article/pii/S0925838812021330)[, ()](https://link.aps.org/doi/10.1103/PhysRev.131.632)[, ()](\doibase 10.1080/08957950412331281057)[**, ()](https://link.aps.org/doi/10.1103/PhysRevB.95.024510)
--- abstract: 'We discuss the results of recent 3D simulations of first structure formation in relationship to the formation of the first stars. On the basis of a new, high-resolution AMR simulation (spatial dynamic range = $3 \times 10^7$), we conclude that the first stars are likely to be massive.' author: - 'Michael L. Norman' - Tom Abel - Greg Bryan title: First Structure Formation and the First Stars --- First Cosmological Objects ========================== Hierarchical theories of structure formation such as Cold Dark Matter (CDM) and its variants have been very successful in accounting for the existence of galaxies, galaxy clusters and cosmological large scale structure in the low redshift universe (cf. Ostriker 1993, Bertschinger 1997). The cynic would say that this is what they were invented to do. However, the models also make definite predictions at earlier epochs and on different mass scales which can be checked observationally. Some proof that these models are [essentially correct]{} is the remarkable agreement achieved between the observed and predicted properties of the Lyman alpha forest at $2 < z < 4$ (e.g., Zhang et al. 1997, Rauch 1998)—something the models were not designed to do. Encouraged by this agreement, we may ask: [what were the first objects to form in such models?]{} Assuming the CDM power spectrum extends to very small mass scales, the first cosmological objects to form are small dark matter halos collapsing from small scale density fluctuations at high redshift ($z \sim 30$) (Couchman & Rees 1986, Tegmark 1997). Halos whose mass is significantly less than the cosmological Jeans mass $M \sim 10^{4-5} M_{\odot} $ would not be able to trap the baryonic fluid, and hence not form astrophysical objects. Conversely, halos of approximately this mass or greater would be expected to form highly condensed objects (stars, black holes, etc.) provided the gas can cool and transport angular momentum. At the temperatures and densities prevailing in such halos, the primordial gas cools inefficiently by collisional excitation of molecules which exist in low abundance. As described in more detail below, forms via nonequilibrium gas phase reactions in which the tiny post-recombination electron abundance acts as a catalyst. This chemistry, combined with the nonlinear dynamics of halo formation and mergers, makes this epoch of structure formation exceedingly complex and interesting. We have termed this epoch [first structure formation]{}, and have explored it numerically in recent papers (Abel et al. 1998a,b; Abel, Bryan & Norman 1998; 2000; Norman, Abel & Bryan 1999). Here we review our numerical methodologies and findings, and discuss their implications to the first stars. Relevance to First Stars ======================== First structure formation is relevant to the theme of this conference as it is likely that the first stars are formed as a byproduct. Note this is not a theoretical certainty since it has been variously argued that the first objects are not stars in the usual sense, but rather objects at both extremes of the mass spectrum, ranging from supermassive black holes (Silk 1998) to Jupiter sized “clumpuscules" (Combes & Pfenniger 1998). Since the mass function of the first objects is not known, much of the theoretical literature consists of making various assumptions about the primordial mass function (PMF) and working out the cosmological consequences. Carr, Bond & Arnett (1984) presented a comprehensive review of the prevailing ideas as of 1984, many of which are still relevant today. Couchman & Rees (1986) placed the matter in a modern cosmological framework. Both papers highlighted the impact of massive stars and possibly very massive objects (VMOs) on heating, ionizing and enriching the pregalactic medium with heavy elements. A few authors have attempted to compute the PMF from first principles: Silk (1983) argued for a flat PMF between $0.1-100 M_{\odot}$ on the basis of a linear analysis of thermal instability in a primordial gas near the critical density for $H_2$ formation via three body reactions. Padoan, Jiminez & Jones (1997) applied the statistical model for star formation of Padoan (1995) to primordial globular cluster formation, and found a PMF slightly shallower than the Miller-Scalo (1979) IMF with a lower cutoff of $0.2 M_{\odot}$. Given the theoretical uncertainties, let us pose the following question: [do stars form in the first nonlinear structures in hierarchical models?]{} Indications from our recent numerical simulations is that they do. In this paper, we follow our historical developments which lead us toward this conclusion. 2 Formation and the Minimum Mass to Cool ======================================== ![The horizontal dark shaded region shows the mass scale for which the virial temperature equals the CMB temperature. The light shaded area (labeled LS83) represents the domain of parameters for which collapsed structures cannot cool. This curve is computed assuming a spherical collapse model as in Tegmark (1997), except we use the Lepp & Shull (1983) $H_2$ cooling function. Only above the light shaded area are structures believed to be able to collapse and cool via 2. The three heavy dashed lines plot the collapse redshift given by linear theory for the standard CDM spectrum for $1\sigma, 3\sigma$, and $4\sigma$ perturbations. The dot-dashed line represents the Jeans mass at 18$\pi^2$ times the background density. The heavy dot-dashed line labeled HM79 is the minimum mass line computed by Tegmark based on the Hollenbach & McKee (1979) cooling function. The circled numbers denote the evolution of the halo discussed in the text.[]{data-label="minimass"}](figure-1.eps){width="80.00000%"} After recombination, the universe is filled with a neutral, metal-free gas of H and He and trace amounts of D and Li. How can such a gas cool to form stars? Long ago it was realized that small amounts of molecular hydrogen will form in a primordial gas via two reaction pathways: the channel (McDowell 1961): H + e\^- H\^- + h, \[f1a\]\ H\^+ + H H\_2 + e\^- . \[f1b\] and the $H_2^+$ channel (Saslaw & Zipoy 1967): H + H\^+ H\_2\^+ + h, \[f2a\]\ H\_2\^+ + H H\_2 + H\^+ . \[f2b\] At high redshifts $z > 100$ the fragile molecule is photodissociated by the CMB, and thus the channel dominates the production of . At lower redshifts, the situation is reversed and the channel dominates. We see from Eq. (1) that free electrons catalyze the formation of via the channel. Where do they come from? Well, the post-recombination primordial gas is not totally neutral. This is because the recombination time is longer than the cooling time in such a diffuse gas, and the electron abundance is out of equilibrium at a given temperature. Detailed nonequilibrium calculations show that there is a residual ionization fraction of about $x_e \sim 10^{-4}$ at $z \sim 100$ (Galli & Palla 1998). This is enough to form with an abundance of $f_{H_2}\equiv n_{H_2}/n_H \sim 10^{-6}$. As gas concentrates in the potential wells of dark matter halos, the fraction increases according to a simple analytic result (Abel 1998a). In the absence of an external UV background at gas temperatures below 6000 K, one can integrate the rate equations for the free electron and 2 fractions in a collapsing gas cloud to get: $$\begin{aligned} f_{H_2}(t)&=&f_{H_2}(t=0)+\frac{k_{PA}}{k_{rec}} ln(x_0 n_H k_{rec} t+1) \nonumber \\ &=&f_{H_2}(t=0)+1.0\times 10^{-8} T_{vir}^{1.53} ln (t/t_{rec}+1),\end{aligned}$$ where $k_{PA}, k_{rec}, t_{rec}, x_0$, and $n_H$ are the rate coefficients of photo-attachment of and recombination to neutral hydrogen, the initial recombination timescale, ionized fraction, and neutral hydrogen number density, respectively. Given the 2 cooling function, one can ask at what concentration will the cooling time be less than the Hubble time for a spherical top hat perturbation of mass M collapsing at redshift z? This analysis was first carried out by Abel (1995) and later by Tegmark (1997), who found a rather universal result: $f_{H_2}(crit) \sim 5 \times 10^{-4}$. Fig. (\[minimass\]) shows the locus of critical cloud masses as a function of redshift for two choices of the cooling function, which is still somewhat uncertain (for a recent discussion of the uncertainties, see Tiné, Lepp & Dalgarno 1998). As can bee seen, extremely rare $4\sigma$ peaks of mass $\gsim 5 \times 10^4 M_{\odot}$ will collapse and cool at $z \sim 40$, whereas more typical $2.5\sigma$ fluctuations of mass $\gsim 5 \times 10^5 M_{\odot}$ will collapse and cool at $z \sim 25$. The evolution of such a peak described in detail below is shown by the circled numbers in Fig. 1. Simulating First Structure Formation ==================================== First structure formation is a well-posed problem both physically and computationally, since for a given cosmogony the thermodynamic properties, baryon content, and chemical composition are specified for the entire universe at high redshifts, at least in a statistical sense. Futhermore, the dominant physical laws are readily identified: the general theory of relativity describing the evolution of the background spacetime geometry and particle geodesics, the Euler equations governing the motion of the baryonic fluid in an expanding universe, and the primordial kinetic rate equations determining the chemical processes. Radiation backgrounds, save the CMB, are entirely absent prior to first structure formation in standard models. The complete set of equations governing first structure formation in a cosmological framework are given in Anninos (1997). Unlike traditional star formation calculations, which are plagued by a lack of knowledge of appropriate initial conditions, the initial conditions for first structure formation calculations are well defined—namely, a Gaussian random field of density fluctuations characterized by a power spectrum whose shape can be computed theoretically. The amplitude of the fluctuations is constrained by observations on a variety of scales and redshifts (Ostriker 1993, Croft et al. 1999, Nusser & Haehnelt 1999). In the calculations described here, we assume a cluster normalized standard cold dark matter (SCDM) model with the following parameters: $H_0=50$ km/s, $\Omega_0=1, \Omega_b=0.06, \sigma_8=0.7$. As in all star formation calculations, resolving the relevant length scales is the principal technical challenge. A simple estimate of what is required is to compare the baryonic Jeans length at a typical collapse redshift of 25 to the solar radius: $$\frac{\lambda_J}{ r_{\odot }}=5.7 pc (1+z)^{-1/2} / r_{\odot} \equiv 5 \times 10^7$$ Adding a couple more decades in scale to model the cosmological environment brings the total spatial dynamic range to around $10^{10}$. This is easily achievable in 1D spherical symmetry with logarithmic or adaptive grids. However, the all-important effects of hierarchical mergers, tidal forces, and fragmentation all demand 3D simulations. Until recently, such dynamic ranges were impossible to achieve in 3D. We have developed two high-resolution, 3D numerical codes for solving the first structure formatiion equations set forth in Anninos (1997). The first, more primitive code HERCULES (Anninos 1994), uses fixed nested Eulerian grids for an effective resolution of $512^3$ in regions of interest. This code was used to carry out the first self-consistent simulations of first structure formation (Abel 1998) in which we verified the predictions of Tegmark (1997) and quantified the cooled mass fraction in primordial halos. The second, more powerful code ENZO (Bryan & Norman 1997, Norman & Bryan 1999), utilizes an adaptive hierarchical mesh of arbitrary depth to achieve unprecedented resolution wherever it is needed. Using adaptive mesh refinement (AMR), we have previously reported the results of a simulation of first structure formation which achieved a spatial dynamic range of 262,144 (Abel, Bryan & Norman 1998, 2000; Norman, Abel & Bryan 1999). Here we update that result with a new simulation which achieves an additional factor of 128 in resolution for a total spatial dynamic range of $3 \times 10^7$. While not $10^{10}$, this is sufficient to characterize the properties of the collapsing protostellar cloud core. The calculation is done in a comoving box 128 kpc on a side. The starting redshift is z=100. At the stopping redshift z=19.1, the range of resolved scales is $6.4 kpc \gsim \ell \gsim 2 \time 10^{-4} pc$. The mass resolution in the initial conditions within the refined region are 0.53(8.96) in the gas (dark matter). The refinement criteria ensure that: (1) the local Jeans length is resolved by at least 4 grid zones, and (2) that no cell contain more than 4 times the initial mass element (0.53 ). Core Formation ============== ![The top panel shows the evolution of the virial mass of the most massive clump as a function of redshift. The remaining panels show the density (both dark and baryonic), the temperature, and the molecular hydrogen mass fraction at the central point of that clump. The central point is defined as the point with the highest baryon density. Finite gas pressure prevents baryons from clumping as much as the dark matter at redshifts $\gsim 23$. Cooling begins in earnest once the 2 fraction reaches a few $\times 10^{-4}$, lowering the central Jeans mass. At $z \sim 19$ a dense core begins to form and collapse with characteristic mass $\sim 200$ .](figure-2.eps "fig:"){width="80.00000%"} \[evolution\] To illustrate the physical mechanisms at work during the formation of the first cosmological object in our simulation, we show the evolution of various quantities in Fig. 2. The top panel of this plot shows the virial mass of the largest object in the simulation volume. We divide the evolution up into four intervals. In the first, before a redshift of about 33 (point 1 in Fig. 1), the Jeans mass in the baryonic component is larger than the mass of any non-linear perturbation. Therefore, the only collapsed objects are dark-matter dominated, and the baryonic field is quite smooth. In the second epoch, $23 < z < 33$, as the non-linear mass increases, gas becomes trapped in the gravitational potential wells of the dark matter halos. However, this gas is not sufficiently dense to cool and the primordial entropy of the gas prevents dense cores from forming. This is shown in the second frame of Fig. 2 by a large gap between the central baryonic and dark matter densities (note that while the central dark matter density is limited by resolution, the baryonic is not, so the true difference is even larger). As mergers continue and the mass of the largest clump increases, its temperature also grows, as shown in the third panel of this figure. The fraction also increases (bottom panel). By $z \sim 23$ (point 2 in Fig. 1), enough has formed (a few $\times 10^{-4}$) that cooling begins to be important. During this third phase, the central temperature decreases and the gas density increases. However, the collapse is quasistatic rather than a runaway freefall because around this point in the evolution, the central density reaches $n \sim 10^4$ cm$^{-2}$, and the excited states of are in LTE. This results in a cooling time which is nearly independent of density rather than in the low–density limit where $t_{cool} \sim \rho^{-1}$ (e.g. Lepp & Shull 1983). Finally, at $z \sim 19$ (point 3 in Fig. 1), a very small dense core forms and reaches the highest resolution that we allowed the code to produce. We follow the collapse to a final central density of $10^{12}$ cm$^{-3}$, well above the critical density for the formation of 2 via 3–body reactions (see Palla 1983). These reactions are included, as well as an escape probability treatment of radiative transfer of the molecular line radiation (Abel 2000) which becomes important at these densities. Finite word length and the lack of an appropriate equation of state forced the termination of the calculation at this point. Core Structure ============== ![Million-fold zoom showing the multiscale structure of a primordial protostellar cloud core in the center of a low mass halo at z=19.1. Plotted is the logarithm of the baryon overdensity on a slice passing through the densest structure on the grid. Zoom proceeds clockwise from upper left. Linear scales are proper. The smallest grid cell in the center of the cloud core in the last frame is $2 \time 10^{-4}$ pc = 43 AU.[]{data-label="first_star"}](figure-3.eps){width="80.00000%"} The structure of the cloud at the end of the calculation is shown in Fig. 4. The quantity plotted is the logarithm of the baryon overdensity $\rho/<\rho>$ on a slice passing through the densest cell at z=19.1. Proceeding from large scales to small scales we see: (6 kpc)–the filamentary cosmological density field, with a dense halo at the intersection of several filaments; (600 pc)–a virialized halo with $M_{vir}\sim 4 \times 10^{5}$ and $r_{vir}\sim 100$ pc; (60pc)–a dense baryonic core in the center of the cooling halo. From scales of a few pc on down, the core is self-gravitationally bound and collapsing. Despite the non-spherical structure of the cloud, mass-weighted spherical averages of various quantities centered on the density maximum provide insight into the physical processes governing the formation and evolution of the core. A schematic based on a careful analysis of these radial profiles is shown in Fig. 4. ![Schematic of the structure of the primordial protostellar cloud core based on spherical averages around the density maximum at z=19.1. Jagged lines demarkate different physical regions.[]{data-label="cartoon"}](figure-4.eps){width="80.00000%"} Jagged lines in Fig. 4 indicate approximate radii where the material makes a kinematic, thermodynamic, or other significant physical transition. Proceeding form large to small radii, we have the virialization shock ($r_{vir} \approx 100$ pc); the cooling radius ($r_{cool} \approx 25$ pc), defined where the gas temperature reaches its maximum value $T_{vir}\sim 1000K$; the radius where the gas reaches the minimum temperature allowed by 2 cooling $\sim$ 200K ($r_{T_{min}} \approx 5$ pc); the radius where the 2level population enters LTE ($r_{LTE} \approx .5$ pc) defined where $n = 10^4$ cm$^{-3}$; and the radius where 3–body 2 formation reactions become important ($r_{3H} \approx 10^{-2}$ pc), defined where $n = 10^8$ cm$^{-3}$. In the settling zone, the number fraction rises from $7\times 10^{-6}$ to near $f_{H_2}(crit)$. Here 2 cooling is negligible. Inside $r_{T_{max}}$, a cooling flow is established reducing the gas temperature to $T_{min}$. At $r_{T_{min}}$, the sound crossing time becomes substantially shorter than the cooling time. This suggests that inside this radius the core is contracting quasi–hydrostatically on the cooling time scale, which approaches its constant high–density value at small radii. This constant cooling time of $\sim 10^5\yrs$ sets the time scale of the evolution of the fragment until it can turn fully molecular via three body associations. Inside $r \sim 0.3$ pc, the enclosed baryonic mass of $\sim 200 M_{\odot}$ exceeds the local Bonnor-Ebert mass, implying this material is gravitationally unstable. However, due to the inefficient cooling, its collapse is subsonic. The radius where $M > M_{BE}$ defines our protostellar cloud core. Fragmentation? ============== Inside $r_{3H}$ atomic hydrogen is completely converted into 2 via the 3–body reaction. This increases the cooling rate sufficiently to initiate a freefall collapse in the central part of the core. What will be the fate of this collapsing cloud? Omukai & Nishi (1998) have simulated the evolution of a collapsing, spherically symmetric (1D) primordial cloud to stellar density including all relevant physical processes. Coincidentally, their initial conditions are very close to our final state. Our results agree with theirs where our solutions overlap. Based on their results, we can say that if the cloud does not fragment, a massive star will be formed. Adding a small amount of angular momentum to the core does not change this conclusion (Bate 1998), who found a dynamical bar mode instability efficiently transports angular momentum outward in a 3D simulation of galactic protostellar collapse. A third possibility is that the cloud breaks up into low mass stars via thermal instability in the quasi-hydrostatic phase. Silk (1983) has argued that, due to the enhanced cooling from the 3–body produced , fragmentation of this core might ensue and continue until individual fragments are opacity limited (i.e. they become opaque to their cooling radiation). We do not see any evidence of this. Our code, which resolves the Jeans mass everywhere and at all times, would not suppress the growth of these instabilities if they exist. Rather, inspection of the density distribution on the smallest resolvable scales shows a centrally concentrated, object of mass $\sim 10$ collapsing supersonically with $v_r \sim$ -4 km/s. The local free fall time is a few hundred years. The protostar is embedded in a flattened nebula with a pronounced one-arm spiral structure, suggesting an operative angular momentum transport mechanism. Implications ============ The direct implications of our results are that the first stars, here defined to be those which form in a gas of zero metallicity, are massive. Here massive means of order the gravitationally unstable core which forms them $\sim 200$ . Where does this mass scale come from? We argue that it comes from the peculiar properties of a gas cooling solely through 2 line excitation, which sets a minimum temperature for the gas of a few hundred K. The other factor is that above $n_{LTE} \approx 10^{4}$ cm$^{-3}$, the cooling time becomes independent of density, meaning that gravitationally bound clouds will evolve quasi-statically until the onset of 3–body production of 2 at much higher densities. The Bonnor-Ebert mass at $n=n_{LTE}$ and $T=T_{min}$ is 240 , close to our simulated core mass. We suggest that cold gas in the centers of primordial low mass halos fragments into cores of this mass, each of which forms a massive star (10-100 ), or possibly a binary. These stars would be strong sources of UV radiation prior to supernova and/or black hole formation. Recent analyses (Haiman, Abel & Rees 1999; Omukai & Nishi 1999) show that the UV radiation from a single O star per halo will destroy 2 both locally and globally, quenching further star formation by this mechanism. Due to the low binding energy of the parent halo, metals ejected by the supernova explosion would be returned to the pre-galactic medium. Pockets of enriched gas collecting in the cores of more massive halos formed by subsequent mergers would form stars of various metallicities in the familiar but complicated way we have been studying for decades in Galactic molecular clouds. A key question, which must await further numerical investigations, is whether there is a pause in the star formation history of the universe, or rather a smooth transition from the formation of the first stars to the “next stars" as suggested by simulations of Ostriker & Gnedin (1996). 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--- abstract: \| A characterization of textured patterns, referred to as the disorder function $\bar\delta(\beta)$, is used to study properties of patterns generated in the Swift-Hohenberg equation (SHE). It is shown to be an intensive, configuration-independent measure. The evolution of random initial states under the SHE exhibits two stages of relaxation. The initial phase, where local striped domains emerge from a noisy background, is quantified by a power law decay $\bar\delta(\beta) \sim t^{-\frac{1}{2} \beta}$. Beyond a sharp transition a slower power law decay of $\bar\delta(\beta)$, which corresponds to the coarsening of striped domains, is observed. The transition between the phases advances as the system is driven further from the onset of patterns, and suitable scaling of time and $\bar\delta(\beta)$ leads to the collapse of distinct curves. The decay of $\bar\delta(\beta)$ during the initial phase remains unchanged when nonvariational terms are added to the underlying equations, suggesting the possibility of observing it in experimental systems. In contrast, the rate of relaxation during domain coarsening increases with the coefficient of the nonvariational term. address: - '$^{1}$ Department of Physics, The University of Houston, Houston, TX 77204' - '$^{2}$ The Institute of Fundamental Studies, Kandy, Sri Lanka' author: - 'Gemunu H. Gunaratne$^{1,2}$, Anuradha Ratnaweera$^{2}$ and K. Tennekone$^{2}$' title: Emergence of Order in Textured Patterns --- [2]{} Introduction ============ The study of spatio-temporal patterns has received considerable impetus from a series of elegant experiments and theoretical developments based on symmetry considerations. Recent experimental studies include those on reaction diffusion chemical systems [@ouyAswi1], convection in fluids [@heuAgol] and gases [@bodAdeb], ferrofluids [@rose], and vibrated layers of granular material [@melAumb]. These results have been supplemented with patterns generated in (relatively) simple model systems [@croAhoh; @jacAmal; @best]. The most complete theoretical treatments of patterns rely on the study of symmetries of the underlying system and those of the patterns [@golAste]. Unfortunately, this analysis is restricted to periodic or quasi-periodic patterns. A theoretical analysis of more complex states requires the identification of suitable “variables" to describe a given pattern. Examples of such variables include the structure factor [@eldAvin], the correlation length [@ouyAswi2; @croAmei; @chrAbra] and the density of topological defects [@houAsas]. In this paper we study properties of another characterization, referred to as the “disorder function" [@gunAjon; @gunAhof]. The patterns studied are generated in physical systems (and models) whose control parameters are uniform in space and time; thus, they result from spontaneous symmetry breaking. The simplest class of nontrivial structures are periodic. They are typically striped, square, triangular or hexagonal patterns that form in perfect, extended arrays [@croAhoh]. To obtain periodic patterns, the initial state of the system and/or the boundary conditions need to be carefully prepared. A second class consist of periodic patterns whose “unit cells" have additional structure [@kudApie; @judAsil]. A field describing periodic arrays can be expanded in a few plane waves. The patterns described above contain a unit cell that is repeated on a “Bravais lattice" to generate a plane-filling structure. The qualitative description of the pattern involves the characterization (in terms of symmetries) of the unit cell and the generators of the Bravais lattice. For example, the unit cell of a honeycomb lattice is $D_6$-symmetric, and the Bravais lattice is generated by two unit vectors $120^o$ apart. Quasi-periodic patterns have also been observed under suitable experimental conditions [@edwAfau]. Their symmetries can be observed in Fourier space. For example, the spectrum of a quasi-crystal is 10-fold symmetric [@levAste]. Quasi-periodic patterns can be described using a few “principal" plane waves along with their nonlinear couplings. The bifurcations to and from a given periodic (or quasi-periodic) state can be studied using the “Landau equations," which once again rely on the symmetries of the physical system and the pattern [@lanAlif]. The information used is that, since the pattern is generated by symmetry breaking, a second pattern obtained under the action of any symmetry of the [physical system]{} has identical features. The imposition of this equivalence (supplemented by the elimination of “higher order" terms) gives the normal form equations of the pattern. They contain information on aspects of dynamics of the pattern and details about its bifurcations [@golAste]. Patterns such as those of Figs. \[set1\] and \[set2\] (which are generated in a model system) do not belong to the classes discussed above. These structures, referred to as “textured" or “natural" patterns [@cros], are observed when the initial states from which they evolve are not controlled. Similar structures are seen in small aspect ratio systems when the boundaries play a significant role in the creation of the pattern [@croAhoh]. There is no (nontrivial) global symmetry of textures; consequently, they cannot be characterized using symmetry groups. Note also that a second realization of the experiment (e.g., starting from a different set of initial conditions) will give a pattern that is vastly different in detail (such as Figs. \[set1\](a) and \[set1\](b)). In spite of these differences, one can clearly recognize similarities between distinct patterns. For example, the correlation length and the density of topological defects of the two textures shown in Fig. \[set1\] are similar. In contrast patterns generated under other external conditions (e.g., Fig. \[set2\]) have different characteristics. A theoretical treatment of textured patterns requires a “configuration independent" description. In Section II, we introduce the disorder function $\bar\delta(\beta)$ whose definition was motivated in part by the argument leading to the derivation of Landau equations; for patterns generated in uniform, extended systems, $\bar\delta(\beta)$ is required to be invariant under rigid motions of a labyrinthine pattern [@gunAjon; @gunAhof]. In section III, we briefly describe the method for evaluating the disorder function from (typically noisy) grid-values of the field. It relies on a method to approximate a continuous function from values given on a grid referred to as the method of “Distributed Approximating Functionals" (DAFs) [@dafref1]. The main results of the paper, which include properties of the disorder function, and its application to provide a quantitative description of the relaxation of the patterns from an initially random state are presented in Sections IV and V. The underlying spatio-temporal dynamics is given by the Swift-Hohenberg equation (SHE) and one of its variants. We first provide evidence to support the claim that the disorder function consists of intensive, configuration independent variables. The use of $\bar\delta(\beta)$ shows that pattern relaxation occurs in two distinct phases separated by a sharp transition. We also study changes in the relaxation profile when the system is driven further away from the onset of patterns. In Section V, we discuss the effects of adding nonvariational terms to the SHE. The Disorder Function ===================== Textured patterns observed in experimental systems [@ouyAswi1; @heuAgol; @bodAdeb; @rose; @melAumb] and those shown in Figs. \[set1\] and \[set2\] can be described by a scalar field $v({\bf x})$ which is smooth, except perhaps at the defect cores. However, unless the patterns are trivial (e.g., perfect stripes, target patterns) the analytical form of the field is unknown. Consequently, it is difficult to determine a set of “configuration independent" characteristics of structures generated under similar conditions. We instead impose a weaker requirement, that the characterizations remain invariant under the action of the symmetries of the underlying physical system; i.e., translations, rotations and reflections [@gunAjon]. Rather surprisingly, the measures so defined have similar values for distinct patterns such as those shown in Fig. \[set1\]. The most significant feature of labyrinthine patterns is that they are locally striped; in a suitable neighborhood $v({\bf x}) \sim sin ({\bf k\cdot x})$, where the modulus $k_0 (\equiv \|{\bf k}\|) $ of the wave vector does not vary significantly over the pattern. Structures generated in experiments and model systems include higher harmonics due to the presence of nonlinearities in the underlying system; they only contribute to the shape of the cross section of stripes. In order to use the simplest characterization of textures, we eliminate the second and higher order harmonics by the use of a suitable window function in Fourier space. For experimental patterns (which do not have periodic boundary conditions) this is a nontrivial task, and a method to implement it is described in Ref. [@hofAgun]. The simplest local field that is derived from $v({\bf x})$ and whose value remains the same under all rigid motions is its Laplacian $\triangle v({\bf x})$. Terms such as $\triangle^n v^m({\bf x})$, though invariant, are difficult to extract from an incompletely sampled field (typically given on a square lattice). The requirement that perfect stripes be assigned a null measure (they are not disordered), coupled with the local sinusoidal form of the (filtered) pattern implies that the lowest-order field relevant for our purpose is $(\triangle + k_0^2) v({\bf x})$. The family of measures, referred to as the disorder function, is defined by $$\delta(\beta) = (2-\beta){ {\int da \|(\triangle + k_0^2) v({\bf x})\|^{\beta}} \over {k_0^{2\beta} <\|v({\bf x})\|>^{\beta}} }, \label{defn}$$ where $<\|v({\bf x})\|>$ denotes the mean of $\|v({\bf x})\|$, and $\delta(\beta)$ has been normalized so that the “intensive variables" ${ {\bar \delta}({\beta}) = \delta(\beta) / \int da }$ are scale invariant. The moment $\beta$ is restricted to lie between 0 and 2 for reasons discussed below. Local deviations of the patterns from stripes (due to curvature of the contour lines [@gunAjon]) contribute to $\delta(\beta)$ through the Laplacian, while variations of the width of the stripes contribute via the choice of a “global" $k_0$. $\bar\delta(\beta)$ depends on the choice of the wave-numer $k_0$ of the “basic" stripes. Analysis of striped patterns $u_{st} ({\bf x}) = A sin ({\bf k\cdot x})$ and target patterns $u_t ({\bf x}) = A cos(k r)$ show that $\bar\delta(1)$ is minimized when $k_0 = k = \|{\bf k}\|$. Studies of textured patterns from model equations indicate (see Fig. \[dvsk\]) the presence of a unique minimum of $\bar\delta(1)$. We use the minimization of $\bar\delta(1)$ as the criterion for the choice of the wave-number $k_0$ in Eqn. (\[defn\]). For patterns generated using the Swift-Hohenberg equation [@croAhoh], $k_0$ is very close to the wave-number obtained by minimizing the “energy" [@pomAman]. We also find that our estimation of $k_0$ is far more robust (i.e., smaller variation between distinct patterns) than that evaluated from the power spectrum. This is presumably because wavelength variations and curvature of contour lines at each location of the pattern contribute to the computation of $\bar\delta(\beta)$. The variation in $k$ over the pattern (traditionally defined to be the half width of the Structure Factor [@tuAcro]) can be estimated using the variation of $\bar\delta(1)$ with the wave-number $k$ (see Fig. \[dvsk\]). In the remainder of the paper we define $\Delta k$ to be the distance between $k$-values for which $\bar\delta(1)$ is twice the minimum value [@footnote2]. Analysis of textures shows that $\Delta k$ is a configuration-independent, intensive variable. =3.0truein 0.05truein Observe that our choice of $k_0$ is arbitrary in one sense; we could have chosen to minimize $\bar\delta(\beta)$ for any fixed $\beta$ to determine $k_0$. However, the observed variations in $k_0$ are insignificant. Alternatively, we could have evaluated [each]{} $\bar\delta(\beta)$ by minimizing it with respect to $k$. We choose not to implement this scheme because of the need to estimate only one free parameter (i.e., the wave-number) for a given pattern. For a perfect set of stripes the function $\bar\delta(\beta)= 0.$ A domain wall contains curvature of the contour lines and variations of the stripe width; consequently it has non-zero disorder. $\delta(\beta)$ for a single domain wall is a monotonically increasing function of the angle $\theta$ between the stripes of the two domains [@jone]. Thus $\delta(\beta)$ provides information absent in characterizations such as the correlation length. The disorder function for a target pattern $v({\bf x}) = a cos(k_0 r) $ is known [@gunAjon], and is used to determine the accuracy of the numerical algorithms. For target patterns, the integral in the numerator diverges as $(2-\beta)^{-1}$, and leads to limiting the range of $\beta (<2)$, and to the introduction of the prefactor in the definition of $\delta(\beta)$. The weights of distinct characteristics of a texture (e.g., domain walls, defects, variations of the stripe-width, etc.) depend on the moment $\beta$. In particular, the contribution to $\delta(\beta)$ from a domain wall vanishes as $\beta \rightarrow 2$, and the limit is proportional to the number of targets in a pattern. The effects of noise on the calculations are minimal. For example, addition of $10\%$ white noise typically changes $\delta(\beta)$ by less than $2\%$. Distributed Approximating Functionals ===================================== The critical requirement for a good estimate of $\bar\delta(\beta)$ is a sufficiently accurate determination of the Laplacian. Calculation of derivatives of a field from values given on a lattice is a delicate task, especially in the presence of noise. A technique, utilizing what are referred to as Distributed Approximating Functionals (DAFs), has been introduced recently, to fit analytically or approximate a continuous function from known values on a discrete grid [@dafref1]. Unlike typical finite difference schemes, it estimates the function and its derivatives using a range of neighboring points ($\sim$ 40 in our case); consequently, the required computations are much less sensitive to noise. The most useful for our application has been a class of DAFs for which the order of accuracy of the fit is the same both on and off the grid. (This is in contrast to interpolation, which forces the fit to be exact on the grid, but always leads to intertwining about the function off the grid, thus leading to less accurate estimation of derivatives.) Alternatively, the method is designed so that there are no special points. If the labels on the grid points are erased after DAF-fitting of a function, it becomes essentially impossible to identify the points that were on the grid. The most general derivation of the DAFs is via a variational principle [@dafref2], yielding $$g_{DAF}(x)=\sum_{k}I(x,x_{k})g(x_{k}), \label{daf0}$$ where $k$ labels the grid points, $g(x_k)$ are the known input values of the function (which may contain noise), and the sum is over $x_k$ such that $\|x_k-x\|/\Delta < R$, $\Delta$ being the lattice spacing. For suitable $I(x,x_{k})$, the function and its derivatives are evaluated to a comparable accuracy [@dafref2]. This proves crucial in the evaluation of the disorder function. The calculations presented are carried out using the “Hermite DAF" [@dafref3], defined by $$I(x,x_k) = I(x-x_k)={\frac{\Delta}{\sigma}}{\frac{e^{-z^2}}{\sqrt{2\pi}}} \sum_{j=0}^{M/2}\left( -\frac{1}{4}\right) ^{j}\frac{1}{j!} H_{2j}(z), \label{daf1}$$ where $z=(x-x_k)/\sigma\sqrt{2}$ and $H_n(z)$ is the $n^{th}$ Hermite polynomial. The Gaussian weight (of width $\sigma$) in Eqn. (\[daf1\]) makes $I(x_l -x_k)$ highly banded, reducing the computational cost of applying the DAF to data. The DAF representation of derivatives of a function known only on a grid is given by $$\left( \frac{d^{l}g}{dx^l}\right)_{DAF}(x)=\sum_{k} \frac{d^l}{dx^l}I(x,x_k)g(x_k), \label{daf2}$$ which can be evaluated either on or off the grid. In the continuum limit, the derivative of the DAF equals (exactly) the DAF of the derivative [@dafref3]. The DAF approximation to a function that is sampled on a square grid $(x_m,y_n)$ can be obtained using the two-dimensional extension $$I((x,y),(x_m,y_n)) = I_X(x,x_m) I_Y(y,y_n) \label{2dDAF}$$ of the approximating kernel [@zhuAhua]. Thus to estimate (say) $\frac {d^2 v}{dx^2}$, Eqn. (\[daf2\]) needs to be applied in the $y$-direction (with $l=0$) and along the $x$-direction (with $l=2$). (The application of the DAF operators in the two directions commute and can be carried out in any order.) As $M\rightarrow \infty$, $I(x,x_k) \rightarrow \delta(x-x_k)$, and the DAF approximation $g_{DAF}(x)$ becomes exact. With finite $M$, the sum on the right side of Eqn. (3) becomes a polynomial (of order $M$), resulting in $g_{DAF}(x)$ being smooth. Note that $g_{DAF}(x)$ can be considered to be a weighted running average of the signal. The Gaussian width $\sigma$ determines the effect of neighboring points in the DAF-approximation. The range $R$ is chosen to be sufficiently large that the terms ignored in Eqn. (2) are negligible. (The Gaussian weight included in the definition of $I(x,x_k)$ guarantees the decay of these terms.) Patterns Generated Using the Swift-Hohenberg equation ===================================================== The patterns analyzed in the paper are obtained from periodic fields $u({\bf x},t)$ generated by integrating random initial states through a modified Swift-Hohenberg equation (SHE) [@swiAhoh; @croAhoh] $$\partial_t u = D \Bigl(\epsilon - (k_0^2+\triangle)^2 \Bigr) u -\gamma u^3 - \nu (\nabla u)^2 . \label{she}$$ The parameters $D$, $k_0$, and $\gamma$ can be eliminated through suitable rescaling of $t$, ${\bf x}$, and $u$ respectively. $\epsilon$ measures the distance from the onset of patterns. The results for the variational case ($\nu=0$) are presented in this Section and those for the nonvariational case ($\nu\ne 0$) are given in the next. The initial fields for the integration were random numbers in a predetermined range. The time evolution is implemented using the Alternating Direction Implicit algorithm [@preAfla]. Each nonlinear term $N\bigl [u({\bf x},t)\bigr ]$ is expanded to first order in $\delta u = u({\bf x},t+\delta t)-u({\bf x},t)$, thus linearizing the equations in $u({\bf x},t+\delta t)$. Updating the field involves the inversion of a penta-diagonal matrix. The typical time step used for the integration, $\Delta t \sim 0.1$, was chosen so that the higher order terms in $\delta u$ are insignificant. We have confirmed the robustness of the integration by comparing (in a few cases) the results with those done for a smaller time-step ($\Delta t \sim 0.001$). Properties of the disorder function ----------------------------------- =3.0truein 0.05truein The form of the measures $\bar\delta(\beta)$ were deduced by requiring invariance under rigid motions of a single pattern. [Are these limited restrictions sufficient to yield characterizations that can delineate the observed “commonality" in distinct patterns generated under identical conditions? ]{} Surprisingly, it appears to be the case. Fig. \[curves\] shows the disorder functions for several patterns. The curves bunched at the bottom show $\bar\delta(\beta)$ for four textures (two of which are shown in Fig. \[set1\]) generated at fixed control parameters. $\bar\delta(\beta)$ appears to have captured the commonality of these distinct patterns. Structures generated in the Gray-Scott model [@graAsco] and in a vibrated layer of granular material [@melAumb] exhibit similar properties [@jone]. The next question is if $\bar\delta(\beta)$ can differentiate between patterns with different visual characteristics. Fig. \[set2\] shows two structures obtained from the SHE for a second set of control parameters. They have characteristics that differ from patterns of Fig. \[set1\]; e.g., they contain smaller domains and a larger density of defects. $\bar\delta(\beta)$ for four such textures are bunched together on the upper curves in Fig. \[curves\]. The significant separation of the two sets of curves (e.g., the values of $\bar\delta(1)$ between the two sets is about 25 times larger than the average difference between curves within a set) confirms the ability of $\bar\delta(\beta)$ to quantify the differences of the two groups of patterns. =2.5truein 0.3truein The disorder function quantifies the characteristics of a labyrinthine pattern using the local curvature of the contour lines and the wavelength variations, which typically increase with the (visual) disorder of a texture. Thus, $\bar\delta(\beta)$ is able to quantify the observation that patterns of Fig. \[set2\] are more disordered that those of Fig. \[set1\]. Next, we provide evidence to substantiate the claim that $\bar\delta(\beta)$ are intensive variables for labyrinthine patterns such as those shown in Figs. \[set1\] and \[set2\] [@excep]. This is done by comparing values of $\bar\delta(\beta)$ for patterns (with periodic boundary conditions) of several sizes. The sizes of the domains chosen are $36\pi\times 36\pi$, $36\pi\times 72\pi$, $72\pi\times 72\pi$, and $144\pi\times 144\pi$, and each pattern is generated by integrating a random initial state (with amplitude between $\pm 10^{-2}$) for a time $T=8000$ under the SHE. The results, shown in Figure \[intensive\], give the mean of 10 patterns for each domain size (except the largest where only 5 patterns were used). The results indicate that $\bar\delta(0.5)$, $\bar\delta(1.0)$, and $\bar\delta(1.9)$ are intensive variables, and the corresponding $\delta(\beta)$ are extensive variables. Relaxation of patterns ---------------------- The characterization of textures using $\bar\delta(\beta)$ finds one useful application in the study of the relaxation from an initially random state. Fig. \[snap-shots\] shows several snapshots of a relaxing pattern. During an initial period ($t<T_0\sim 800$) the local domains emerge out of the random background and the mean intensity $<\|u({\bf x},t)\|>$ nearly reaches its final value. The subsequent evolution due to domain coarsening is very slow. These qualitative features are repeated in multiple runs under the same control parameters. Figure \[df\_one\] shows the behavior of $\Delta k$, $\bar\delta(0.5)$, $\bar\delta(1.0)$ and $\bar\delta(1.9)$ for the evolution shown in Fig. \[snap-shots\]. The curves remain identical (except for small statistical fluctuations) for different realizations of the experiment; i.e., the disorder function captures configuration independent aspects of the organization of patterns. The relaxation clearly consists of two stages, with a sharp transition in $\bar\delta(\beta)$ at $t=T_0$ [@eldAvin; @croAmei]. =5.5truein 0.75truein [2]{} During the initial phase, the time evolution of $\bar\delta(1)$ changes smoothly from a logarithmic decay to a power law $\bar\delta(1) \sim t^{-\gamma_1}$, where $\gamma_1 \approx 0.5$. Corresponding $t^{-\frac{1}{2}}$ decay has been observed in the width of the structure factor [@eldAvin]. The scaling is “trivial" in the sense that for other “moments" $\bar\delta(\beta) \sim t^{-\frac{1}{2} \beta}$ [@footnote1]. The decay of $\bar\delta(\beta)$ appears to be associated with the $L \sim t^{\frac{1}{2}}$ growth of domains in non-conserved systems [@rutAbra]. The second phase of the relaxation (due to domain coarsening) exhibits a more complex behavior. The moments $\bar\delta(0.5)$, $\bar\delta(1)$ and $\bar\delta(1.9)$ behave approximately as $t^{-0.09}$, $t^{-0.15}$ and $t^{-0.20}$ respectively, indicating the presence of “non-trivial" scaling [@chrAbra]. Since the relative contribution of isolated defects increases with $\beta$ (Section II), the slower decay of $\bar\delta(1.9)$ (compared to $\bar\delta(1)^{1.9}$) suggests that changes in the density of defects is less significant than the reduction of curvature of the contour lines [@eldAvin]. =3.0truein 0.05truein Changes in the relaxation with $\epsilon$ ----------------------------------------- Figure \[df\_all\] shows the behavior of $\bar\delta(1)$ during the relaxation of random initial states under the SHE for several values of $\epsilon$, all other parameters being fixed. The initial decay of $\bar\delta(1)$ and the rate of decay during the second phase are seen to be independent of $\epsilon$. Furthermore, the transition between the two phases advances with increasing $\epsilon$. Similar results are observed for all values of the moments $\beta$. =3.0truein 0.05truein Suitable scaling of variables, including $t \rightarrow \epsilon t$ can be used to eliminate $\epsilon$ from the SHE. Hence we expect that the rescaling $t \rightarrow \epsilon t$ and $\bar\delta(1) \rightarrow \epsilon^{-\frac{1}{2}} \bar\delta(1)$ will lead to collapse of the curves shown in Fig. \[df\_all\]. This is indeed the case as seen from the scaling function (Fig. \[scale\_all\]). Relaxation in Nonvariational Systems ==================================== In this Section we discuss properties of $\bar\delta(\beta)$ when the spatio-temporal dynamics is nonvariational. The absence of an underlying “energy" of the dynamics suggests a faster relaxation, since the system cannot be constrained by “metastable states" during the evolution. The behavior of the disorder function confirms this expectation. Figure \[nv\_one\] shows the behavior of $\bar\delta(1)$ for the organization of a random field under a nonvariational SHE (i.e., $\nu \ne 0$). The decay of $\bar\delta(1)$ remains the same (as the analogous variational dynamics) during the initial relaxation (Fig. \[df\_one\]) and becomes faster during the coarsening phase. As the coefficient $\nu$ of the “nonvariational term" in Eqn. (\[she\]) increases (the value of the remaining coefficients remaining the same), so does the relaxation rate during the coarsening phase (Fig. \[nv\_all\]). The wave-number (obtained by minimizing $\bar\delta(1)$ over $k_0$) relaxes to a value ($k_0=0.61$) that is larger than the corresponding one for the variational case ($k_0=0.59$). Such a deviation was observed earlier in Ref. [@croAmei] where it was suggested that $k_0$ (for the nonvariational case) corresponds to the zero-climb velocity of isolated dislocation defects. =3.0truein 0.05truein Discussion ========== We have used the disorder function $\bar\delta(\beta)$ to characterize properties of textured patterns and their relaxation from initially random states. The disorder function was defined by requiring its invariance under rigid motions of a single texture. It was found to be identical for multiple patterns generated under similar external conditions; i.e., $\bar\delta(\beta)$ is configuration independent. We provided evidence to confirm that the moments are intensive variables. In addition, the disorder function can differentiate between patterns with distinct characteristics. The evolution of initially random states under the Swift-Hohenberg equation is conveniently described using $\bar\delta(\beta)$. The relaxation consists of two distinct stages separated by a sharp transition. During the initial phase, local striped domains emerge out of the noisy background and their amplitudes saturate close to their final value. This behavior is described by a logarithmic decay followed by a power law decay of the disorder $\bar\delta(1) \sim t^{-\frac{1}{2}}$. The scaling is “trivial" in the sense that the decay of the remaining moments satisfy $\bar\delta(\beta) \sim \bar\delta(1)^{\beta}$. The second phase of the relaxation corresponds to domain coarsening and is a much slower process. The scaling during this phase is nontrivial. As the system is driven further from the onset of patterns (as measured by the parameter $\epsilon$) the duration of the initial phase is reduced. However, the rates of decay of the disorder function for the two phases remain unchanged and rescaling of time by $\epsilon$ and of $\bar\delta(\beta)$ by $\epsilon^{-\frac{1}{2} \beta}$ leads to a scaling collapse. =3.0truein 0.05truein The addition of nonvariational terms to the spatio-temporal dynamics leads to several interesting observations. The decay of disorder during the initial phase is unchanged, and appears to be a model independent feature. Thus, one may expect to observe it during relaxation of patterns in experimental systems. The expectation of a faster relaxation in nonvariational systems (due to the absence of “potential minima") is seen only during domain coarsening. This rate of relaxation is system dependent and increases as the coefficient of the nonvariational term. There is very little theoretical understanding of the observed behavior of the disorder function. 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--- abstract: 'Motivated by applications arising from large scale optimization and machine learning, we consider stochastic quasi-Newton (SQN) methods for solving unconstrained convex optimization problems. Much of the convergence analysis of SQN methods, in both full and limited-memory regimes, requires the objective function to be strongly convex. However, this assumption is fairly restrictive and does not hold in many applications. To the best of our knowledge, no rate statements currently exist for SQN methods in the absence of such an assumption. Also, among the existing first-order methods for addressing stochastic optimization problems with merely convex objectives, those equipped with provable convergence rates employ averaging. However, this averaging technique has a detrimental impact on inducing sparsity. Motivated by these gaps, we consider optimization problems with non-strongly convex objectives with Lipschitz but possibly unbounded gradients. The main contributions of the paper are as follows: (i) To address large scale stochastic optimization problems, we develop an iteratively regularized stochastic limited-memory BFGS (IRS-LBFGS) algorithm, where the stepsize, regularization parameter, and the Hessian inverse [[approximation]{}]{} are updated iteratively. We establish convergence of the iterates (with no averaging) to an optimal solution of the original problem both in an almost-sure sense and in a mean sense. The convergence rate is derived in terms of the objective function values and is shown to be $\mathcal{O}\left(1/k^{\left({1}/{3}-\e\right)}\right)$, where $\e$ is an arbitrary small positive scalar; (ii) In deterministic regimes, we show that [[the algorithm]{}]{} displays a rate [[$\mathcal{O}({1}/{k^{1-\e}})$. We present numerical experiments]{}]{} performed on a large-scale text classification problem [[and compare IRS-LBFGS with standard SQN methods as well as first-order methods such as SAGA and IAG]{}]{}.' author: - 'Farzad Yousefian[^1]' - 'Angelia Nedi['' c]{}[^2]' - 'Uday V. Shanbhag[^3]' bibliography: - 'reference.bib' title: 'On stochastic and deterministic quasi-Newton methods for non-strongly convex optimization: Asymptotic convergence and rate analysis[^4] ' --- stochastic optimization, quasi-Newton, regularization, large scale optimization 65K05, 90C06, 90C30, 90C53 Introduction {#sec:intro} ============ [[We]{}]{} consider the following stochastic optimization problem: $$\label{eqn:problem} \min_{x \in \Real^n} f(x){{\color{black}\ \triangleq \ }}{\mathsf{E}\!\left[F(x,\xi({{\color{black}\omega}}))\right] },\tag{SO}$$ where $F: \Real^n\times{{\color{black}\Real^d}}\to\mathbb{R}$ is a [[real-valued]{}]{} function, the random vector $\xi$ is defined as [$\xi:\Omega \rightarrow \Real^d$]{}, [[$(\Omega,{\cal F}, \mathbb{P})$ denotes the associated probability space[[,]{}]{} and the expectation [[${\mathsf{E}\!\left[F(x,\xi)\right] }$]{}]{} is taken with respect to $\mathbb{P}$]{}]{}. Problem provides a general framework that can capture a wide range of applications in operations research, machine learning, statistics and control to name a few [[([[cf.]{}]{}[@bottou-2010; @nocedal15]). Addressing problem ]{}]{} has led to significant [[progress via]{}]{} Monte[[-]{}]{}Carlo sampling techniques. [[, stochastic approximation (SA) methods [@robbins51sa; @nemirovski_robust_2009] have [[proved particularly]{}]{} popular.]{}]{} [[The standard SA method, introduced by Robbins and Monro [@robbins51sa], [[for solving]{}]{} , produces a sequence $\{x_k\}$ using the following update rule]{}]{} $$\label{eqn:SA}\tag{SA} x_{k+1}:=x_k-\g_k\nabla {{\color{black}F}}(x_k,\xi_k), \quad \hbox{for }k\geq 0,$$ where [[$x_0 \in \Real^n$ is a randomly generated initial point,]{}]{} $\g_k>0$ denotes the stepsize and $\nabla F(x_k,\xi_k)$ denotes [[a [[sampled]{}]{} gradient of $f$]{}]{} with respect to $x$ at $x_k$. [[SA]{}]{} schemes are characterized by several disadvantages, including [[a]{}]{} poorer rate of convergence (than their deterministic counterparts) and the detrimental impact of conditioning on their performance. [[In deterministic regimes, the BFGS]{}]{} method, named after Broyden, Fletcher, Goldfarb, and Shanno, is [[amongst the [[most]{}]{} popular quasi-Newton methods]{}]{} [@goldfarb70; @fletcher70], [[displaying]{}]{} a [[superlinear]{}]{} convergence rate without requiring [[second-order]{}]{} information. Addressing large scale deterministic problems, the limited-memory variant of the BFGS method, denoted by LBFGS, was developed [[and attains]{}]{} an ${{\color{black}R}}$-linear convergence rate under strong convexity of the objective function (see Theorem 6.1 in [@nocedal89]). Recently, there has been a growing interest in applying [[stochastic quasi-Newton (SQN) methods]{}]{} for solving large-scale optimization and machine learning problems. In these methods, $x_k$ is [[updated]{}]{} by the following rule: $$\label{eqn:SQN}\tag{SQN} x_{k+1}:=x_k-\g_kH_k\nabla {{\color{black}F}}(x_k,\xi_k), \quad \hbox{for }k\geq 0,$$ where $H_k \succeq 0$ is an approximation of the inverse of the [[Hessian]{}]{} at iteration $k$ that incorporates the curvature information of the objective function [[within]{}]{} the algorithm. The convergence of this class of algorithms can be derived under a careful choice of [[$H_k$]{}]{} and the stepsize sequence [[$\{\g_k\}$]{}]{}. In particular, [[the]{}]{} boundedness of the eigenvalues of $H_k$ is an important factor in achieving global convergence in convex and nonconvex problems (cf. [@Fukushima01; @Bottou09]). While in [@Schraudolph07] the performance of SQN methods [[was found]{}]{} to be favorable in solving high dimensional problems, Mokhtari et al. [@mokh14] considered stochastic optimization problems with strongly convex objectives and developed a regularized BFGS method (RES) [[by updating $H_k$ according to]{}]{} a modified version of BFGS update rule to assure convergence. To address large scale applications, limited-memory variants were employed to ascertain scalability in terms of the number of variables [@Mokhtari15; @nocedal15]. [[Recent extensions have included]{}]{} a stochastic quasi-Newton method [@wang14] for solving nonconvex stochastic optimization problems [[and a constant stepsize]{}]{} variance reduced SQN method [[[@Lucchi15] for smooth strongly convex problems characterized by a linear convergence rate]{}]{}. [[, an incremental quasi-Newton (IQN) method with a local superlinear convergence rate has been recently developed for addressing the sum of a large number of strongly convex functions [@Iqn17].]{}]{} [Motivation:]{} [[In]{}]{} [[both the full and limited memory variants of the SQN methods [[in the literature]{}]{} [@nocedal15; @Mokhtari15; @Iqn17]]{}]{}, it is uniformly assumed that the objective function is strongly convex. This assumption plays an important role in deriving the rate of convergence of the algorithm. However, in many applications, the objective function is convex, but not strongly convex [[such as, [[when considering]{}]{} the logistic regression function.]{}]{} While [[a]{}]{} lack of strong convexity might lead to [[slower]{}]{} convergence [[in practice]{}]{}, no [[rigorous support for]{}]{} the convergence rate is [[currently]{}]{} [[available]{}]{} in the literature of SQN methods. A [[simple]{}]{} remedy to address this challenge is to regularize the objective function with the term $\frac{1}{2}\mu\\|x\\|^2$ and solve the approximate problem of the form [[$\min_{x \in \Real^n} f(x)+\frac{\mu}{2}\\|x\\|^2$]{}]{}, where $\mu>0$ is the regularization parameter. Several challenges arise in applying this technique. A drawback of this technique is that the optimal solution to the [[regularized problem]{}]{} is not [[necessarily]{}]{} an optimal of the original problem . [[nother challenge arises from]{}]{} the choice of $\mu$. While [[larger]{}]{} values of $\mu$ may result in large deviations from the true optimal [[solution(s)]{}]{}, choosing a small $\mu$ leads to a deterioration of [[the constant factor]{}]{} in the convergence rate of the algorithm. [[This issue has been addressed to some extent with the help of averaging techniques. In particular, [[under mere convexity, most]{}]{} first-order methods admit convergence rate guarantees under averaging. For example, averaging SA schemes achieve a]{}]{} rate of $\mathcal{O}\left(\frac{M}{\sqrt{k}}\right)$, where $M$ is an upper bound on the norm of the subgradient [[(see [@nemirovski_robust_2009; @Nedic2014]).]{}]{} In past few years, fast incremental gradient methods with improved rates of convergence have been developed (see [@Saga14; @Sag17; @Piag16; @Iag17]). Of these, addressing the merely convex case, SAGA with averaging achieves a sublinear convergence rate $\mathcal{O}\left(\frac{N}{k}\right)$ [[where $N$ denotes the number of blocks]{}]{}, while in the presence of strong convexity, non-averaging variants of SAGA and IAG admit a linear convergence rate assuming that the function satisfies some smoothness conditions. A [[crucial concern that plagues]{}]{} the aforementioned schemes is that the averaging technique has a detrimental impact on inducing sparsity. In the case of incremental methods such as SAGA and IAG, despite the fast convergence speed, the application of these methods is [[impaired by]{}]{} the excessive memory requirements. For standard SAGA and IAG, the memory requirements are $\mathcal{O}\left(nN\right)$. [[Accordingly, in this paper, our main goal lies in addressing such shortcomings in absence of strong convexity and developing a first-order method equipped with a rate of convergence for the generated non-averaged iterates]{}]{}. Related research on regularization: In [[optimization]{}]{}, in order to obtain solutions with desirable properties, it is common to regularize the problem as follows $$\label{eqn:problemReg2} \min_{x \in \Real^n} f_\mu(x):=f(x)+\mu R(x),$$ where $R:\Real^n\to \Real$ is a proper convex function and the scalar $\mu>0$ is the regularization parameter. The properties of the regularized problem and its relation [[to]{}]{} the original problem have been investigated by different researchers. Mangasarian and his colleagues appear among the first researchers who studied exact regularization of linear and nonlinear programs [@Mangasarian79nonlinearperturbation; @Mangasarian91]. A regularization is said to be exact when an optimal solution of , is also optimal for problem if $\mu$ is small enough. Tseng et al. [@Tseng08; @Tseng09] established the necessary and sufficient conditions of exact regularization for convex programs and derived error bounds for inexact regularized convex problems. [[In a similar veing]{}]{}, exact regularization of variational [[inequality problems]{}]{} has been studied in [@Charitha17]. A challenging question is concerned about the choice of the regularization parameter $\mu$. A common approach to find an acceptable value for $\mu$ is through a two-loop scheme where in the inner loop, problem is solved for a fixed value of $\mu$, while [[$\mu$ is tuned in the outer loop]{}]{}. The main drawback of this approach is that[[,]{}]{} in general, there is no guidance on the tuning rule for $\mu$. In addition, this approach is computationally inefficient. [[Furthermore]{}]{}, tuning rules may result in losing the desired properties of the sample path of the solutions to regularized problems. [[In this work, we address this issue through employing]{}]{} an iterative single-loop algorithm where we update the regularization parameter $\mu$ at each iteration of the scheme and reduce it iteratively to converge to zero [@Farzad3; @koshal13]. [Contributions:]{} We consider stochastic optimization problems with non-strongly convex objective functions and Lipschitz but possibly unbounded gradient mappings. Our main contributions are as follows:\ (i) [[Asymptotic convergence]{}]{}: We develop [[an iteratively]{}]{} regularized SQN method [[where]{}]{} the stepsize, regularization parameter, and the [[Hessian inverse approximation denoted by $H_k$]{}]{} are updated iteratively. We assume that $H_k$ satisfies a set of general assumptions on its eigenvalues and its dependency [[on]{}]{} the uncertainty. The [[asymptotic]{}]{} convergence of the [[method]{}]{} is established under a [[suitable]{}]{} choice of [[an error]{}]{} function. [[For the sequence of the iterates $\{x_k\}$ produced by the algorithm]{}]{}, we obtain a set of suitable conditions on the stepsize and regularization sequences for which $f(x_k)$ converges to the optimal objective value, i.e., $f^$, of [[both]{}]{} in an almost sure sense and in [[a mean sense]{}]{}. We also derive an upper bound for [[$f(x_k)-f^$]{}]{}.\ (ii) [[Rate of convergence for regularized LBFGS methods]{}]{}: To address large scale stochastic optimization problems, motivated by our earlier work [@FarzadCDC16] on SQN methods for small scale stochastic optimization problems with non-strongly convex objectives, [[we develop an iteratively regularized stochastic limited-memory BFGS scheme (see Algorithm \[algorithm:IR-S-BFGS\])]{}]{}. We show that under a careful choice of the [[update rules for the stepsize and]{}]{} regularization parameter, [[Algorithm \[algorithm:IR-S-BFGS\]]{}]{} displays a convergence rate $\mathcal{O}\left(k^{-\left(\frac{1}{3}-\e\right)}\right)$ in terms of the objective [[function]{}]{} values, where $\e$ is an arbitrary small positive scalar. [[Similar to standard stochastic LBFGS schemes, the memory requirement is independent of $N$ and is [[$\mathcal{O}\left(mn\right)$]{}]{}, where $m \ll n$ [[denotes]{}]{} the memory parameter [[in the LBFGS scheme]{}]{}. In the deterministic case, we]{}]{} show that the convergence rate [[improves]{}]{} to $\mathcal{O}\left(\frac{1}{k^{1-\e}}\right)$. [[Both of these convergence rates appear to be new for the class of deterministic and stochastic quasi-Newton methods.]{}]{}\ Outline of the paper: The rest of the paper is organized as follows. Section \[sec:alg\] presents the general framework of the proposed SQN algorithm and the sets of main assumptions. In Section \[sec:conv\], we [[prove the asymptotic convergence of the iterates produced by]{}]{} the scheme in both almost sure and [[a mean sense]{}]{} and derive the a general error bound. In Section \[sec:BFGS\], we develop [[an iteratively]{}]{} regularized stochastic LBFGS method ([[Algorithm \[algorithm:IR-S-BFGS\]]{}]{}) and derive [[its]{}]{} convergence rate. The rate analysis is also provided for the deterministic variant of this scheme. We [[then]{}]{} present the [[numerical experiments]{}]{} performed on a large scale classification problem in Section \[sec:num\]. The paper ends with some concluding remarks in Section \[sec:conc\]. Notation: A vector $x$ is assumed to be a column vector and $x^T$ denotes its transpose, [[while]{}]{} $\\|x\\|$ [denotes]{} the Euclidean vector norm, i.e., $\\|x\\|=\sqrt{x^Tx}$. We write a.s. as the abbreviation for “almost surely”. [[For a symmetric matrix $B$, $\lambda_{\min}(B)$ and $\lambda_{\max}(B)$ [[denotes the smallest and largest eigenvalue of $B$]{}]{}, respectively.]{}]{} We use ${\mathsf{E}\!\left[z\right] }$ to denote the expectation of a random variable $z$. A function $f:X \subset \mathbb{R}^n\rightarrow \mathbb{R}$ is said to be strongly convex with parameter $\mu>0$, if $f(y)\geq f(x)+\nabla f(x)^T(y-x)+\frac{\mu}{2}\\|x-y\\|^2,$ for any $x,y \in X$. A mapping $F:X \subset \mathbb{R}^n\rightarrow \mathbb{R}$ is [[said to be]{}]{} Lipschitz continuous with parameter $L>0$ if for any $x, y \in X$, we have $\\|F(x)-F(y)\\|\leq L\\|x-y\\|$. For a continuously differentiable function $f$ with Lipschitz gradients with parameter $L>0$, we have $f(y)\leq f(x)+\nabla f(x)^T(y-x)+\frac{L}{2}\\|x-y\\|^2,$ for any $x,y \in X$. For a vector $x \in \Real^n$ and a [[nonempty]{}]{} set $X \subset \Real^n$, the Euclidean distance of $x$ from $X$ is denoted by $dist(x,X)$. We denote the optimal objective value of problem by $f^$ and the set of the optimal solutions by $X^$. Outline of the SQN scheme {#sec:alg} ========================= In this section, we describe a general SQN scheme for solving problem and present the main assumptions. Let $x_0\in \Real^n$ be an arbitrary initial point, and $x_k$ be generated by the following recursive rule $$\begin{aligned} \label{eqn:LM-cyclic-reg-BFGS}\tag{IR-SQN} x_{k+1}:=x_k -\gamma_kH_k\left(\nabla F(x_k,\xi_k)+ \mu_k(x_k-x_0)\right), \quad \hbox{for all } k \geq 0.\end{aligned}$$ Here, $\g_k$ and $\mu_k$ are the steplength and the regularization parameter, respectively. $H_k \in \Real^{n\times n}$ is a matrix that [[contains]{}]{} the curvature information of the objective function. The scheme can be seen as a regularized variant of [[the]{}]{} classical stochastic SQN [[method]{}]{}. Here we regularize the gradient map by the term $\mu_k(x_k-x_0)$ to [[induce]{}]{} the strong monotonicity property. In [[the]{}]{} absence of strong convexity of $f$, unlike the classical schemes where $\mu_k$ is maintained fixed, we let $\mu_k$ be updated and [[decreased]{}]{} to zero. Throughout, we let $\sF_k$ denote the history of the method up to time $k$, i.e., $\sF_k{{\color{black}\ \triangleq \ }}\{x_0,\xi_0,\xi_1,\ldots,\xi_{k-1}\}$ for $k\ge 1$, and $\sF_0\triangleq\{x_0\}$. \[assum:convex\] Consider problem . Let the following hold: 1. [[The function $f(x)$ is convex over $\Real^n$.]{}]{} 2. $f(x)$ is continuously differentiable [[with Lipschitz continuous gradients]{}]{} over $\Real^n$ with parameter $L>0$. 3. The optimal solution set of [[the problem]{}]{} is nonempty. Next, we state the assumptions on the random variable $\xi$ and the properties of the stochastic estimator of the gradient mapping. \[assum:main\] 1. [[Vectors $\xi_k \in \mathbb{R}^d$ are i.i.d. realizations of the random variable $\xi$]{}]{} for any $k \geq 0$; 2. The stochastic gradient mapping $\nabla F(x, \xi)$ is an unbiased estimator of $\nabla f(x)$, i.e. ${\mathsf{E}\!\left[\nabla F(x,\xi)\right] }=\nabla f(x)$ for all $x$, and has a bounded variance, i.e., there exists a [[scalar]{}]{} $\nu>0$ such that ${\mathsf{E}\!\left[\\|\nabla F(x,\xi)-\nabla f(x)\\|^2\right] } \leq \nu^2,$ for all $x \in \mathbb{R}^n$. The next assumption [pertains to]{} the properties of $H_k$. \[assump:Hk\] Let the following hold for all $k\geq 0$:\ (a) The matrix $H_k\in \Real^{n\times n}$ is $\sF_{k}$-measurable, i.e., ${\mathsf{E}\!\left[H_k \mid \sF_{k}\right] }=H_k$.\ (b) Matrix $H_k$ is symmetric and positive definite and satisfies the following condition: There exist positive scalars $\lambda_{\min},\lambda$ and scalar $\alpha\leq 0$ such that $$\lambda_{\min}\mathbf{I} \preceq H_k \preceq \lambda \mu_k^\alpha\mathbf{I},\qquad \hbox{for all }k\geq 0,$$ [[where $\mu_k$ is the regularization parameter in .]{}]{} Assumption \[assump:Hk\] holds for the stochastic gradient method where $H_k$ is the identity matrix, $\lambda_{\min}=\lambda=1$ and $\alpha=0$. In the case of employing an appropriate LBFGS update rule that will be discussed in Section \[sec:BFGS\], the maximum eigenvalue is obtained in terms of the regularization parameter. Convergence analysis {#sec:conv} ==================== In this section, we present the convergence analysis of the method. Our discussion starts by some preliminary definitions and properties. After obtaining a recursive error bound for the method in Lemma \[lemma:main-ineq\], we show a.s. convergence in Proposition \[prop:a.s\], establish convergence in mean, and derive an error bound in Proposition \[prop:mean\]. \[def:regularizedF\] Consider the sequence $\{\mu_k\}$ of positive [[scalars]{}]{} and the starting point of the algorithm , i.e., $x_0$. The regularized function $f_k$ and its gradient are defined as follows [[for all $k\geq 0$]{}]{}: $$\begin{aligned} f_k(x)\triangleq f(x)+\frac{\mu_k}{2}{\\|x-x_0\\|^2},\quad \nabla f_k(x)\triangleq\nabla f(x)+\mu_k(x-x_0).\end{aligned}$$ In a similar way, we denote the regularized stochastic function $F(x,\xi)$ and its gradient with $F_k$ and $\nabla F_k$ for any $\xi$, respectively. \[proper:propsfk\] We have: - $f_k$ is strongly convex with a parameter $\mu_k$. - $f_k$ has Lipschitzian gradients with parameter $L+\mu_k$. - $f_k$ has a unique minimizer over $\Real^n$, denoted by $x^_k$. Moreover, for any $x \in \Real^n$, $$2\mu_k (f_k(x)-f_k(x^_k)) \leq \\|\nabla f_k(x)\\|^2\leq 2(L+\mu_k) (f_k(x)-f_k(x^_k)).$$ The existence and uniqueness of $x^_k$ in Property \[proper:propsfk\](c) is due to the strong convexity of the function $f_k$ (see, for example, Section 1.3.2 in [@Polyak87]), while the relation for the gradient is known to hold for a strongly convex function with a parameter $\mu$ that also has Lipschitz gradients with a parameter $L$ (see Lemma 1 on page 23 in [@Polyak87]). In the convergence analysis, we make use of the following result, which can be found in [@Polyak87] [[(see Lemma 10 on page 49)]{}]{}. \[lemma:probabilistic\_bound\_polyak\] Let $\{v_k\}$ be a sequence of nonnegative random variables, where ${\mathsf{E}\!\left[v_0\right] } < \infty$, and let $\{\a_k\}$ and $\{\beta_k\}$ be deterministic scalar sequences such that: $$\begin{aligned} & {\mathsf{E}\!\left[v_{k+1}\|v_0,\ldots, v_k\right] } \leq (1-\alpha_k)v_k+\beta_k \quad a.s. \ \hbox{for all }k\ge0, \end{aligned}$$ where $0 \leq \alpha_k \leq 1$, $\beta_k \geq 0$, $\sum_{k=0}^\infty \alpha_k =\infty$, $\sum_{k=0}^\infty \beta_k < \infty$, and $\lim_{k\to\infty}\,\frac{\beta_k}{\alpha_k} = 0.$ Then, $v_k \rightarrow 0$ almost surely. Throughout, we denote the stochastic error of the gradient estimator by $$\begin{aligned} \label{def:wk}w_k\triangleq \nabla F(x_k,\xi_k)-\nabla f(x_k), {{\color{black}\quad\hbox{for all }k\ge0.}} \end{aligned}$$ Note that under Assumption \[assum:main\], from the definition of $w_k$ in , we obtain ${\mathsf{E}\!\left[w_k\mid \sF_k\right] }=0$ and ${\mathsf{E}\!\left[\\|w_k\\|^2\mid \sF_k\right] }{{\color{black}\leq}} \nu^2$. [[The following result plays a key role in the convergence and rate analysis of the proposed schemes.]{}]{} \[lemma:main-ineq\] Consider the method and suppose Assumptions \[assum:convex\], \[assum:main\], and \[assump:Hk\] hold. Also, assume $\mu_k$ is a non-increasing sequence and let $$\begin{aligned} \label{mainLemmaCond}\g_k\mu_k^{2\alpha} \leq \frac{\lambda_{\min}}{\lambda^2(L+\mu_0)},\quad \hbox{for all }k\geq 0.\end{aligned}$$Then, the following inequality holds [[for all $k\geq 0$]{}]{}: $$\begin{aligned} \label{ineq:cond-recursive-F-k} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})\mid\sF_k\right] }-f^* &\leq (1-\lambda_{\min}\mu_k\g_k)(f_k(x_k)-f^)+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\mu_k^2\g_k\notag \\ &+\frac{ (L+\mu_k)\lambda^2\nu^2}{2}\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ The Lipschitzian property of $\nabla f_k$ and the update rule imply that $$\begin{aligned} f_k(x_{k+1}) &\leq f_k(x_k)+\nabla f_k(x_k)^T(x_{k+1}-x_k)+\frac{ (L+\mu_k)}{2}\\|x_{k+1}-x_k\\|^2 \cr &= f_k(x_k)-\g_k\nabla f_k(x_k)^TH_k\left(\nabla F(x_k,\xi_k) +\mu_k(x_k-x_0)\right)\\&+ \frac{ (L+\mu_k)}{2}\g_k^2\\|H_k\left(\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)\right)\\|^2.\end{aligned}$$ [[Invoking the definition of the stochastic error $w_k$ in]{}]{} [[and]{}]{} Definition \[def:regularizedF\], we obtain $$\begin{aligned} \label{ineq:term1-2} f_k(x_{k+1}) &\leq f_k(x_k)-\g_k\nabla f_k(x_k)^TH_k(\nabla f(x_k)+w_k +\mu_k(x_k-x_0))\notag\\&+ \frac{ (L+\mu_k)}{2}\g_k^2\\|H_k(\nabla f(x_k)+w_k+\mu_k(x_k-x_0))\\|^2\\ \notag = f_k(x_k)&-\g_k\underbrace{\nabla f_k(x_k)^TH_k(\nabla f_k(x_k)+w_k)}_{\tiny\hbox{Term } 1}+ \frac{ (L+\mu_k)}{2}\g_k^2\underbrace{\\|H_k(\nabla f_k(x_k)+w_k)\\|^2}_{\tiny\hbox{ Term } 2},\end{aligned}$$ where in the last [[equation,]{}]{} we used the definition of $f_k$. Next, we estimate the conditional expectation of Term 1 and 2. From Assumption \[assump:Hk\], we have $$\begin{aligned} \hbox{Term }1 &= \nabla f_k(x_k)^TH_k\nabla f_k(x_k)+\nabla f_k(x_k)^TH_kw_k \\&\geq \lambda_{\min}\\|\nabla f_k(x_k)\\|^2+\nabla f_k(x_k)^TH_kw_k.\end{aligned}$$ Taking expectations conditioned on $\sF_k$, from the preceding inequality, we obtain $$\begin{aligned} \label{equ:Term1} {\mathsf{E}\!\left[\hbox{Term } 1\mid\sF_k\right] } &\geq \lambda_{\min}\\|\nabla f_k(x_k)\\|^2+{\mathsf{E}\!\left[\nabla f_k(x_k)^TH_kw_k\mid\sF_k\right] }\\ \notag&=\lambda_{\min}\\|\nabla f_k(x_k)\\|^2+\nabla f_k(x_k)^TH_k{\mathsf{E}\!\left[w_k\mid\sF_k\right] } =\lambda_{\min}\\|\nabla f_k(x_k)\\|^2,\end{aligned}$$ where we recall that ${\mathsf{E}\!\left[w_k\mid\sF_k\right] }=0$ and ${\mathsf{E}\!\left[H_k\mid\sF_k\right] }=H_k$. Similarly, in Term 2, invoking Assumption \[assump:Hk\](b), we may write $$\begin{aligned} \hbox{Term } 2&= (\nabla f_k(x_k)+w_k)^TH_k^2(\nabla f_k(x_k)+w_k)\leq (\lambda\mu^{\alpha}_k)^2\\|\nabla f_k(x_k)+w_k\\|^2 \\ &=\lambda^2\mu^{2\alpha}_k\left(\\|\nabla f_k(x_k)\\|^2+\\|w_k\\|^2+2\nabla f_k(x_k)^Tw_k\right).\end{aligned}$$ Taking conditional expectations from the preceding inequality, and using Assumption \[assum:main\], we obtain $$\begin{aligned} \label{equ:Term2} {\mathsf{E}\!\left[\hbox{Term } 2\mid\sF_k\right] }&\leq\lambda^2\mu^{2\alpha}_k\left(\\|\nabla f_k(x_k)\\|^2+{\mathsf{E}\!\left[\\|w_k\\|^2\mid\sF_k\right] }+2\nabla f_k(x_k)^T{\mathsf{E}\!\left[w_k\mid\sF_k\right] }\right)\notag\\ & \leq \lambda^2\mu^{2\alpha}_k\left(\\|\nabla f_k(x_k)\\|^2+\nu^2\right).\end{aligned}$$ Next, taking conditional expectations in , and using and , we obtain $$\begin{aligned} &{\mathsf{E}\!\left[f_k(x_{k+1})\mid\sF_k\right] } \leq f_k(x_k)-\g_k\lambda_{\min}\\|\nabla f_k(x_k)\\|^2\notag \\ &+\lambda^2\mu^{2\alpha}_k\frac{ (L+\mu_k)}{2}\g_k^2\left(\\|\nabla f_k(x_k)\\|^2+\nu^2\right)\notag \\ &\leq f_k(x_k)-\frac{\g_k\lambda_{\min}}{2}\\|\nabla f_k(x_k)\\|^2\left(2-\frac{\lambda^2\mu_k^{2\alpha}\g_k(L+\mu_k)}{\lambda_{\min}}\right)+\lambda^2\mu^{2\alpha}_k\frac{ (L+\mu_k)}{2}\g_k^2\nu^2.\end{aligned}$$ From the assumption that $\g_k$ and $\mu_k$ satisfy $\g_k\mu^{2\alpha}_k \leq \frac{\lambda_{\min}}{\lambda^2(L+\mu_0)}$ for any $k \geq 0$ and that $\mu_k$ is non-increasing, we have $\g_k\mu^{2\alpha}_k \leq \frac{\lambda_{\min}}{\lambda^2(L+\mu_k)}$. As a consequence, we get $2-\frac{\lambda^2\mu_k^{2\alpha}\g_k(L+\mu_k)}{\lambda_{\min}}\geq 1$. Therefore, from [[the]{}]{} preceding inequality, we obtain $$\begin{aligned} {\mathsf{E}\!\left[f_k(x_{k+1}) \mid\sF_k\right] }&\leq f_k(x_k)-\frac{\g_k\lambda_{\min}}{2}\\|\nabla f_k(x_k)\\|^2+\lambda^2\mu^{2\alpha}_k\frac{ (L+\mu_k)}{2}\g_k^2\nu^2.\end{aligned}$$ Employing Property \[proper:propsfk\](c), we have $$\begin{aligned} {\mathsf{E}\!\left[f_k(x_{k+1})\mid\sF_k\right] } &\leq f_k(x_k)-\lambda_{\min}\mu_k\g_k(f_k(x_k)-f_k(x^_k))+\lambda^2\mu^{2\alpha}_k\frac{ (L+\mu_k)}{2}\g_k^2\nu^2.\end{aligned}$$ Note that, since $\mu_k$ is a non-increasing sequence, Definition \[def:regularizedF\] implies that $${\mathsf{E}\!\left[f_{k+1}(x_{k+1})\mid \sF_k\right] } \leq{\mathsf{E}\!\left[f_{k}(x_{k+1})\mid \sF_k\right] }.$$ Therefore, we obtain $$\begin{aligned} \label{ineq:lemmaLastIneq} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})\mid\sF_k\right] } &\leq f_k(x_k)-\lambda_{\min}\mu_k\g_k(\underbrace{f_k(x_k)-f_k(x^_k)}_{\tiny{\hbox{Term } 3}})+\lambda^2\mu^{2\alpha}_k\frac{ (L+\mu_k)}{2}\g_k^2\nu^2.\end{aligned}$$ Next, we derive a lower bound for Term 3. Since $x_k^$ is the unique minimizer of $f_k$, we have $f_k(x_k^) \leq f_k(x^)$. Therefore, invoking Definition \[def:regularizedF\], for an arbitrary optimal solution $x^ \in X^$, we have $$f_k(x_k)-f_k(x^_k)\geq f_k(x_k)-f_k(x^)=f_k(x_k)-f^-\frac{\mu_k}{2}\\|x^-x_0\\|^2.$$ From the preceding relation and , we have $$\begin{aligned} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})\mid\sF_k\right] } &\leq f_k(x_k)-\lambda_{\min}\mu_k\g_k(f_k(x_k)-f^)+\frac{\lambda_{\min}\\|x^-x_0\\|^2}{2}\mu_k^2\g_k\\ &+\frac{(L+\mu_k)\lambda^2\nu^2}{2}\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Since $x^$ is an arbitrary optimal solution, taking minimum from the right-hand side of the preceding inequality over $X^$, we can replace $\\|x^-x_0\\|$ by $\mbox{dist}(x_0,X^)$. Then, subtracting $f^$ from both sides of the resulting relation yields the desired inequality. Next, we show the convergence of the scheme. [In order to apply Lemma \[lemma:probabilistic\_bound\_polyak\] to inequality and prove the almost sure convergence, we use the following definitions: $$\begin{aligned} \label{def:lemma3} & v_k := f_k(x_k)-f^, \quad \alpha_k := \lambda_{\min}\g_k\mu_k, \notag\\ &\beta_k := \frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\mu_k^2\g_k+\frac{(L+\mu_k)\lambda^2\nu^2}{2}\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ [[To satisfy the conditions of Lemma \[lemma:probabilistic\_bound\_polyak\], we identify a set of sufficient conditions on $\{\gamma_k\}$ and $\{\mu_k\}$ in the following assumption.]{}]{} Later in the subsequent sections, for each class of algorithms, we provide a set of sequences that meet these assumptions.]{} \[assum:sequences\] Let the sequences $\{\gamma_k\}$ and $\{\mu_k\}$ be positive and satisfy the following conditions: [lll]{} (a) [[\_[k ]{}\_k\_k\^[2-1]{} =0;]{}]{}& &(b) {\_k}\_k 0;\ (c) \_\_k\_k 1k 0; && (d) \_[k=0]{}\^\_k\_k =;\ (e) \_[k=0]{}\^\_k\^2\_k <;&&(f) \_[k=0]{}\^\_k\^2\_k\^[2]{} <[[.]{}]{} \[prop:a.s\] Consider the scheme. Suppose Assumptions \[assum:convex\], \[assum:main\], \[assump:Hk\], and \[assum:sequences\] hold. Then, $\lim_{k \to \infty }f(x_k)= f^$ almost surely. First, [[note that from Assumption \[assum:sequences\](a,b), we have $\lim_{k \to \infty }\g_k\mu_k^{2\alpha} =0$]{}]{}. Thus, there exists $K\geq 1$ such that for any $k\geq K$, we have $\g_k\mu_k^{2\alpha} \leq \frac{\lambda_{\min}}{\lambda^2(L+\mu_0)}$ implying that [[condition of Lemma \[lemma:main-ineq\] holds for all $k\geq K$. Hence, relation holds for any $k\geq K$.]{}]{} Next, we apply Lemma \[lemma:probabilistic\_bound\_polyak\] to prove a.s. convergence of the scheme. [Consider the definitions in for any $k \geq K$.]{} The non-negativity of $\alpha_k$ and $\beta_k$ is implied by the definition and that $\lambda_{\min}$, $\g_k$ and $\mu_k$ are positive. From , we have $$\begin{aligned} & {\mathsf{E}\!\left[v_{k+1}\mid \sF_k\right] } \leq (1-\alpha_k)v_k+\beta_k \quad \hbox{for all }k\geq K.\end{aligned}$$ Since $f^ \leq f(x)$ for any $x \in \Real^n$, we can write $v_k= (f(x_k)-f^)+\frac{\mu_k}{2}\\|x_k-x_0\\|^2 \geq 0.$ From Assumption \[assum:sequences\](c), we obtain $\alpha_k \leq 1$. Also, from Assumption \[assum:sequences\](d), we get $\sum_{k=K}^\infty \alpha_k =\infty$. [[Using Assumption \[assum:sequences\](b,e,f) and the definition of $\beta_k$ in , for an arbitrary solution $x^$, we [[may prove the summability of $\beta_k$ as follows.]{}]{}]{}]{} $$\begin{aligned} &\sum_{k=K}^\infty \beta_k\leq \frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\sum_{k=K}^\infty \mu_k^2\g_k+\frac{(L+\mu_0)\lambda^2\nu^2}{2}\sum_{k=K}^\infty\mu^{2\alpha}_k\g_k^2 <\infty.\end{aligned}$$ Similarly, we can write $$\begin{aligned} \lim_{k\to \infty}\frac{\beta_k}{\alpha_k}& \leq \frac{\mbox{dist}^2(x_0,X^)}{2}\lim_{k\to \infty}{\mu_k}+ \frac{(L+\mu_0)\lambda^2\nu^2}{2}\lim_{k\to \infty}\mu^{2\alpha-1}_k\g_k =0, \end{aligned}$$ where the last equation is implied [[by Assumption \[assum:sequences\](a,b).]{}]{} Therefore, all conditions of Lemma \[lemma:probabilistic\_bound\_polyak\] hold [[(with an index shift)]{}]{} and we conclude that [[$v_k:=f_k(x_k)-f^$ converges to 0 a.s.]{}]{} Let us define $v'_k:= f(x_k)-f^$ and $v''_k := \frac{\mu_k}{2}\\|x_k-x_0\\|^2$, [[ so that $v_k=v'_k+v''_k$. Since $v'_k$ and $v''_k$ are nonnegative, and $v_k \to 0$ a.s., it follows that $v'_k \to 0$ and $v''_k \to 0$ a.s., implying that $\lim_{k \to \infty }f(x_k)= f^$ a.s.]{}]{} In the following, our goal is to state the assumptions on the sequences $\{\g_k\}$ and $\{\mu_k\}$ under which we can show the convergence in mean. \[assum:sequences-ms-convergence\] Let the sequences $\{\gamma_k\}$ and $\{\mu_k\}$ be positive and satisfy the following conditions: - $\lim_{k \to \infty }\g_k\mu_k^{2\alpha-1} =0$; - $\{\mu_k\}\hbox{ is non-increasing and }\mu_k \to 0$; - $\lambda_{\min}\g_k\mu_{k} \leq 1$ for $k \geq 0$; - There exist $K_0$ and $0<\beta<1$ such that $$\begin{aligned} \g_{k-1}\mu_{k-1}^{2\alpha-1}\leq \g_{k}\mu_{k}^{2\alpha-1}(1+\beta \lambda_{\min}\g_{k}\mu_{k}),\quad \hbox{for all }k \geq K_0;\end{aligned}$$ - There exists a scalar $\rho>0$ such that $ \mu_k^{2-2\alpha} \leq \rho\g_k$ for all $k \geq 0$[[.]{}]{} Next, we use Assumption \[assum:sequences-ms-convergence\] to establish the convergence in mean. \[prop:mean\] Consider the scheme. Suppose Assumptions \[assum:convex\], \[assum:main\], \[assump:Hk\], and \[assum:sequences-ms-convergence\] hold. Then, there exists $K\geq 1$ such that:$$\begin{aligned} \label{ineq:bound} {\mathsf{E}\!\left[f(x_{k+1})\right] }-f^\leq \theta\g_k\mu_k^{2\alpha-1}, \quad \hbox{for all } k \geq K, \end{aligned}$$ where $f^$ is the optimal value of problem [[and]{}]{} $$\begin{aligned} \label{def:theta}\theta := \max \bigg\{\frac{{\mathsf{E}\!\left[f_{K+1}(x_{K+1})\right] }-f^}{\g_K\mu_K^{2\alpha-1}},\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2{{\color{black}\lambda_{\min}}}(1-\beta)}\bigg\}.\end{aligned}$$ Moreover, $\lim_{k\to \infty}{\mathsf{E}\!\left[f(x_{k})\right] }=f^.$ Note that Assumption \[assum:sequences-ms-convergence\](a,b) imply that holds for a large enough $k$, say after $\hat K$. Then, since the conditions of Lemma \[lemma:main-ineq\] are met [[(with an index shift)]{}]{}, taking expectations on both sides of , we obtain for any $k\geq \hat K$: $$\begin{aligned} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } &\leq (1-\lambda_{\min}\mu_k\g_k){\mathsf{E}\!\left[f_k(x_k)-f^\right] }+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\mu_k^2\g_k\\ &+\frac{ (L+\mu_0)\lambda^2\nu^2}{2}\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Using Assumption \[assum:sequences-ms-convergence\](e), we have $\mu_k^2\g_k\leq \rho \g_k^2\mu_k^{2\alpha}$. Thus, we obtain $$\begin{aligned} \label{ineq:cond-recursive-F-k-expected2} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } &\leq (1-\lambda_{\min}\mu_k\g_k){\mathsf{E}\!\left[f_k(x_k)-f^\right] }\notag\\ &+\left(\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2}\right)\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Let us define [[$K\triangleq\max\{\hat K, K_0\}$, where $K_0$ is from Assumption \[assum:sequences-ms-convergence\](d)]{}]{}. Using the preceding relation and by induction on $k$, we show the desired result. To show , we show the following relation first: $$\begin{aligned} \label{ineq:bound_v2} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})\right] }-f^\leq \theta\g_k\mu_k^{2\alpha-1}, \quad \hbox{for all } k \geq K, \end{aligned}$$ Note that [[implies]{}]{} the relation since we have $ {\mathsf{E}\!\left[f(x_{k+1})\right] }\leq {\mathsf{E}\!\left[f_{k+1}(x_{k+1})\right] }$. First, we show that holds for [[$k=K$]{}]{}. Consider the term ${\mathsf{E}\!\left[f_{K+1}(x_{K+1})\right] }-f^$. Multiplying and dividing by $\g_K\mu_K^{2\alpha-1}$, we obtain $$\begin{aligned} {\mathsf{E}\!\left[f_{K+1}(x_{K+1})\right] }-f^=\left(\frac{{\mathsf{E}\!\left[f_{K+1}(x_{K+1})\right] }-f^}{\g_k\mu_K^{2\alpha-1}}\right)\g_K\mu_K^{2\alpha-1} \leq \theta \g_K\mu_K^{2\alpha-1},\end{aligned}$$ where the last inequality is obtained by invoking the definition of $\theta$ in . This implies that holds for $k=K$. Now assume that holds for some $k \geq K$. We show that it also holds for $k+1$. From the induction hypothesis and we have $$\begin{aligned} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } &\leq (1-\lambda_{\min}\mu_k\g_k)\theta\g_{k-1}\mu_{k-1}^{2\alpha-1}\\ &+\left(\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2}\right)\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Using Assumption \[assum:sequences-ms-convergence\](d) we obtain $$\begin{aligned} \label{ineq:thm2-rel1} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } &\leq \theta\g_{k}\mu_{k}^{2\alpha-1}(1-\lambda_{\min}\mu_k\g_k)(1+\beta \lambda_{\min}\g_{k}\mu_{k})\notag \\ &+\left(\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2}\right)\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Next we find an upper bound for the term $(1-\lambda_{\min}\mu_k\g_k)(1+\beta \lambda_{\min}\g_{k}\mu_{k}) $ as follows $$\begin{aligned} (1-\lambda_{\min}\mu_k\g_k)(1+\beta \lambda_{\min}\g_{k}\mu_{k}) &= 1-\lambda_{\min}\mu_k\g_k+\beta\lambda_{\min}\mu_k\g_k-{{\color{black}\beta\lambda_{\min}^2\mu_k^2\g_k^2}}\\ & \leq 1-(1-\beta)\lambda_{\min}\mu_k\g_k.\end{aligned}$$ Combining this relation with , it follows [[that]{}]{} $$\begin{aligned} &{\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } \leq \theta\g_{k}\mu_{k}^{2\alpha-1} -\theta{{\color{black}\lambda_{\min}}}(1-\beta)\mu_k^{2\alpha}\g_k^2 \\ &+\left(\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2}\right)\mu^{2\alpha}_k\g_k^2\\ & =\theta\g_{k}\mu_{k}^{2\alpha-1}-\left(\underbrace{\theta{{\color{black}\lambda_{min}}}(1-\beta)-\frac{\rho\lambda_{\min}\mbox{dist}^2(x_0,X^)+ (L+\mu_0)\lambda^2\nu^2}{2}}_{\tiny{\hbox{Term }1}}\right)\mu^{2\alpha}_k\g_k^2.\end{aligned}$$ Note that the definition of $\theta$ in implies that Term 1 is nonnegative. Therefore, $$\begin{aligned} {\mathsf{E}\!\left[f_{k+1}(x_{k+1})-f^\right] } &\leq \theta\g_{k}\mu_{k}^{2\alpha-1}.\end{aligned}$$ Hence, the induction statement holds for $k+1$. We conclude that holds for all $k\geq K$. As a consequence, holds for all $k\geq K$ as well. To complete the proof, we need to show $\lim_{k\to \infty}{\mathsf{E}\!\left[f(x_{k})\right] }=f^$. This is an immediate result of and Assumption \[assum:sequences-ms-convergence\](a). Iteratively regularized stochastic and deterministic LBFGS methods {#sec:BFGS} ================================================================== [[In this section, our main goal is to develop an efficient update rule for matrix $H_k$ of]{}]{} the scheme [[and establish a convergence rate result]{}]{}. Background {#sec:LBFGS-background} ---------- Stochastic gradient methods are known to be sensitive to the choice of stepsizes. In our [[prior]{}]{} work [@Farzad1; @Farzad2; @FarzadTAC19], we address this challenge [[in part]{}]{} by developing self-tuned stepsizes [[under the strong convexity assumption]{}]{}. Another avenue to enhance the robustness of [[this]{}]{} scheme [[lies in incorporating]{}]{} curvature information of the objective function. A well-known updating rule for the matrix $H_k$ that uses the curvature estimates is [[the BFGS update]{}]{}. The deterministic BFGS [[scheme]{}]{}, achieves a superlinear convergence rate (cf. Theorem 8.6 [@Nocedal2006NO]) outperforming the deterministic gradient/subgradient method. In the classical deterministic BFGS scheme, the curvature information is incorporated within the algorithm using two terms: the first term is the displacement factor $s_k=x_{k+1}-x_k$, while the other is the change in the gradient mapping, $y_k=\nabla f(x_{k+1})-\nabla f(x_k)$, where $\nabla f$ denotes the gradient mapping of the deterministic objective function [[$f$]{}]{}. To have a well-defined update rule, it is essential that at each iteration, the [[curvature condition]{}]{} [[$s_k^Ty_k>0$]{}]{} is satisfied. By maintaining this condition at each iteration, the positive definiteness of the [[approximate Hessian]{}]{}, denoted by $B_k$, is preserved. The BFGS update rule in deterministic regime also ensures that $B_k$ satisfies a secant equation given by $B_{k+1}s_k=y_k$[[, which ensures]{}]{} that the [[approximate Hessian]{}]{} maps $s_k$ into $y_k$. To address optimization problems in [[the]{}]{} stochastic [[regime]{}]{}, a regularized BFGS update rule, namely (RES), [[was]{}]{} developed for problem under the strong convexity assumption [@mokh14]. [[In problems with a large dimension (see [@nocedal15] for some examples), the implementation of this scheme becomes challenging]{}]{}. This is because the computation [[of]{}]{} $B_k$ and its inverse become [[expensive]{}]{}. Moreover, at each iteration, a matrix of size $n\times n$ needs to be stored. To address these issues in [[large scale]{}]{} optimization problems, limited-memory variants of stochastic BFGS scheme, denoted by stochastic LBFGS, have been developed [@Mokhtari15; @nocedal15]. The key idea in LBFGS update rule is that instead of storing the full $n\times n$ matrix at each iteration, a fixed number of vectors of size $n$ are stored and used to update the [[approximate Hessian inverse]{}]{}. Outline of the stochastic LBFGS scheme {#sec:LBFGS-outline} -------------------------------------- [[The]{}]{} strong convexity property assumed in [@Mokhtari15; @nocedal15] plays a key role in developing the LBFGS update rules and establishing the convergence. Note that in [[the]{}]{} absence of strong convexity, the [[curvature condition]{}]{} does not hold. To address this issue, a standard approach is to employ a damped variant of the BFGS update rule [@Nocedal2006NO]. A drawback of this class of update rules is that there is no guarantee on the rate statements under such rules. Here, we resolve this issue through employing the properties of the regularized gradient map. This is carried out in by adding the regularization term $\mu_k(x_k-x_0)$ to the stochastic gradient mapping $\nabla F(x_k,\xi_k)$. [[To maintain the curvature condition]{}]{}, [[we consider updating]{}]{} the matrix $H_k$ and the parameter $\mu_k$ [[in alternate steps]{}]{}. Keeping the regularization parameter constant in one iteration [[allows for maintaining the [[curvature condition]{}]{}]{}]{}. After [[updating]{}]{} $H_k$, in the [[subsequent]{}]{} iteration, we keep this matrix fixed and drop the value of the regularized parameter. [[Accordingly, the update rule for]{}]{} the regularization parameter $\mu_k$ [[is based on the]{}]{} following general procedure: $$\begin{aligned} \label{eqn:mu-k} \begin{cases} \mu_{k}{{\color{black}:=}}\mu_{k-1}, & \text{if } k \text{ is odd},\\ \mu_{k}<\mu_{k-1}, & \text{otherwise}{{\color{black}.}} \end{cases}\end{aligned}$$ Note that we allow for updating the stepsize sequence at each iteration. We construct the update rule in terms of the following two factors defined for any odd [[$k\geq 1$:$$\begin{aligned} \label{equ:siyi-LBFGS}&s_{\lceil k/2\rceil}:= x_{k}-x_{k-1},\cr &y_{\lceil k/2\rceil}:= \nabla F(x_{k},\xi_{k-1}) -\nabla F(x_{k-1},\xi_{k-1}) + {{\color{black}\tau}}\mu_k^\delta s_{\lceil k/2\rceil},\end{aligned}$$]{}]{}where [[$\tau >0$ and]{}]{} $0<\delta\leq 1$ [[are parameters to control]{}]{} the level of regularization in the matrix $H_k$. Here, $\delta$ only controls the regularization for matrix $H_k$, but not that of the gradient direction. It is assumed that $\delta>0$ to [[ensure]{}]{} that the perturbation term [[$\mu_k^\delta s_{\lceil k/2\rceil} \to 0$, as $k\to\infty$]{}]{}. The update policy for $H_k$ is [[defined]{}]{} as follows: $$\begin{aligned} \label{eqn:H-k}H_{k}{{\color{black}\triangleq}} \begin{cases} H_{k,m}, & \text{if } k \text{ is odd}, \\ H_{k-1}, & \text{otherwise}, \end{cases}\end{aligned}$$ where $m<n$ (in the large scale settings, [[$m\ll n$]{}]{}) is [[the memory parameter and represents]{}]{} the number of pairs [[$(s_i,y_i)$]{}]{} to be [[stored]{}]{} to estimate $H_k$. Matrix $H_{k,m}$, for any [[odd $k\geq 2m-1$]{}]{}, is updated using the following recursive formula: $$\begin{aligned} \label{eqn:H-k-m} H_{k,j}:=\left(\mathbf{I}-\frac{y_is_i^T}{{y_i^Ts_i}}\right)^TH_{k,j-1}\left(\mathbf{I}-\frac{y_is_i^T}{y_i^Ts_i}\right)+\frac{s_is_i^T}{y_i^Ts_i}, \quad 1 \leq j\leq m,\end{aligned}$$ where [[$i\triangleq \lceil k/2\rceil-(m-j)$ and we set $H_{k,0}:=\frac{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}{y_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}\mathbf{I}$]{}]{}. Here, at odd iterations, matrix $H_k$ is obtained recursively from $H_{k,0},H_{k,1},\ldots, H_{k,m-1}$. Note that [[computation of]{}]{} $H_k$ at an odd $k$ [[needs]{}]{} $m$ pairs [[of]{}]{} [[$(s_i,y_i)$]{}]{}. More precisely, $H_k$ uses the following curvature information: $\left\{(s_i,y_i)\mid i=\lceil k/2\rceil-m+1, \lceil k/2\rceil-m+2,\ldots,\lceil k/2\rceil\right\}.$ For convenience, in the first [[$2m-2$]{}]{} iterations, we [[let $H_k$ be the identity matrix]{}]{}. [[This allows for collecting the first set of $m$ pairs $(s_i,y_i)$, where $i=1,2,\ldots,m$, that is used at iteration $k:=2m-1$ to obtain $H_{2m-1}$]{}]{}. [[The main differences between update rule and that of the standard SQN schemes [@Mokhtari15; @nocedal15] are as follows:]{}]{} (i) The first distinction is with respect to the definition of $y_i$ in . Here the term ${{\color{black}\mu_k^\delta s_{\lceil k/2\rceil}}}$ [[compensates]{}]{} for the lack of strong monotonicity of the gradient mapping and [[aids in]{}]{} establishing the [[curvature condition]{}]{}. (ii) Second, instead of obtaining the pair $(s_i,y_i)$ at every iteration, we [[evaluate these terms only at]{}]{} odd iterations to allow for updating the regularization parameter satisfying . [[Implementation of this stochastic LBFGS scheme requires computing the term $H_k\nabla F_k(x_k,\xi_k)$ at the $k$th iteration.]{}]{} This can be performed through a two-loop recursion with $\mathcal{O}\left(mn\right)$ number of operations (see Ch. 7, Pg. 178 in [@Nocedal2006NO]). [[This will be shown for Algorithm \[algorithm:IR-S-BFGS\] in Theorem \[thm:rate\](b).]{}]{} In this section, we consider a stronger variant of Assumption \[assum:convex\] stated as follows: \[assum:convex2\] 1. [[The function $F(x,\xi)$ is convex over $\Real^n$ for any $\xi \in \Omega$.]{}]{} 2. For any $\xi \in \Omega$, $F(\cdot,\xi)$ is continuously differentiable [[with Lipschitz continuous gradients]{}]{} over $\Real^n$ with parameter $L_\xi>0$. Moreover, $L:=\sup_{\xi \in \Omega}L_\xi < \infty$. 3. The optimal solution set $X^$ of problem is nonempty. Next, in Lemma \[LBFGS-matrix\], we derive bounds on the eigenvalues of the matrix $H_k$ and show that at iterations where $H_k$ is updated, both the [[curvature condition]{}]{} and the secant equation hold. In [[the]{}]{} proof of Lemma \[LBFGS-matrix\], we will make use of the following result. \[sumProductBounds\] Let $0 < a_1 \leq a_2 \leq \ldots \leq a_n$, and $P$ and $S$ be positive scalars such that [[$\sum_{i=1}^n a_i \leq S$ and $\prod_{i=1}^na_i \geq P$. Then, we have $a_1 \geq (n-1)!P/S^{n-1}$.]{}]{} See Appendix \[app:sumProductBounds\]. \[LBFGS-matrix\] Consider the method. Let $H_k$ be given by the update rule -, where $s_i$ and $y_i$ are defined in and $\mu_k$ is updated according to the procedure . Let [[Assumption \[assum:convex2\](a,b)]{}]{} hold. Then, the following results hold: - For any odd [[$k \geq 2m-1$]{}]{}, the [[curvature condition]{}]{} holds, i.e., [[$s_{\lceil k/2\rceil}^T{y_{\lceil k/2\rceil}} >0$]{}]{}. - For any odd [[$k \geq 2m-1$]{}]{}, the secant equation holds, i.e., [[$H_{k}{y}_{\lceil k/2\rceil}=s_{\lceil k/2\rceil}$]{}]{}. - For any [[$k \geq 2m-1$]{}]{}, $H_k$ satisfies Assumption \[assump:Hk\] [[with]{}]{} the following values: $$\begin{aligned} \label{equ:valuesForAssumH_k}&\lambda_{\min}=\frac{1}{(m+n)\left(L+{{\color{black}\tau}}\mu_0^\delta\right)}, \quad {{\color{black}\lambda= \frac{(m+n)^{n+m-1}\left(L+{{\color{black}\tau}}\mu_0^\delta\right)^{n+m-1}}{(n-1)!{{\color{black}\tau^{(n+m)}}}}}}, \notag\\ & {{\color{black}\hbox{and }\alpha=-\delta(n+m).}}\end{aligned}$$ More precisely, $H_k$ is symmetric, ${\mathsf{E}\!\left[H_k\mid\sF_k\right] }=H_k$ and $$\begin{aligned} \label{proof:H_kbounds} \frac{1}{(m+n)\left(L+{{\color{black}\tau}}\mu_0^\delta\right)}\mathbf{I} \preceq H_{k} \preceq \frac{(m+n)^{n+m-1}\left(L+{{\color{black}\tau}}\mu_0^\delta\right)^{n+m-1}}{(n-1)!{{\color{black}\left(\tau\mu_k^\delta\right)^{(n+m)}}}}\ \mathbf{I}.\end{aligned}$$ See Appendix \[app:lemmaLBFGSmatrix\]. [[In the following two lemmas, we provide update rules for the stepsize and the regularization parameter to ensure [[convergence in [[both]{}]{} an a.s. and mean sense]{}]{} for the proposed LBFGS scheme.]{}]{} \[lemma:a.s.sequences\] Let the sequences $\g_k$ and $\mu_k$ be given by the following rules: $$\begin{aligned} \label{equ:seq} \g_k=\frac{\g_0}{(k+1)^a}, \quad \mu_{k}=\frac{{{\color{black}2^b\mu_0}}}{\left(k+\kappa\right)^b},\end{aligned}$$ where $\kappa=2$ if $k$ is even and $\kappa=1$ otherwise, $\g_0$and $\mu_0$ are positive scalars such that $\g_0\mu_0 \leq L(m+n)$, [[and $a$, $b$, and $\delta\leq 1$ are positive scalars [[satisfying]{}]{}: $$\begin{aligned} & {{\color{black}\frac{a}{b}>1+2\delta(n+m)}}, \quad a+b \leq 1,\quad a+2b > 1, \quad\hbox{and}\quad a-\delta b(m+n)>0.5.\end{aligned}$$ Then, $\g_k$ and $\mu_k$ satisfy Assumption \[assum:sequences\] with $\lambda_{\min}$ and $\alpha$ in and $\mu_k$ satisfies .]{}]{} See Appendix \[app:feasibleSeqASconv\]. The conditions on parameters $a$, $b$, $\g_0$, and $\mu_0$ in Lemma \[lemma:a.s.sequences\] hold for $\g_0=\mu_0\leq\sqrt{L}$, $a=\frac{5}{6}$ and $b=\frac{1}{6}$, and $\delta=\frac{1}{m+n}$. \[lemma:mean-sequences\] Let the sequences $\g_k$ and $\mu_k$ be given by , where $\g_0$ and $\mu_0$ are positive scalars such that $\g_0\mu_0 \leq L(m+n)$. [[Let, $0 <\delta \leq 1$ $a>0$ and $b>0$ [[satisfying]{}]{}: $$\begin{aligned} & \frac{a}{b}>1+2\delta\left(m+n\right), \quad a+b < 1, \quad \frac{a}{b} \leq 2\left(1+\delta\left(m+n\right)\right).\end{aligned}$$Then, [[$\mu_k$ satisfies ]{}]{} and $\g_k$ and $\mu_k$ satisfy Assumption \[assum:sequences-ms-convergence\] with any arbitrary $0<\beta<1$, $\rho\triangleq \g_0^{-1}\left(\mu_02^b\right)^{2+2\delta(m+n)}$, and with $\lambda_{\min}$ and $\alpha$ given by .]{}]{} See Appendix \[app:feasibleSeqMeanConv\]. An efficient implementation with rate analysis {#sec:LBFGS-rate} ---------------------------------------------- Input: LBFGS memory parameter $m\geq 1$, Lipschitzian parameter $L>0$, random initial point $x_0 \in \Real^n$, initial stepsize $\g_0>0$, and initial regularization parameter $\mu_0>0$ such that $\g_0\mu_0 \leq (m+n) L$, [[scalars $0<\epsilon < \frac{1}{3}$, $\delta \in \left(0,\frac{1.5\epsilon}{n+m}\right)$, and $\tau>0$]{}]{}; Set $a:=\frac{2}{3}-\epsilon+\frac{2\delta(n+m)}{3}$, $b:=\frac{1}{3}$; Compute $\g_k :=\frac{\g_0}{(k+1)^a}$ and $\mu_k:=\frac{\mu_02^b}{\left(k+1+\text{mod}(k+1,2)\right)^b}$; Compute index $i:=\lceil k/2\rceil$; Compute vector $s_i:=x_k-x_{k-1}$; Compute vector $y_i:= \nabla F(x_k,\xi_{k-1})-\nabla F(x_{k-1},\xi_{k-1})+{{\color{black}\tau}}\mu_k^\delta s_i$; Discard the vector pair $\{s_{i-m},y_{i-m}\}$ from storage; Update solution iterate $x_{k+1}:=x_k-\g_k\left(\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)\right)$; Initialize Hessian inverse $H_{k,0}:=\frac{s_i^Ty_i}{y_i^Ty_i}\mathbf{I}$; Initialize $q:=\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)$; Compute scalar $\alpha_{i-t+1}:=\frac{s_t^Tq}{s_t^Ty_t}$; Update vector $q:=q-\alpha_{i-t+1}y_t$; Initialize vector $r:=H_{k,0}q$; Update vector $r:=r+\left(\alpha_{i-t+1}-\frac{y_t^Tr}{s_t^Ty_t}\right)s_t$; Update solution iterate $x_{k+1}:=x_k-\g_kr$; ($(right)!(top.north)!($(right)-(0,1)$)$) – ($(right)!(bottom.south)!($(right)-(0,1)$)$) node \[align=center, text width=4.5cm, pos=0.5, anchor=west\] ; Algorithm \[algorithm:IR-S-BFGS\] presents an efficient implementation of the proposed stochastic LBFGS scheme. Note that update rules for the stepsize and regularization parameter are specified in line $\#2$ and $\#4$. Before presenting the complexity analysis in Theorem \[thm:rate\], we make some comments on the choice of parameter $\tau$. As mentioned, the parameter $\tau>0$ in line \#9 in Algorithm \[algorithm:IR-S-BFGS\] is used to control the level of the iterative regularization employed in the computation of matrix $H_k$. Intuitively, it may seem that a [[small]{}]{} choice for $\tau$ [[may]{}]{} reduce the distortion caused by the term $\mu_k^\delta s_i$ in approximating the Hessian inverse, and consequently, improve the performance of the algorithm. However, this may not be always the case. To see this, first note that the relation shows the dependency of eigenvalues of $H_k$ on the choice of $\tau$. Recall that the relation is a key assumption used in the convergence and rate analysis of the proposed method. It can be seen that when $\tau \to 0$, the right-hand side of will decrease to zero. This indicates that a small $\tau$ enforces a small value for the term $\gamma_0/\mu_0^{2\delta(n+m)}$. For example, assuming a fixed value for $\mu_0$, this would lead to a small $\g_0$. This may have a negative impact on the performance of the algorithm. As such, it is not clear if a small $\tau$ can be always beneficial. A closer look into this trade off calls for a more detailed analysis of the finite-time performance of the algorithm, which is not the focus of our current work and remains as a future direction to our study. In Theorem \[thm:rate\](a), we establish the convergence rate of Algorithm \[algorithm:IR-S-BFGS\]. Moreover, in Theorem \[thm:rate\](b), we show that the term $H_k\nabla F_k(x_k,\xi_k)$ is computed efficiently using [[the LBFGS two-loop recursion]{}]{} in the algorithm with $\mathcal{O}\left(mn\right)$ complexity per iteration. \[thm:rate\] [[Consider Algorithm \[algorithm:IR-S-BFGS\]. The following statements hold:]{}]{} - Suppose Assumptions \[assum:main\] and \[assum:convex2\] hold. Then, there exists [[$K\geq2m-1$]{}]{} such that $${\mathsf{E}\!\left[f(x_{k })\right] }-f^\leq \left(\frac{\theta\g_0}{\left(\mu_0\sqrt[3]{2}\right)^{1-2\alpha}}\right)\frac{1}{k^{\frac{1}{3}-\e}}, \quad \hbox{for all } k > K,$$ where $\theta$ is given by , and $\lambda_{\min}$, $\lambda$, and $\alpha$ are given by . - [[Let the scheme be at the $k$th iteration where $k\geq 2m-1$. Then, by the end of the LBFGS [[two-loop]{}]{} recursion, i.e., line $\#$26 in Algorithm \[algorithm:IR-S-BFGS\], we have $$\begin{aligned} \label{eqn:r_Lemma} r=H_k\left(\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)\right),\end{aligned}$$ where $H_{k}$ is defined by .]{}]{} \(a) First, we show that the conditions of Proposition \[prop:mean\] are satisfied. Assumption \[assum:convex\] holds as a consequence of Assumption \[assum:convex2\]. From Lemma \[LBFGS-matrix\](c), Assumption \[assump:Hk\] holds for any [[$k \geq 2m-1$]{}]{} as well. To show that Assumption \[assum:sequences-ms-convergence\] holds, we apply Lemma \[lemma:mean-sequences\]. We have $$\frac{a}{b}=\frac{\frac{2}{3}-\e+\frac{2\delta(n+m)}{3}}{1/3}=2-3\e+2\delta(n+m)>1+2\delta\left(m+n\right),$$ where we used $\e<\frac{1}{3}$. Moreover, since $\delta < \frac{1.5\e}{n+m}$, we have $a+b=1-\e +\frac{2\delta(n+m)}{3}<1$. Also, from the values of $a$ and $b$ we have $2b\left(1+\delta\left(m+n\right)\right)=a+\e>a.$ Thus, the conditions of Lemma \[lemma:mean-sequences\] hold. This implies that there exists $K_0>0$ such that for any $k\geq K_0$, the sequences $\g_k$ and $\mu_k$ satisfy Assumption \[assum:sequences-ms-convergence\] with any arbitrary $0<\beta<1$ and for $\rho=\g_0^{-1}\left(\mu_02^b\right)^{2+2\delta(m+n)}$, and with $\lambda_{\min}$, $\lambda$ and $\alpha$ given by . Let us define $K:=\max\{K_0,2m-1\}$. Since all conditions in Proposition \[prop:mean\] are met, [[from , , and by substituting values of $a$, $b$, and $\alpha$, for any $k \geq K$ we obtain]{}]{} $$\begin{aligned} {\mathsf{E}\!\left[f(x_{k+1})\right] }-f^&\leq \theta\g_{k-1}\mu_{k-1}^{2\alpha-1}= \frac{\theta\g_0(k+\kappa-1)^{(1-2\alpha)/3}}{\left(\mu_0\sqrt[3]{2}\right)^{1-2\alpha}k^{2/3-\e-\frac{2}{3}\alpha}}\\ &\leq \frac{\theta\g_0(k+1)^{(1-2\alpha)/3}}{\left(\mu_0\sqrt[3]{2}\right)^{1-2\alpha}(k+1)^{2/3-\e-\frac{2}{3}\alpha}}=\left(\frac{\theta\g_0}{\left(\mu_0\sqrt[3]{2}\right)^{1-2\alpha}}\right)\frac{1}{(k+1)^{\frac{1}{3}-\e}}.\end{aligned}$$ Through a change of variable from $k+1$ to $k$, we conclude the result.\ [[(b) To show , it suffices to show that $$\begin{aligned} r=\begin{cases} H_{k,m}\left(\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)\right), & \text{if } k \text{ is odd}, \\ H_{k-1,m}\left(\nabla F(x_k,\xi_k)+\mu_k(x_k-x_0)\right), & \text{otherwise}, \end{cases}\end{aligned}$$ where $H_{k,m}$ is defined by the recursion for an odd $k$. First, consider the case that $k\geq 2m-1$ is an odd number. As such, at the $k$th iteration, from line $\#$7, we have $i:=\lceil k/2 \rceil$. For clarity of the presentation, throughout [[this]{}]{} proof, we use $K$ (instead of $i$), i.e., $K\triangleq \lceil k/2 \rceil$ [[and]{}]{} $q_{K-t+1}$ [[is used to]{}]{} denote the value of the vector $q \in \mathbb{R}^n$ after being updated at iteration $t$ in line $\#$21. Similarly, we use $r_{t-K+m}$ to denote the value of the vector $r \in \mathbb{R}^n$ after being updated at iteration $t$ in line $\#$25. Also, we use the following definitions: $$\begin{aligned} & q_0\triangleq \nabla F(x_k,\xi_k)+\mu_k(x_k-x_0), \quad r_0\triangleq H_{k,0}q_{m},\cr &\rho_j\triangleq \frac{1}{y_j^Ts_j}, \hbox{ and }V_j\triangleq \mathbf{I}-\rho_{j}y_js_j^T, \quad \hbox{for all } j=K-(m-1),\ldots, K.\end{aligned}$$ Consider relation . By applying this recursive relation repeatedly, we obtain $$\begin{aligned} \label{equ:H_kClosedForm} H_{k,m} &=\left(\prod_{j=1}^{m}V_{K-(m-j)}\right)^TH_{k,0}\left(\prod_{j=1}^{m}V_{K-(m-j)}\right)\\ &+ \rho_{K-m+1}\left(\prod_{j=2}^{m}V_{K-(m-j)}\right)^Ts_{K-m+1}s^T_{K-m+1}\left(\prod_{j=2}^{m}V_{K-(m-j)}\right) \notag\\ & + \rho_{K-m+2}\left(\prod_{j=3}^{m}V_{K-(m-j)}\right)^Ts_{K-m+2}s^T_{K-m+2}\left(\prod_{j=3}^{m}V_{K-(m-j)}\right)\notag\\ & + \ldots\notag\\ & +\rho_{K-1}V^T_{K}s_{K-1}s_{K-1}^TV_{K}\notag\\ &+\rho_{K}s_{K}s_{K}^T.\notag\end{aligned}$$ Next, we derive a formula for $q_t$. From lines $\#$20-21 in the algorithm, we have $$\begin{aligned} q_{K-t+1}&=q_{K-t} -\alpha_{K-t+1}y_t =q_{K-t} - \rho_t\left(s_t^Tq_{K-t}\right)y_t = q_{K-t} - \rho_t\left(y_ts_t^T\right)q_{K-t} \\ &= \left( \mathbf{I}-\rho_{t}y_ts_t^T\right)q_{K-t}=V_tq_{K-t}, \quad \hbox{for all } t= K, K-1, \ldots, K-m+1.\end{aligned}$$ From the preceding relation, we obtain $$\begin{aligned} \label{equ:q_ell} q_{\ell}=\left(\prod_{j=m-\ell+1}^{m}V_{K-(m-j)}\right)q_0, \quad \hbox{for all } \ell= 1,2, \ldots, m.\end{aligned}$$ From the update rule for $\alpha_{i-t+1}$ in line $\#$20, using the definition of $\rho_t$, and applying the previous relation, we have $\alpha_1=\rho_{K}s_K^Tq_0$ and $$\begin{aligned} \label{equ:alpha_ell} \alpha_{\ell}=\rho_{K-\ell+1}s_{K-\ell+1}^T\left(\prod_{j=m-\ell+2}^{m}V_{K-(m-j)}\right)q_0, \quad \hbox{for all } \ell= 2,3, \ldots, m.\end{aligned}$$ [[Multiplying both sides of]{}]{} by $q_0$ and employing and , we obtain $$\begin{aligned} \label{equ:H_kq_0} H_{k,m}q_0 &=\left(\prod_{j=1}^{m}V_{K-(m-j)}\right)^TH_{k,0}q_m + \left(\prod_{j=2}^{m}V_{K-(m-j)}\right)^Ts_{K-m+1}\alpha_m \\ & + \left(\prod_{j=3}^{m}V_{K-(m-j)}\right)^Ts_{K-m+2}\alpha_{m-1} +\ldots+V^T_{K}s_{K-1}\alpha_2+s_{K}\alpha_1.\notag\end{aligned}$$ Next, we derive a formula for $r_t$. From line $\#$25 in the algorithm, we have $$\begin{aligned} r_{t-K+m}&=r_{t-K+m-1}+\left(\alpha_{K-t+1}-\rho_ty_t^Tr_{t-K+m-1}\right)s_t \\ &= r_{t-K+m-1}-\rho_ts_t y_t^Tr_{t-K+m-1}+\alpha_{K-t+1}s_t\\ & = V_t^Tr_{t-K+m-1}+\alpha_{K-t+1}s_t, \quad \hbox{for all } t= K-m+1,\ldots,K-1, K.\end{aligned}$$ Combining the preceding two relations, we obtain $$\begin{aligned} r_{\ell}&= V_{K-(m-\ell)}^Tr_{\ell-1}+\alpha_{m-\ell+1}s_{K-(m-\ell)}, \quad \hbox{for all } \ell= 1,2,\ldots,m.\end{aligned}$$ Using the preceding equation repeatedly, we obtain $$\begin{aligned} \label{equ:r_m} r_m &=\left(\prod_{j=1}^{m}V_{K-(m-j)}\right)^Tr_0 +\alpha_m \left(\prod_{j=2}^{m}V_{K-(m-j)}\right)^Ts_{K-m+1} \\ & + \alpha_{m-1} \left(\prod_{j=3}^{m}V_{K-(m-j)}\right)^Ts_{K-m+2} +\ldots+\alpha_2V^T_{K}s_{K-1}+\alpha_1s_{K}.\notag\end{aligned}$$ From and , and the definition of $r_0$, we obtain $r_m=H_{k,m}q_0$. Taking to account the definition of $q_0$, the desired result holds for any odd $k\geq 2m-1$. Now, consider the case where $k\geq 2m-1$ is an even number. This implies that the “if” condition in line $\#$6 is skipped and as such, the value of $i$ is not updated from the iteration $k-1$, i.e., $i= \lceil (k-1)/2 \rceil$. Consequently, the LBFGS two-loop recursion at an even $k$ uses the following pairs $$\left\{(s_\ell,y_\ell)\mid \ell=\lceil (k-1)/2\rceil-m+1, \lceil (k-1)/2\rceil-m+2,\ldots,\lceil (k-1)/2\rceil\right\}.$$ [[Now,]{}]{} considering the definition for $k-1$, the desired relation can be shown following the same steps discussed for the case the iteration number is odd.]{}]{} Analysis [[of]{}]{} the deterministic case {#sec:LBFGS-deterministic} ------------------------------------------ Our goal in the remainder of this section lies in establishing the convergence and rate statement for the deterministic LBFGS scheme. Consider the following regularized deterministic LBFGS method: $$\begin{aligned} \label{eqn:R-L-BFGS}\tag{IR-LBFGS} x_{k+1}:=x_k -\gamma_kH_k\left(\nabla f(x_k)+ \mu_k \left(x_k-x_0\right)\right), \quad \hbox{for all } k \geq 0, \end{aligned}$$ where $H_k$ is given by the update rule , $\mu_k$ is updated according to [[, and for an odd $k \geq 1$ we set]{}]{} $$\begin{aligned} \label{equ:siyi-DLBFGS}&s_{\lceil k/2\rceil}:= x_{k}-x_{k-1},\cr &{y_{\lceil k/2\rceil}}:= \nabla f(x_{k}) -\nabla f(x_{k-1}) + {{\color{black}\tau}}\mu_k^\delta s_{\lceil k/2\rceil}.\end{aligned}$$ Let $x_k$ be generated by the \[eqn:R-L-BFGS\] method. Suppose Assumption \[assum:convex\] holds. Let $\lambda_{\min}$, $\lambda$ and $\alpha$ be given by . Then [[the following hold.]{}]{} \(a) Let $\mu_k$ satisfies . If $\g_k$ and $\mu_k$ satisfy the following relation: $$\begin{aligned} \label{DLBFGScond}\g_k\mu_k^{2\alpha} \leq \frac{\lambda_{\min}}{\lambda^2(L+\mu_0)},\quad \hbox{for all }k\geq 0,\end{aligned}$$then, for any $k \geq 0$, we have $$\begin{aligned} \label{ineq:DLBFGSbound} f_{k+1}(x_{k+1})-f^* &\leq (1-\lambda_{\min}\mu_k\g_k)(f_k(x_k)-f^)+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\mu_k^2\g_k.\end{aligned}$$ (b) Let $\g_k$ and $\mu_k$ be given by the update rule where [[$a, b>0$ and $0<\delta \leq 1$ satisfy $$\frac{a}{b}>2\delta(n+m) \quad a+b\leq 1, \quad a+2b>1.$$Then, $\lim_{k\to \infty}f(x_k)=f^$. Specifically, for $a=\frac{4}{5}$, $b=\frac{1}{5}$, and $\delta=\frac{1}{m+n}$, this result holds.]{}]{} \(c) Let $\e \in (0,1)$ be an arbitrary small scalar. Let $\g_k$ and $\mu_k$ be given by the update rule where $a=\e$, $b=1-\e$. Also, assume [[$\delta \in \left(0,\frac{\e}{2(n+m)(1-\e)}\right)$.]{}]{} Let $\g_0$ and $\mu_0$ satisfy the following condition: $$\begin{aligned} \label{DLBFGScondmu0gamma0}\g_0\mu_0\geq (n+m)\left(L+{{\color{black}\tau}}\mu_0^\delta\right).\end{aligned}$$ Then, there exists $K$ such that $$\begin{aligned} \label{rateRDLBGFS} f(x_k)-f^\leq \frac{\Gamma}{(k+1)^{1-\e}}, \quad \hbox{for all } k \geq K,\end{aligned}$$ where $\Gamma \triangleq \max\bigg\{(K+1)^{1-\e}\left(f_K(x_K)-f^\right), \frac{\lambda_{\min}\g_0\mu_0^2{\mbox{dist}}^2(x_0,X^)}{4^{a}(\lambda_{\min}\g_0\mu_0-b)} \bigg\}.$ (a) The conditions of Lemma \[LBFGS-matrix\] are met indicating that Assumption \[LBFGS-matrix\] holds. Assumption \[assum:main\] is clearly met with $\nu=0$ as the problem is deterministic. Therefore, all of the conditions of Lemma \[lemma:main-ineq\] are satisfied and thus holds. Substituting $\nu=0$ in and eliminating the expectation operator yields the desired inequality.\ (b) First, we show that holds. We can write $$\begin{aligned} \g_k\mu_k^{2\alpha} & =\frac{\g_0}{(2^b\mu_0)^{-2\alpha}}(k+1)^{-a}(k+\kappa)^{-2\alpha b}\leq \frac{\g_0}{(2^b\mu_0)^{-2\alpha}}(k+1)^{-a-2\alpha b}.\end{aligned}$$ [[Note that the assumption that $a > 2b\delta(n+m)$, implies that $-a-2\alpha b<0$]{}]{}. Therefore, $\g_k\mu_k^{2\alpha}\to 0$ showing that there exists $K_0$ such that for any $k\geq K_0$, holds. We apply Lemma \[lemma:probabilistic\_bound\_polyak\] to the inequality by setting $$\alpha_k:=\lambda_{\min}\g_0\mu_0, \quad \beta_k:=\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)}{2}\mu_k^2\g_k, \quad v_k:=f_k(x_k)-f^.$$ [[From]{}]{} $a+b\leq 1$, we have $\sum_{k=0}^\infty \alpha_k =\infty$. Also, $a+2b >1$ indicates that $\sum_{k=0}^\infty \beta_k <\infty$. Since all conditions of Lemma \[lemma:probabilistic\_bound\_polyak\] are met, we have $f_k(x_k)\to f^$. Recalling Definition \[def:regularizedF\], this implies that $f(x_k)\to f^$.\ (c) First, we show that by the given update rules for $\g_k$ and $\mu_k$, relation holds. Note that [[$\alpha=-\delta(n+m)$]{}]{}. Therefore, we can write [[ $$\begin{aligned} \label{RDBFGSeps}\g_k\mu_k^{2\alpha}&=\frac{\g_0(k+\kappa)^{2(m+n)\delta b}}{\left(\mu_02^b\right)^{2(m+n)\delta }(k+1)^a}\leq \frac{\g_0(k+2)^{2(m+n)\delta b}}{\left(\mu_02^b\right)^{2(m+n)\delta }(k+1)^a}\notag \\&= \frac{\g_0(1+\frac{1}{k+1})^{2(m+n)\delta b}}{\left(\mu_02^b\right)^{2(m+n)\delta}(k+1)^{a-2(m+n)\delta b}}.\end{aligned}$$ Using the condition on $\delta$, we have $a-2(m+n)b\delta= \e-2(1-\e)\delta(m+n)>0.$]{}]{} Thus, relation indicates that there exists $K_1$ such that for any $k\geq K_1$, holds. Besides, since $a$ and $b$ are positive, there exits $K_2$ such that for any $k\geq K_2$, we have $(1-\lambda_{\min}\g_k\mu_k)>0$. Let us now define [[$K:=\max\{K_1,K_2,2m-1\}$]{}]{}. Next, we use induction on $k$ to show . For $k=K$, it clearly holds. Let us assume holds for $k>K$. Let $e_k$ denote $f_k(x_k)-f^$. From and the update rules of $\g_k$ and $\mu_k$ we can write $$\begin{aligned} \label{ineq:DLBFGSbound2} e_k & \leq \left(1-\frac{\lambda_{\min}\g_0\mu_02^b}{k^a(k+\kappa-1)^b}\right)e_{k-1}+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)\g_0\mu_0^22^{2b-1}}{k^a(k+\kappa-1)^{2b}} \notag\\ & \leq \left(1-\frac{\lambda_{\min}\g_0\mu_02^b}{k^a(k+1)^b}\right)e_{k-1}+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)\g_0\mu_0^22^{2b-1}}{k^{a+2b}}\notag\\ & \leq \left(1-\frac{\lambda_{\min}\g_0\mu_0}{k}\right)e_{k-1}+\frac{\lambda_{\min}\mbox{dist}^2(x_0,X^)\g_0\mu_0^22^{2b-1}}{k^{a+2b}},\end{aligned}$$ where $\kappa$ is defined in , and the last inequality is implied by $\frac{k^a(k+1)^b}{k^{a+b}}\leq 2^b$ for $k\geq 1$. Note that since $k\geq K_2$, the term $ \left(1-\frac{\lambda_{\min}\g_0\mu_0}{k^{a+b}}\right)$ in is nonnegative. Therefore, we can replace $e_{k-1}$ by its upper bound [[$\frac{\Gamma}{k^b}$]{}]{} in . Doing so and noticing that $a+b=1$, we obtain $$\begin{aligned} \label{ineq:DLBFGSbound3} e_k & \leq \left(1-\frac{C_1}{k}\right)\frac{\Gamma}{k^b}+\frac{C_2}{k^{b+1}},\end{aligned}$$ where we [[define]{}]{} $C_1{{\color{black}\triangleq}}\lambda_{\min}\g_0\mu_0$ and $ C_2{{\color{black}\triangleq}} \lambda_{\min}\mbox{dist}^2(x_0,X^)\g_0\mu_0^22^{2b-1}.$ Using , to show that $e_{k}\leq \frac{\Gamma}{(k+1)^{1-a}}$, it is enough to show that $$\Gamma\left(\frac{1}{k^b}-\frac{1}{(k+1)^b}\right)\leq \frac{C_1\Gamma -C_2}{k^{b+1}}.$$ Rearranging the terms, we need to verify that $\Gamma \geq \frac{C_2}{C_1-C_3}$ and $C_3<C_1$, where [[$C_3$ is an upper bound on]{}]{} $\sup_{k\geq1}\bigg\{k^{b+1}\left(\frac{1}{k^b}-\frac{1}{(k+1)^b}\right)\bigg\}$. We claim that [[$C_3:=b$ is a feasible choice]{}]{}. To prove this, we need to show that $k^{b+1}\left(\frac{1}{k^b}-\frac{1}{(k+1)^b}\right)\leq b$, [[or equivalently,]{}]{} $$\left(1-\frac{1}{k+1}\right)^b\geq 1-\frac{b}{k}, \quad \hbox{for all } k\geq 1.$$ Consider the function $g(x):=(1-\frac{1}{1+x})^b+\frac{b}{x}-1$ for $x\geq 1$. we have $$g'(x)=\frac{b}{(1+x)^2}\left(1-\frac{1}{1+x}\right)^{1-b}-\frac{b}{x^2}=\frac{b}{(1+x)^2}\left(\left(\frac{x+1}{x}\right)^{1-b}-\left(\frac{x+1}{x}\right)^{2}\right)\leq 0,$$ due to $0<b<1$. Hence, $g$ is non-increasing implying that it suffices to show $g(1)\geq0$, i.e., $2^b(1-b)\leq 1$. Let us define $h(x):=2^x(1-x)$ for $0<x<1$. We have $h'(x)=2^x\left(\ln(2)(1-x)-1\right)$. This indicates that $h'(x)<0$ over $x\in(0,1)$, implying that $h(b)\leq h(0)=1$. Hence, we conclude that $C_3:=b$ [[is a feasible choice]{}]{}. To show that $C_3<C_1$ holds, we need to verify that $C_1>b$. This is true due to . To complete the proof we need to show $\Gamma \geq \frac{C_2}{C_1-b}$. This holds from the definition of $\Gamma$. Numerical experiments {#sec:num} ===================== In this section, we present the implementation results of [[Algorithm \[algorithm:IR-S-BFGS\]]{}]{} [[on]{}]{} a [[classification]{}]{} application. The Reuters Corpus Volume I (RCV1) data set [@lewis2004rcv1] is a collection of news-wire stories produced by Reuters. After the tokenization process, each article is converted to a sparse binary vector, in that 1 denotes the existence and 0 denotes nonexistence of a token in the corresponding article. We consider a subset of the data with [[$N=100,000$]{}]{} articles and $n=138,921$ tokens. The articles are categorized into [[different]{}]{} hierarchical groups. Here we focus our attention on the binary classification of the articles with respect to the [[Markets]{}]{} class. We consider the logistic regression loss minimization problem given as follows: $$\begin{aligned} \label{logistic}\tag{LRM} \min_{x \in \Real^n} f(x):=\frac{1}{N}\sum_{i=1}^N\ln \left(1+\exp\left(-u_i^Txv_i\right)\right),\end{aligned}$$ where $u_i \in \Real^n$ is the input binary vector associated with article $i$ [[and]{}]{} $v_i \in \{-1,1\}$ [[denotes]{}]{} the class of the $i$th article. We run three experiments. Of these, in Section \[sec:num-reg\], we compare the performance of Algorithm \[algorithm:IR-S-BFGS\] [[(with $\tau=1$)]{}]{} with that of standard SQN methods. [[In Sections \[sec:num-SAGA\] and \[sec:num-IAG\], we provide comparisons of Algorithm \[algorithm:IR-S-BFGS\] with SAGA [@Saga14] and with IAG [@Iag17] applied to regularized problems, respectively.]{}]{} \[m2\] [m[1cm]{} \|\| c c c]{} $N$ & initial cond. 1& initial cond. 2& initial cond. 3\ \ $10^3$ & ![image](figures/plot1run2.pdf) & ![image](figures/plot2run2.pdf) & ![image](figures/plot3run2.pdf) \ $10^4$ & ![image](figures/plot4run2.pdf) & ![image](figures/plot5run2.pdf) & ![image](figures/plot6run2.pdf) \ $10^5$ & ![image](figures/plot7run2.pdf) & ![image](figures/plot8run2.pdf) & ![image](figures/plot9run2.pdf) \[tableSaga\] [m[1cm]{} \|\| c c c]{} $N$ & $\mu_{IAG}=0.1$&$\mu_{IAG}=0.01$& $\mu_{IAG}=0.001$\ \ $1000$ & ![image](figures/plot1revisedStepsizes.pdf) & ![image](figures/plot2revisedStepsizes.pdf) & ![image](figures/plot3revisedStepsizes.pdf) \ $2000$ & ![image](figures/plot4revisedStepsizes.pdf) & ![image](figures/plot5revisedStepsizes.pdf) & ![image](figures/plot6revisedStepsizes.pdf) \ $5000$ & ![image](figures/plot7revisedStepsizes.pdf) & ![image](figures/plot8revisedStepsizes.pdf) & ![image](figures/plot9revisedStepsizes.pdf) \[tableIAG\] Comparison with standard SQN schemes {#sec:num-reg} ------------------------------------ [[To solve problem , the standard LBFGS]{}]{} methods in [@Mokhtari15; @nocedal15] solve an approximate problem of the form $$\begin{aligned} \label{logisticReg}\tag{Regularized LRM} \min_{x \in \Real^n} f(x):=\frac{1}{N}\sum_{i=1}^N\ln \left(1+\exp\left(-u_i^Txv_i\right)\right)+{{\color{black}\frac{\eta}{2}\\|x\\|^2}},\end{aligned}$$ where [[$\eta>0$]{}]{} is an arbitrary regularization parameter. To perform the first experiment, we consider comparison of Algorithm \[algorithm:IR-S-BFGS\] with three variants of the standard LBFGS schemes, all denoted by RS-LBFGS (see Figure \[m2\]). In RS-LBFGS schemes, we use the stepsize of the form $\gamma_k=\frac{\g_0}{{k+1}}$ and drop $\eta$ at epochs of 400 iterations using a decay factor denoted by $\rho \in (0,1]$. Of these, in the first scheme, we assume $\rho=1$, meaning that $\eta$ is kept constant throughout the implementation of the SQN scheme. In the second scheme, we use $\rho=0.5$. This means for example, after every $400$ iterations, we set $\eta:=0.5\eta$. In the third scheme, we use $\rho=0.3$. [[We let $\g_0 \in \{10,0.5,0.1\}$, $\eta_0 \in \{1,0.5,0.01\}$, $m \in \{2,5\}$, $N=10^4$, and $x_0$ be the origin.]{}]{} In all cases, we use five sample paths to calculate the average value of the objective function in . Insights:* We observe that Algorithm \[algorithm:IR-S-BFGS\] performs uniformly better than the three variants of the standard SQN scheme under different tuning rules for the regularization parameter. This suggests that for merely convex stochastic optimization, SQN schemes using the tuning rules for the stepsize and regularization parameter given as $\g_k \approx 1/\sqrt[3]{k^2}$ and $\mu_k\approx1/\sqrt[3]{k}$ have a faster convergence speed. Comparison with SAGA [[on merely convex problems]{}]{} {#sec:num-SAGA} ------------------------------------------------------ Recall that in addressing the finite-sum minimization problems with merely convex objectives, employing averaging and under a constant stepsize, SAGA admits a sublinear convergence rate of $\mathcal{O}\left(\frac{N}{k}\right)$ [@Saga14]. The simulation results are provided in Figure \[tableSaga\]. These results include different sample sizes, i.e., $N \in \{10^3,10^4,10^5\}$, different initial conditions for SAGA, and different choices of the stepsizes and the initial regularization parameter for Algorithm \[algorithm:IR-S-BFGS\]. SAGA uses the evaluation of the gradient map of the component [[functions]{}]{} at the starting point. [[Here, we use three different values for the evaluated gradient maps]{}]{} at the starting point, i.e., the origin, to [[study]{}]{} the sensitivity of SAGA with respect to the initial conditions. Of these, in initial condition 3, we use the exact value of the gradient maps, while in initial condition 2, we perturb values of the gradient maps. This perturbation is increased in initial condition 1. Insights: From Figure \[tableSaga\], we observe that Algorithm \[algorithm:IR-S-BFGS\] competes well with SAGA. We discuss the comparisons as follows: (i) A computational burden in implementation of SAGA is the memory requirement of this scheme. Generally speaking, SAGA requires storing a matrix of $\mathcal{O}(Nn)$ at each iteration. Exceptions include the case where the objective function is in terms of a linear regression model function (e.g., in ). This is in contrast with Algorithm \[algorithm:IR-S-BFGS\] where the memory requirement is $\mathcal{O}(mn)$. (ii) As expected, the performance of SAGA deteriorates when the sample size increases. However, the performance of Algorithm \[algorithm:IR-S-BFGS\] seems to be more robust with respect to the increase in the sample size. (iii) The performance of SAGA seems to be moderately sensitive to the initial conditions. Comparison with IAG {#sec:num-IAG} ------------------- Recall that in solving finite-sum minimization problems with $\mu$-strongly convex objectives, using a constant stepsize, (non-averaging) IAG admits a linear convergence rate of $\mathcal{O}\left(\left(1-(\mu/N)^2\right)^{2k}\right)$ where $N$ is the number of component functions (cf. [@Iag17]). Accordingly, to do the numerical comparisons with IAG, we regularize problem with a constant $\mu_{IAG}>0$. Figure \[tableIAG\] shows the simulation results for different choices of $\mu_{IAG}$, $N$, IAG stepsize, and the initial stepzie and regularization parameter of Algorithm \[algorithm:IR-S-BFGS\]. Insights: (i) Due to the excessive memory requirements of $\mathcal{O}(nN)$ [[associated with]{}]{} IAG, [[such a scheme becomes challenging to implement when $n$ becomes large as in this case where]{}]{} $n=138,921$. Consequently, [[we use a sample size]{}]{} $N \in \{1000,2000,5000\}$. However, Algorithm \[algorithm:IR-S-BFGS\] only requires memory of $\mathcal{O}(nm)$, allowing for implementations with large values of $N$. (ii) Similar to SAGA, the performance of IAG is deteriorated when the sample size increases. However, the performance of Algorithm \[algorithm:IR-S-BFGS\] seems to be more robust with changes in the sample size. (iii) [[For]{}]{} each fixed value of $N$, despite the change in the value of $\mu_{IAG}$, the performance of IAG in terms of the true objective function in does not necessarily improve. [[Importantly, this observation suggests that in the standard regularization approach, tuning the regularization parameter could be computationally expensive.]{}]{} Concluding remarks {#sec:conc} ================== We consider stochastic quasi-Newton (SQN) methods for solving large scale stochastic optimization problems with smooth but unbounded gradients. Much of the past research on convergence rates of these algorithms relies on the strong convexity of the objective function. [[We employ an iterative regularization scheme where the regularization parameter is updated iteratively within the algorithm. We establish the convergence in an a.s. sense and a mean sense. Moreover, ]{}]{}we prove that the iterates generated by the [[iteratively]{}]{} regularized stochastic LBFGS scheme converges to an optimal solution at the rate $\mathcal{O}\left(\frac{1}{k^{1/3-\e}}\right)$ for arbitrary small $\e>0$. The deterministic variant of this algorithm achieves the rate $\mathcal{O}\left(\frac{1}{k^{1-\e}}\right)$. [[The numerical experiments performed on a large scale classification problem indicate that the proposed LBFGS scheme performs well compared to methods such as standard SQN schemes, and other first-order schemes such as SAGA and IAG]{}]{}. Appendix {#App} ======== Proof of Lemma \[sumProductBounds\] {#app:sumProductBounds} ----------------------------------- From $0<a_1 \leq \ldots \leq a_n$, we can write $$\begin{aligned} (n-(i-1))a_i \leq \sum_{j=1}^na_j, \qquad \hbox{for all } i \in \{1,\ldots,n\}.\end{aligned}$$ Invoking $\sum_{i=1}^n{a_i} \leq S$, we obtain $a_i \leq \frac{S}{n-(i-1)}$, for all $i \in \{1,\ldots,n\}$. From the preceding relation and that $\prod_{j=1}^na_j \geq P$, we can obtain $a_1 \geq (n-1)!P/S^{n-1}.$ Proof of Lemma \[LBFGS-matrix\] {#app:lemmaLBFGSmatrix} ------------------------------- [[Throughout, we let $\lambda_{k,\min}$, $\lambda_{k,\max}$, and $B_k$ denote the minimum eigenvalue, maximum eigenvalue, and inverse of matrix $H_k$ in , respectively.]{}]{} It can be seen, by induction on $k$, that [[$H_k$ is symmetric and $\sF_k$ measurable.]{}]{} We use induction on odd values of [[$k\geq 2m-1$ to show that parts (a), (b), and (c) hold.]{}]{} Suppose [[$k\geq 2m-1$]{}]{} is odd and for any [[odd]{}]{} $t<k$, we have [[$s_{\lceil t/2\rceil}^T{y_{\lceil t/2\rceil}} >0$, $H_{t}{y}_{\lceil t/2\rceil}=s_{\lceil t/2\rceil}$]{}]{}, and for $t$. We show that these statements also hold for $k$ [[as well]{}]{}. First, we [[show]{}]{} that the [[curvature condition]{}]{} holds. We can write $$\begin{aligned} {{\color{black}s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}}&=(x_{k}-x_{k-1})^T(\nabla F(x_k,\xi_{k-1})- \nabla F(x_{k-1},\xi_{k-1})+{{\color{black}\tau}}\mu_k^\delta(x_k-x_{k-1}))\\ &=(x_{k}-x_{k-1})^T(\nabla F(x_k,\xi_{k-1})- \nabla F(x_{k-1},\xi_{k-1}))+{{\color{black}\tau}}\mu_k^\delta\\|x_k-x_{k-1}\\|^2\\ &\geq {{\color{black}\tau}}\mu_k^\delta\\|x_k-x_{k-1}\\|^2,\end{aligned}$$ where the inequality follows from the monotonicity of the gradient map $\nabla F(\cdot,\xi)$. [[Next, we show that $\\|x_k-x_{k-1}\\|^2 >0$.]{}]{} From the induction hypothesis [[and that $k-2$ is odd]{}]{}, $H_{k-2}$ is positive definite. [[Moreover, from the update rule and that $k-2$ is odd, we have $H_{k-1}=H_{k-2}$.]{}]{} Therefore, $H_{k-1}$ is [[also]{}]{} positive definite. [[Without loss of generality, we assume $\nabla F(x_{k-1},\xi_{k-1})+\mu_{k-1}(x_{k-1}-x_0)\neq 0$]{}]{}[^5]. Since $H_{k-1}$ is positive definite, we have $$H_{k-1}\left(\nabla F(x_{k-1},\xi_{k-1})+\mu_{k-1}{{\color{black}(x_{k-1}-x_0)}}\right) \neq 0,$$ implying that $x_{k} \neq x_{k-1}$. Hence ${{\color{black}s_{\lceil k/2\rceil}^T{y_{\lceil k/2\rceil}} }}\geq {{\color{black}\tau}}\mu_k^\delta\\|x_k-x_{k-1}\\|^2 >0,$ where [[the second inequality is a consequence of]{}]{} ${{\color{black}\tau}},\mu_k>0$. Thus, the [[curvature condition]{}]{} holds. Next, we show that holds for $k$. It is well-known that using the Sherman-Morrison-Woodbury formula, $B_k$ is equal to $B_{k,m}$ given by $$\begin{aligned} \label{equ:B_kLimited} B_{k,j}=B_{k,j-1}-\frac{B_{k,j-1}s_is_i^TB_{k,j-1}}{s_i^TB_{k,j-1}s_i}+\frac{y_iy_i^T}{y_i^Ts_i}, \quad i:={{\color{black}{\lceil k/2\rceil}-(m-j)}}, \quad 1 \leq j \leq m,\end{aligned}$$ where $s_i$ and $y_i$ are defined by and [[$B_{k,0}=\frac{y_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}\mathbf{I}$]{}]{}. [[Note that with $j$ varying between $1$ to $m$, the index $i$ takes values in $\left\{\lceil k/2\rceil-m+1,\lceil k/2\rceil-m+2,\ldots,\lceil k/2\rceil\right\}$.]{}]{} First, we show that for any $i$ [[in this range]{}]{}, $$\begin{aligned} \label{equ:boundsForB0} {{\color{black}\tau}}\mu_k^\delta \leq \frac{\\|y_i\\|^2}{y_i^Ts_i} \leq L+{{\color{black}\tau}}\mu_k^\delta,\end{aligned}$$ where $L$ is the Lipschitzian parameter of the gradient mapping $\nabla F$ [[given by Assumption \[assum:convex2\](b)]{}]{}. Let us [[define the function $h(x)\triangleq F(x,\xi_{i-1})+\tau\frac{\mu_k^\delta}{2}\\|x-x_0\\|^2$]{}]{} for fixed $i$ and $k$. Note that this function is strongly convex and has a gradient mapping of the form $\nabla F+{{\color{black}\tau}}\mu_k^\delta(\mathbf{I}-x_0)$ that is Lipschitz with parameter $L+{{\color{black}\tau}}\mu_k^\delta$. For a convex function $h$ with Lipschitz gradient with parameter $L+{{\color{black}\tau}}\mu_k^\delta$, the following inequality, referred to as co-coercivity property, holds for any $x_1,x_2 \in \Real^n$ [[(see]{}]{}[@Polyak87], Pg. 24 , Lemma 2): $$\\|\nabla h(x_2)-\nabla h(x_1)\\|^2 \leq \left(L+{{\color{black}\tau}}\mu_k^\delta\right)(x_2-x_1)^T(\nabla h(x_2)-\nabla h(x_1)).$$ Substituting $x_2$ by $x_i$, $x_1$ by $x_{i-1}$, and recalling , the preceding inequality yields $$\begin{aligned} \label{ineq:boundsForB0-1}\\|y_i\\|^2 \leq \left(L+{{\color{black}\tau}}\mu_k^\delta\right)s_i^Ty_i.\end{aligned}$$ Note that function $h$ is strongly convex [[with]{}]{} parameter ${{\color{black}\tau}}\mu_k^\delta$. Applying the Cauchy-Schwarz inequality, we can write $$\frac{\\|y_i\\|^2}{s_i^Ty_i} \geq \frac{\\|y_i\\|^2}{\\|s_i\\|\\|y_i\\|} =\frac{\\|y_i\\|}{\\|s_i\\|}\geq \frac{\\|y_i\\|\\|s_i\\|}{\\|s_i\\|^2} \geq \frac{y_i^Ts_i}{\\|s_i\\|^2}\geq {{\color{black}\tau}}\mu_k^\delta.$$ Combining this relation with , we obtain . Next, we show that the maximum eigenvalue of $B_k$ is bounded. Let $Trace(\cdot)$ denote the trace of a matrix. Taking trace from both sides of and summing up over index $j$, we obtain [[for $i:={\lceil k/2\rceil}-(m-j)$,]{}]{} $$\begin{aligned} \label{ineq:trace} Trace(B_{k,m})&=Trace(B_{k,0})-\sum_{j=1}^m Trace\left(\frac{B_{k,j-1}s_is_i^TB_{k,j-1}}{s_i^TB_{k,j-1}s_i}\right)+\sum_{j=1}^m Trace\left(\frac{y_iy_i^T}{y_i^Ts_i}\right)\cr & =Trace{{\color{black}\left(\frac{\\|y_{\lceil k/2\rceil}\\|^2}{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}\mathbf{I}\right)}} - \sum_{j=1}^m \frac{\\|B_{k,j-1}s_i\\|^2}{s_i^TB_{k,j-1}s_i} + \sum_{j=1}^m \frac{\\|y_i\\|^2}{y_i^Ts_i}\cr &\leq n {{\color{black}\frac{\\|y_{\lceil k/2\rceil}\\|^2}{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}}} +\sum_{j=1}^m \left(L+{{\color{black}\tau}}\mu_k^\delta\right) {{\color{black}\leq}} (m+n)\left(L+{{\color{black}\tau}}\mu_k^\delta\right),\end{aligned}$$ where the third relation is obtained by positive-definiteness of $B_k$ (this can be seen by induction on [[$j$]{}]{}, and using and $B_{k,0}\succ 0$). Since $B_k=B_{k,m}$, the maximum eigenvalue of the matrix $B_k$ is bounded [[by $(m+n)\left(L+\tau\mu_k^\delta\right)$]{}]{}. As a result, $$\begin{aligned} \label{proof:lowerbound} \lambda_{k,\min}\geq \frac{1}{(m+n)\left(L+{{\color{black}\tau}}\mu_k^\delta\right)}.\end{aligned}$$ In the next part of the proof, we establish the bound for $\lambda_{k,\max}$. The following relation can be shown (e.g., see Lemma 3 in [@Mokhtari15]) $$\begin{aligned} det(B_{k,m})=det(B_{k,0})\prod_{j=1}^m\frac{s_i^Ty_i}{s_i^TB_{k,j-1}s_i}{{\color{black}, \quad \hbox{for }i:={\lceil k/2\rceil}-(m-j)}}.\end{aligned}$$ Multiplying and dividing by $s_i^Ts_i$, using the strong convexity of the function $h$, and invoking and the result of , we obtain $$\begin{aligned} \label{ineq:detBk} det(B_{k})&={{\color{black}det\left(\frac{y_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}\mathbf{I}\right)}}\prod_{j=1}^m\left(\frac{s_i^Ty_i}{s_i^Ts_i}\right)\left(\frac{s_i^Ts_i}{s_i^TB_{k,j-1}s_i}\right)\cr & \geq{{\color{black}\left(\frac{y_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}{s_{\lceil k/2\rceil}^Ty_{\lceil k/2\rceil}}\right)^n}}\prod_{j=1}^m{{\color{black}\tau}}\mu_k^\delta\left(\frac{s_i^Ts_i}{s_i^TB_{k,j-1}s_i}\right)\cr & \geq {{\color{black}\left(\tau\mu_k^\delta\right)^{(n+m)}}} \prod_{j=1}^m \frac{1}{(m+n)\left(L+{{\color{black}\tau}}\mu_k^\delta\right)} = \frac{{{\color{black}\left(\tau\mu_k^\delta\right)^{(n+m)}}}}{(m+n)^{m}\left(L+{{\color{black}\tau}}\mu_k^\delta\right)^m}.\end{aligned}$$ Let [[$\alpha_{k,1}\leq \alpha_{k,2}\leq\ldots\leq\alpha_{k,n}$]{}]{} be the eigenvalues of $B_k$ sorted non-decreasingly. Note that since $B_k\succ0$, all the eigenvalues are positive. Taking and into account, and employing Lemma \[sumProductBounds\], we obtain $$\alpha_{1,k} {{\color{black}\geq}} \frac{(n-1)!{{\color{black}\left(\tau\mu_k^\delta\right)^{(n+m)}}}}{(m+n)^{n+m-1}\left(L+{{\color{black}\tau}}\mu_k^\delta\right)^{n+m-1}}.$$ This relation and that $\alpha_{k,1}=\lambda_{k,\max}^{-1}$ imply that $$\begin{aligned} \label{proof:upperbound} \lambda_{k,\max}\leq \frac{(m+n)^{n+m-1}\left(L+{{\color{black}\tau}}\mu_k^\delta\right)^{n+m-1}}{(n-1)!{{\color{black}\left(\tau\mu_k^\delta\right)^{(n+m)}}}}.\end{aligned}$$ Therefore, from and and that $\mu_k$ is non-increasing, we conclude that holds for $k$ as well. Next, we show $H_ky_{\lceil k/2\rceil}=s_{\lceil k/2\rceil}$. From , for $j=m$ we obtain $$B_{k,m}=B_{k,m-1}-{{\color{black}\frac{B_{k,m-1}s_{\lceil k/2\rceil}s_{\lceil k/2\rceil}^TB_{k,m-1}}{s_{\lceil k/2\rceil}^TB_{k,m-1}s_{\lceil k/2\rceil}}+\frac{y_{\lceil k/2\rceil}y_{\lceil k/2\rceil}^T}{y_{\lceil k/2\rceil}^Ts_{\lceil k/2\rceil}}}},$$ where we used [[$i={\lceil k/2\rceil}-(m-m)={\lceil k/2\rceil}$]{}]{}. Multiplying both sides of the preceding equation by [[$s_{\lceil k/2\rceil}$]{}]{}, and using $B_k=B_{k,m}$, we have [[$B_{k}s_k=B_{k,m-1}s_{\lceil k/2\rceil}-B_{k,m-1}s_{\lceil k/2\rceil}+y_{\lceil k/2\rceil}=y_{\lceil k/2\rceil}$]{}]{}. Multiplying both sides of the preceding relation by $H_k$ and invoking $H_k=B_k^{-1}$, we conclude that [[$H_ky_{\lceil k/2\rceil}=s_{\lceil k/2\rceil}$]{}]{}. Therefore, we showed that the statements of (a), (b), and (c) hold for [[an odd $k$]{}]{}, assuming that they hold for any odd [[$t<k$]{}]{}. In a similar fashion to this analysis, it can be seen that the statements hold for [[$t=2m-1$]{}]{}. Thus, by induction, we conclude that the statements hold for any odd [[$k\geq 2m-1$]{}]{}. To complete the proof, it is enough to show that holds for any even [[$k\geq 2m$]{}]{}. Let $t=k-1$. Since [[$t$]{}]{} is odd, relation holds. Writing for $k-1$, and taking into account that $H_k=H_{k-1}$, and $\mu_k<\mu_{k-1}$, we can conclude that holds for any even [[$k\geq 2m-1$]{}]{} and this completes the proof. Proof of Lemma \[lemma:a.s.sequences\] {#app:feasibleSeqASconv} -------------------------------------- In the following, we show that the presented class of sequences satisfy each of the conditions listed in Assumption \[assum:sequences\]. Throughout, we let $\alpha$ denote [[$-(m+n)\delta$]{}]{}.\ (a) Replacing the sequences by their given rules, we obtain [[$$\begin{aligned} \g_k\mu_k^{2\alpha-1} & =\frac{\g_0}{(2^b\mu_0)^{1-2\alpha}}(k+1)^{-a}(k+\kappa)^{(1-2\alpha) b}\leq \frac{\g_0}{(2^b\mu_0)^{1-2\alpha}}(k+1)^{-a+(1-2\alpha) b}.\end{aligned}$$]{}]{} [[From the assumption that $\frac{a}{b}>1+2\delta(m+n)$, we obtain$-a+(1-2\alpha) b<0$]{}]{}. Thus, the preceding term goes to zero verifying Assumption \[assum:sequences\](a).\ (b) Let $k$ be an even number. Thus, $\kappa=2$. From we have $\mu_{k}=\mu_{k+1}=\frac{\mu_02^b}{\left(k+2\right)^b}$. Now, let $k$ be an odd number. Again, according to can write $$\mu_{k+1}=\frac{\mu_02^b}{\left((k+1)+2\right)^b }<\frac{\mu_02^b}{\left(k+1\right)^b}= \frac{\mu_02^b}{\left(k+\kappa\right)^b}=\mu_k.$$ Therefore, $\mu_k$ given by satisfies . Also, from we have $\mu_k \to 0$. Thus, Assumption \[assum:sequences\](b) holds.\ (c) The given rules imply that $\g_k$ and $\mu_k$ are both non-increasing sequences. Therefore, we have $\g_k\mu_{k} \leq \g_0\mu_{0}$ for any $k\geq 0$. So, to show that Assumption \[assum:sequences\](c) holds, it is enough to show that $\lambda_{\min}\g_0\mu_{0} \leq 1$ where $\lambda_{\min}$ is given by . Since we assumed that $\g_0\mu_0 \leq L(m+n)$, for any $\delta \in (0,1]$, we have $\g_0\mu_0 \leq (m+n)(L+\mu_0^\delta)$, implying that $\lambda_{\min}\g_0\mu_{0} \leq 1$ and that Assumption \[assum:sequences\](c) holds.\ (d) From , we can write $$\sum_{k=0}^\infty \g_k\mu_k =\g_0\mu_02^b\sum_{k=0}^\infty (k+1)^{-a}(k+\kappa)^{-b} \geq \g_0\mu_02^b\sum_{k=0}^\infty (k+2)^{-(a+b)}= \infty,$$ where the last relation is due to $a+b\leq 1$. Therefore, Assumption \[assum:sequences\](d) holds.\ (e) Using , it follows [[$$\begin{aligned} &\sum_{k=0}^\infty \g_k\mu_k^2 = \g_0\mu_0^24^b\sum_{k=0}^\infty (k+1)^{-a}(k+\kappa)^{-2b}\leq \g_0\mu_0^24^b\sum_{k=0}^\infty (k+1)^{-(a+2b)}<\infty,\end{aligned}$$]{}]{} where the last inequality is due to [[$a+2b>1$]{}]{}. Therefore, Assumption \[assum:sequences\](e) holds.\ (f) From , we have $$\begin{aligned} &\sum_{k=0}^\infty \g_k^2\mu_k^{2\alpha} \leq \g_0^2(\mu_02^b)^{2\alpha}\left(\sum_{k=0}^1 \frac{(k+\kappa)^{-2\alpha b}}{(k+1)^{2a}}+\sum_{k=2}^\infty \frac{(2k)^{-2\alpha b}}{k^{2a}}\right)<\infty $$ where in the first inequality, we use $\alpha <0$ and in the last inequality, we note that $a+\alpha b=a-\delta\left(m+n\right)b>0.5$. Therefore, Assumption \[assum:sequences\](f) is verified. Proof of Lemma \[lemma:mean-sequences\] {#app:feasibleSeqMeanConv} --------------------------------------- [[Throughout, we let $\alpha$ denote $-\delta(m+n)$.]{}]{} [[Assumption \[assum:sequences-ms-convergence\](a, b, c) and have been already shown in parts (a, b, c) of the proof of Lemma \[lemma:a.s.sequences\].]{}]{}\ (d) It suffices to show there exists [[$K_0$]{}]{} such that for any $k \geq K_0$ and $0<\beta<1$, $$\begin{aligned} \label{ineq:partd}\frac{\g_{k-1}}{\g_k}\frac{\mu_{k}^{1-2\alpha}}{\mu_{k-1}^{1-2\alpha}}-1\leq \beta \lambda_{\min}\g_k\mu_k.\end{aligned}$$ From [[and the definition of $\alpha$]{}]{}, we obtain $$\begin{aligned} &\frac{\g_{k-1}}{\g_k}\frac{\mu_{k}^{1-2\alpha}}{\mu_{k-1}^{1-2\alpha}}-1 \leq \frac{\g_{k-1}}{\g_k}-1=\left(1+\frac{1}{k}\right)^a-1= 1+\frac{a}{k}+{{\color{black}{o}\left(\frac{1}{k}\right)}}-1=\mathcal{O}\left(\frac{1}{k}\right),\end{aligned}$$ where the first inequality is implied due to $\{\mu_k\}$ is non-increasing, and in the second equation, we used the Taylor’s expansion of $\left(1+\frac{1}{k}\right)^a$. Therefore, since the right-hand side of is of the order $\frac{1}{k^{a+b}}$ and that $a+b<1$, the preceding inequality shows that such $K_0$ exists for which Assumption \[assum:sequences-ms-convergence\](d) holds for all $0<\beta<1$.\ (e) From , we have $$\begin{aligned} \frac{\mu_k^{2-2\alpha}}{\g_k}& ={{\color{black}\g_0}}^{-1}\left(\mu_02^b\right)^{2-2\alpha}(k+\kappa)^{-b(2-2\alpha)}(k+1)^{a} \leq\frac{{{\color{black}\gamma_0}}^{-1}\left(\mu_02^b\right)^{2-2\alpha}}{(k+1)^{-a+(2-2\alpha)b}}\\&\leq {{\color{black}\gamma_0}}^{-1}\left(\mu_02^b\right)^{2-2\alpha}=\rho, \end{aligned}$$ where the first inequality is due to [[$\alpha <0$, and the second inequality follows by the assumption $a\leq 2b(1+\delta(m+n))$]{}]{}. Therefore, Assumption \[assum:sequences-ms-convergence\](e) is satisfied. [^1]: School of Industrial Engineering & Management, Oklahoma State University, Stillwater, OK 74078, USA (, <https://sites.google.com/site/farzad1yousefian>). [^2]: School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA (, <https://ecee.engineering.asu.edu/project/angelia-nedich>). Nedić gratefully acknowledges the support of the NSF through grant CCF-1717391. [^3]: Industrial & Manufacturing Engineering, Pennsylvania State University, University Park, State College, PA 16802, USA (, <http://www.personal.psu.edu/vvs3>). [^4]: A preliminary version of this paper appeared in the IEEE $55^\text{th}$ Conference on Decision and Control (see reference [@FarzadCDC16]). [^5]: If $\nabla F(x_{k},\xi_k)+\mu_{k}(x_k-x_0)=0$[[,]{}]{} then we can draw a new sample of $\xi_k$ to satisfy the relation.
--- abstract: 'The Doppler tracking data of the Chang’e 3 lunar mission is used to constrain the stochastic background of gravitational wave in cosmology within the 1 mHz to 0.05 Hz frequency band. Our result improves on the upper bound on the energy density of the stochastic background of gravitational wave in the 0.02 Hz to 0.05 Hz band obtained by the Apollo missions, with the improvement reaching almost one order of magnitude at around 0.05 Hz. Detailed noise analysis of the Doppler tracking data is also presented, with the prospect that these noise sources will be mitigated in future Chinese deep space missions. A feasibility study is also undertaken to understand the scientific capability of the Chang’e 4 mission, due to be launched in 2018, in relation to the stochastic gravitational wave background around 0.01 Hz. The study indicates that the upper bound on the energy density may be further improved by another order of magnitude from the Chang’e 3 mission, which will fill the gap in the frequency band from 0.02 Hz to 0.1 Hz in the foreseeable future.' address: - 'Science and Technology on Aerospace Flight Dynamics Laboratory, Beijing Aerospace Control Center, Beijing, China.' - 'Institute of Applied Mathematics, Morningside Center of Mathematics and LESC, Institute of Computational Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, China.' - 'Department of Aerospace Guidance Navigation and Control, School of Astronautics, Beihang University, Beijing, China.' - 'Beijing Institute of Aerospace Control Devices, Beijing, China.' - 'National astronomical observatories, Chinese Academy of Sciences, Beijing, China.' - 'University of Chinese Academy of Sciences, Beijing, China.' author: - Wenlin Tang - Peng Xu - Songjie Hu - Jianfeng Cao - Peng Dong - Yanlong Bu - Lue Chen - Songtao Han - Xuefei Gong - Wenxiao Li - Jinsong Ping - 'Yun-Kau Lau' - Geshi Tang title: 'Chang’e 3 lunar mission and upper limit on stochastic background of gravitational wave around the 0.01 Hz band' --- Chang’e lunar mission; Doppler tracking data; stochastic background of gravitational waves. Introduction {#Introduction} ============ Chang’e 3 is an unmanned lunar exploration mission operated by the China National Space Administration. As part of the second phase of the Chinese Lunar Exploration Program, it was landed on the Moon on 14 December 2013, becoming the first spacecraft to soft-land on the Moon since the Soviet Union’s Luna 24 in 1976. At present it is located on the Lunar surface at about $44.12^{\circ}$ N, $19.51^{\circ}$ W and -2640 m in elevation(Cao et al., 2014; Ping, 2014). The tracking of the lander by the Chinese deep space network is still ongoing. Every day the lander is tracked continuously for about two to four hours by two ground stations located at Kashi and Jiamusi within China by means of X band radio waves (uplink and downlink at 8.47 GHz). For the Jiamusi station, the two-way Doppler tracking can reach the measurement accuracy of about 0.2mm/s, with sampling time of one second. The high precision Doppler tracking data of Chang’e 3 encodes information concerning the dynamics of the motion of the Moon relative to the Earth and it is worth understanding better whether it is feasible to extract useful science from the data. As a starting point of our investigation in this direction, the present work aims to understand possible upper bound on the isotropic stochastic background of gravitational waves (SBGWs) imposed by the Chang’e 3 Doppler tracking data. The stochastic background is of cosmological significance as it contains a component of primordial gravitational waves generated during the beginning stage of our Universe(Maggiore, 2000; Sathyaprakash et al., 2009). The structure of the Chang’e 3 data suggests that the frequency window around 0.01 Hz would be the appropriate window to be looked at. Further, with Chang’e 4 to be launched at around 2018 and the prospect of deep space exploration beyond the Earth-Moon system after Chang’e 4, it is anticipated that more Doppler tracking data with higher precision will be available to the scientific community in future, the present work then also serves the dual purpose of being a pilot study of Doppler tracking data analysis for future Chinese deep space missions. With this prospect in mind, we undertake meticulous noise analysis for the Chang’e 3 Doppler tracking data, in order to understand the prospect of mitigating various noise sources in the Doppler tracking data in future deep space Chinese missions. In addition, a feasibility study is also undertaken to understand the scientific potential of the Chang’e 4 mission in relation to the stochastic gravitational wave background. Currently, over the band from $10^{-6}$ Hz up to 1 Hz, the Cassini spacecraft(Armstrong et al., 2003) gives the best constraint on the energy density ( $\Omega_{gw}(f)$ ) of the SBGWs from $1.2\times10^{-6}$ Hz up to $10^{-4}$ Hz, the ULYSSES spacecraft(Bertotti et al., 1995) together with the normal modes of the Earth(Coughlin et al., 2014a) give the best constraint on $\Omega_{gw}(f)$ of the SBGWs from $2.3\times10^{-4}$ Hz up to 0.02 Hz, the Apollo missions(Aoyama et al., 2014) gives the best constraint on $\Omega_{gw}(f)$ from 0.02 Hz up to 0.05 Hz, the Earth’s seismic data(Coughlin et al., 2014b) gives the best constraint on $\Omega_{gw}(f)$ from 0.05 Hz up to 0.1 Hz, and the Lunar seismic data(Coughlin et al., 2014c) gives the best constraint on $\Omega_{gw}(f)$ from 0.1 Hz up to 1 Hz. See Figure 10 for an illustration of the sensitivity limits obtained by previous missions or other detection methods in different frequency windows. Upon comparison, we find that the constraint on $\Omega_{gw}(f)$ imposed by the Apollo missions from 0.02 Hz up to 0.1 Hz is by far worse than others in the frequency band ranges from $10^{-4}$ Hz to 1 Hz. As the frequency window of Chang’e 3 overlaps with that of the Apollo missions and at the same time the measurement accuracy of the Doppler tracking data of Chang’e 3 is better than that of the Apollo missions, it is not surprising to obtain that the upper bound on $\Omega_{gw}(f)$ imposed by Chang’e 3 improves on that given by the Apollo missions. The outline of this paper is as follows. Section 2 describes the algorithm for the data analysis and we work out the power spectral density of the noise in the measured Doppler tracking data of Chang’e 3. Section 3 presents the main results concerning the upper bound on the energy density $\Omega_{gw}(f)$ of the SBGWs in the band from 1 mHz to 0.05 Hz by using the Doppler tracking data of Chang’e 3. A detailed noise analysis of the Doppler tracking data of the Chang’e 3 mission is given in Section 4. Section 5 presents the feasibility study of constraining the SBGWs in the frequency band from 1 mHz to 0.1 Hz using the future Doppler tracking data of Chang’e 4. Some brief remarks are then made to conclude this paper in the final section. Data analysis {#Sec-dataandanalysis} ============= The Doppler tracking data used is the two-way range rate data recorded at the Jiamusi station, it is taken from UTC 13:22:25.0 to 15:01:49.0 on 17 December 2013. The time series has sampling time of 1 s and it composes of 6000 data points, with the standard deviation about 0.2 mm/s, as shown in Fig. \[Fig-omc532023202\_56643\_RRate\_obs\]. The round-trip time between the lander and the station is about 2.65 s. The Sun-Earth-Moon angle was about 180 degrees and therefore the lander was in the solar opposite direction. As we shall see later, the Sun-Earth-Moon angle is an important factor when it comes to the estimate of the tropospheric and ionospheric delay noises in the Doppler tracking. -- -- -- -- The algorithm for data analysis is illustrated in the flow chart displayed in Fig. \[Fig-CE3\_DataProc\_FlowChart\]. In this work, we will focus on constraining the SBGWs in the frequency band from 1 mHz to 0.05 Hz using the Doppler tacking data of Chang’e 3. According to the Nyquist criterion, the data used to constrain the SBGWs must have a sampling interval smaller than 10 s. Further, the random fluctuations in a time series can be suppressed by smoothing it to a new time series with longer sampling interval. To this end, we smooth the Doppler tracking data to generate a new time series with sampling time 9 s. Then we evaluate the residual of the new time series and estimate its power spectral density in the band from 1 mHz to 0.05 Hz. Finally, we give the upper bound on the energy density $\Omega_{gw}$ of the SBGWs in the frequency band from 1 mHz to 0.05 Hz. ---------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-CE3\_DataProc\_FlowChart\] The flow chart for the data analysis. ](CE3_DataProc_FlowChart.eps "fig:"){width="40.00000%"} ---------------------------------------------------------------------------------------------------------------------------------- In the Chang’e 3 mission, the radio signal is transmitted from the station at time $t_1$ to the lander at time $t_2$ and then reflected from the lander at time $t_2$ to the station at time $t_3$. All dates $t_1$, $t_2$ and $t_3$ are in Universal Time Coordinated (UTC). For the Doppler tracking, one observation consists in $t_3$, the date of the radio signal arrived at the station in Universal Time Coordinated (UTC), and $v_o$, the two-way range rate, which is obtained by $v_o=c\cdot \triangle f_o$, where $\triangle f_o$ is the relative variation of the frequency of the radio signal. The other data supplied are the frequency of the radio signal, the signal/noise rate, and the temperature, pressure and humidity of the atmosphere. Denote by $\{v(t_i)\}_{i=1}^n$ the range rate data, where n is its length. The range rate is smoothed to generate a new time series $\{\bar{v}(s_i)\}_{i=1}^m$ with sampling time 9 s, where $m$ is the largest integer smaller than $n/9$ and $s_i = t_{1+9(i-1)}$. Denote by $\{\epsilon(s_i)\}_{i=1}^m$ the residuals in the time series $\{\bar{v}(s_i)\}_{i=1}^m$ defined by $$\label{Eq-Residual_9sInterval} \epsilon(s_i) \equiv \bar{v}(s_i) - v_c(s_i), \quad i = 1,2,\cdots,m,$$ where $v_c(t)$ is the theoretical value of the two-way range rate at epoch $t$, which is evaluated as $$\label{Calculated_TwoWayRangeRate} v_c(t) = \frac{\rho^{2w}(t) - \rho^{2w}(t-\triangle t)}{\triangle t}$$ from the two-way range $\rho^{2w}$ by difference, where in our calculations, $\triangle t = 1$s. The calculation of the two-way range in Chang’e 3 is introduced as follows. We show in Fig. \[Fig-ChangE3measure\] the schematic diagram of the lunar radio ranging of Chang’e 3. Compare with the lunar laser ranging (LLR) measurement(Chapront et al., 2006), we find that the role of the lander is similar as that of the reflectors in the LLR. Thus we may follow the same procedure(Chapront et al., 2006) in calculating the round-trip time of light in LLR to calculate the two-way range in Chang’e 3. The theoretical value of the two-way range can be evaluated by $\rho^{2w}=c \cdot \triangle t$ from the duration $\triangle t$ of the round trip travel of the radio signal in atomic time (TAI, Temps Atomique International), where $c$ is the speed of light. To model the duration $\triangle t$, we need to take into account the relativistic curvature of the signal, and the influence of the troposphere and ionosphere. The computation of the duration $\triangle t$ should be given in the frame of General Relativity theory, and will depend on the barycentric positions of $T$, $L$ and $S$, respectively the center of the mass of the Earth, of the Moon and of the Sun. Thus in the calculation all coordinated are in the celestial barycentric reference system. The theoretical value of $\triangle t$ is given by(Chapront et al., 2006) $$\label{Calculated_Duration} \triangle t = [t_3 - \triangle T_1(t_3)] - [t_1 - \triangle T_1(t_1)],$$ which is the same as that used in the calculation of the round trip time of the light, where $\triangle T_1$ is the relativistic correction on the time scale, it transforms the time in Barycentric Dynamical Time (TDB) to that in TAI, whose calculation can see the reference(Soffel et al., 2013). However, since the observation data contains only the date $t_3$, $t_1$ (or $\triangle t$) should be given by iteration from $t_3$ and the ephemerids of the Earth, the Moon and the Sun, such as $$\begin{aligned} % \nonumber to remove numbering (before each equation) t_3 &=& t_2 + \frac{1}{c} \|\textbf{BR}(t_2)-\textbf{BO}(t_3)\|+\triangle T_{grav}+\triangle T_{trop}+\triangle T_{iono},\\ t_2 &=& t_1 + \frac{1}{c} \|\textbf{BR}(t_2)-\textbf{BO}(t_1)\|+\triangle T_{grav}+\triangle T_{trop}+\triangle T_{iono},\end{aligned}$$ where $B$ is the barycenter of the solar system, $\textbf{BR}$ and $\textbf{BO}$ are respectively the coordinates of the lander and the station with respect to $B$, $\triangle T_{grav}$ is the time contribution due to the gravitational curvature of the signal, $\triangle T_{trop}$ is the atmospheric delay and the $\triangle T_{iono}$ is the ionospheric delay. The vectors $\textbf{BR}$ and $\textbf{BO}$ are given by(Chapront et al., 2006) $$\textbf{BR}(t) = \textbf{BG}(t)+\frac{m_T}{m_T+m_L}\textbf{TL}(t)+\textbf{LR}(t)$$ and $$\textbf{BO}(t) = \textbf{BG}(t)-\frac{m_L}{m_T+m_L}\textbf{TL}(t)+\textbf{TO}(t),$$ where $m_T$ and $m_L$ are respectively the Earth and Moon masses, $\textbf{BG}(t)$ is the coordinates of the Earth-Moon barycenter $G$ with respect to $B$, $\textbf{TL}(t)$ is the position vector from the center of mass of the Earth to that of the Moon, $\textbf{LR}(t)$ is the position vector from the center of mass of the Moon to the lander, and $\textbf{TO}(t)$ is the position vector from the center of mass of the Earth to the station. In our paper, the vectors $\textbf{BG}(t)$ and $\textbf{TL}(t)$ are provided by the JPL planetary ephemeris DE421. The vector $\textbf{LR}$ is dependent of the lander coordinates, which are $(1173217.870, -416319.429, 1208153.007) (m)$(Wagner et al., 2014) in a selenocentric frame defined by the principal axes of inertia of the Moon. The coordinates of the lander should be transformed from the selenocentric frame to the celestial barycentric reference system(Chapront et al., 2006). The vector $\textbf{TO}$ are primarily defined in the International Terrestrial Reference Frame (ITRF) and are subject to various corrections due to the Earth deformations: terrestrial and oceanic tides and pressure anomaly. The transformation from the ITRF to the celestial barycentric reference system involves the Earth rotation parameters, the precession, the nutation, the obliquity $\epsilon$ in J2000.0 and the arc $\phi$ separating the inertial equinox and the origin of the right ascensions on the equator of J2000.0(Chapront et al., 2006). The correction $\triangle T_{grav}$ due to the gravitations of the Moon, Earth and Sun can be given according to the one-body light time equation(Moyer, 2003). The tropospheric delay $\triangle T_{trop}$ and ionospheric delay $\triangle T_{iono}$ on the radio signal are evaluated according to the tropospheric model and ionospheric model recommended by IERS convention No.36(Petit et al., 2010). To calculate the ionospheric delay $\triangle T_{iono}$, the total electron content (TEC) used is provided by International GNSS Service (IGS) associate analysis centers (see http://cddis.nasa.gov). ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![\[Fig-ChangE3measure\] The Lunar Radio Ranging in the Chang’e 3 mission. It measures the distance between the station on the Earth and the lander on the Moon. This measurement principle is the same as that of the Lunar Laser Ranging, which measures the distance between the station on the Earth and the reflectors on the Moon (Williams et al., 2009).](ChangE3measure2.eps "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ When we obtain the theoretical value of the two-way range, the theoretical value of the two-way range rate can be obtained from the Eq. (\[Calculated\_TwoWayRangeRate\]). The final residual is shown in Fig. \[Fig-omc532023202\_56643\_vel\_Chebfit\_9sInterval\]. Its root mean square is about $5.92\times 10^{-5}$ m/s, which is consistent with the measurement accuracy of the original two-way range rate data in Chang’e 3. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_vel\_Chebfit\_9sInterval\] The residual of the time series $\{\bar{v}(s_i)\}_{i=1}^m$ used to constrain the SBGWs in the frequency band from 1 mHz to 0.05 Hz. Its mean is zero and its root mean square is about $5.92\times 10^{-5}$ m/s.](omc532023202_56643_vel_Chebfit_9sInterval_v2.eps "fig:"){width="42.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The power spectral density of the residual is estimated as follows. Our approach is to fit the residual by a standard autoregressive moving average (ARMV) process(Peter et al., 1991). Then estimate the power spectral density of the residual by that of the fitted process. The sample autocorrelation coefficients of the residual are evaluated as shown in Fig. \[Fig-omc532023202\_56643\_vel\_autocorr\_II\_9sInterval\]. From this figure, we find that for all nonzero time lags only $\rho(1) = -0.1639$ exceed the bounds given by the two blue lines in the figure, where the two lines represent the 2$\sigma$ uncertainties if we estimate the autocorrelation coefficients of the residual by the sample autocorrelation coefficients. Here $\sigma$ is about 0.03. Thus we may fit the residual with the standard moving average processing MV(1) model(Peter et al., 1991): $$\label{MV1Model} X_t = Z_t + \theta Z_{t - 1},$$ where $\theta$ is a constant, and $Z_t$ is a Gaussian process with mean zero and variance $\sigma_z^2$. The values of $\theta$ and $\sigma_z$ can be estimated from the root mean square and the sample correlation coefficient $\rho(1)$ of the residual as $$\begin{aligned} % \nonumber to remove numbering (before each equation) \theta &=& -0.1639, \label{MV1ThetaEstVal} \\ \sigma_z &=& 5.84 \times 10^{-5}. \label{MV1SigmaEstVal}\end{aligned}$$ Therefore the noise in the new range rate data is estimated by the MV(1) process (\[MV1Model\]), where the parameter $\theta$ and the root mean square $\sigma_z$ of the Gauss process $Z_t$ are given by Eq. (\[MV1ThetaEstVal\]) and Eq. (\[MV1SigmaEstVal\]) respectively. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_vel\_autocorr\_II\_9sInterval\] The sample autocorrelation coefficients of the residual. The lower and upper bounds( the two blue lines) in the figure represent the 2$\sigma$ bounds of the estimated autocorrelation coefficients of the residual. The residual of the range rate data measured by the Deep Space Station at Jiamusi. The data is obtained at modified Julian date 56643.](omc532023202_56643_vel_autocorr_II_9sInterval.eps "fig:"){width="42.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- For the MV(1) process given in Eq. (\[MV1Model\]), from its spectral density(Peter et al., 1991) $$\label{PsdOfMV1Process} S(f) = \frac{\sigma_z^2}{2\pi}[1 + 2\theta \cos(2\pi f) + \theta^2],$$ we may estimate the power spectral density of the noise in the new range rate in the band from 0.1 mHz to 0.05 Hz to be $$\label{PsdOfNoiseInRR} S_{vel}(f) = 5.528\times 10^{-10}\cdot\left[1 - 0.3278\cos(2\pi f) + 0.0269\right].$$ Since the Doppler shift $\triangle f / f$ is obtained from the two-way range rate $v^{2w}$ by $\triangle f / f = v^{2w}/c $, the power spectral density of the noise in the Doppler shift in the band from 0.1 mHz to 0.05 Hz is estimated from Eq. (\[PsdOfMV1Process\]) as $$\label{PsdOfNoiseInDS} S_{DS}(f) = 6.039\times 10^{-27}\cdot\left[1 - 0.3278\cos(2\pi f) + 0.0269\right],$$ and it is shown in Fig. \[Fig-omc532023202\_56643\_DS\_PSD\_9sInterval\]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_DS\_PSD\_9sInterval\] The power spectral density of the noise in Doppler shift. The data is obtained at modified Julian date 56643 and at the deep space station at Jiamusi.](omc532023202_56643_DS_PSD_9sInterval.eps "fig:"){width="42.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Stochastic background of gravitational waves’ upper bound in the 0.02 to 0.05 Hz band {#sec_SBGWUB} ===================================================================================== Using spacecraft Doppler tracking to detect gravitational waves was first proposed by Estabrook and Wahlquist in 1975(Estabrook et al., 1975). Let $S^{gw}_{y2}(f)$ be the power spectrum of the two-way fractional Doppler fluctuations generated by the isotropic gravitational wave background. It is related to the power spectrum $S_h(f)$ of the stochastic background of gravitational waves by $S^{gw}_{y_2}(f) = \bar{R}_2(f)S_h(f)$, where $\bar{R}_2(f)$ is the transfer function. For each Fourier component of the SBGWs, the transfer function is given as(Estabrook et al., 1975) $$\begin{aligned} \label{RespFuncFreqSBGW_EandW} % \nonumber to remove numbering (before each equation) \bar{R}_2(f) &=& 1 - \frac{1}{3}\cos(2\pi f T_2) -\frac{(3 + \cos(2\pi f T_2))}{(\pi f T_2)^2}\nonumber\\ &&+ \frac{2\sin(2\pi f T_2) }{(\pi f T_2)^3},\end{aligned}$$ where $T_2$ is the round-trip time of the radio signal transmitted from the station to the lander and then back to the station. Input the mean round-trip time $T_2 = 2.65$ s into the above transfer function and the result is shown in Fig. \[Fig-TransferFunShow1e-421e-1\]. ----------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-TransferFunShow1e-421e-1\] Transfer function of Doppler shift w.r.t SBGW when $T_2 = 2.65 $ s.](TransferFunShow1e-421e-1.eps "fig:"){width="42.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------- The spectrum of the isotropic SBGWs can be characterized by its dimensionless energy density $\Omega_{gw}(f)$, or the characteristic rms strain $S_h(f)$ of the wave, which are defined respectively by(Maggiore, 2000; Armstrong et al., 2003) $$\label{rms_strain} S_h(f) = \frac{S^{gw}_{y_2}(f)}{\bar{R}_2(f)},$$ and $$\label{ENergydensityOmega} \Omega_{gw}(f) = \frac{8\pi^2 f^3}{3H_0^2}S_h(f),$$ where the $H_0$ denotes the Hubble constant. At this level of measurement precision of the Doppler tracking data of the Chang’e 3 mission, the fluctuation power generated by the stochastic background of gravitational waves is expected to be submerged under the observed fluctuation power, namely, $$\label{UpperBoundOnS_gw} S^{gw}_{y2}(f) \leq S_{DS}(f).$$ From Eqs. (\[rms\_strain\]) and (\[ENergydensityOmega\]), this implies that $$\label{UpperBoundOnS_h} S_h(f) \leq \frac{S_{DS}(f)}{\bar{R}_2(f)}$$ and $$\label{UpperBoundOnOmega_gw} \Omega_{gw}(f) \leq \frac{8\pi^2 f^3}{3H_0^2}\cdot\frac{S_{DS}(f)}{\bar{R}_2(f)}.$$ From the power spectral density $S_{DS}(f)$ of the noise in the Doppler shift given in Eq. (\[PsdOfNoiseInDS\]), the upper bounds on the characteristic rms strain $S_h(f)$ and the dimensionless energy density $\Omega_{gw}(f)$ are worked out and shown respectively in Fig. \[Fig-omc532023202\_56643\_UBonSh\_9sInterval\] and Fig. \[Fig-omc532023202\_56643\_UBonOmega\_9sInterval\]. For the dimensionless energy density $\Omega_{gw}(f)$, the results give the upper bound on the $\Omega_{gw}$ ranging from $6.04 \times 10^5\cdot h_{75}^{-2}$ at 1 mHz to $ 3.24\times 10^7\cdot h_{75}^{-2}$ at 0.05Hz, where $h_{75}$ is the Hubble constant in units of $75 km\cdot s^{-1}\cdot Mpc^{-1}$. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_UBonSh\_9sInterval\] The upper bound on the characteristic rms strain $S_h(f)$ of the stochastic background of gravitational wave in the frequency band from 1 mHz to 0.05 Hz.](omc532023202_56643_UBonSh_9sInterval_EandW.eps "fig:"){width="42.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_UBonOmega\_9sInterval\] The upper bound on the energy density $\Omega_{gw}$ of the stochastic background of gravitational wave in the frequency band from 1 mHz to 0.05 Hz.](omc532023202_56643_UBonOmega_9sInterval_EandW.eps "fig:"){width="42.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Comparing (see Fig. \[Fig-omc532023202\_56643\_UBonOmega\_Comparing\_I\_9sInterval\]) the upper bound of the energy density $\Omega_{gw}$ obtained from Chang’e 3 with those given by other missions in the low frequency band from 0.1 mHz to 1 Hz, we find that Chang’e 3 gives the best upper bound in the frequency band from 0.02 to 0.05 Hz. This improves the results given by the Apollo missions(Aoyama et al., 2014), with almost one order of improvement at around 0.05 Hz. Further, at about 0.05 Hz, our result is slightly better than that given by the Earth’s seismic data(Coughlin et al., 2014b). ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-omc532023202\_56643\_UBonOmega\_Comparing\_I\_9sInterval\]Summary of the upper bounds on the energy density of SBGWs from $10^{-6}$Hz to $10^4$ Hz band. It includes the results obtained from the Cassini spacecraft(Armstrong et al., 2003), the ULYSSES spacecraft(Bertotti et al., 1995), the normal modes of the Earth(Coughlin et al., 2014a), the Apollo missions(Aoyama et al., 2014), the Earth’s seismic data(Coughlin et al., 2014b), the Lunar seismic data(Coughlin et al., 2014c), the torsion-bar antenna(Shoda et al., 2013), the LIGO mission(Aasi et al., 2014)(corresponds to four lines) and the Chang’e 3 mission (red line). ](omc532023202_56643_UBonOmega_9sInterval.eps "fig:"){width="42.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Noise analysis ============== In this section we will carry out detailed noise analysis of Chang’e 3. It should be remarked that in the Chang’e 3 mission, the Doppler tracking data was originally used for the determination of the position of the lander on the lunar surface. Thus no detailed calibrations on individual noise source have been made. Only a total noise budget for the Doppler tracking data was considered. What we do is to estimate these individual noise sources indirectly through some auxiliary data, together with the characteristics of the noises and some previous noise analysis of the Doppler tracking data in other missions, such as the Cassini mission. The expectation is that the detailed noise analysis will help us understand the prospect of mitigating various noise sources in the Doppler tracking data in the upcoming Chang’e 4 mission and future Chinese deep space missions. The main noises in the Doppler tracking in Chang’e 3 contain the propagation noises and the instrumental noises, as listed in Table \[Tab-CE3\_Noise\_Analysis\]. These noise terms include the tropospheric noise $y^{trop}$ and the ionospheric noise $y^{iono}$ due to the phase scintillation when the radio signal propagates through the neutral atmosphere and the ionosphere; the antenna noise $y^{ant}$ due to the antenna mechanical motion; the clock noise $y^{FTS}(t)$ due to the instability of the frequency standard; the unmodeled mechanical motion of the lander $y^{ldr}$; the transponder noise $y^{trans}$; the thermal noise $y^{ther}$ in the receiver due to the finite signal-to-noise ratio on the downlink and the ground electrical noise $y^{groundelec}$. These noises enter into the two-way fractional Doppler $y_2(t)$ in a way given by $$\begin{aligned} \label{CE-3NoiseTransfer} % \nonumber to remove numbering (before each equation) y_2(t) &=& y_2^{gw}(t) + y^{trop}(t) + y^{trop}(t - T_2) + y^{iono}(t) + y^{iono}(t - T_2)\nonumber\\ && + y^{FTS}(t) - y^{FTS}(t - T_2) + y^{ant}(t) + y^{ant}(t - T_2) + y^{ldr}\left(t - \frac{T_2}{2} \right) \nonumber \\ && + y^{trans}\left(t - \frac{T_2}{2} \right) + y^{ther}(t) + y^{groundelec}(t).\end{aligned}$$ For any noise, its statistics can be given by the Allan deviation $\sigma_y(\tau)$(Barnes et al., 1971), where $\tau$ is the integration time. The Allan deviation may be evaluated from the spectrum $S_y(f)$ of the noise by(Barnes et al., 1971) $\sigma_y^2(\tau)=\int_{-\infty}^{\infty}df 2S_y(f)\sin^4(\pi f\tau)/(\pi f\tau)^2$. -------------------- -------------------------------------------------- Noise Allan deviation $\sigma_y$ ($\tau = 9$s) Clock Estimated to be smaller than $1.0\times10^{-13}$ Thermal Estimated to be smaller than $5.5\times10^{-15}$ Troposphere Estimated to be smaller than $5.0\times10^{-14}$ Ionosphere Estimated to be smaller than $9.1\times10^{-15}$ Lander Estimated to be smaller than $1.0\times10^{-16}$ Transponder Estimated to be smaller than $1.0\times10^{-13}$ Antenna No explicitly positive correlation exists in the residual, thus expected to be smaller than the noise in the Doppler tracking Ground electronics Estimated not to contribute significantly to the noise in the Doppler tracking -------------------- -------------------------------------------------- : Main noise sources and the Allan deviations of their contributions to the residual of the Doppler tracking data of the Chang’e 3 mission. []{data-label="Tab-CE3_Noise_Analysis"} In this work, we will concentrate on the Doppler noise spectra in the frequency band from 1 mHz to 0.1 Hz. Within this frequency band, the instrumental noises are dominated by the noise in the clock, the thermal noise in the receiver, and the antenna mechanical noise(Tinto, 2002; Asmar et al., 2005). The clock noise $y^{FTS}$ is fundamental to the radio observation. It enters into the Doppler shift by $y^{FTS}(t) - y^{FTS}(t-T_2)$. In Chang’e 3, the clock used is the hydrogen masers, its stability can be better than $5\times 10^{-14}$ when $\tau = 9$s. Thus the Allan deviation of the clock noise in the Doppler shift is at most $1.0\times 10^{-13}$. It is smaller than that of the noise of the Doppler shift, because from the power spectral density $S_{DS}(f)$ given in Eq.(\[PsdOfNoiseInDS\]), the Allan deviation of the noise in the Doppler shift is evaluated to be close to $2.13\times 10^{-13}$ when the parameter $\tau$ is 9 s. The thermal noise is white in phase, which is determined essentially by the finite effective temperature of the receiver and the finite intensity of the signal. The Allan deviation for white phase noise associated with the finite signal-to-noise ratio (SNR) thermal noise component is given by $\sigma_{y}(\tau) \approx \sqrt{3BS_{\phi}}/(2\pi f_0 \tau)$, where $B$ is the bandwidth of the phase detector, $f_0$ is the frequency of the radio signal, and $S_{\phi}$ is the one-sided phase noise spectral density, which is approximated by $1/$(SNR in a 1-Hz bandwidth)(Barnes et al., 1971). For the Chang’e 3 mission, using the X-band observation at the Jiamusi station, its signal-to-noise ratio in the 1 Hz band is larger than 59 dB. Thus its Allan deviation is smaller than $5.5\times 10^{-15}$ when $\tau$ is 9s. According to Eq. (\[CE-3NoiseTransfer\]), the Allan deviation for the noise in the Doppler shift induced by the thermal noise is the same as that of the thermal noise itself. It is about two orders in magnitude smaller than that of the noise in the Doppler shift. The antenna mechanical noise of the antenna is not previously measured and calculated, thus we cannot estimate its Allan deviation. Here, we will borrow from the experience of the Cassini mission, though the antenna mechanical noise is not the same. The antenna mechanical noise is a random process which may have the positive correlation at the two-way light time(Armstrong et al., 2003). But from the autocorrelation of the residual as shown in Fig. \[Fig-omc532023202\_56643\_vel\_autocorr\_II\_9sInterval\], there is no explicitly positive correlation in the residual, thus we may expect from the Cassini experiment(Asmar et al., 2005) that the Allan deviation of the Doppler shift noise induced by the antenna mechanical noise is smaller than that of the noise in Doppler shift. For the transponder noise and the ground electronics noise, they are also not tested. From the analysis of these noises in the Cassini experiment(Asmar et al., 2005), the Allan deviation of the transponder noise may be smaller than $1\times 10^{-13}$. If we assume the same form of the power spectrum $S_{groundelec} \propto f^2$ of the ground electronics noise as that in the Cassini mission, then the Allan deviation of the ground electrical noise is smaller than $3\times10^{-14}$ when $\tau = 9$s. Thus we expect that the ground electronics noise of the Chang’e 3 mission will not contribute significantly to the noise in the Doppler shift. The unmodeled mechanical motion of the lander can also give rise to noise in the Doppler tracking data. The mechanical motion of the lander is mainly generated by the lunar seismic shaking and the solid-body tide on the Moon due to the attraction of the Earth. For the lunar seismic shaking, from the analysis of the events collected from the seismometers of the Apollo program, it was predicted that a ground motion of magnitude larger than 22.5 nm may occur at most once in one year(Mendell, 1998). The time-varying tidal displacements mainly contains a constant term and two dominant periodic terms with periods 27.55 d and 27.21 d, whose amplitudes are all smaller than 0.1 m(Williams et al., 1996). Therefore the noise due to the unmodelled motion of the lander in the Doppler shift is very small. Its Allan deviation is estimated to be smaller than $1\times 10^{-16}$. Among the propagation noises, the tropospheric delay noise in the Doppler tracking is the most important. At microwave frequencies, tropospheric refractive index fluctuations are non-dispersive and dominated by the water vapor fluctuation. Since the data used was obtained on 17 December, 2013, which was winter time at the Jiamusi station, the elevation angles were larger than 20 degrees, the diameter of the antenna was 65m and the Sun-Earth-Moon angle was nearly 180 degrees, thus the Allan deviation $\sigma_{y}(\tau)$ for the Doppler shift noise induced by the troposphere is estimated to be smaller than $5.0\times10^{-14}$(Linfield, 1998) when $\tau = 9$s. The ionospheric delay noise is another source of propagation noise. The ionospheric correction is small in the X-band ($\approx8.47$ GHz ), which is of the order of 10 $\mu m/s$ in magnitude. Since the power spectral density of the ionospheric noises is of the form $S_y(f) \propto f^{-2/3}$(Asmar et al., 2005), and the Sun-Earth-Moon angle is nearly 180 degrees, we may estimate that the Allan deviation $\sigma_{y}(\tau)$ for the Doppler shift noise due to the ionosphere is smaller than $9.1\times 10^{-15}$ with $\tau = 9$s. Thus it is of one to two orders in magnitude smaller than the Allan deviation of the noise in the Doppler shift. A feasibility study of using the future Chang’e 4 to constrain the SBGWs around 0.01 Hz ======================================================================================= Chang’e 4 was originally built as a backup to the Chang’e 3 mission. After the successful landing of the Chang’e 3 mission on the lunar surface, Chang’e 4 is redefined to land on the far side of the Moon and due to be launched by the end of 2018. To maintain communication between the lander and the ground station, a tracking and data relay satellite (TDRS) will be launched and located at the Earth-Moon L2 point. With this mission design, the Doppler tracking of two possible coherent radio links, with frequency standard referenced to a very stable hydrogen clock on ground, will be considered in what follows. The first link is the station-TDRS-station link, as shown in Fig. \[Fig-CE-4\_Scheme\_I\]. In this link, the radio signal is transmitted from the station to the TDRS and then phase coherently sent back to the station. The second link is the station-TDRS-lander-TDRS-station link, as shown in Fig. \[Fig-CE-4\_link\_II\]. In this link, the radio signal is transmitted from the station to the TDRS, then it is phase coherently transmitted from the TDRS to the lander. At the lander, the signal is phase coherently sent back to the TDRS and then sent back to the station. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![\[Fig-CE-4\_Scheme\_I\]The station-TDRS-station link. The radio signal is transmitted from the station to the TDRS and then phase coherently sent back to the station.](CE-4_Scheme_I.eps "fig:"){width="70.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-CE-4\_link\_II\]The station-TDRS-lander-TDRS-station link. The radio signal is transmitted from the station to the TDRS, then it is phase coherently transmitted from the TDRS to the lander. At the lander, the signal is phase coherently sent back to the TDRS and then sent back to the station.](CE-4_link_II.eps "fig:"){width="70.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- From each link, with the upgrade of the radio measurement system of the deep space network, especially with higher precision ultra stable oscillators, the Doppler tracking data with higher precision is expected to be available. Given the upper bound on the SBGWs within the frequency band from 1 mHz to 0.05 Hz attained by the Chang’e 3 mission, it is likely that the Doppler tracking data of the Chang’e 4 mission may yield a better upper bound on the SBGWs in the frequency band from 1 mHz to 0.1 Hz. The first scheme to constrain the SBGWs ---------------------------------------- In this scheme, the Doppler tracking data is obtained from the first link, which is denoted by $\dot{\rho}_E^{2w}$. The corresponding baseline is the distance between the station and the TDRS, as shown in Fig. \[Fig-CE-4\_Scheme\_I\], which is about $4.2\times 10^5$ km. The corresponding round-trip time, denoted by $T_2^E$, of the radio signal along the first link is about 2.80 s. After subtracting the theoretical values from the observations, the residual $y_I$ in the Doppler shift $\dot{\rho}_E^{2w}/c$ ( $c$ is the speed of light) may be modelled as $$\begin{aligned} \label{NoiseModelCE4_I} % \nonumber to remove numbering (before each equation) y_I(t) &=& y_{2,E}^{gw}(t) + y^{trop}(t) + y^{trop}(t - T_2^E) + y^{iono}(t)+y^{iono}(t-T_2^E) \nonumber\\ &&+ y^{ant}(t) + y^{ant}(t-T_2^E) + y^{FTS}(t) - y^{FTS}(t-T_2^E) \nonumber\\ &&+ y^{TDRS}(t-T_2^E/2) + y^{ther}(t) + y^{groundelec}(t),\end{aligned}$$ where $y_{2,E}^{gw}$ is the contribution of the SBGWs to the Doppler variability, the other terms are the main noise sources of variability in the Doppler shift. The meanings of the noise terms $y^{trop}$, $y^{iono}$, $y^{ant}$, $y^{FTS}(t)$, $y^{ther}$ and $y^{groundelec}$ are the same as those given in Eq.(\[CE-3NoiseTransfer\]). The noise $y^{TDRS}$ represents the transponder noise in the TDRS. In the spectral domain, the power spectrum $S_{y_I}(f)$ of $y_I$ is given by $$\begin{aligned} \label{NoiseModelCE4PSD_I} % \nonumber to remove numbering (before each equation) S_{y_I}(f) &=& S_{y_{2,E}}^{gw}(f) + 4\cos^2(\pi T_2^E f)\left[S_{y}^{trop}(f) + S_{y}^{iono}(f)+ S_{y}^{ant}(f) \right] \nonumber\\ && + 4\sin^2(\pi T_2^E f)S_{y}^{FTS}(f)+ S_{y}^{TDRS}(f) + S_{y}^{groundelec}(f)+ S_{y}^{ther}(f), \nonumber\\\end{aligned}$$ where $S_{y_{2,E}}^{gw}$ is the power spectrum of the term $y_{2,E}^{gw}$ due to the SBGWs and the other terms are the corresponding power spectrums of the noises in Eq.(\[NoiseModelCE4\_I\]). Suppose the power spectrum $S_{y_I}(f)$ of $y_I$ is white in the band from 1 mHz to 0.1 Hz. It may be modelled as $S_{y_I}(f) = \frac{\sigma_{I}^2}{2\pi}$, where $\sigma_{I}$ represents the root mean square of the residual $y_I$. For the power spectrum $S_{y_I}(f)$ of $y_I$ to give an improved upper bound on the SBGWs, the main noises must satisfy certain requirements, which are easily evaluated from Eq. (\[NoiseModelCE4PSD\_I\]) and listed in the Table \[Tab-RequirementOnNoise\_I\]. ----------------- ------------------------------------------------------- Noise Requirement on psd Troposphere $ \leq \frac{\sigma_{I}^2}{8\pi\cos^2(\pi T_2^E f )}$ Ionosphere $\leq \frac{\sigma_{I}^2}{8\pi\cos^2(\pi T_2^E f )}$ Antenna $\leq \frac{\sigma_{I}^2}{8\pi\cos^2(\pi T_2^E f )}$ Clock $\leq \frac{\sigma_{I}^2}{8\pi\sin^2(\pi T_2^E f )}$ TDRS $\leq \frac{\sigma_{I}^2}{2\pi}$ Ground electric $\leq \frac{\sigma_{I}^2}{2\pi}$ Thermal $\leq \frac{\sigma_{I}^2}{2\pi}$ ----------------- ------------------------------------------------------- : Requirements on the power spectral density of the noises in the first scheme. []{data-label="Tab-RequirementOnNoise_I"} The second scheme to constrain the SBGWs ----------------------------------------- In Chang’e 4, the simultaneously obtained two coherent radio links will share the same uplink carrier wave, and for the downlink waves, they will pass the same space path between the relay satellite and the ground tracking antenna, as well as the receiver, the low noise amplifier, and the Doppler counter. Thus it may be possible to reduce or cancel out many common noise fluctuations by the combination of the two kinds of Doppler tracking. Let $\dot{\rho}^{4w}(t)$ denote the range rate given from the second link. The residual $y_4$ in the Doppler shift $\dot{\rho}^{4w}(t)/c$ may be modelled as $$\begin{aligned} \label{NoiseModel4wRR} % \nonumber to remove numbering (before each equation) y_4(t) &=& y_4^{gw}(t) + y^{trop}(t) + y^{trop}(t - T_2^E-T_2^M) + y^{iono}(t)+y^{iono}(t-T_2^E-T_2^M) \nonumber\\ &&+ y^{ant}(t)+y^{ant}(t-T_2^E-T_2^M)+y^{FTS}(t)-y^{FTS}(t-T_2^E-T_2^M) \nonumber\\ && + y^{TDRS}(t-T_2^E/2) + y^{TDRS}(t-T_2^E/2-T_2^M) \nonumber\\ &&+ y^{ldr}(t-T_2^E/2-T_2^M/2)+y^{groundelec}(t) + y^{ther}(t),\end{aligned}$$ where $T_2^M$ is the round-trip time of the signal from the TDRS to the lander and then return back to the TDRS, the $y_4^{gw}$ is the contribution of the SBGWs to the Doppler variability, $y^{ldr}$ is the noise in the Doppler shift due to the transponder noise of the lander. The meanings of the other terms are the same as those in Eq. (\[NoiseModelCE4\_I\]). From the noise analysis of Chang’e 3, we know that the main noises are the clock noise, the tropospheric noise and the antenna mechanical noise. It follows from Eq. (\[NoiseModelCE4\_I\]) and Eq. (\[NoiseModel4wRR\]) that to reduce or cancel out their influences, we construct a new observable $O_2(t)$ as $O_2(t)=\dot{\rho}^{4w}(t)-\dot{\rho}_E^{2w}(t)$. If we denote $y_{II}(t)$ as the residual in the Doppler shift $O_2(t)/c$, then it equals to the differential of $y_4$ and $y_I$. The residual $y_{II}(t)$ may be modelled from Eq. (\[NoiseModelCE4\_I\]) and Eq. (\[NoiseModel4wRR\]) as $$\begin{aligned} \label{NoiseModelCE4_II} % \nonumber to remove numbering (before each equation) y_{II}(t) &=& y_4^{gw}(t) - y_{2,E}^{gw}(t) - y^{trop}(t - T_2^E) + y^{trop}(t - T_2^E-T_2^M) \nonumber \\ && - y^{iono}(t - T_2^E)+y^{iono}(t-T_2^E-T_2^M) - y^{ant}(t - T_2^E) \nonumber\\ && + y^{ant}(t-T_2^E-T_2^M)+y^{FTS}(t - T_2^E)-y^{FTS}(t-T_2^E-T_2^M) \nonumber\\ && + y^{TDRS}(t-T_2^E/2-T_2^M) + y^{ldr}(t-T_2^E/2-T_2^M/2).\end{aligned}$$ Obviously, the influence of the SBGWs to the Doppler shift is $y_4^{gw}(t) - y_{2,E}^{gw}(t)$, which is in fact equivalent to the influence of the SBGWs to the Doppler shift given from the two-way range rate between the lander and the TDRS, thus we may denote by $y_{2,M}^{gw}$ the $y_4^{gw}(t) - y_{2,E}^{gw}(t)$. In this scheme, the distance between the lander and the TDRS is about $6.5\times 10^4$ km. So the round-trip time $T_2^M$ is about 0.43 s. The power spectrum $S_{y_{II}}$ of $y_{II}$ may be evaluated as $$\begin{aligned} \label{NoiseModelCE4PSD_II} % \nonumber to remove numbering (before each equation) S_{y_{II}}(f) &=& S_{y_{2,M}}^{gw}(f) + 4\sin^2(\pi T_2^M f)\left[S_{y}^{trop}(f) + S_{y}^{iono}(f)+ S_{y}^{ant}(f) \right] \nonumber\\ && + 4\sin^2(\pi T_2^M f)S_{y}^{FTS}(f)+ S_{y}^{TDRS}(f) + S_{y}^{ldr}(f),\end{aligned}$$ where $S_{y_{2,M}}^{gw}(f)$ is the power spectrum of $y_{2,M}^{gw}$ and $S_{y}^{ldr}(f)$ is the power spectrum of $y^{ldr}$. Assume that the power spectrum $S_{y_{II}}(f)$ of the residual $y_{II}(t)$ is constant in the frequency band from 1 mHz to 0.1 Hz. It may be written as $S_{y_{II}}(f) = \frac{\sigma_{II}^2}{2\pi}$, where $\sigma_{II}$ is the root mean square of the residual $y_{II}$. The requirements on the main noises are obtained from Eq. (\[NoiseModelCE4PSD\_II\]), which are listed in the Table \[Tab-RequirementOnNoise\_II\]. ------------- ------------------------------------------------------ Noise Requirement on psd Troposphere $\leq \frac{\sigma_{II}^2}{8\pi\sin^2(\pi T_2^M f)}$ Ionosphere $\leq \frac{\sigma_{II}^2}{8\pi\sin^2(\pi T_2^M f)}$ Antenna $\leq \frac{\sigma_{II}^2}{8\pi\sin^2(\pi T_2^M f)}$ Clock $\leq \frac{\sigma_{II}^2}{8\pi\sin^2(\pi T_2^M f)}$ TDRS $\leq \frac{\sigma_{II}^2}{2\pi}$ Lander $\leq \frac{\sigma_{II}^2}{2\pi}$ ------------- ------------------------------------------------------ : Requirements on the power spectral density of the noises in the second scheme. []{data-label="Tab-RequirementOnNoise_II"} Comparisons of two schemes -------------------------- It follows from the Tables \[Tab-RequirementOnNoise\_I\] and \[Tab-RequirementOnNoise\_II\] that these two schemes have different noise requirements. In the first scheme, there is a common noise requirement on the tropospheric noise, the ionospheric noise and the antenna mechanical noise, namely, their power spectrums should be smaller than $\frac{\sigma_{I}^2}{8\pi\cos^2(\pi T_2^E f )}$. The power spectrum of the clock should be smaller than $\frac{\sigma_{I}^2}{8\pi\sin^2(\pi T_2^E f )}$. In the second scheme, the noise requirements on the tropospheric noise, the ionospheric noise, the antenna mechanical noise and the clock noise are identical. Their power spectrums should be smaller than $\frac{\sigma_{II}^2}{8\pi\sin^2(\pi T_2^M f)}$. For comparison, we show in Fig. \[Fig-CE4\_specialFun\] the functions $1/(4\cos^2(\pi T_2^E f))$, $1/(4\sin^2(\pi T_2^E f))$ and $1/(4\sin^2(\pi T_2^M f))$. We find that when $\sigma_{I} = \sigma_{II}$, the requirements on the tropospheric noise, the ionospheric noise and the antenna mechanical noise in the first scheme are about 1.5 orders in magnitude higher than those in the second scheme. Further the requirement on the clock noise in the first scheme is also much higher than that in the second scheme. For example, the requirement in the first scheme is about 3.5 orders in magnitude higher than that in the second scheme at 0.01 Hz. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-CE4\_specialFun\]The plot for the functions $1/(4\cos^2(\pi T_2^E f))$, $1/(4\sin^2(\pi T_2^E f))$ and $1/(4\sin^2(\pi T_2^M f))$.](CE4_specialFun.eps "fig:"){width="70.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- From the noise analysis of Chang’e 3, the main noises limiting the improvement of the measurement accuracy are the propagation noises and the antenna mechanical noise. Thus the second scheme is better than the first scheme. Further, in the second scheme, we do not have to consider the ground electronic noise and thermal noise. Thus, the second scheme is recommended for future Doppler tracking of Chang’e 4 to constrain the SBGWs. In the following, we will further elaborate on certain practical aspects of the main noises when we try to implement the measurement scheme. In the Chang’e 4 mission, the measurement accuracy of the range rate is expected to reach about 15$\mu$m/s with the sampling time to be 1 s. Since we are interested in constraining the SBGWs in the frequency band from 1 mHz to 0.1 Hz, the original Doppler tracking can be smoothed to a new time series with the sampling time to be 5 s. Thus in principle the root mean squares $\sigma_{I}$ and $\sigma_{II}$ are about $2.3\times 10^{-14}$. It then follows from the Table \[Tab-RequirementOnNoise\_II\] that the explicitly requirements on the main noises are listed in the Table \[Tab-RequirementOnNoise\_II\_15\]. For clarity, we plot the functions $2.1\times10^{-29}/\sin^2(\pi T_2^M f )$ and $8.5\times10^{-29}$ in the Fig. \[Fig-CE4\_specialFun\_II\_NoiseReq\]. ------------- ---------------------------------------------- Noise Requirement on psd Troposphere $\leq 2.1\times10^{-29}/\sin^2(\pi T_2^M f)$ Ionosphere $\leq 2.1\times10^{-29}/\sin^2(\pi T_2^M f)$ Antenna $\leq 2.1\times10^{-29}/\sin^2(\pi T_2^M f)$ Clock $\leq 2.1\times10^{-29}/\sin^2(\pi T_2^M f)$ TDRS $\leq 8.5\times10^{-29}$ Lander $\leq 8.5\times10^{-29}$ ------------- ---------------------------------------------- : Requirements on the power spectral density of the noises in the second scheme when $\sigma_{II} = 2.3\times 10^{-14}$. []{data-label="Tab-RequirementOnNoise_II_15"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![\[Fig-CE4\_specialFun\_II\_NoiseReq\]The plot for the functions $2.1\times10^{-29}/\sin^2(\pi T_2^M f )$ and $8.5\times10^{-29}$.](CE4_specialFun_II_NoiseReq.eps "fig:"){width="70.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ For the tropospheric noise, its power spectral density is(see Linfield, 1998) $$\begin{aligned} \label{psd_Troposphere} % \nonumber to remove numbering (before each equation) S_{y}^{trop}(f) & = & 1.4\times10^{-27}f^{-2/5} \textrm{Hz}^{-1}, \quad 10^{-5}\leq f \leq 10^{-2} \textrm{Hz},\\ & = & 2.2\times10^{-30}f^{-3} \textrm{Hz}^{-1}, \quad 10^{-2} \leq f \leq 1 \textrm{Hz}.\end{aligned}$$ It is smaller than that requirement listed in the Table \[Tab-RequirementOnNoise\_II\_15\] when the frequency is smaller than 0.01 Hz. But it is about 0.5 to 1 order in magnitude larger than the requirement from 0.01 to 1 Hz. Since it is well known that the tropospheric noise depends on the elevation angle, the season, time of day, and weather conditions, it is likely we may lower the tropospheric noise to reach the requirement. For the ionospheric noise, for the X-band($\approx$ 8.47GHz ), its power spectrum $S_{y}^{iono}(f)$ is approximated by $7.79\times10^{-29}f^{-2/3} \textrm{Hz}^{-1}$(Tinto, 2002). Thus it is smaller than that in the Table \[Tab-RequirementOnNoise\_II\_15\]. For the ground master clock and the frequency and timing distribution, if we use a clock with the one-sided power spectral density given as(Tinto et al., 2009) $$\begin{aligned} % \nonumber to remove numbering (before each equation) S_{y}^{FTS}(f) &=& 6.2\times[10^{-28}f + 10^{-33}f^{-1}] + 1.3\times10^{-28}f^2 \quad \textrm{Hz}^{-1},\end{aligned}$$ then the clock noise satisfies the requirement listed in the Table \[Tab-RequirementOnNoise\_II\_15\]. As far as antenna mechanical noise is considered, if we convert from the required power spectrum of the antenna mechanical noise to the Allan deviation $\sigma_{ant}(\tau)$, we have that $\sigma_{ant} \approx 3.2\times10^{-13}$ when $\tau = 5$s. From the experience of Cassini, it may be reached if the observation is under some favorable conditions(Asmar et al., 2005), which we will study carefully in future. For the TDRS and the lander, the main noises are the transponder noise(Asmar et al., 2005). In the Cassini spacecraft, the one-sided power spectral density $S_{y}^{TR}(f)$ is(Riley et al., 1990) $$\begin{aligned} % \nonumber to remove numbering (before each equation) S_{y}^{TR}(f) &=& 1.6\times10^{-26}f \quad \textrm{Hz}^{-1} .\end{aligned}$$ This power spectrum is $1.6\times10^{-29}$ at about 1 mHz and then increases linearly to $1.6\times10^{-27}$ at 0.1 Hz which is about 1.2 orders in magnitude higher than the required $8.5\times10^{-29}$. Thus it is possible to improve the transponder noise on the TDRS and the lander. Subject to the requirements on the noises in the second scheme, it is expected that the best measurement accuracy of the data is about 15$\mu$m/s, then the upper bound on the energy density $\Omega_{gw}$ of SBGWs in the band from 1 mHz to 0.1 Hz is calculated and shown in Fig. \[Fig-CE4\_SBGW\_Omega\_comparison\_VII\_10s\_EandW\]. It indicates that the upper bound on $\Omega_{gw}$ will be improved nearly one order in a wider band when compared with that of Chang’e 3. Further, it will be improved by nearly 1.3 orders at about 0.1 Hz when compared with those from the Earth’s seismic data. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[Fig-CE4\_SBGW\_Omega\_comparison\_VII\_10s\_EandW\]The expected upper bound on SBGWs from Chang’e 4 (green dash line). The other constraints are listed here for comparison. These constrains include the results obtained from the Cassini spacecraft(Armstrong et al., 2003), the ULYSSES spacecraft(Bertotti et al., 1995), the normal modes of the Earth(Coughlin et al., 2014a), the Apollo missions(Aoyama et al., 2014), the Earth’s seismic data(Coughlin et al., 2014b), the Lunar seismic data(Coughlin et al., 2014c), the torsion-bar antenna(Shoda et al., 2013), the LIGO mission(Aasi et al., 2014)(corresponds to four lines) and the Chang’e 3 mission (red line).](CE4_SBGW_Omega_comparison_VII_10s_EandW.eps "fig:"){width="40.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Conclusion ========== A detailed analysis has been presented on the range rate data of the Chang’e 3 lunar mission. Apart from giving an improved upper bound on the SBGWs in the narrow frequency band of 0.02 to 0.05 Hz, the feasibility of improving on the upper bound in the upcoming Chang’e 4 mission is also discussed. By making use of the differential coherent Doppler measurement in Chang’e 4, the effects due to the interplanetary plasma, ionosphere, troposphere, and the effects due to the ground instruments and clock can be removed considerably. The upper bound on the SBGWs may then be improved by nearly one order in a wider band when compared with that of Chang’e 3. With the promise in science for future lunar as well as deep space programs in China, we hope our work constitutes a modest beginning on this front as far as experimental tests of general relativity are concerned. Acknowledgment {#acknowledgment .unnumbered} ============== The present research is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences��Grant No.XDB23030100, National Natural Science Foundation of China (project number 11173005, 11171329, 61304233 and 41590851), and also in part by the National Basic Research Program of China under Grant 2015CB857101. [00]{} Aasi J., Abbott B. P., et al. Improved Upper Limits on the Stochastic Gravitational-Wave Background from 2009-2010 LIGO and VIRGO Data\[J\]. Physical Review Letters, 2014, 113(23): 231101 Aoyama S, Tazai R, and Ichiki K. Upper limit on the amplitude of gravitational waves around 0.1 Hz from the Global Positioning System\[J\]. Phys.rev.d, 2014, 89(6): 196-204. Armstrong J W, Iess L, Tortora P, et al. Stochastic Gravitational Wave Background: Upper Limits in the 10-6 to 10-3 Hz Band\[J\]. Astrophysical Journal, 2003, 599(2): 806-813. Armstrong J W, Estabrook F B, Asmar S W, et al. 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--- abstract: 'The impossibility of perfect cloning and state estimation are two fundamental results in Quantum Mechanics. It has been conjectured that quantum cloning becomes equivalent to state estimation in the asymptotic regime where the number of clones tends to infinity. We prove this conjecture using two known results of Quantum Information Theory: the monogamy of quantum correlations and the properties of entanglement breaking channels.' author: - Joonwoo Bae and Antonio Acín title: Asymptotic quantum cloning is state estimation --- The impossibility of perfect state estimation is a major consequence of the nonorthogonality of quantum states: the state of a single quantum system cannot be perfectly measured. In other words, a measurement on a system in order to acquire information on its quantum state perturbs the system itself. The full reconstruction of the state is only possible by computing statistical averages of different observables on a large number of identically prepared systems. Thus, any measurement at the single-copy level only provides partial information. The fact that state estimation is in general imperfect leads in a natural way to the problem of building optimal measurements. Being a perfect reconstruction impossible, it is relevant to find the measurement strategy that maximizes the gain of information about the unknown state. A standard approach to this problem in Quantum Information Theory (QIT) is to quantify the quality of a measurement by means of the so-called fidelity [@MP]. This quantity is defined as follows. Consider the situation in which a quantum state $\ket{\psi}$ is chosen from the ensemble $\{p_i,\ket{\psi_i}\}$, i.e. $\ket{\psi}$ can be equal to $\ket{\psi_i}$ with probability $p_i$. A measurement, defined by $N_M$ positive operators, $M_j\geq 0$, summing up to the identity, $\sum_j M_j=\one$, is applied on this unknown state. For each obtained outcome $j$, a guess $\ket{\phi_j}$ for the input state is made. The overlap between the guessed state and the input state, $\|\braket{\psi_i}{\phi_j}\|^2$, quantifies the quality of the estimation process. The averaged fidelity of the measurement then reads $$\label{avfidmeas} \bar F_M=\sum_{i,j} p_i\, \tr(M_j\proj{\psi_i})\,\|\braket{\psi_i}{\phi_j}\|^2 .$$ A measurement is optimal according to the fidelity criterion when it provides the largest possible value of $\bar F_M$, denoted in what follows by $F_M$. The No-cloning theorem [@WZ], one of the cornerstones of QIT [@reviews], represents another known consequence of the nonorthogonality of quantum states. It proves that given a quantum system in an unknown state $\ket{\psi}$, it is impossible to design a device producing two identical copies, $\ket{\psi}\ket{\psi}$. Indeed, two nonorthogonal quantum states suffice to prove the no-cloning theorem. As it happens for state estimation, the impossibility of perfect cloning leads to the characterization of optimal cloning machines [@BH]. In this case, one looks for the quantum map $\L$ that, given a state $\ket{\psi}$ chosen from an ensemble $\{p_i,\ket{\psi_i}\}$ in $\compl^d$, produces a state $\L(\psi)=\rho_{C_1\ldots C_N}$ in $(\compl^d)^{\otimes N}$, such that each individual clone $\rho_{C_k}=\tr_{\bar k}(\rho_{C_1\ldots C_N})$ resembles as much as possible the input state, where $\tr_{\bar k}$ denotes the trace with respect to all the systems $C_1,\ldots,C_N$ but $C_k$. The average fidelity of the cloning process is then $$\label{avfidcl} \bar F_C(N)=\sum_{i,k} p_i\, \frac{1}{N}\bra{\psi_i}\tr_{\bar k}\L(\psi_i)\ket{\psi_i} .$$ The goal of the optimal machine is to maximize this quantity, this optimal value being denoted by $F_C(N)$. One can easily realize that the no-cloning theorem and the impossibility of perfect state estimation are closely related. On the one hand, if perfect state estimation was possible, one could use it to prepare any number of clones of a given state, just by measurement and preparation. On the other hand, if perfect cloning was possible, one could perfectly estimate the unknown state of a quantum system by preparing infinite clones of it and then measuring them. Beyond these qualitative arguments, the connection between state estimation and cloning was strengthened in [@GM; @BEM]. The results of these works suggested that asymptotic cloning, i.e. the optimal cloning process when $N\to\infty$, is equivalent to state estimation, in the sense that $$\label{conj} F_C=F_C(N\to\infty)=F_M .$$ Later, this equality was rigorously shown to hold in the cases of (i) universal cloning [@KW], where the initial ensemble consists of an arbitrary pure state in $\compl^d$, chosen with uniform probability, and (ii) phase covariant qubit cloning [@BCDM], where the initial ensemble corresponds to a state in $\compl^2$ lying on one of the equators of the Bloch sphere. Since then, the validity of this equality for any ensemble has been conjectured and, indeed, has been identified as one of the open problems in QIT [@web]. In this work, we show that the fidelities of optimal asymptotic cloning and of state estimation are indeed equal for any initial ensemble of pure states. Actually, we prove that asymptotic cloning does effectively correspond to state estimation, from which the equality of the two fidelities trivially follows. The proof of this equivalence is based on two known results of QIT: the monogamy of quantum correlations and the properties of the so-called entanglement breaking channels (EBC). It is easy to prove that $F_M\leq F_C$. Indeed, given the initial state $\ket{\psi}$, a possible asymptotic cloning map, not necessarily optimal, consists of first applying state estimation and then preparing infinite copies of the guessed state. It is sometimes said that the opposite has to be true since “asymptotic cloning cannot represent a way of circumventing optimal state estimation". As already mentioned in [@web], this reasoning is too naive, since it neglects the role correlations play in state estimation. For instance, take the simplest case of universal cloning of a qubit, i.e. a state in $\compl^2$ isotropically distributed over the Bloch sphere. The optimal cloning machines produces $N$ approximate clones pointing in the same direction in the Bloch sphere as the input state, but with a shrunk Bloch vector [@KW]. If the output of the asymptotic cloning machine was in a product form, it would be possible to perfectly estimate the direction of the local Bloch vector, whatever the shrinking was. Then, a perfect estimation of the initial state would be possible. And of course, after the perfect estimation one could prepare an infinite number of perfect clones! This simple reasoning shows that the correlations between the clones play an important role in the discussion. Actually, it has recently been shown that the correlations present in the output of the universal cloning machine are the worst for the estimation of the reduced density matrix [@rafael]. As announced, the proof of the conjecture is based on two known results of QIT: the monogamy of entanglement and the properties of EBC. For the sake of completeness, we state here these results, without proof. Quantum correlations, or entanglement, represent a monogamous resource, in the sense that they cannot be arbitrarily shared. One of the strongest results in this direction was obtained by Werner in 1989 [@Werner]. There, it was shown that the only states that can be arbitrarily shared are the separable ones. Recall that a bipartite quantum state $\rho_{AC}$ in $\compl^d\otimes\compl^d$ is said to be $N$-shareable when it is possible to find a quantum state $\rho_{AC_1\ldots C_N}$ in $\compl^d\otimes(\compl^d)^{\otimes N}$ such that $\rho_{AC_k}=\tr_{\bar k}\rho_{AC_1\ldots C_N}=\rho_{AC},\,\forall k$. The state $\rho_{AC_1\ldots C_N}$ is then said to be an $N$-extension of $\rho_{AC}$. The initial correlations between subsystems $A$ and $C$ are now shared between $A$ and each of the $N$ subsystems $C_i$, see Fig. \[entshar\]. It is straightforward to see that $$\rho_{AC_1\ldots C_N}=\sum_i q_i\proj{\alpha_i}\otimes \proj{\gamma_i}^{\otimes N}$$ gives a valid $N$-extension of a separable state $\rho_{AC}^s=\sum_i q_i\proj{\alpha_i}\otimes\proj{\gamma_i}$ for all $N$. As proven by Werner, if the state is entangled, there exists a finite $N$ where no valid extension can be found. ![The state $\rho_{AC}$ is said to be $N$-shareable when there exists a global state $\rho_{AC_1\ldots C_N}$ such that the local state shared between $A$ and $C_i$ is equal to $\rho_{AC}$, for all $i$.[]{data-label="entshar"}](rhoAcccc "fig:"){width="8cm"}\ The second ingredient needed in what follows are the properties of EBC. A channel $\Upsilon$ is said to be entanglement breaking when it cannot be used to distribute entanglement. In Ref. [@HSR] it was proven that the following three statements are equivalent: (1) $\Upsilon$ is entanglement breaking, (2) $\Upsilon$ can be written in the form $\Upsilon(\rho)=\sum_j\tr(M_j\rho)\rho_j$, where $\rho_j$ are quantum states and $\{M_j\}$ defines a measurement and (3) $(\one\otimes\Upsilon)\ket{\Phi^+}$ is a separable state, where $\ket{\Phi^+}=\sum_i \ket{ii}/\sqrt d$ is a maximally entangled state in $\compl^d\otimes\compl^d$. The equivalence of (1) and (2) simply means that any EBC can be understood as the measurement of the input state, $\rho$, followed by the preparation of a new state $\rho_j$ depending on the obtained outcome. The equivalence of (1) and (3) reflects that the intuitive strategy for entanglement distribution where half of a maximally entangled state is sent through the channel is enough to detect if $\Upsilon$ is entanglement breaking. After collecting all these results, we are now ready to prove the following [Theorem:]{} Asymptotic cloning corresponds to state estimation. Thus, $F_M=F_C$ for any ensemble of states. [Proof:]{} The idea of the proof is to characterize the quantum maps $\L$ associated to asymptotic cloning machines. First of all, note that, for any number of clones, we can restrict our considerations to symmetric cloning machines, $\L^s$, where the clones are all in the same state. Indeed, given a machine where this is not the case, one can construct a symmetric machine achieving the same fidelity $F_C(N)$, just by making a convex combination of all the permutations of the $N$ clones [@note]. Now, denote by $\L^c$ the effective cloning map consisting of, first, the application of a symmetric machine $\L^s$ and then tracing all but one of the clones, say the first one. The cloning problem can be rephrased as, see Eq. (\[avfidcl\]), $$\label{asclon} \max_{\L^c}\sum_ip_i\bra{\psi_i}\L^c(\psi_i)\ket{\psi_i} .$$ Note that this maximization runs over all channels that can be written as $\L^c=\tr_{\bar 1}\L^s$. For instance, the identity map, where $\psi\to\psi,\,\forall\psi$, does not satisfy this constraint. If the $N$-cloning map is applied to half of a maximally entangled state, the resulting state, $$\rho_{AC_1\ldots C_N}=(\one_A\otimes\L^s_B)\ket{\Phi^+}_{AB},$$ is such that, for all $i$, $$\label{rhoac} \rho_{AC_i}=(\one\otimes\L^c)\ket{\Phi^+}=\rho_{AC}.$$ That is, the output of the $N$-cloning machine acting on half of a maximally entangled state is a valid $N$-extension of $\rho_{AC}$. When taking the limit of an infinite number of clones, and because of the monogamy of entanglement, this implies that $\rho_{AC}$ has to be separable and, thus, $\L^c$ is entanglement breaking (\[rhoac\]). Since any EBC can be seen as measurement followed by state preparation, asymptotic cloning (\[asclon\]) can be written as [@note2] $$\max_{\{M_j,\phi_j\}}\sum_{i,j} p_i\, \tr(M_j\proj{\psi_i})\,\|\braket{\psi_i}{\phi_j}\|^2 ,$$ which defines the optimal state estimation problem. Therefore, $F_M=F_C$ for any ensemble of states. $\Box$ The same argument applies to the case in which $L$ copies of the initial state $\ket{\psi}$ are given. The measurement and cloning fidelities now read, see Eqs. (\[avfidmeas\]) and (\[avfidcl\]), $$\begin{aligned} % \nonumber to remove numbering (before each equation) \bar F_M(L) &=& \sum_{i,j} p_i\, \tr(M_j\proj{\psi_i}^{\otimes L})\,\| \braket{\psi_i}{\phi_j}\|^2 \nonumber\\ \bar F_C(N,L) &=& \sum_{i,k} p_i\, \frac{1}{N}\bra{\psi_i} \tr_{\bar k}\L(\psi_i^{\otimes L})\ket{\psi_i} .\end{aligned}$$ Using the same ideas as in the previous Theorem, it is straightforward to prove that $$\label{eqfidL} F_M(L)=F_C(N\to\infty,L),$$ where $F_M(L)$ and $F_C(N,L)$ denote the optimal values of $\bar F_M(L)$ and $\bar F_C(N,L)$, as above. One can also extend this result to asymmetric scenarios. An asymmetric cloning machine [@asclon], given an initial input state $\ket{\psi}$, produces $N_A$ clones of fidelity $F_C(N_A)$ and $N_B$ clones of fidelity $F_C(N_B)$. The machine is optimal when it gives the largest $F_C(N_B)$ for fixed $F_C(N_A)$. In the case of measurement, we are thinking of measurement strategies where the goal is to obtain information on an unknown state introducing the minimal disturbance. As above, we consider that a guess for the input state is done depending on the measurement outcome. The information vs disturbance trade-off can again be expressed in terms of fidelities [@banaszek]: the information gain is given by the overlap, $G$, between the initial and the guessed state, while the disturbance is quantified by the overlap, $F$, between the state after the measurement and the initial state. A measurement is optimal when for fixed disturbance, $F$, it provides the largest value of $G$. The optimal trade-off between $F$ and $G$ has been derived in [@banaszek] for the case in which the input ensemble consists of any pure state in $\compl^d$ with uniform probability. As it happens for the symmetric case, a connection between this state estimation problem and asymmetric cloning machines can be expected when $N_A=1$ and $N_B\to\infty$. Indeed, the previous measurement strategy gives a possible realization of this asymptotic and asymmetric cloning machine, not necessarily optimal. Actually, when the input state is any pure state, with uniform probability, the optimal measurement strategy of [@banaszek] turns out to saturate the optimal cloning $1\to N_A+N_B$ fidelities of [@IACFFG], with $N_A=1$ and $N_B\to\infty$. Now, the equality between the measurement and asymptotic cloning fidelities in the asymmetric scenario for any ensemble of input states can be proven using the same arguments as above: one has to symmetrize the $N_B$ clones and then use the connection with entanglement shareability and EBC when $N_B\to\infty$. From a more speculative point of view, there exist several works relating the impossibility of perfect cloning to the no-signaling principle, namely the impossibility of having faster-than-light communication (see for instance [@gisin]). Actually, a no-cloning theorem can be derived just from the no-signaling principle, without invoking any additional quantum feature [@ncsign]. In view of the strong connection between cloning and state estimation, it would be interesting to study whether a similar link between the no-signaling principle and the impossibility of perfect state estimation could also be established. To conclude, this work proves the long-standing conjecture on the equivalence between asymptotic cloning and state estimation. It represents the strongest link between two fundamental no-go theorems of Quantum Mechanics, namely the impossibilities of perfect cloning and state estimation. We thank Emili Bagan, John Calsamiglia, Sofyan Iblisdir and Ramon Muñoz-Tapia for discussion. This work is supported by the Spanish MCyT, under “Ramón y Cajal" grant, and the Generalitat de Catalunya, 2006FIR-000082 grant. S. Massar and S. Popescu, Phys. Rev. Lett. [74]{}, 1259 (1995). W. K. Wootters and W. H. Zurek, Nature [299]{}, 802 (1982). Two independent reviews on the no-cloning theorem have recently appeared: V. Scarani, S. Iblisdir, N. Gisin and A. Acín, Rev. Mod. Phys. [77]{}, 1225 (2005); N. J. Cerf and J. Fiurášek, quant-ph/0512172. V. Bužek and M. Hillery, Phys. Rev. A [54]{}, 1844 (1996). N. Gisin and S. Massar, Phys. Rev. Lett. [79]{}, 2153 (1997). D. Bruß, A. Ekert and C. Macchiavello, Phys. Rev. Lett. [81]{}, 2598 (1998). M. Keyl and R. F. Werner, J. Math. Phys. [40]{}, 3283 (1999). D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello, Phys. Rev. A [62]{}, 12302 (2000). See problem 22 in http://www.imaph.tu-bs.de/qi/problems/. R. Demkowicz-Dobrzanski, Phys. Rev. A [71]{}, 062321 (2005). R. F. Werner, Lett. Math. Phys. [17]{}, 359 (1989); another, and somehow extended, proof of this result can also be found in A. C. Doherty, P. A. Parrilo and F. M. Spedalieri, Phys. Rev. A [69]{}, 022308 (2004). M. Horodecki, P. W. Shor and M. B. Ruskai, Rev. Math. Phys [15]{}, 629 (2003). Notice that this does not mean that the output of the cloning machine lives in the symmetric subspace. We can already restrict the guessed states to be pure, without any loss of optimality. K. Banaszek, Phys. Rev. Lett. [86]{}, 1366 (2001). C.-S. Niu and R. B. Griffiths, Phys. Rev. A [58]{}, 4377 (1998); N. J. Cerf, Acta Phys. Slov. [48]{}, 115 (1998); V. Bužek, M. Hillery and M. Bendik, [ibid]{}, 177 (1998); N. J. Cerf, J. Mod. Opt. [47]{}, 187 (2000). S. Iblisdir, A. Acín, N. J. Cerf, J. Fiurášek, R. Filip and N. Gisin, Phys. Rev. A [72]{}, 042328 (2005). N. Gisin, Phys. Lett. A [242]{}, 1 (1998). Ll. Masanes, A. Acín and N. Gisin, Phys. Rev. A [73]{}, 012112 (2006); J. Barrett, quant-ph/0508211.
--- abstract: 'We study oxygen $K$-edge x-ray absorption spectroscopy (XAS) and investigate the validity of the Zhang-Rice singlet (ZRS) picture in overdoped cuprate superconductors. Using large-scale exact diagonalization of the three-orbital Hubbard model, we observe the effect of strong correlations manifesting in a dynamical spectral weight transfer from the upper Hubbard band to the ZRS band. The quantitative agreement between theory and experiment highlights an additional spectral weight reshuffling due to core-hole interaction. Our results confirm the important correlated nature of the cuprates and elucidate the changing orbital character of the low-energy quasi-particles, but also demonstrate the continued relevance of the ZRS even in the overdoped region.' address: - '$^1$Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA' - '$^2$Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA' - '$^3$Department of Physics, Stanford University, Stanford, California 94305, USA' - '$^4$Department of Applied Physics, Stanford University, Stanford, California 94305, USA' - '$^7$Department of Physics and Astrophysics, University of North Dakota, Grand Forks, North Dakota 58202, USA' - '$^8$Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA' - '$^9$Center for Electronic Correlations and Magnetism, Theoretical Physics III, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany' author: - 'C.-C. Chen$^{1,2}$' - 'M. Sentef$^{1}$' - 'Y. F. Kung$^{1,3}$' - 'C. J. Jia$^{1,4}$' - 'R. Thomale$^{3,5,6}$' - 'B. Moritz$^{1,7,8}$' - 'A. P. Kampf$^9$' - 'T. P. Devereaux$^{1}$' title: 'Doping Evolution of Oxygen $K$-edge X-ray Absorption Spectra in Cuprate Superconductors' --- Introduction ============ Unraveling the nature of low-energy quasi-particles in cuprate superconductors is crucial to understanding their unconventional superconducting mechanism. Despite two decades of studies, however, the minimal model for describing the quasi-particle band which emerges upon doping remains unclear. Previous theoretical work has focused on the single-band Hubbard model[@singleband_Hubbard] or the $t-J$ model obtained by projecting out doubly-occupied charge configurations.[@tJ_Spalek; @ZRS_tJ] In these down-folded Hamiltonians, the fundamental quasi-particle has been assigned to the so-called Zhang-Rice singlet (ZRS):[@ZRS_tJ; @ZRS_George] a locally bound $d^9$ copper $3d_{x^2-y^2}$ hole hybridized with a doped ligand hole ($L$) distributed on the planar oxygen $2p_{x,y}$ orbitals \[Fig. \[fig:ClusterandZRS\](a)\]. The relevance of the ZRS in cuprate materials is supported by various spectroscopies: spin-resolved photoemission confirms the singlet character of the first ionization state,[@ZRS_SpinPhotoemission] and other probes show that doped holes in undoped or lightly-doped cuprates reside primarily on oxygens.[@PES_ZXShen; @XAS_Tranquada; @EELS_Nucker; @Compton] The doping evolution of oxygen content has been studied extensively by core-level x-ray absorption spectroscopy (XAS).[@XAS_CTChen; @XAS_CTChen_pz; @XAS_Pellegrin] In the insulating parent compounds, early oxygen $K$-edge XAS experiments (oxygen core-electron $1s\rightarrow 2p$ transition) show that below the main absorption edge a single “pre-peak" exists near 530 eV,[@XAS_CTChen] attributed to excitations into the upper Hubbard band (UHB). This nonzero projection of the UHB onto the oxygen XAS spectra mainly results from a strong hybridization between copper and oxygen, and the mixing of $d^9$ and $d^{10}L$ character in the ground state. Upon hole-doping, a lower-energy ($d^9L$) peak emerges near 528.5 eV and grows linearly with hole concentration, while the intensity of the higher-energy UHB decreases. Polarized x-ray studies indicate that the lower-energy peak in the underdoped regime shows almost no oxygen $2p_z$ content, but is dominated by planar oxygen $2p_{x,y}$ contributions, [@XAS_CTChen_pz; @XAS_Pellegrin] lending further credence to the ZRS picture. However, recent measurements report that while the low-energy ZRS weight increases linearly with doping over a wide doping range, it deviates from a linear trend and exhibits a weak doping dependence beyond $\sim20\%$.[@XAS_saturation_Schneider; @XAS_Peets] It was suggested that this “anomalous" behavior implies the inapplicability of the single-band Hubbard model and a breakdown of the ZRS picture.[@XAS_Peets] But cluster dynamical mean-field theory (DMFT) calculations argue that the single-band Hubbard model could explain the experimental finding, by showing a change in slope with doping of the integrated unoccupied density of states (DOS) beyond a certain doping level.[@DMFT_Comments] On the other hand, calculations for the more sophisticated three-orbital (copper-oxygen) Hubbard model with single-site DMFT find that the low-energy weight in the oxygen-projected DOS simply increases linearly with doping; the result contrasts with experiment and challenges even the validity of the three-orbital model.[@XAS_Wang] Currently, the debate continues and questions remain.[@XAS_Peets; @XAS_Wang; @DMFT_Comments; @DMFT_Liebsch; @XAS_Reply_1; @XAS_Reply_2] Can oxygen $K$-edge XAS be fully described by the Hubbard models? Are these model Hamiltonians and the ZRS picture suitable for the cuprates in the overdoped region? ![ (Color online) (a) Cartoon picture of a Zhang-Rice singlet formed by a copper $3d_{x^2-y^2}$ hole hybridized with a hole on its neighboring oxygen $2p_{x,y}$ orbitals. (b) The Cu$_8$O$_{16}$ cluster with periodic boundary conditions considered in the exact diagonalization calculations. (c) Schematic of the oxygen $K$-edge x-ray absorption process, highlighting additional spectral weight reshuffling due to core-hole interaction. []{data-label="fig:ClusterandZRS"}](ClusterandZRS.eps){width="\columnwidth"} In this work, we investigate these questions by tracking the doping evolution of photoemission and oxygen $K$-edge XAS spectra in the three-orbital Hubbard model. Using large-scale exact diagonalization, we treat many-body correlations exactly and include the effect of core-hole interaction explicitly, which is essential in the proper evaluation of the XAS cross sections. Our calculations reproduce the experimental data in a quantitative way and explain the crossover or apparent saturation in ZRS intensity in the overdoped regime, highlighting the dynamical spectral weight transfer from the UHB to the ZRS band expected for strongly correlated systems. The slope of the ZRS weight versus doping further decreases due to the presence of core-hole interaction. These results confirm the correlated, multi-orbital nature of the low-energy quasi-particles in the cuprates, but also indicate that the ZRS picture may indeed survive to high doping levels, albeit with an evolving orbital character. Methods ======= We consider the three-orbital Hubbard model on a square lattice of planar CuO$_2$ plaquettes containing copper $3d_{x^2-y^2}$ and oxygen $2p_{x,y}$ orbitals:[@Mattheiss; @Varma_threeband; @Emery_model] $$\begin{aligned} H&=&-\sum_{\langle i,j\rangle\sigma} t^{ij}_{pd}(d^\dagger_{i\sigma} p_{j\sigma}+h.c.) -\sum_{\langle j,j'\rangle\sigma} t^{jj'}_{pp}(p^\dagger_{j\sigma} p_{j'\sigma}+h.c.) \nonumber\\ &+&\sum_{i\sigma} \epsilon_d n^d_{i\sigma} + \sum_{j\sigma} \epsilon_p n^p_{j\sigma}+ \sum_i U_d n^d_{i\uparrow} n^d_{i\downarrow}+ \sum_j U_p n^p_{j\uparrow} n^p_{j\downarrow}. \nonumber\end{aligned}$$ Here $d^\dagger_{i\sigma}$ ($p^\dagger_{j\sigma}$) creates a hole with spin $\sigma$ at a copper site $i$ (oxygen site $j$), and $n^d_{i\sigma}$ ($n^p_{j\sigma}$) is the copper (oxygen) hole number operator. The first two terms of the Hamiltonian represent the nearest-neighbor copper-oxygen and oxygen-oxygen hybridization. The hopping integrals ($ t^{ij}_{pd}$ and $t^{jj'}_{pp}$) can change sign depending on the phases of the overlapping wavefunctions \[Fig. \[fig:ClusterandZRS\](a)\].[@Emery_model; @threeband_VCA] The third and fourth terms are the copper and oxygen site energies, and the last two terms are the on-site Hubbard interactions on copper and oxygen, respectively. We use the following parameters (in units of eV): $\epsilon_p-\epsilon_d=3.23$; $U_d=8.5$, $U_p=4.1$; $\|t_{pd}\|=1.13$, $\|t_{pp}\|=0.49$. These parameters have been employed to study the cuprate material La$_2$CuO$_4$ and produce various spectral features in good agreement with experiment.[@Ohta_parameters; @LCO_RIXS] Using exact diagonalization for a Cu$_8$O$_{16}$ cluster with periodic boundary conditions \[Fig. \[fig:ClusterandZRS\](b)\], we obtain the ground state and calculate the photoemission and XAS cross sections for four doping levels: 0% (undoped), 12.5% (underdoped), 25% (overdoped), and 37.5% (heavily overdoped). To calculate the spectral functions, we exploit translational symmetries of the problem and diagonalize the Hamiltonian matrices in momentum space. For oxygen $K$-edge XAS, the presence of local core-hole interaction breaks translational symmetry, and we perform the calculations in real space with a larger matrix size. The largest Hilbert space considered in this work contains $\sim5.7\times 10^9$ basis states.[@EDalgorithm] Results and Discussion ====================== Spectral Functions and Photoemission Spectra -------------------------------------------- Oxygen $K$-edge XAS promotes $1s$ electrons into empty $2p$ states \[Fig. \[fig:ClusterandZRS\](c)\], and the resulting spectrum is related to the unoccupied oxygen projected-DOS measured in inverse photoemission spectroscopy (IPES). To capture the doping evolution of electronic structures, we first discuss the orbitally-resolved spectral functions in Fig. \[fig:AKW\], where energy zero is defined by the valence-electron ground-state energy at the corresponding filling. At $0\%$ doping \[Fig. \[fig:AKW\](a)\], the spectra show an indirect gap of size $\sim$1.5 eV, related to charge-transfer excitation of moving a hole from copper to oxygen.[@CTinsulator_ZSA] Without doping, the first electron removal state appears at $(\pi/2,\pi/2)$, while the first electron addition state occurs at $(\pi,0)$. This indirect gap and its size are consistent with angle-resolved photoemission (ARPES) experiments.[@indirectgap_NCCO] In an indirect gap system, optical conductivity measurements would obtain a gap larger than the actual band gap, as no net momentum is transferred in the process. We find an optical gap $\sim1.7$ eV, also in agreement with the observed gaps (1.5-2.0 eV) in insulating cuprate parent compounds.[@OPC_Tokura; @OPC_Cooper; @OPC_Uchida; @OPC_Falck] ![ (Color online) Spectral functions calculated using exact diagonalization on a Cu$_8$O$_{16}$ cluster at (a) half-filling, (b) 12.5$\%$, and (c) 25$\%$ hole dopings. The ground state energy of the valence electrons at the corresponding filling is defined as zero energy. The solid and the dashed lines represent the electron occupied and unoccupied states, respectively. The spectra are broadened with a 0.1 eV Lorentizian. []{data-label="fig:AKW"}](AKW.eps){width="\columnwidth"} Upon hole-doping, low-energy peaks above the Fermi level emerge especially at momentum points along the magnetic Brillouin zone boundary. This emergent band \[near zero energy in Figs. \[fig:AKW\](b) and \[fig:AKW\](c)\] consisting of both copper and oxygen are associated with the ZRS; its orbital composition, spectral intensity, and energy position show strong momentum-dependence. In particular, the ZRS band contains vanishing oxygen weight at the $\Gamma$ point, due to a cancellation in the phase factor of the ZRS wavefunction,[@ZRSweight_Unger; @ZRSweight_Pothuizen; @ZRSweight_Yin] which is well-defined only away from the $\Gamma$ point. This is not captured in the single-band model and stresses the need for considering the multi-orbital nature in describing the ZRS doping evolution at a quantitative level. On the occupied side \[solid lines in Fig. \[fig:AKW\]\], the spectral features at $(\pi,0)$ and $(\pi/2,\pi/2)$ show distinct doping dependences: While the weight of the highest occupied state at $(\pi/2, \pi/2)$ gradually decreases upon hole-doping, the intensity increases for the highest occupied state at $(\pi,0)$. Moreover, the first electron removal state changes from momentum $(\pi/2,\pi/2)$ in the underdoped region to $(\pi,0)$ on the overdoped side,[@ARPES_SCOC_Kim; @ARPES_LCO_Ino] reminiscent of the observations from ARPES.[@Waterfall_footnote] This doping dependence cannot be accounted for by a simple chemical-potential shift in a rigid-band model, but is a manifestation of spectral weight redistribution due to correlation effects.[@dynamical_weight_singleband; @dynamical_weight_Eskes; @XAS_singleband; @dynamical_weight_Meinders; @dynamical_weight_PES] The occupied bands between roughly $-3$ to $-5$ eV are related to oxygen bands dispersing with $t_{pp}$[@NB_Mizuno] and the Zhang-Rice triplet.[@ZRT_Learmonth] In real cuprate materials, other orbitals (such as copper $3d_{3z^2-r^2}$ and non-bonding oxygens) also contribute to the spectral weight in this energy range.[@DOS_DMFT_Weber] Below -6 eV \[not shown\], we find a highly incoherent band containing mostly copper character, associated with the lower Hubbard band. These bands at deeper binding-energies show a relatively weak doping dependence in the doping range considered in this study. ![ (Color online) Calculations of oxygen $K$-edge XAS (solid red lines) and IPES measurements of unoccupied oxygen-projected DOS (black dotted lines) at various dopings. The yellow shaded area represents the emergent low-energy ZRS band. The XAS and IPES spectra are lined up in energy by their peak maximum positions. []{data-label="fig:OKXAS"}](OKXAS.eps){width="\columnwidth"} Oxygen $K$-edge X-ray Absorption Spectra ---------------------------------------- We next discuss in Fig. \[fig:OKXAS\] the oxygen $K$-edge XAS calculations, which include an additional oxygen $1s-2p$ core-hole interaction ($\sum_{i\sigma\sigma'} U_c n^{1s}_{i\sigma} n^{2p}_{i\sigma'}$) with $U_c=6$ eV.[@coreholestrength; @UQonXAS_Eskes; @OKXAS_1DUQ_Okada] The spectral peaks at $0\%$ doping are closely related to the UHB \[near 1-2 eV in Fig. \[fig:AKW\]\], which upon doping becomes more incoherent and shifts its weight to the lower-energy ZRS band. The energy separation between these two bands increases systematically with doping: the UHB disperses toward higher energy and the ZRS band moves in the opposite direction, in agreement with experiment.[@XAS_CTChen] Figure \[fig:OKXAS\] also shows the IPES calculations of the oxygen unoccupied DOS. The $K$-edge XAS lineshape resembles closely that of IPES, because the only core-hole interaction is a monopole term forming a simple charge density attraction due to the isotropic $1s$ orbital.[@LedgeXAS] However, the two spectra still vary quantitatively \[Fig. \[fig:XASZRSweight\]\].[@RIXS_Jia] In Fig. \[fig:XASZRSweight\] we compare our theory with oxygen $K$-edge XAS experiments on La$_{2-x}$Sr$_x$CuO$_4$ (LSCO).[@XAS_Peets] LSCO has a relatively simple crystal structure and is suitable for systematic studies of electronic properties over a wide doping range. As shown in Fig. \[fig:XASZRSweight\], the experimental ZRS weight increases roughly linearly with doping in the underdoped region. Near optimal doping, the rate of increase changes and exhibits a weaker doping dependence (smaller slope). It was suggested that the ZRS intensity may even saturate on the overdoped side ($\geq20\%$).[@XAS_saturation_Schneider; @XAS_Peets] ![ (Color online) Doping dependence of the ZRS spectral weight. The La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) XAS data are from Refs. (Chen), (Pellegrin), and (Peets). The XAS theory (red circle) is obtained by integrating the yellow shaded area in Fig. \[fig:OKXAS\]. The black circle represents the ZRS weight from the IPES calculation of the unoccupied DOS. Both theory curves are rescaled by normalizing their respective 12.5% ZRS weight to the straight line suggested by experiment, [@XAS_Peets] and the ZRS intensity at $0\%$ doping is defined as zero . []{data-label="fig:XASZRSweight"}](XASZRSweight.eps){width="\columnwidth"} For comparison, in Fig. \[fig:XASZRSweight\] we also plot the ZRS weight of the XAS and IPES calculations. We define the ZRS intensity as zero at $0\%$ doping and rescale both theory curves by normalizing their respective 12.5% ZRS weight to the straight line suggested by experiment.[@XAS_Peets] Neither calculation shows a saturation, but only a linear increase at low dopings and a change of slope at a higher doping level. This crossover in slope is related to the spectral weight transfer from the UHB to the ZRS band, which can be illustrated by considering the single-band Hubbard model in the atomic limit.[@dynamical_weight_singleband; @dynamical_weight_Eskes; @XAS_singleband; @dynamical_weight_Meinders; @dynamical_weight_PES] On a half-filled $N$-site lattice, the electron removal spectrum (associated with singly occupied states) and addition spectrum (associated with the unoccupied UHB) both have a spectral weight equal to $N$. Upon doping $x$ holes, there are $N-x$ singly occupied states, and the UHB weight also becomes $N-x$. Due to the conserved sum of the occupied and unoccupied states ($2N$), the low-energy spectral weight which emerges upon doping $x$ holes is thereby increased by 2$x$. In contrast, for an uncorrelated system the emergent spectral weight equals $x$ when doped with $x$ holes. We also note that an effective weakening in correlation strength and spectral weight transfer upon doping is absent in the $t-J$ model, which always behaves as a single-band Hubbard model in the atomic limit.[@dynamical_weight_Eskes; @OKXAS_Bansil] Although we do not find an apparent saturation of the ZRS weight with doping, our XAS calculation seems to capture well the experimental findings. In particular, the XAS theory curve matches the experimental data within error bars near optimal doping and in the overdoped regime; our theory also predicts the doping dependence of the heavily overdoped ($x\sim0.35$) LSCO sample. Compared with IPES ($U_c=0$), the core-hole attraction ($U_c=6$ eV) effectively overcomes the energy cost of double occupancy and shifts roughly an additional $\sim10\%$ of the UHB weight to lower energies. This effect is more prominent at low doping where a substantial UHB weight still remains, leading to a more pronounced change of slope in ZRS intensity in XAS. The IPES calculations underestimate the low-energy spectral weight at low dopings and overestimate the slope of the ZRS intensity with doping in the overdoped region. We note that Liebsch[@DMFT_Liebsch] and also Peets et al.[@XAS_Reply_2] have pointed out the importance of choosing a proper energy integration window for the ZRS band when comparing theory and experiment. Here we use a $\sim$2 eV energy window whose upper limit changes with doping and resides in the minimum between the centroids of the UHB and the ZRS band (see Fig. \[fig:OKXAS\]), consistent with experiment.[@XAS_Peets; @XAS_Reply_2] As mentioned previously, the doping evolution of the low-energy ZRS spectral weight has also been studied by various theoretical techniques.[@XAS_Wang; @DMFT_Comments; @DMFT_Liebsch; @XAS_singleband; @OKXAS_Bansil; @OKXAS_Okada; @XAS_FEFF] In particular, single-site DMFT using the three-orbital model shows a linear increase with doping of the low-energy weight.[@XAS_Wang] Cluster DMFT of the single-band model demonstrates a change in slope of the ZRS weight beyond a certain doping level,[@DMFT_Liebsch; @DMFT_Comments] which resembles the behavior of our calculations for the unoccupied DOS in the three-orbital model. However, the slope on the overdoped side will be overestimated in the single-band case,[@dynamical_weight_Meinders; @XAS_singleband; @DMFT_Comments] as there the UHB completely disappears close to 100% hole doping, while in the three-orbital model the $d^8$ double occupancy stays nonzero. Therefore, if the multi-orbital nature of the problem is neglected, an effective doping-dependent Coulomb repulsion needs to be imposed in the single-band model in order to quantitatively capture the spectral weight transfer in the overdoped region.[@OKXAS_Bansil] Conclusion ========== To summarize, our exact diagonalization calculation of the three-orbital Hubbard model agrees with the XAS experiments at a quantitative level, which relies crucially on the combined effect of correlation-induced spectral weight transfer and an additional spectral weight redistribution due to core-hole interaction. The change of slope in ZRS intensity as a function of doping manifests clearly the presence of strong electronic correlation. We also find a strong doping and momentum dependence of the orbital composition in the ZRS band, which cannot be fully described by the single-band Hubbard model especially in the overdoped region. Our results potentially confirm the continuation of the ZRS picture even in overdoped cuprates. Further measurements would be helpful to illuminate the nature of the emergent low-energy quasi-particles in cuprate superconductors. The authors acknowledge discussions with M. A. van Veenendaal, J. Fernandez-Rodriguez, and C.-Y. Mou. This work is supported by the U.S. Department of Energy (DOE), Basic Energy Sciences, Office of Science, under Contracts No. DE-AC02-76SF00515 and No. DE-AC02-06CH11357. C.C.C. is supported by the Aneesur Rahman Postdoctoral Fellowship at ANL. 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--- abstract: 'The interface between two band insulators LaAlO$_3$ and SrTiO$_3$ exhibits low-temperature superconductivity coexisting with an in-plane ferromagnetic order. We show that topological superconductivity hosting Majorana bound states can be induced at the interface by applying a magnetic field perpendicular to the interface. We find that the dephasing effect of the in-plane magnetization on the topological superconducting state can be overcomed by tuning a gate-voltage. We analyze the vortex-core excitations showing the zero-energy Majorana bound states and the effect of non-magnetic disorder on them. Finally, we propose an experimental geometry where such topological excitations can be realized.' author: - 'N. Mohanta' - 'A. Taraphder' title: 'Topological superconductivity and Majorana bound states at the LaAlO$_3$/SrTiO$_3$ interface' --- =1 Introduction ============ Majorana fermion, which often arises as a quasi-particle excitation in condensed matter systems, is being studied intensively due to its indispensable utility in defect-free topological quantum computation [@Alicea2011]. The Majorana bound state (MBS) naturally exists in spin-triplet chiral $p$-wave superconductivity in superfluid He$^3$ (A-phase) [@RevModPhys.69.645] and Sr$_2$RuO$_4$ [@Ishida1998] and in the fractional quantum Hall state at $5/2$ filling [@Moore1991362]. Also, there have been several proposals of experimentally feasible systems that host MBS such as quantum wire coupled to $s$-wave superconductor [@PhysRevLett.105.177002], semiconductor-superconductor heterostructure [@PhysRevLett.105.077001; @Mourik25052012], proximity-induced superconductor at the surface of topological insulator [@PhysRevLett.100.096407], 2DEG at semiconducting quantum well [@PhysRevB.81.125318], and the more promising Al-InSb nanowire topological superconductor [@Das2012]. Also there is a trend to realize topological orders in fermionic $s$-wave superfluids of ultracold atoms in optical lattices [@PhysRevLett.101.160401]. However, due to the lack of convincing experimental evidence so far, Majorana fermion still remains as elusive and its search, therefore, should expand on to uncharted routes, new systems and novel experimental designs. The two-dimensional electron liquid (2DEL) at the LaAlO$_3$/SrTiO$_3$ interface is formed as a result of an intrinsic electronic transfer mechanism known as the polar catastrophe in which half an electronic charge is transferred from the top of the polar LaAlO$_3$ to the terminating TiO$_2$ layer on the non-polar SrTiO$_3$ side to avoid a charge discontinuity at the interface [@Ohtomo2004; @Nakagawa2006]. The 2DEL becomes superconducting below $200$mK [@Reyren31082007; @GariglioJPCM2009] along with large magnetic moment ($\sim$0.3-0.4 $\mu_B$) aligned parallel to the interface plane [@Li2011; @Bert2011; @PhysRevLett.107.056802]. Suggestions for the mechanism responsible for the novel ferromagnetism include a RKKY interaction [@Michaeli_PRL2012], a double exchange process [@Banerjee2013] and Oxygen vacancies [@PhysRevB.86.064431] developed at the interface during the deposition process. On the other hand, there are proposals for phonon-mediated electron-pairing [@Michaeli_PRL2012; @Banerjee2013; @PhysRevB.89.184514] as well as unconventional superconductivity [@schmalian_arxiv2014; @chetan_PRB2013; @caprara_arXiv2013]. Another important feature of the interface is the Rashba spin-orbit interaction (SOI) which arises because of the broken mirror symmetry along perpendicular to the interface. A back-gate voltage can tune both the electron concentration and the Rashba SOI and therefore can drive a superconductor-insulator transition [@Caviglia2008; @CavigliaPRL2010] making the system a potential candidate for novel electronic devices [@Mannhart26032010]. Here we show that a magnetic field, applied perpendicular to the interface plane, can induce topological superconductivity that harbours gapless edge states and MBS at the core of a vortex. The intrinsic in-plane magnetization favours a finite momentum pairing and, therefore, weakens the topological superconducting phase. We show that by tuning the Rashba SOI (i.e. the back-gate voltage) the topological superconducting phase can be stabilized against the deterrent effect of the in-plane magnetization. We study the in-gap excitations and find that the zero-energy MBS located at the vortex-core is accompanied by low-energy particle-hole symmetric in-gap states of electronic origin. We study the effect of non-magnetic disorder on the low-energy excitations and observe that the MBS vanishes with moderate disorder. We propose an experimental set-up where the existence of MBS can be tested experimentally and discuss about some future directions. The large Rashba spin-orbit interaction (RSOI), arising from the broken inversion symmetry along the $\hat{z}$-direction, converts the $s$-wave superconductivity into an effective p$_x$ $\pm$ ip$_y$ one. The in-plane magnetization $h_x$ introduces asymmetry in the two-sheeted Fermi surface leading to finite-momentum pairing of electrons [@mohantaJPCM]. The main idea to get a topological superconductivity in two-dimensional $s$-wave superconductor with RSOI [@PhysRevB.77.220501] is to apply a large perpendicular Zeeman field $h_z$ to essentially remove one of the helicities of the RSOI-induced p$_x$ $\pm$ ip$_y$ states. One has to circumvent the deterrent effect of the in-plane magnetization to stabilize topological superconductivity in this system. In the present work, we predict that regardless of the asymmetry around the $\Gamma$-point in the Fermi surface, the single species p$_x$ $+$ ip$_y$ superconductivity still harbours a single MBS at the core of a vortex. We show that the tunable RSOI competes with the in-plane magnetization and restores the topological phase. However, we find that the excitation at the vortex core is highly sensitive to non-magnetic disorder; even a moderate disorder can destroy the MBS as the magnetization breaks time-reversal symmetry explicitly. As the interface in LAO/STO possesses intrinsic disorders like Oxygen vacancies, developed during the deposition process, it is indeed quite challenging to detect a Majorana fermion here. Some remedies and possible experimental requirements for realizing the MBS in this system are discussed. Model for interface 2DEL ======================== The electrons in the 2DEL occupy the three $t_{2g}$-bands (viz. d$_{xy}$, d$_{yz}$ and d$_{xz}$) of Ti atoms in the terminating TiO$_2$ plane giving rise to a quarter-filled ground state. Excess electrons supplied by the Oxygen vacancies or the back-gating accumulate on the next TiO$_2$ layer below the interface. Electrons in the $d_{xy}$ band are mostly localized at the interface sites due to Coulomb correlation. The electrons in the itinerant bands, in the TiO$_2$ layer just below the interface couple to the localized moments via ferromagnetic exchange leading to an in-plane spin-ordering of the interface electrons. The temperature variation of the gap is found to be BCS-like, $2\Delta_0/{k_BT_{gap}\approx3.4}$, where $\Delta_0$ is the pairing-gap amplitude at $T=0$, $k_{B}$ is the Boltzmann constant and $T_{gap}$ is the gap-closing temperature [@RichterNature2013] (the transition to the superconducting state is of BKT-type [@Caviglia2008]); it is therefore generally assumed that the itinerant electrons at the interface undergo conventional $s$-wave pairing, although there are suggestions of unconventional pairing as well [@schmalian_arxiv2014; @chetan_PRB2013; @caprara_arXiv2013]. The simple model describing the interface electrons, at the mean-field level, is $$\vspace{-1em} \begin{split} {\cal H}&=\sum_{\mathbf{k},\sigma}(\epsilon_{\mathbf{k}}-\mu) c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma}+\alpha \sum_{\mathbf{k},\sigma,\sigma^{\prime}} [\mathbf{g_k}\cdot \mathbf{\sigma}]_{\sigma,\sigma^{\prime}} c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma^{\prime}}\\ &-\sum_{\mathbf{k},\sigma,\sigma^{\prime}}[h_x \sigma_x]_{\sigma,\sigma^{\prime}}c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma^{\prime}}+\sum_{\mathbf{k}} (\Delta c_{\mathbf{k}\uparrow}^\dagger c_{-\mathbf{k}\downarrow}^\dagger+h.c.) \end{split} \label{model}\vspace{-1em}$$ where $\epsilon_{\mathbf{k}}$ = - 2$t$(cos $k_x$ + cos $k_y$) is the energy band dispersion with the hopping amplitude $t$ and chemical potential $\mu$, $\mathbf{g_k}=(\sin k_y, -\sin k_x)$ describes the RSOI of strength $\alpha$ and $\Delta=-<c_{\mathbf{k}\uparrow}c_{-\mathbf{k}\downarrow}>$ is the pairing gap and $\mathbf{\sigma}=[\sigma_x, \sigma_y, \sigma_z]$ being the Pauli matrices. Inducing topological superconductivity ====================================== The RSOI breaks the spin-degeneracy of the original bands and creates two new electronic bands while the in-plane magnetization $h_x$ shifts the Berry curvature from the $\Gamma$-point to P (0, $-h_x/\alpha$) point, thus making it energetically favorable for the electrons to pair up at finite center-of-mass momentum proportional to $h_x$. In the diagonal basis of the Rashba Hamiltonian, one essentially obtains p$_x$ $\pm$ ip$_y$ pairing symmetry of the superconductivity [@mohantaJPCM]. When an external Zeeman field, perpendicular to the interface 2DEL, ${\cal H}_Z=-h_z\sum_{\mathbf{k}}[c_{\mathbf{k}\uparrow}^{\dagger}c_{\mathbf{k}\uparrow} - c_{\mathbf{k}\downarrow}^{\dagger}c_{\mathbf{k}\downarrow}]$ is applied, a gap is opened at the point P. The pairing amplitudes in the newly created bands, $\epsilon_{\pm}(\mathbf{k})=\epsilon_{\mathbf{k}}^{\prime} \pm \xi$ are given by $$\vspace{-1em} \begin{split} &\Delta_{\pm \pm}=-\frac{\alpha}{2\xi} \Delta \left(\sin k_y \pm i\sin k_x \right): \text{intraband \textit{p}-wave}\\ &\Delta_{+-}=\frac{h_z}{\xi} \Delta: \text{interband \textit{s}-wave} \end{split}$$ where $\xi=\left(\alpha^2\|\mathbf{g_k}\|^2+h^2-2 \alpha h_x \sin k_y\right)^{1/2}$, $h^2=h_x^2+h_y^2$ and $\epsilon^{\prime}_{\mathbf{k}}=\epsilon_{\mathbf{k}}-\mu$. As shown in FIG. \[band\_fs\], when $h_z$ increases beyond a critical field $h_{zc}$, there is only one Fermi surface (i.e., one of the two helicities p$_x$ $\pm$ ip$_y$ is removed) and the superconductivity is transformed into a topological superconductivity. The effective Hamiltonian ${\cal H}_{eff}={\cal H}+{\cal H}_Z$ of the system can be written in the usual Nambu basis $\Psi(\mathbf{k})=[c_{\mathbf{k}\uparrow}, c_{\mathbf{k}\downarrow}, c_{-\mathbf{k}\downarrow}^{\dagger}, -c_{-\mathbf{k}\uparrow}^{\dagger}]$ as $$\vspace{-0.5em} \begin{pmatrix} \begin{array}{cc} {\cal H}_0(\mathbf{k}) & \Delta \\ \Delta & -\sigma_y {\cal H}_0(\mathbf{k}) \sigma_y^* \end{array} \end{pmatrix} \Psi(\mathbf{k}) = E_{\pm}(\mathbf{k}) \Psi(\mathbf{k})$$ where ${\cal H}_0(\mathbf{k})=\epsilon^{\prime}_{\mathbf{k}}+\alpha \mathbf{g_k \cdot \sigma}-h_z\sigma_z-h_x\sigma_x$ and we obtain the bulk spectrum $E_{\pm}^2(\mathbf{k})= \left( \epsilon^{\prime 2}_{\mathbf{k}}+\Delta^2+\xi^2 \right) \pm \zeta$ where $\zeta=\left[ \Delta^2h_z^2+\epsilon^{\prime 2}_{\mathbf{k}} \xi^2 \right]^{1/2}$. The transition to the topological state occurs only when the gap of the bulk spectrum closes i.e., when $\zeta= \epsilon_{\mathbf{k}}^{\prime 2}+\Delta^2+\xi^2$. This is satisfied (with $\Delta\not=0$) when either $\xi^2=h_z^2$ or $\epsilon_{\mathbf{k}}^{\prime 2}+\Delta^2=h_z^2$ which essentially reduces to the familiar relation $h_z = \sqrt{\Delta^2+\mu^2}$ [@PhysRevB.81.125318] when $h_x=0$. As $h_z$ increases beyond this transition point, a topologically protected excitation gap $E_g$ (given by the minimum of $E_-(\mathbf{k})$), proportional to the Rashba coupling strength $\alpha$ in the limit of small $h_z$, is induced. Therefore, $E_g$ keeps track of the quantum phase transition from ordinary superconductivity to topological superconductivity. The phase diagrams of the system, revealing the parameter regime in which the topological superconductivity can be achieved, is shown in FIG. \[phase\]. There are two competing energy gaps, the minimum of which tries to destroy the topological state; one is the induced bulk excitation gap at the Fermi level ($\mathbf{k}=\mathbf{k_F}$), $\Delta_{FS}=2\Delta_{--}$, the other is the Zeeman gap at the point P, given by $\Delta_z=E_-(0,-h_x/{\alpha})$. In addition, the in-plane magnetization, by introducing a finite-momentum pairing, weakens the topological superconductivity. However, the RSOI competes with the in-plane magnetization to restore the topological state, which is clearly visible in the phase diagram, FIG. \[phase\]. Since the RSOI in LaAlO$_3$/SrTiO$_3$ interface is tunable by external gate voltage [@CavigliaPRL2010], it can be tuned to stabilize the topological state. Vortex core excitations ======================= The induced topological superconductivity exhibits edge states and zero-energy MBS at the core of a vortex [@Nagai_JPSJ2014]. Since Majorana fermions are essentially half an ordinary fermion, they always come in pair, generally located in different vortex-cores. A system having only one vortex, hosts the second Majorana fermion at the boundary. The topological property of the gapless excitation at the boundary is connected to the bulk topological state as a consequence of the “bulk-boundary correspondence”. In the following we study the vortex core states by solving the following effective BdG Hamiltonian in real space. $$\vspace{-1.5em} \begin{split} {\cal H}&=-t^{\prime}\hspace{-0.8em}\sum_{<ij>,\sigma}c_{i\sigma}^\dagger c_{j\sigma}-\mu\sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma} -\hspace{-0.5em}\sum_{i,\sigma,\sigma^{\prime}}(\mathbf{h \cdot \sigma})_{\sigma \sigma^{\prime}}c_{i\sigma}^\dagger c_{i\sigma^{\prime}}\\ &-i\frac{\alpha}{2}\hspace{-0.5em}\sum_{<ij>,\sigma,\sigma^{\prime}}\hspace{-0.8em}(\mathbf{\sigma}_{\sigma \sigma^{\prime}} \times \mathbf{\hat{d}}_{ij})_z c_{i\sigma}^{\dagger}c_{j\sigma^{\prime}} + \sum_{i}(\Delta_i c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}+h.c.) \end{split}\vspace{-1em} \label{HBdG}$$ where $t^{\prime}$ is the hopping amplitude of electrons on a square lattice, $\mathbf{h}=(h_x, 0, h_z)$, $\mathbf{\hat{d}}_{ij}$ is unit vector between sites $i$ and $j$, and $\Delta_i=-U (c_{i\uparrow}c_{i\downarrow})$ is the onsite pairing amplitude with $U$, the attractive pair-potential. The Hamiltonian (\[HBdG\]) is diagonalized via a spin-generalized Bogoliubov-Valatin transformation $\hat{c}_{i\sigma}(r_i)=\sum_{i,\sigma^{\prime}}u_{n\sigma\sigma^{\prime}}(r_i)\hat{\gamma}_{n\sigma^{\prime}}+v_{n\sigma\sigma^{\prime}}^(r_i)\hat{\gamma}^{\dagger}_{n\sigma^{\prime}}$ and the quasi-particle amplitudes $u_{n\sigma}(r_i)$ and $v_{n\sigma}(r_i)$ are determined by solving the BdG equations: ${\cal H}\phi_n(r_i)=\epsilon_n\phi_n(r_i)$ where $\phi_n=[u_{n\uparrow}(r_i),u_{n\downarrow}(r_i),v_{n\uparrow}(r_i),v_{n\downarrow}(r_i)]$. To model a vortex, we use open boundary conditions and solve self-consistently the BdG equations by taking an initial ansatz for the gap as $\Delta_j=\Delta_0r_je^{i\phi_j}/\sqrt{r_j^2 + 2r_c^2}$, where ($r_j, \phi_j$) are the polar coordinates of site $j$ with respect to the core at the center of the 2D plane, $\Delta_0$ and $r_c$ are respectively the depth and size of the vortex core. The results, presented here, are for a $40\times40$ system with vortex-size $r_c=1$. For the rest of the paper, we set $\mu=0$, $t^{\prime}=1$ and $U=4.0$, unless explicitly specified. In FIG. \[vortex\](a), we plot the local density of states (LDOS) at the vortex core, given by $\rho(\epsilon,r_i)=\frac{1}{N}\sum_{n,\sigma}[\|u_{n\sigma}(r_i)\|^2\delta (\epsilon-\epsilon_n)+\|v_{n\sigma}(r_i)\|^2\delta(\epsilon+\epsilon_n)]$ for various in-plane field $h_x$, $N$ being the total number of lattice sites. Evidently, the zero-bias Majorana mode is accompanied by other low-energy vortex-bound-states which generally appear at a vortex core in conventional superconductors and are known as the Caroli-de Gennes Matricon states [@Caroli_1964]. These states exist in both the normal and topological superconducting phase. With increasing $h_x$, the fermionic states move away and mix with the bulk bands and the MBS vanishes suddenly as $h_x$ reaches the critical value for transition to the trivial superconducting state. The Majorana excitation at the vortex core is shown in FIG. \[vortex\](b). As shown in FIG. \[vortex\](c), the bulk band-gap reduces slowly with increasing $\alpha$ and the low-energy fermionic excitations merge with the MBS. Remarkably, similar features of the LDOS, for various gate voltages, have been seen in the tunneling spectra obtained at 50 mK using Au-gated LAO/STO tunnel device [@Richter_thesis]. FIG. \[vortex\](d)-(e) show three regimes: regimes I, II and III are all superconducting: region II has a magnetized vortex and only region III has non-trivial superconductivity and the Majorana modes (two red lines at zero energy for the vortex-core and edge excitations). Region II is quite interesting as it has a zero energy fermionic excitation close to the boundary of I which arises in this trivial superconducting state from the competition between the Zeeman field that magnetizes the vortex and the superconductivity, thereby changing the vortex structure [@SchafferPRB2013]. The spacing between the in-gap bound states is typically of the order of $\Delta^2/E_F$, where $\Delta$ is the gap amplitude, $E_F$ is the Fermi energy. In the present situation, the critical temperature of the real system is not proportional to the mean-field ’gap’ we are using [@RichterNature2013]: it is dictated primarily by the coupling between the superconducting grains and therefor much smaller (200 mK) than would otherwise appear from the gap values. In the tunneling spectrum, the position of these in-gap bound states is thus not completely determined by the real T$_c$ (200mK). However, while tuning the perpendicular magnetic field ($h_z$) or the gate-voltage (i.e., Rashba SOI strength $\alpha$), the abrupt transition from region II to region III, in FIG.3(d)-(e), can also give rise to a stronger, unique experimental signature of Majorana bound states. Influence of disorder ===================== The topological excitations are, however, quite fragile against the imperfections of the host system. The LaAlO$_3$/SrTiO$_3$ interface contains intrinsic disorder such as Oxygen vacancies, known to have significant effects on the interface electrons [@PhysRevB.86.064431; @MohantaVacancyJPCM2014]. It is, therefore, imperative to study the robustness of the MBS against non-magnetic disorder, introduced through an onsite random potential $V_d$ in the Hamiltonian (\[HBdG\]) by ${\cal H}_{dis}=V_d \sum_{i,\sigma} c_{i\sigma}^\dagger c_{i\sigma}$, where, $V_d \in [-W, W]$ uniformly. In FIG. \[dis\], we plot the density of states $\rho(\epsilon)=\frac{1}{N}\sum_{n,r_i,\sigma}[\|u_{n\sigma}(r_i)\|^2\delta (\epsilon-\epsilon_n)+\|v_{n\sigma}(r_i)\|^2\delta(\epsilon+\epsilon_n)]$ for disorder realizations of various disorder strength $W$. The MBS is quickly destroyed as disorder increases and other in-gap excitations appear within the bulk gap due to the defects. The MBS is, in fact, not expected to be robust against perturbations like disorder in this system since the time-reversal symmetry is already broken explicitly [@PhysRevB.83.184520]. As shown in FIG. \[dis\], the vulnerability of the zero-energy MBS is worse in presence of larger magnetic fields. In other words, the MBS survives against larger strength of disorder when $h_z$ is smaller (provided, $h_z$ should always be greater than the critical value $h_{zc}$ to ensure topological regime). In the present system, the degree of vulnerability is severe due to the in-plane magnetization which, in reality, weakens the topological state. Hence the low-energy excitations are destroyed even when there is a finite superconducting gap. In FIG. \[dis\], we show a situation where disorder of random strength (up to $W$) is present at all sites. We also consider a diluted situation by putting disorder at some random sites. FIG. \[dis1\](a) describes how the low-energy in-gap excitations are affected as the disorder concentration ($N_d$) is varied. We find that the results are not different qualitatively from that in FIG. \[dis\]. The zero-energy Majorana excitation remains unaffected unless a defect potential appears exactly at the vortex-core and, as in FIG. \[dis\], new states appear within the bulk-gap. To check where these defect-induced states are localized, we plot, in FIG. \[dis1\], the quasi-particle density and LDOS spectra at a defect site. We find that, as reported previously [@Nagai_arXiv2014], the new in-gap states are actually located at the defect sites. Therefore in the tunneling conductance measurement, getting an excitation at zero-energy does not necessarily confirm a Majorana particle. One has to be very careful in disentangling the Majorana fermion from defect induced states or Andreev bound states [@Stanescu_PRB2013]. Experimental aspects ==================== For the experimental realization of topological superconductivity and MBS in the interface 2DEL, an external Zeeman field $h_z$ is required. This can be achieved by mounting the LaAlO$_3$/SrTiO$_3$ interface on top of a ferromagnetic insulator whose spins are aligned to the $\hat{z}$-direction, as shown in FIG. \[design\]. The two major obstacles towards a realization of the topological state however, are (i) the intrinsic in-plane magnetization and (ii) the intrinsic defects. To stabilize the topological state, the effect of the in-plane magnetization can be partially offset by tuning the RSOI via the gate voltage. To reduce the laser-induced defects (in pulsed laser deposition method) and intrinsic disorder such as the Oxygen vacancies, molecular beam epitaxy techniques are used instead. Ozone may be used as oxidant instead of Oxygen during the annealing process as suggested by Warusawithana et al.* [@Warusawithana2013]. High resolution scanning tunneling microscopy (STM) or point-contact spectroscopy may identify a single MBS at the vortex core. Typical experimental resolution in the tunneling experiments is about 2$\mu$V and should be sufficient to identify the zero-bias peak of the Majorana BS from the nearby excitations. Though the mean-field gap is known to be a gross over-estimation, a rescaling of the gap in FIG. \[vortex\](c) on to the experimentally observed gap, about 80 $\mu$V, gives a value of the resolution-limit about 13 $\mu$V, close to the experimental values ($\sim 9.5\mu$V) observed [@Richter_thesis]. It is worth mentioning that the usual temperature range, in which the thermal fluctuation is small for the detection of the MBS, is less than 100 mK [@Mourik25052012; @Das2012] which is far below the Curie temperature (200 mK) of this interface superconductivity. Recently, it has been shown that superconductivity is possible in quasi-1D structures, grown at the LaAlO$_3$/SrTiO$_3$ interface [@2010arXiv1009.2424C; @2012arXiv1210.3606V] and may support Majorana zero modes at the ends of such quantum wires [@PhysRevB.84.195436]. These are the steps towards developing qubits, using this interface, which is to be used as the building blocks of a topological quantum simulator. 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--- abstract: 'The topological approach to baryon-antibaryon production in the chiral phase transition is numerically simulated for rapidly expanding hadronic systems. For that purpose the dynamics of the effective chiral field is implemented on a space - rapidity lattice. The essential features of evolutions from initial ’hot’ configurations into final ensembles of (anti-)baryons embedded in the chiral condensate are studied in proper time of comoving frames. Typical times for onset and completion of the roll-down and exponents for the growth of correlations are extracted. Meson and baryon-antibaryon yields are estimated. For standard assumptions about initial coherence lengths they are compatible with experimental results.' address: 'Siegen University, 57068 Siegen, Germany' author: - 'G. Holzwarth[^1]' title: 'The topological approach to baryon-antibaryon and meson production in rapidly expanding Bjorken rods.' --- Introduction ============ The topological approach to baryon structure and dynamics in the framework of an effective action for mesonic chiral fields has achieved a number of remarkable successes. The soliton concept [@Skyrme] for baryons provides an impressive account of spectrum and properties of baryon resonances (essentially without numerous ’missing resonances’) [@Schwesinger], with predictive power that recently has even led to the first indications for pentaquarks [@Nakano]. Model-independent relations between T-matrix elements for meson-baryon scattering [@Hayashi] and explicit results for specific channels are well supported by experimental data [@Mattis]. The matrix element of the axial singlet current related to the spin content of the proton is naturally of the observed order of magnitude [@Brodsky]. The ’unexpected’ behaviour recently found [@Jones] in the ratio of electric and magnetic proton form factors has been predicted in this approach long ago [@Ho96]. The underlying chiral effective action is profoundly based on the $1/N_c$-expansion of QCD [@Witten], preserving all relevant symmetries. Efforts to include next-to-leading order quantum corrections have brought substantial improvement as expected [@Mouss]. The manifestations of a chiral phase transition pose another natural challenge for an effectice theory with a ground state that is characterized by spontaneously broken symmetry. The possible formation of disoriented domains [@Bjorken] during the growth of the chiral condensate has been in the focus of interest for some time. But signatures in terms of anomalous multiplicity ratios for differently charged pions have not been observed [@Bearden], in accordance with theoretical conclusions [@Gavin; @HoKl02]. Anomalies in anti-baryon production were very early recognized as possible signals for interesting dynamics [@DeGrand] in that phase transition, and the concept to consider baryons as topological solitons in a chiral condensate should lead to quite definite expectations for this process. Meanwhile, in relativistic heavy-ion collisions at RHIC, very high energy densities are being produced in extended spatial regions which are essentially baryon free and well separated in rapidity from the nuclear slabs receding from the collision volume. The experimental values found in the central rapidity region for the ratio of the integrated ${\bar p}$ to $\pi^-$ yields lies between 0.065 and 0.075 [@phenix]. This is still too close to the thermal equilibrium ${\bar p}/\pi^-$ ratio (for a typical plasma temperature of $\Te \sim 200$ MeV), /\^- \~2 ((m\_-m\_p)/)=0.035 to constitute a clear indication for interesting underlying physics. Still, although the experimental result does not look very exciting, it still poses a constraint for the possible validity of the soliton concept, because any conceivable dynamical production process must be able to produce a comparable number. In the topological approach the number of baryon-antibaryon pairs produced during the chiral phase transition depends on two factors: the first is the modulus $\|\rho\|$ of the average winding density present in the initial ’hot’ field configuration. In analogy to applications in cosmology [@Kibble] and condensed matter systems [@Zurek] this quantity is closely related to the coherence length for the local orientations of the chiral field $\Bphi$. Without detailed knowledge about the initial field configurations this coherence length enters as a parameter and takes away stringent predictive power from the approach. However, different conjectures about the nature of the initial field ensemble suggest typical ranges for the coherence lengths which then may be discriminated by the experimentally observed abundancies. The second factor is the reduction of the initially present total $n_i=\int \|\rho\| dV$ through the dynamical ordering process, which finally leads to the formation of stable soliton structures embedded in the topologically trivial ordered chiral condensate of the ’cold’ system. The solitons or antisolitons evolve from topological obstacles which are met by the aligning field orientations, and develop into their stable ’cold’ form during the course of the evolution. At the end, the same integral $n_f=\int \|\rho\| dV$ counts the number of finally surviving nontrivial separate structures, so it is identified with the number of baryons and antibaryons created in the process. The decrease of $n$ during the roll-down is reasonably well represented by a power law $(\tau/\tau_0)^{-\gamma}$ and the exponent $\gamma$ can be measured in numerical simulations. Evidently, the initial time $\tau_0$ which marks the onset of the evolution, enters here as a second parameter which further reduces the predictive power of the approach. Fortunately, it turns out that $\gamma$ is rather small, so the dependence on $\tau_0$ is only weak. Measuring $\gamma$ and the time $\tau_f$ when the roll-down is completed, presents a typical task for numerical simulations once the equation of motion (EOM) which governs the field evolutions is implemented on a lattice. The underlying effectice chiral action is known from other applications, so no additional parameters enter at this point. In condensed matter applications, a phase transition is generally driven by an externally imposed quench, or by a dissipative term included in the EOM. In cosmology or in our present heavy-ion application it is the rapid expansion of the hot volume which drives the cooling process. This expansion is efficiently implemented [@Huang] by transforming to rapidity - proper-time coordinates, i.e. by boosting to the local comoving frame. This is especially convenient if we consider a system that expands only in one (longitudinal) direction with its transverse scales unchanged, (the Bjorken rod). The resulting dilution of the longitudinal gradients drives the system towards its global minimum. However, as there is no genuine dissipation in the system, the total energy approaches a constant which resides in the chiral fluctuations around the global minimum. Thus, the simulations also allow to estimate pion- or sigma- meson abundancies. Naturally, before the field configurations can roll down towards the global minimum, the potential $V(\Phi^2,\Te)$ which underlies the EOM must have changed from the ’hot’ chirally symmetric form to its ’cold’ symmetry-violating form. But, during the early stages, the evolutions are dominated by local aligning of the field orientation $\hphi$. During this phase the form of the potential is not important. So its time dependence can be replaced by a sudden quench where the ’hot’ field configuration is exposed to the ’cold’ potential $V(\Phi^2,\Te=0)$, from the outset at initial time $\tau_0$ . In the following, for definiteness we make use of this sudden quench approximation, (although the simulations, of course, allow to study other cases as well). For the sake of simplicity we first discuss all relevant features for the case of the 2-dimensional $O(3)$-model, with only one spatial dimension transverse to the longitudinal rapidity coordinate. Except for computational complexity the extension to the 3-dimensional $O(4)$-field presents no essential new features. The effective action, its transformation to the Bjorken frame, and the resulting EOM are presented in section II. It is important for the choice of the initial ensemble of field configurations that it allows in a convenient way to monitor the initial coherence lengths because they are the crucial parameters for the final baryon-antibaryon multiplicities. We choose an isotropic Gaussian random ensemble of field fluctuations in momentum space which is characterized by a temperature-like parameter to be able to compare with other approaches. Of course, this is not necessary. In fact, even at initial time $\tau_0$ the longitudinally expanding Bjorken rod need not be an isotropic system, and it may be physically justified to distinguish already in the initial ensemble two different, longitudinal and transverse correlation lengths. This is easy to incorporate, but in section III we present initial conditions which are locally isotropic. As discussed elsewhere [@Ho03] stable solitons shrink in a spatially expanding frame. Therefore, lattice implementations of their dynamics will necessarily involve lattice artifacts after some time. These are discussed in section IV. They can be isolated and subtracted from the physically interesting quantities. In section V the essential features of typical evolutions are discussed. Estimates for the times of onset and completion of the roll-down are obtained, and the dynamical exponents for the growth of correlation lengths and decrease of defect number are established and compared. The spectrum of the fluctuations remaining after the roll-down is considered and finally the mesonic and baryonic multiplicities are obtained. The extension to the physically interesting 3+1-dimensional $O(4)$-field is discussed in section VI. The topological generalization is well known, the additional transverse dimension is of little influence for the growth exponents. However, the coupling constants in the effective action here are related to physical quantities, so they are known (except for some uncertainty concerning the $\sigma$-mass), and the results can be compared with experimentally determined abundance ratios. Of course, it would be desirable to obtain a very definite answer whether the topological approach to antibaryon production in a chiral phase transition is validated or ruled out by the data. However, with our poor knowledge about the initial conditions in the hot plasma after a heavy-ion collision, we cannot expect much more than allowed ranges for the relevant parameters, which hopefully overlap with standard ideas about coherence lengths and formation times. The effective action in the Bjorken frame. ========================================== For simplicity we first discuss the 2+1 dimensional $O(3)$ model. It is defined in terms of the dimensionless $3$-component field $\Bphi = \Phi \hphi$ with unit-vector field $\hphi$, ($\hphi \cdot \hphi =1$), and modulus (’bag’-)field $\Phi$, with the following lagrangian density in $2+1$ dimensions $(x,z,t)$ \[lag\] = f\_\^2(\^[(2)]{}+\^[(4)]{}+\^[(0)]{}) ($f_\pi^2$ is an overall constant of dimension \[mass$^1$\], so the physical fields $f_\pi\Bphi$ are of mass-dimension \[mass$^{1/2}$\]). The second-order part ${\cal{L}}^{(2)}$ comprises the kinetic terms of the linear $\sigma$ model \[L2\] \^[(2)]{}= \_\^, ${\cal{L}}^{(4)}$ is the four-derivative ’Skyrme’-term (which involves only the unit-vector field $\hphi$) defined in terms of the topological current $\rho_\mu$ \[top\] \^= \^ ( \_\_) , (which satisfies $\partial_\mu \rho^\mu = 0$) \[L4\] \^[(4)]{}=-\^2 \_\^= -, and ${\cal{L}}^{(0)}$ contains the $\Phi^4$ potential and an explicit symmetry-breaker in 3-direction \[pot\] \^[(0)]{}=-V(,)=-( ( \^2-f()\^2 )\^2 - H \_3 ) - with dimensionless coupling constants $\lambda$ and $H$, and \[f2\] f\^2()= f\_0\^2()-. This choice ensures that the global minimum of the potential $V(\Bphi,\Te)$ is always located at $\Bphi_0=(0,0,f_0(\Te))$. Generically, the function $f_0^2(\Te)$ decreases from $f_0^2=1$ at $\Te=0$ towards zero for large $\Te$. The constant in the potential (\[pot\]) is chosen such that the value of the potential $V$ at $\Bphi=0$ is independent of $\Te$, (given by the constant $V(0,\Te)=(\lambda +2H)/(4\ell^2)$), and at the $(\Te=0)$-minimum $\Bphi=\Bphi_0=(0,0,1)$ we have $V(\Bphi_0,\Te=0)=0$. The masses of the $\pi$- and $\sigma$-fluctuations $(\pi_1,\pi_2, f_0+\sigma)$ around this minimum are \[masses\] m\_\^2=, m\_\^2= + m\_\^2. Without explicit symmetry breaking, $H=0$, we assume that $f^2(\Te)$ changes sign at $\Te=\Te_c$, such that $\Bphi_0=(0,0,0)$ and $m_\sigma^2=m_\pi^2=m^2=\lambda \|f^2\| / \ell^2$ for $\Te>\Te_c$. The parameter $\ell$ (with dimension of a length) which we have separated out from the coupling constants of potential and Skyrme term can be absorbed into the spatial coordinates $\Bx$. So it characterizes the spatial radius of stable extended solutions (which scales like $1/\sqrt{f^2}$). As $\ell$ simply sets the spatial scale, it could be put equal to one, as long as no other (physical or artificial) length scales are relevant. For lattice implementations, however, the lattice constant $a$ and the size of the lattice $(Na)$ set (usually unphysical) scales. To avoid artificial scaling violations we have to ensure that the size of physical structures (like solitons) is large as compared to the lattice constant $a$ and small as compared to the lattice size $Na$. So, for numerical simulations we have to choose $1\ll \ell/a \ll N$. It has been shown in ref.[@HoKl01] that for solitons which extend over more than at least 4-5 lattice units the energy $E_B$ is independent of $\ell/a$. So, in the following we will adopt $\ell/a\sim 5$ as sufficiently large. This appears also as physically reasonable, if we consider typical lattice constants of 0.2 fm and baryon radii of about 1 fm. On the other hand this will require lattice sizes of at least $N \sim 50$ to avoid boundary effects for the structure of individual solitons. Unfortunately, in the Bjorken frame which we shall use in the following, the longitudinal extension of stable solitons shrinks like $\tau^{-1}$ as function of proper time $\tau$. This means that after times of order $\ell$ the simulations will be influenced by lattice artifacts, which may even dominate for large times. For rapid expansion in (longitudinal) $z$-direction we perform the transformation from $(z,t)$ to locally comoving frames ($\eta$,$\tau$) with proper time $\tau$ and rapidity $\eta$, defined through $$\begin{aligned} \label{trafo} t&=&\tau \cosh\eta,~~~~~~~~~~\tau~=~\sqrt{t^2-z^2},~~~~~~~~~~~~~ \partial_t~=~\cosh\eta\; \dt -\frac{\sinh\eta}{\tau}\dz \nonumber\\ z&=&\tau \sinh\eta,~~~~~~~~~~\eta~=~\mbox{atanh}\left(\frac{z}{t}\right),~~~~~~~~~~~~ \partial_z~=~-\sinh\eta\; \dt +\frac{\cosh\eta}{\tau}\dz.\end{aligned}$$ Inserting (\[trafo\]) into (\[L2\]) and (\[L4\]) leaves the form of ${\cal{L}}^{(2)}$ and ${\cal{L}}^{(4)}$ invariant, with $\partial_t$ replaced by $\dt$, and $\partial_z$ replaced by $\frac{1}{\tau}\dz$. The specific structure of the Skyrme term again eliminates all terms with four $\tau$- or $\eta$-derivatives. For the effective action we take the integration boundaries from $-\infty$ to $+\infty$ for rapidity $\eta$ and for the transverse coordinate $x$. The 3-dimensional space-time volume element $dx\:dz\:dt$ is replaced by $\tau \:dx \:d\eta \:d\tau$. Therefore, in a separation of the action ${\cal S}$ in kinetic terms $T$, gradient terms $L$, and the potential $U$, \[S\] [S]{}=d\_[-]{}\^[+]{} ddx =(T\_+T\_-L\_-L\_-U) dthe longitudinal $\\|$-terms involving rapidity gradients carry a factor $1/\tau$, while all other terms carry a factor $\tau$. So we have T\_&=&{ () +\^2}ddx,\[int1\]\ T\_&=& { \^2}ddx,\[int2\]\ L\_&=&{ (\_x\_x) }ddx,\[int3\]\ L\_&=& { () +\^2}ddx, \[int4\]\ U&=&{( \^2-f\^2 )\^2 - \_3 +}ddx. \[int5\] Variation of ${\cal S}$ with respect to $\Bphi$ leads to the equation of motion (EOM). The contributions of ${\cal{L}}^{(4)}$ to the longitudinal and transverse parts $T_{\\|}$ and $T_{\bot}$ of the kinetic energy cause certain numerical difficulties for the implementation of the EOM on a lattice. They require at every timestep the inversion of matrices which depend on gradients of the unit-vectors $\hphi$, which multiply first and second time-derivatives of the chiral field. This can be troublesome in areas where the unit-vectors are aligned, and can be poorly defined in regions where the unit-vectors vary almost randomly for next-neighbour lattice points (i.e. for initially random configurations, or near the center of defects). In any case, stabilizing the evolutions requires extremely small timesteps and leads to very time-consuming procedures. Although these problems can be handled, we have compared the results with evolutions where the kinetic energy is taken from ${\cal{L}}^{(2)}$ alone. For coupling strenghths $\lambda\ell^2$ within reasonable limits, we find that the resulting differences do not justify the large additional expense caused by the fourth-order kinetic contributions. Evidently, the reason is, that the EOM determines the field-velocities (depending on the functional form of the kinetic energy) in such a way that the numerical value of the total kinetic energy is not very sensitive to its functional form. We therefore use in the following an effective action where the kinetic terms (\[int1\]) and (\[int2\]) are replaced by \[Tkin\] T\_=()ddx, T\_=0. With this simplification the EOM is \[EOM\] \_+\_-\_[xx]{}- \_+(\^2-f\^2)- \_3 +=0. This form has the big advantage that we can make use of the geometrical meaning of the winding density $\rho_0$ as the area of a spherical triangle, bounded by three geodesics on a 2-dimensional spherical surface. In closed form it is expressed through the unit-vectors pointing to its corners, and does not involve gradients. So this allows for a very accurate and fast lattice implementation of the last term in the EOM. Initial configurations ====================== We assume that at an initial proper time $\tau_0$ the system consists of a hadronic fireball with energy density $\varepsilon_0$ stored in a random ensemble of hadronic field fluctuations. Subsequently, for $\tau > \tau_0$, it is subject to EOM (\[EOM\]). The initial condition and the symmetry of the action imply boost invariance, i.e. the system looks the same in all locally comoving frames, so it is sufficient to consider its dynamics in a rapidity slice of size $\Delta\eta$ near midrapidity $\eta=0$, which constitutes a section of the initially created Bjorken rod with transverse extension ${\cal A}$. The energy $E=T+L+U$ in this slice then is given by an $\eta$-integral which extends over the finite rapidity interval $\Delta\eta$ and represents the energy contained in a comoving volume ${\cal V}= \tau \Delta\eta {\cal A}$. Due to the symmetry of the initial condition this comoving volume grows with increasing proper time $\tau$ into spatial regions with high energy density, therefore $E$ contains contributions which increase with $\tau$. The average energy density $\varepsilon=E/{\cal V}$ satisfies $d\varepsilon/d\tau \leq 0$. For numerical simulations we implement the configurations $\Bphi(x,\eta,\tau)$ on a rectangular lattice $(x,\eta) = (ia,jb)$ ($i,j=1...N$) with lattice constants $a$ for the transverse coordinate and $b$ for the rapidity lattice. We define the initial configurations $\Bphi_{ij}$ at the lattice sites ($i,j$) as Fourier transforms of configurations $\tilde {\Bphi}_{kl}$ on a momentum lattice \[Fourier\] \_[ij]{}=\_[k,l=-N/2+1]{}\^[N/2]{} ( e\^[(ik +jl)]{} \_[kl]{}+ c.c.), with $\tilde{\Bphi}^_{kl}=\tilde{\Bphi}_{-k-l}$. Inversely, the real parts $\Bal_{kl}$ and the imaginary parts $\Bet_{kl}$ of $\tilde {\Bphi}_{kl}$ are obtained from the real configuration $\Bphi_{ij}$ through \[albet\] \_[kl]{}& =&\_[i,j=1]{}\^[N]{}(i k +j l)\_[ij]{}=\_[-k-l]{}\ \_[kl]{} &=&-\_[i,j=1]{}\^[N]{}(i k +j l)\_[ij]{}=-\_[-k-l]{}, so we obtain the spectral power $P_{pq}$ of the configurations (or a specific component of it) at any time $\tau$ from P\_[pq]{}=\_[kl]{}\_[kl]{}\^\=\_[kl]{}\_[kl]{}+\_[kl]{}\_[kl]{} for any transverse or longitudinal momentum $(p,q)= \frac{2\pi}{aN}(k,l)$, for $(k,l = -N/2+1,..., N/2)$. For the initial configurations at $\tau=\tau_0$ the real and imaginary parts of each of the three components of $\tilde {\Bphi}_{kl}$ at each momentum-lattice point ($p,q$) are chosen randomly from a Gaussian deviate $G_{kl}(\tilde{\Phi})$ with $kl$-dependent width $\sigma_{kl}$, \[Gauss\] G\_[kl]{}()= (-), \_[kl]{}\^2=(-), with normalization $Z$ chosen in such a way that \[modsum\] \_[k,l=-N/2+1]{}\^[N/2]{}\_[kl]{}\^2 = N\^2\_0\^2. (In the continuum limit $(a\rightarrow 0$, $N\rightarrow\infty)$ we have $Z=\frac{\Te^2}{2\pi}\left(1+\frac{m}{\Te}\right)e^{-m/\Te}.)$ In other words, we choose a Boltzmann distribution for the average occupation numbers $n_{kl} =\langle\langle \tilde\Phi_{kl}\tilde\Phi_{kl}^\rangle\rangle = \sigma_{kl}^2$ for each field component, as for relativistic (non-interacting) particles with mass $m$. Here the mass $m^2$ is defined by the absolute value \[mass\] m\^2()= \|f\^2()\| for the fluctuations around $\Bphi=0$ in the symmetric potential (\[pot\]) at the initially high temperature $\Te=\Te_0$, where $f^2(\Te)$ is negative. The amplitude $\sigma_0^2$ plays the role of a fugacity \_0\^2=(-/) for negative chemical potential $\mu$. In the temperature range which we consider $(0.05<a\Te<0.8)$ (cf. fig.(\[fig3\])) a suitable value for $\mu$ is $a\mu\sim -0.6$. (With this choice the average amplitude of the chiral field is not subject to abrupt deviations from its initial value immediately after the onset of the dynamical evolution). We assume isotropy of the initial ensemble with respect to rotations in $O(3)$-space such that the three components of the field fluctuations $\tilde\Phi^\alpha_{kl}$ ($\alpha=1,2,3$) have the same average square amplitude $\sigma_{kl}^2$. By picking each component independently at each point $(k,l)$ from the Gaussian ensemble, different components are uncorrelated and equal components at different points (on the momentum lattice) are also uncorrelated, \^\_[kl]{} \^\_[k’l’]{} &=& \^\_[kl]{} \^\_[k’l’]{} + \^\_[kl]{} \^\_[k’l’]{}\ &=&\_[kl]{}\^2 \_ ( (\_[k k’]{}\_[l l’]{} +\_[-k k’]{}\_[-l l’]{}) +(\_[k k’]{}\_[l l’]{} -\_[-k k’]{}\_[-l l’]{})) =\_[kl]{}\^2 \_\_[k k’]{}\_[l l’]{}. Together with (\[Fourier\]) this leads to the fluctuation in the real field configurations \[fluc\] \^\_[ij]{}\^\_[ij]{} =\_\_[k,l=-N/2+1]{}\^[N/2]{} \_[kl]{}\^2 = \_ \_0\^2 which is, of course, independent of the lattice point $(i,j)$. Its magnitude is controlled by the constant $\sigma_0^2$ in (\[Gauss\]). It should be sufficiently small to keep the amplitudes of the average initial fluctuations small. On the lattice the upper limit for the momenta $p$,$q$ is $\frac{\pi}{a}$, (i.e. $k,l=N/2$). So, as long as \[LamT\] , the lattice cut-off (upper limit momentum) imposed by the finite lattice constant is unimportant because the corresponding states are almost unoccupied. Note that periodicity and antisymmetry of the imaginary parts in (\[albet\]) requires that $\Bet_{kl}$ vanishes if both $k$ and $l$ are multiples of $N/2$. With the condition (\[LamT\]) satisfied, this holds with good accuracy also for the initial configuration picked randomly from the ensemble (\[Gauss\]). The average number of topological defects in a random ensemble of vector configurations is closely related to the characteristic angular coherence length in that ensemble. Therefore, it will be necessary to measure the (equal-time) correlation functions for the unit-vector fields $\hphi$ for the evolving ensembles. In order to have an analytical result at least for the initial configurations (where length and orientation of the 3-vectors are uncorrelated), it is easier to consider the correlations among the full vectors $\Bphi$. Therefore, we define normalized transverse and longitudinal correlation functions \[corrfct\] C\_(i)&=& ,\ C\_(i)&=& , with transverse coherence lengths $R_\bot$ and longitudinal (dimensionless) coherence rapidity $R_\\|$ defined through \[cut\] C(i)< i > , i > respectively. For the initial ensemble (\[Gauss\]) the correlations are, of course, isotropic on the lattice, i.e. \[initcorr\] == with initial spatial coherence length $R_0$. In the continuum limit $(a\rightarrow 0$, $N\rightarrow\infty)$, we obtain $C(r)$ as function of the spatial distance $r$ (or rapidity $\eta=r (b/a)$) \[corrinit\] C(r)= (). Specifically, putting $m=0$, the coherence length $R$ as defined in (\[cut\]) is \[Rm0\] R= . This allows to put limits on the range of temperatures which can be reasonably represented on the lattice. Typically, for lattice size of $N\sim 100$, $\Te$ should lie within the range from about 0.02 to about 0.8 inverse lattice units. For smaller values the inital coherence length already covers more than half of the lattice so almost no defects will fit on the lattice, for larger values the correlation lengths approach the lattice constant. It may be noted that with (\[mass\]), for $(\ell/a)\sim 5$ and $(a\Te)\sim 0.1$, the ratio $m/\Te$ is not very small, so generally we expect appreciable deviations from the $\Te^{-1}$ scaling in (\[Rm0\]), (e.g. for $(\ell/a)=4$ we find $(R/a)\sim (a\Te)^{-0.8}$, cf. fig.\[fig3\]). During the evolution in the Bjorken frame the correlations rapidly become anisotropic. We then conveniently define an average coherence length $\bar R$ through \[avcorr\] =( +). This may be compared to the coherence radius obtained from the angular-averaged correlation function \[avcorrfct\] \|C(r) = where the $i,j$-sum indicates an average over all lattice points in a narrow circular ring with radius $r$ around the lattice point $mn$. The essential characteristics of the evolutions are not very sensitive to the choice of the initial time derivatives. (They can as well be put to zero.) The equations of motion very quickly establish appropriate velocities. Of course, the absolute value of the total energy depends on that choice. For the simulations presented in the following we construct in analogy to the initial configurations (\[Fourier\]) an initial ensemble of time derivatives through (\_)\_[ij]{}=\_[k,l=-N/2+1]{}\^[N/2]{} ( e\^[(ik +jl)]{} \_[kl]{}- c.c.), \_[kl]{}=. The Fourier coefficients $\tilde{\Bphi}_{kl} $ again are picked randomly from the same Gaussian deviate (\[Gauss\]). Shrinking solitons in comoving frames ===================================== Let $\Bphi^{(s)}(x,z)$ be a static soliton solution of the model (\[lag\]) in its $(x,z)$ rest frame, which minimizes the static energy $E=L+U$ with a finite value for the soliton energy $E=E_0$. After the transformation to the Bjorken frame, the configuration $\Bphi_\tau^{(s)}(x,\eta)=\Bphi^{(s)}(x,\tau\eta)$ then describes a static solution of the action in the comoving $(x,\eta)$ frame at proper time $\tau$ (where $\partial_z$ is replaced by $(1/\tau) \partial_\eta$), for the same value of $E_0$. It represents a soliton with the same finite radius in transverse $x$-direction as before, but with its radius in longitudinal $\eta$-direction shrinking like $1/\tau$ with increasing proper time $\tau$. The total energy $E_0$ of this shrinking soliton is, of course, independent of $\tau$. (Naturally, this consideration strictly applies only to the adiabatic case, where $\tau$ is considered as a parameter. In the dynamical ordering process the evolution of the solitons towards their static form may appreciably lag behind the actual progress of proper time.) For lattice implementations, with the typical spatial radius of the stable solitons given by $\ell$, the longitudinal extension of the solitons for times $\tau \gg \ell $ has shrunk down to (dimensionless rapidity-)lattice-unit size and longitudinally adjacent solitons no longer interact. In transverse direction, however, the solitons develop their stable size of $\ell/a$ lattice units, they keep interacting, attracting close neighbours or annihilating with overlapping antisolitons (cf. fig.\[fig1\]). For solitons shrinking longitudinally down to lattice-unit size the energy will begin to deviate from the value $E_0$ as soon as the longitudinal extent covers merely a few lattice units. To get an approximate idea for the energy limit let us assume that a single separate soliton finally degenerates into a transverse string of $2\ell+1$ lattice points, on which $\|\Bphi\|$ varies from nearly zero (in its center) to the surrounding vacuum configuration $\Bphi_0=(0,0,1)$, i.e. $\Bphi=(0,0, \|i\|/\ell)$ for $-\ell\leq i\leq\ell$, on that string of lattice points. Then we find for the contributions of a single soliton \[shrilat\] L\^[(2)]{}\_\~, L\^[(2)]{}\_\~, U\~. So, apparently, solitons shrinking on a lattice contribute to the energy terms which rise linearly with proper time $\tau$ which (as lattice artifact) will dominate the total energy for large $\tau$. We expect the winding density of the squeezed defect to be located on $\nu$ lattice squares near its center. This implies for the fourth-order term \[e4lim\] L\^[(4)]{}\_=. The winding density is determined by the orientation of the field unit-vectors alone, so it sufficient to consider the unit-vectors $\hphi$. We expect the squeezed defect to consist of just one unit-vector $\hphi=(0,0,-1)$ at the soliton center looking into the direction opposite to all surrounding unit-vectors $(0,0,1)$. That lattice point is the top of four adjacent rectangular triangles (with the diagonals connecting the four nearest neighbour points as bases) which together cover an area of two lattice squares. So we expect a winding density $\rho=1/\nu$ with $\nu\sim 2$. This dominance of lattice artifacts for $\tau \gg \ell $ is illustrated in fig.(\[fig2\]) which shows a typical evolution on a $50\times 50$ lattice for $\ell=4$. The total winding number is $B=-1$. After the roll-down the number of solitons stabilizes at $n=9$ (cf. fig.\[fig1\]). Apparently, both $U$ and $L^{(2)}_\bot$, approach a linearly rising limit for $\tau \gg 10^2$, approximately like $\sim \frac{1}{2}n (\tau/\ell)$ which dominates the total energy, but does not affect the (essentially constant) kinetic energy. Longitudinal contributions drop off like $\tau^{-1}$, so they are irrelevant. It appears from fig.(\[fig2\]) that for this evolution the roll-down (where the average of $\Phi$ aproaches the vacuum value $\Phi=1$) takes place during the time interval $2\ell <\tau < 4 \ell$, i.e. long before artificial lattice effects dominate the energy. It is also by the end of the roll-down that the number of created defects stabilizes. So we would conclude that results obtained from lattice simulations for baryon-antibaryon production during the chiral phase transition in a rapidly expanding chiral gas are not severely affected by lattice artifacts. On the other hand, to follow the evolutions beyond the end of the roll-down, which comprise small ’$\sigma$’ and ’$\pi$’-oscillations of $\Bphi$ around the true vacuum, interfering with small oscillations of the bag profiles (resonances), will require to subtract the lattice artifacts. Evolution until freeze-out ========================== In this chapter we will follow typical evolutions of the chiral field after a sudden quench in more detail and try to analyze their characteristic features up to the end of the roll-down. Immediately before the sudden quench at $\tau=\tau_0$ the initial ensemble is prepared as described in section II. The average length of one component of the chiral field is given by $\sigma_0$ (cf. eq. (\[fluc\])), the potential in (\[pot\]) is characterized by a negative value of $f^2$. So, for sufficiently small $\sigma_0^2$ we have at $\tau=\tau_0$ \[Uth0\] U\_0= \_0 (f\^4+2 \|f\^2\|\^2) dxd=(C\_0+C\_2)[V]{}\_0, where ${\cal V}_0=\tau_0\Delta\eta {\cal A}$ is the inital volume of the Bjorken slice, and the constants are \[const1\] C\_0=f\^4, C\_2=\|f\^2\| \_0\^2. For the derivatives at lattice points $(i,j)$ Eq.(\[Fourier\]) implies \[deriv\] (\_x)\_[ij]{}=\_[k,l=-N/2+1]{}\^[N/2]{} () e\^[(ik +jl)]{} \_[kl]{}, (\_)\_[ij]{}=\_[k,l=-N/2+1]{}\^[N/2]{} ( ) e\^[(ik +jl)]{} \_[kl]{}. Again replacing the integrands in (\[int3\]),(\[int4\]) by ensemble averages leads to the second-order gradients contribution at $\tau=\tau_0$ \[L20\] L\_0\^[(2)]{}=C\^[(2)]{}\_0. For the constant $C^{(2)}$ we have in the continuum limit \[C2\] C\^[(2)]{}=\_0\^2 \^2 (1+). Similarly, one may obtain a rough estimate for $L^{(4)}$ averaged over the initial ensemble by replacing in (\[L4\]) the unitvectors $\hphi$ by $\Bphi / \sigma_0$. During the very early phase of an evolution in proper time the initially random ensemble of fluctuations will essentially stay random. This means that the integrals in (\[int3\])-(\[int5\]) will remain constant, given by their initial values. Therefore, the time dependence of the different contributions (\[int3\])-(\[int5\]) to the total energy is given by the kinematical factors ($\tau/\tau_0$) or ($\tau_0/\tau$) alone, with the integrals approximated by replacing the integrands through their averages in the initial ensemble. After the quench, $f^2$ is positive, so for sufficiently small $\sigma_0^2$ we have \[Uth\] U= (f\^2-2\^2) dxd=(C\_0-C\_2)[V]{}\_0, and \[linear\] L\_\^[(2)]{}= C\^[(2)]{}\_0, L\_\^[(2)]{}=C\^[(2)]{}\_0. In fig.(\[fig2\]) both straight lines, (\[linear\]) with (\[C2\]), are included for comparison. It may be observed that the integral $L_{\\|}^{(2)}$ involving the longitudinal gradients follows the straight line decrease almost until the onset of the roll-down. This means that the rapidity gradients basically stay random. On the other hand, the integral $L_{\bot}^{(2)}$ follows the linear rise only for about one unit of proper time after the onset of the evolution. Already near $\tau/\tau_0 \sim 2$, the transverse gradients are strongly affected by the dynamics and interfere with the kinetic energy. Due to the relative factor of $1/\tau^2$ of $L_{\\|}$ as compared to $L_{\bot}$ the dynamics quickly gets dominated by the transverse gradients alone, such that the average kinetic energy follows the average transverse-gradient energy $L^{(2)}_\bot$, while the rapidity gradients (in $L_{\\|}^{(2)}$ and $L_{\\|}^{(4)}$) which decrease like $1/\tau$ are no longer relevant for the overall dynamical evolution. Disregarding rapidity gradients altogether, the EOM (\[EOM\]) reduces to \_+\_-\_[xx]{} -m\^2=0, which describes wave propagation in transverse direction, $A(\tau) \exp(ipx)$. Here the mass $m^2$ again characterizes the fluctuations around $\Bphi=0$, m\^2=f\^2/\^2 so $m^2$ is negative for negative $f^2$ (where $\Bphi=0$ is the stable minimum), and it is positive for potentials which actually do have a lower symmetry-breaking minimum. The amplitudes $A(\tau)$ generically are Bessel functions, \[waves\] A() &\~& J\_0() p\^2-m\^2 >0,\ A() &\~& I\_0() p\^2-m\^2 <0 . For large values of their arguments the amplitudes of $J_0$ decrease like $1/\sqrt{\tau}$, while $I_0$ contains exponentially rising parts. Modes with large transverse wave numbers contribute most to $L^{(2)}_\bot$. Therefore, with their amplitudes decreasing like $1/\sqrt{\tau}$, the kinematical factor $\tau$ in $L^{(2)}_\bot$ is compensated. So we expect that the linear rise of $L^{(2)}_\bot$ ends as soon as the dynamics is dominated by the transverse gradients and is followed by a phase where T \~L\^[(2)]{}\_\~\|\_. For negative $m^2$ no amplification occurs. After the quench, however, when $f^2$ has become positive, a few modes with small transverse wave numbers will start to get amplified. Typically, for wave numbers $p=2\pi k/N$, with $k$ integer ($0\leq k\leq N/2$), waves with $k/N < \sqrt{\lambda}f/(2\pi\ell)$ get amplified, e.g. the lowest three or four out of $N=100$ for $\ell\sim 5$ (for $\lambda=1$ and $f^2=1$). At first, the rate of amplification is slow because the exponential rise is compensated by a decreasing function for small arguments in $I_0(x)$. These low-$k$ modes do not contribute much to $L^{(2)}_\bot$. In fact, the $k=0$ mode, which experiences the largest rate of amplification, does not contribute at all. While the amplification effect is not very pronounced for $L^{(2)}_\bot$, the few slowly exponentially rising contributions from the lowest-momentum transverse waves cause a noticeable rise of the condensate $\langle\langle \Phi^2 \rangle\rangle$ after some time. This enters into the fluctuating part $C_2$ of the potential $U$ and drives it away from its linear rise given by (\[Uth\]). Then also the fourth-order terms in the potential become important and the dynamical evolution subsequently is dominated by the local potential. This initiates the roll-down of the field configuration at the majority of the lattice points into the true vacuum $\Bphi_0=(0,0,1)$. The transition into the symmetry-violating configuration takes place, with formation of bags and solitons in those regions where the winding density happens to be high. To estimate the time $\tau_1$ for the onset of the roll-down we consider the $k=0$ mode with amplitude $I_0(\tau m)$. Amplification of this amplitude by a factor $e$ in the time interval from $\tau_0$ to $\tau_1$ requires \[tau1\] I\_0(\_1 m)= 1+ I\_0(\_0 m). The r.h.s. depends only very weakly on $\tau_0$, as long as $(\tau_0 m)\leq 1$. In fact, $(\tau_1 m)$ varies only from 2.26 to 2.55 for $0\leq(\tau_0 m)\leq 1$. So, for convenience we simply take $(\tau_1 m)\approx 5/2$ if $(\tau_0 m)$ is of the order of 1 or less. Otherwise, for larger values of $(\tau_0 m)$, $\tau_1$ has to be obtained more accurately from (\[tau1\]). Typically, therefore, the transition from the gradient-dominated to the potential-dominated phase, happens near \[onset\] \_1 \~. However, up to this time $\tau_1$ of the onset of the roll-down, i.e. throughout the whole gradient-dominated phase the potential plays no significant role. The overall evolution proceeds practically independently from the (positive or negative) value of $f^2$ in the $\Phi^4$-potential (\[pot\]). This also implies that the quenchtime (the timescale for changes in $f^2$) is irrelevant as long as it is smaller than the time during which the gradient terms dominate the evolution and it justifies the use of the sudden quench approximation where we impose the ’cold’ ($\Te=0$) potential from the outset at $\tau>\tau_0$. With $\ell/\tau_0 > 1$, the ratio $(\tau_1/\tau_0)^2$ is sufficiently large to render all longitudinal (rapidity) gradients unimportant as compared to the potential. This means that during the subsequent roll-down different rapidity slices become effectively decoupled, and begin to evolve independently from each other, while in longitudinal directions the solitons contract to lattice unit size. Within these rapidity slices, $\pi$- and $\sigma$-modes propagate transversely, and eventual further annihilations of soliton-antisoliton pairs take place while the transverse shapes of the sqeezed bags are established. By the end of the roll-down the remaining nontrivial and sufficiently separate structures have essentially reached their stable form. Apart from small fluctuations, the integral (or sum) over the absolute values of the winding density $n=\int \!\|\rho\| \;dx dz$ then stabilizes and counts the (integer) number of these defects. Therefore we identify the end-of-the-roll-down time with the (chemical) freeze-out time $\tau_f$ when the numbers of baryons and antibaryons created are fixed. A rough estimate for $\tau_f$ may be obtained if we follow the further amplification of the amplitudes $I_0(\tau m)$ of the $k=0$ modes beyond $\tau_1$. For large arguments the increase in $I_0(\tau m)$ is mainly due to the exponential $\exp(\tau m)$, so we obtain \[freeze\] \_f \~\_1 + () For a typical amplification ratio of 5 to 10 during roll-down we then find an approximate freeze-out time of \[tauf\] \_f \~. This certainly represents a lower limit for the duration of the roll-down, because the increasing $\Phi^4$ contributions to the potential will slow down the symmetry-breaking motion. The numerical simulations confirm this simple argument and indicate that $(m \tau_f) \sim 4-5$ provides a reasonably accurate estimate for the freeze-out time (as long as $(\tau_0 m) \leq 1$). After the quench, when $f_0^2$ has assumed its $(\Te=0)$-value $f_0^2=1$, we may neglect the small contributions of the explicit symmetry-breaking $H$ to $f^2$ and to the $\sigma$-mass $m_\sigma^2$ in eqs.(\[f2\]) and (\[masses\]) and rewrite (\[tauf\]) in the form \[taufreeze\] . The typical example for an evolution given in fig.\[fig2\] shows how during the roll-down the configurations pick up an appreciable amount of kinetic energy until the potential starts to deviate from its linear rise and interferes with $\langle\langle \,T\rangle\rangle$. Subsequently, $\langle\langle U\rangle\rangle$ starts to pick up the unphysical linearly rising lattice contributions (\[shrilat\]) of the shrinking solitons, while the time-averaged $\langle\langle \,T\rangle\rangle$ remains basically constant. As the heavy solitons carry no kinetic energy, $\langle\langle \,T\rangle\rangle$ then resides in small transversely propagating fluctuations which eventually are emitted as $\sigma$ and $\pi$ mesons. Correlation lengths and defect numbers -------------------------------------- In contrast to the integer net-baryon number $B=\int \!\rho \;d^2x$, the integral (or lattice sum) over the absolute values of the local winding density $\|\rho\|$ \[defnum\] n=\|\| dxdz generally is not integer. The ensemble average of $n$ is closely related to the coherence length $R$ for the field unit-vectors in the statistical ensemble of $O(3)$-field configurations. If an $O(n)$-field is implemented on a $d$-dimensional cubic lattice with lattice constant $a$, then the field orientations on the vertices of a sublattice with lattice unit $R/a$ can be considered as statistically independent. Then the average $\langle\langle \; n \; \rangle\rangle$ expected on an $N^d$ lattice is \[Kibble\] n =\_d (aN/R)\^d where $\nu_d$ is the average fraction of the surface of the sphere ${\bf S}^d$ covered by the image of the sublattice unit (this is the very definition of a winding density). The number $\nu_d$ can be estimated for different manifolds [@Kibble]. For the map (compactified-)${\bf R}^2\rightarrow {\bf S}^2$ defined by the unit-vectors $\hphi(x,z)$ of the $O(3)$-field in $d=2$ dimensions it is $\nu_2=1/4$, (i.e. $1/2^{d+1}$ for each of two triangles which make up each square sublattice unit cell). Inserting our result (\[Rm0\]) (obtained for $m=0$) into this estimate for $d=2$ dimensions, leads to $ \langle\langle \; n \; \rangle\rangle _{\tau=\tau_0} \propto \Te^2$. In fig.\[fig3\]a this is compared with numbers $n$ and coherence lengths $R$ measured for several initial configurations on an $N\times N$ lattice for different temperatures (for $N=100$ and mass $m=0$). Evidently, the finiteness of the lattice causes a small systematic deviation from this $\Te^2$-dependence, especially for large values of $\langle\langle \, n \, \rangle\rangle$, as the coherence length approaches the lattice constant. The measured numbers $\langle\langle \; n \; \rangle\rangle$ follow (\[Kibble\]) with satisfactory accuracy for $\nu\approx 1/5$. In fig.\[fig3\]b the same comparison is shown for non-vanishing mass $m=1/\ell$, for $\ell/a=4$. For $m\ne 0$ the coherence length $R$ can be obtained from (\[corrinit\]) and compared to the measured values. Fig.\[fig3\]b shows that they are reasonably well described by $R\propto\Te^{-0.8}$. The corresponding measured numbers $n$ follow (\[Kibble\]) with good accuracy for $\nu\approx 1/6$. Of course, small changes of $\nu$ could be absorbed into a slightly redefined coherence length (note that the correlation function (\[corrinit\]) for $m=0$ does not decrease exponentially). We shall, however, keep the definition (\[cut\]). The above considerations apply to random configurations which need not contain any fully developed solitons but may consist of only small fluctuating local winding densities which cover small fractions of the image sphere. However, if the configurations finally have evolved into an ensemble of well-separated solitons or antisolitons embedded in a topologically trivial vacuum with only small fluctuations in the local winding density, then the integral (\[defnum\]) counts the number of these embedded baryons-plus-antibaryons. We therefore adopt the notion ’number of defects’ for $\langle\langle \; n \; \rangle\rangle$, irrespective whether configurations comprise only small local winding densities, or partial or complete solitons. For a typical evolution (see e.g. fig.\[fig2\]) the number of defects measured as function of proper time shows a slow decrease which follows approximately a power law \[powlaw\] n\~n\_[(=\_0)]{}()\^[-]{} . By the end of the roll-down at freeze-out time $\tau_f$ this decrease levels off and $n$ settles near the constant which counts the number of the finally surviving fully developed solitons-plus-antisolitons (cf. figs.\[fig1\]). The decrease in $n$ reflects the slow increase in the average coherence length ${\bar R}$ up till the end of the roll-down. The longitudinal coherence length $R_\\|$ grows very slowly because rapidity gradients are suppressed with $1/\tau$ in the Bjorken frame. This leads to an effective decoupling of field-vectors in longitudinal direction and subdues the drive for aligning field orientations in adjacent rapidity bins. On the other hand, the transverse coherence length $R_\bot$ grows rapidly. For $R_\bot\gg R_\\|$, the average radius ${\bar R}$ obtained from (\[avcorr\]) is dominated by $R_\\|$. A typical example is shown in fig.\[fig5\] for an evolution which starts at $\tau_0/a=1$. The average ${\bar R}$ grows with an exponent of $\alpha \approx 0.25$. The statistical argument in (\[Kibble\]) then leads to $n\sim \tau^{-2\alpha}$, with $2\alpha=\gamma\approx 0.5$. This is slightly steeper than the measured decrease in $n$. But as the growth in the coherence radii sets in only after one or two units of $\tau$ after the onset of the evolution, the final number of surviving defects is reasonably well reproduced by the statistical expression (\[Kibble\]) (we adopt $\nu=1/6$ from fig.\[fig3\]b). Altogether, we typically find exponents $\gamma\sim 0.4\pm 0.05$ for the decrease (\[powlaw\]) of the number of defects. Then, with (\[taufreeze\]) for the typical freeze-out time, we have \[decrease\] n \_[\|=\_f]{}= n \_[\|=\_0]{} ()\^[0.4]{} for the reduction of the number of defects from its initial value at the onset of the evolution until the end of the roll-down. With $(\tau_0\,m_\sigma)$ of the order of 0.5 to 1, we find reduction factors of 1/3 to 1/2, which is not even one order of magnitude. So, this is not a dramatic result. The reason is, evidently, that in the expanding Bjorken frame the gradient coupling in rapidity direction quickly gets suppressed. It should be noted that all numerically measured exponents are independent of the choice of the lattice constants, because scaling $x\rightarrow ax$, (i.e.$\ell\rightarrow a\ell$), $\eta\rightarrow b\eta$, $\tau\rightarrow (a/b)\tau$ leaves the EOM (\[EOM\]) invariant. The length unit $a$ only serves to define the resolution with which the spatial structure of the field configurations is analyzed and all physical results should be independent of this scale. On the other hand, the initial time $\tau_0$ denotes the physical point in time when the system begins its evolution in terms of hadronic degrees of freedom with a sudden or rapid quench in the relevant potential. So, physical results generally will depend on $\tau_0$, as is evident from the reduction factor obtained in (\[decrease\]). Small explicit symmetry breaking ($H\ne 0$) accelerates the decrease of $n$ during roll-down, but at the same time it reduces the freeze-out time, such that the final number of $n$ remains essentially unaffected by small non-zero values of $H$. Fig.\[fig4\] shows a number of evolutions for two different strengths $H$ of explicit symmetry breaking. The same is true if additional damping is introduced into the EOM (\[EOM\]) by adding a term $\kappa\partial_\tau\Bphi$ with damping constant $\kappa$ to account for the fact that the field fluctuations are actually emitted from the expanding Bjorken rod, carrying away energy. Through this dissipative dynamics the evolutions are slowed down, the roll-down times may be retarded by an order of magnitude, but the overall reduction factor in the number of surviving defects remains unaffected. Of course, all fluctuations then are damped away during the course of the evolution, and the integrals (\[int3\]) to (\[int5\]) finally are determined by the remaining ensemble of squeezed solitons alone, while the kinetic energy goes to zero. Meson spectrum -------------- For times long after the roll-down the average kinetic energy $\langle\langle \; T \; \rangle\rangle$, and the potential energy parts $\langle\langle \; L_\bot+U \; \rangle\rangle$ (after subtraction of the linearly rising (lattice) contributions from the squeezed solitons (\[shrilat\])) converge towards the same constant $E_f/2$. Their sum $E_f$ represents the average total energy stored in the mesonic field fluctuations after the roll-down. Both averages show residual fluctuations around their smooth background with opposite phases, such that their sum $E_f$ is smooth. Analysing the spectral density of either $\langle\langle \; T \; \rangle\rangle$, or $\langle\langle \; L_\bot+U \; \rangle\rangle$, (after subtracting the background), tells us about the spectral distribution of pions and $\sigma$-mesons that will eventually be emitted from the expanding Bjorken rod. We consider the Fourier-transforms c()+i s()=\_[\_a]{}\^[\_b]{} T() -[\|T]{}() e\^[i ]{} d, where the integral covers times long after the roll-down, e.g. $\tau_a/\tau_0\sim 100 $, $\tau_b/\tau_0\sim 1000$, and ${\bar T}(\tau)$ subtracts the smooth background. The absolute value, $\epsilon(\omega)= \sqrt{c^2+s^2}$, represents a spectral energy density, from which we may extract the spectral particle number density n()= = \_[ij]{}n\_[ij]{}\^[()]{}(-2\_[ij]{}\^[()]{})+ \_[ij]{}n\_[ij]{}\^[()]{}(-2\_[ij]{}\^[()]{}) + . The $ij$-sum with $i,j=0,1,2,...N/2$ covers all frequencies on the lattice for pions and $\sigma$-mesons with masses $m_\pi$ and $m_\sigma$ given in (\[masses\]) \[omega\] \_[ij]{}\^[(/)]{}= . Generically, $T(\tau)$ contains contributions $\sim (\cos(\omega_{ij}^{(\pi)}\tau))^2$ from the pionic fluctuations, and $\sim (\cos(\omega_{ij}^{(\sigma)}\tau)+ c)^2$ from the $\sigma$-fluctuations around some nonvanishing average $c$. Therefore, the spectral functions $\epsilon(\omega)$ and $n(\omega)$ will, in addition to the double frequencies $2\omega_{ij}^{(\sigma)}$, also contain contributions for the $\sigma$-mesons at the single frequencies $\omega_{ij}^{(\sigma)}$. Figure \[fig6\] shows the spectral density $n(\omega)$ as obtained from the residual fluctuations in the average kinetic energy. The long vertical arrows point to the first four $2\omega_{ij}^{(\pi)}$ pionic frequencies (\[omega\]) for $ij=00,10,20,30$, with $m_\pi^2=H/\ell^2$, ($H=0.1,\ell/a=4,f_0=1$). It may be seen that the overwhelming part of the strength resides in the lowest and first excited pionic modes. The strength decreases rapidly with excitation energy, approximately like $\exp(-12a\omega)$. The same is true for the strength of the $\sigma$-modes. (The short arrows in fig.\[fig6\] point to the first three modes with $ij=00,10,20$, with single frequencies $\omega_{ij}^{(\sigma)}$ and double frequencies $2\omega_{ij}^{(\sigma)}$). However, the number density $\sum n_{ij}^{(\sigma)}$ for the sigmas, which we may extract from the strength located at the double frequencies $2\omega_{ij}^{(\sigma)}$, is only about $5\%$ of the pionic strength residing in the first three pionic modes. For an order-of-magnitude estimate of the pionic multiplicities we therefore ignore the $\sigma$-contributions. Meson and baryon multiplicities ------------------------------- To obtain a simple estimate for the energy $E_f$ finally available for meson production we consider the time of the onset of the roll-down $\tau_1$ in (\[onset\]) which marks the transition from the gradient-dominated to the potential-dominated phase. At this time, for sufficiently small $\sigma_0^2$, the total energy is dominated by the linearly rising term $(\tau/\tau_0)C_0 {\cal V}_0$ in the potential (\[Uth\]). With the onset of the roll-down the average potential $\langle\langle \; U \; \rangle\rangle$ starts to deviate from this linear rise and bends down to interfere with $\langle\langle \; T \; \rangle\rangle$ and $\langle\langle \; L \; \rangle\rangle$, (cf. fig. \[fig2\]). In the numerical simulations the large-time limit of $\langle\langle \; U \; \rangle\rangle$ and $\langle\langle \; L \; \rangle\rangle$ is masked by the (lattice-artificial) rise of the soliton contributions. But the asymptotic $\langle\langle \; T \; \rangle\rangle$ is free of these artifacts and (apart from residual fluctuations) approaches a constant value, which is well represented by the linearly rising $\frac{1}{2}(\tau/\tau_0)C_0 {\cal V}_0$ taken at $\tau=\tau_1$. Approximating $\tau_1$ by $5/(\sqrt{2}m_\sigma)$ as given in (\[onset\]), we then have (with $f^2=1$) \[Ef\] E\_f = f\_\^2C\_0[V]{}\_0f\_\^2(ab N\^2), where we have again neglected the small contribution of the explicit symmetry-breaking $H$ to the $\sigma$-mass. Within this level of accuracy we can also ignore that about $30\%$ of the pions carry the energy $\omega_{10}^{(\pi)}$ (instead of $m_\pi$), and obtain the pion multiplicity $n_\pi$ from dividing (\[Ef\]) by $m_\pi$, n\_= f\_\^2 (ab N\^2). This number may be compared with the baryon-plus-antibaryon multiplicity given in (\[decrease\]). We use for $\langle\langle \; n \; \rangle\rangle_{\|\tau=\tau_0}$ the statistical result (\[Kibble\]), with initial spatial coherence length $R_0$. Then we have n = ()\^2 ()\^. The last factor relies on the estimates (\[onset\]) and (\[tauf\]) for the times $\tau_1$ and $\tau_f$, which are valid as long as $(\tau_0 m)\leq 1$, (otherwise they have to be obtained more accurately from (\[tau1\]) and (\[freeze\])). The evolutions described above have been performed for initial configurations selected with netbaryon number $B=0$. So, the average number $n_{\bar p}$ of antibaryons created during the phase transition is $\langle\langle \; n \; \rangle\rangle/2$. With typical values $\nu\sim 1/4$, $\gamma\sim 0.4$ we find for the multiplicity ratio of antibaryons to pions n\_[\|p]{}/n\_0.14 . With an overall energy scale $f_\pi^2$ of the order of the pion mass $m_\pi$, and $R_0$ of the order of $m^{-1}=\sqrt{2} m_\sigma^{-1}$, this ratio is n\_[\|p]{}/n\_0.07 (m\_) (\_0 m\_)\^. The ratio $a/b$ of the spatial and rapidity lattice constants which appears in this result has a physical meaning: according to (\[initcorr\]) it is equal to the ratio of the (transverse) spatial coherence length $R_0$ and the (longitudinal) rapidity coherence distance $R_{\\| 0}$ in the initial configuration. Naturally, this ratio is of the order of $\tau_0$. So, for initial times $\tau_0$ typically of the order of the inverse $\sigma$-mass we find antibaryon-to-pion multiplicity ratios of the order of $0.05$ to $0.1$. Generalization to 3-d O(4) ========================== For the generalization to the 3+1 dimensional O(4)-model we keep the parametrization as given in eqs.(\[lag\]), (\[L2\]), and (\[pot\]). In this case $f_\pi^2$ is an overall constant of dimension \[mass$^2$\], so the physical fields $f_\pi\Bphi$ are of mass-dimension one. The winding density is no longer given by (\[top\]), but we keep ${\cal{L}}^{(4)}$ as defined by the second equality in eq.(\[L4\]). Conventionally, the strength of the ${\cal{L}}^{(4)}$-term in (\[L4\]) is given in terms of the Skyrme parameter $e$ as \[Skyrme\] . In this case the typical spatial radius of a stable skyrmion in its rest frame is mainly determined by the balance between ${\cal{L}}^{(2)}$ and ${\cal{L}}^{(4)}$, so it is of the order of $(ef_\pi)^{-1}$. For the map (compactified-)${\bf R}^3\rightarrow {\bf S}^3$ defined by the unit vectors of the $O(4)$-field in 3 spatial dimensions the statistical result (\[Kibble\]) for the average number of defects found on a $(aN)^3$ lattice for initial configurations with coherence length $R_0$ generalizes as \[Kibble3\] n \|\_[=\_0]{}= (aN/R\_0)\^3. (We again use $a=b\tau_0$ for the lattice constants). The factor 5 counts the number of 3-simplices (tetraeders) which make up a cubic sub-lattice cell of size $R_0^3$, the factor $(1/2^{d+1})$ with $d=3$ is the (absolute value of the) average surface area covered by the image of one 3-simplex on the image sphere ${\bf S}^3$. So the factor $5/16$ counts the average ’number of defects’ associated with a cubic lattice cell with lattice constant given by the initial coherence length $R_0$. A certain arbitrariness in the definition of the coherence length may translate into modifications of this factor $5/16$; (e.g. for a random lattice of 3-simplices the factor 5 is replaced by ($24\pi^2/35 \sim 6.8$) [@Kibble]). In any case we do not expect order-of-magnitude changes in this factor as compared to the $d=2$ case, where we had $2/2^{2+1}$. However, through the cubic power the result is now very sensitive to the actual value of $R_0$ in the initial ensemble. Different concepts about the physical nature of the initial configurations will imply quite different ways to arrive at the appropriate initial coherence lengths $R_0$. For an initial ensemble which is characterized by a temperature $\Te$ we could proceed as in (\[corrinit\]) and relate $R_0$ to the temperature, or to the mass $m^2(\Te)=\lambda \|f^2(\Te)\|/\ell^2$ of the field fluctuations; but it has also been suggested [@Ellis] to tie $R_0$ to the parton density which makes it independent of the temperature concept. So, for the moment it seems appropriate to keep the initial coherence length $R_0$ as a parameter. Adding a second transverse dimension does not change the result (\[avcorr\]) for the average of the transverse and longitudinal coherence lengths. The growth in the resulting ${\bar R}$ again is dominated by the slow increase of the longitudial coherence length $R_\\|$ which is unaffected by additional transverse dimensions. The estimates (\[onset\]) and (\[tauf\]) for the times $\tau_1$ and $\tau_f$ of the onset and end of the roll-down also remain unaffected, as they only rely on the amplitudes $A(\tau)$ of the transverse waves (\[waves\]), irrespective of the number of spatial dimensions. (In this case, ${\cal{L}}^{(4)}$ now contributes to $L_\bot$ with a term containing four transverse gradients, acting on the direction of the O(4) field. The roll-down, however, takes place in areas which are topologically trivial, i.e. with small angular gradients, so we do not expect a strong effect on the roll-down times. Within the approximations which led to (\[decrease\]), we then find for the average number of baryons+antibaryons present after the roll-down \[defects\] n \_[\|=\_f]{}= ()\^3 ()\^[3]{} with $\alpha\sim 0.2$ to 0.25. We denote the transverse area $(aN)^2$ of the Bjorken rod by ${\cal A}$, and replace the ratio $(a/b)$ of the lattice constants again by the initial time $\tau_0$. Then we obtain for the rapidity density of antiprotons ($n_{\bar p}=\frac{1}{4}\langle\langle \; n \; \rangle\rangle$) \[pbar\] = ()\^[3]{}. (At this point we count all baryons as nucleons, assuming that excitational fluctuations and rotations contribute to the surrounding pionic fluctuations). Although the strength of the Skyrme term does not appear explicitly in (\[pbar\]), the presence of the ${\cal{L}}^{(4)}$-term is essential for the formation of the solitons, because during the evolution it transforms the sum over the absolute values of average winding densities into the average number of fully developed solitons and antisolitons. So it is hidden in the growth law characterized by $\alpha$. For meson production we adopt the considerations which led to the estimate (\[Ef\]). Counting again all mesons as zero momentum pions we have \[npi\] n\_= = C\_0[V]{}\_0 =f\_\^2([A]{}b N). The rapidity density of negatively charged pions $n_{\pi^-}=\frac{1}{3}n_\pi$ then is \[piminus\] = f\_\^2. In a heavy-ion collision the transverse area ${\cal A}$ of the Bjorken rod will correspond to the spatial overlap of the colliding relativistic nuclear slabs. As we have assumed spatially homogeneous initial conditions we have to consider slabs with constant nucleon (area-)density. In order to account for the number A of nucleons contained in one slab, its radius must be taken as $r_0$A$^{1/2}$, with $r_0\approx 1.2$ fm. Then, as function of centrality, $(d n_{\pi^-}/d \eta)$ is directly proportional to the number of participants $N_p$, which is one of the basic experimental results in relativistic heavy-ion collisions. For central collisions of A-nucleon slabs we have $N_p = 2$A, so we find for the $\pi^-$-rapidity density per $N_p/2$ participants \[ppp\] = (r\_0 f\_)\^2. This is an interesting result because all parameters have been absorbed into physical quantities. There are, however, several caveats: We have used for this result the form of the potential (\[pot\]) [after]{} the quench (only this enters into the calculations). This means that differences between the average potential energy [before]{} the quench (\[Uth0\]) and immediately [after]{} the quench (\[Uth\]) are left out. However, this difference is of the order $\sigma_0^2$, which has been neglected in (\[ppp\]) anyway. But generally, the result (\[ppp\]) should be considered as a lower limit. It should further be noted that the result (\[ppp\]) depends linearly on the time $\tau_1$ for the onset of the roll-down. The definition of $\tau_1$ in (\[onset\]) is not very stringent and may be subject to changes by $\pm 20\%$. The estimate (cf. eq.(\[onset\])) we used for the time $\tau_1$ required $(\tau_0 m_\sigma)\leq\sqrt{2}$ , i.e. with $m_\sigma \sim 3-5$ fm$^{-1}$, the initial time $\tau_0$ should not exceed $0.3-0.5$ fm. Another unsatisfactory feature of the homogeneous Bjorken rod is that the inhomogeneity in the nucleon (area-)density of real relativistic nuclear slabs with transverse radii $r_0$A$^{1/3}$ has to be represented through radii $r_0$A$^{1/2}$ for homogeneous slabs. The experimental value [@phenix] for the $\pi^-$-rapidity density per $N_p/2$ lies between 1.2 and 1.5 for $N_p$ increasing up to 350. With $r_0 f_\pi=0.57$, $m_\sigma/m_\pi\approx 5-8$, our result (\[ppp\]) leads to \[pipp\] 0.75-1.2 . In the light of the reservations discussed above, this is quite satisfactory. In contrast to the parameter-free pion multiplicities, the result for baryon-antibaryon creation depends on two parameters: the time $\tau_0$ when the initial hadronic field ensemble is established and begins its expansion, and the initial coherence length $R_0$ within that ensemble. From (\[pbar\]) and (\[piminus\]) we have (with $\alpha\sim 0.2$ to 0.25) \[ratio\] 0.15 (). The experimental value for the ratio of integrated ${\bar p}$ to $\pi^-$ multiplicities lies between 0.065 and 0.085 [@phenix] for varying numbers of participants. With $m_\pi/f_\pi^2=3.0$ fm, and a typical $\sigma$-mass of $m_\sigma \approx$ 3 fm$^{-1}$, the experimentally observed multiplicity ratios are reproduced if $R_0$ and $\tau_0$ (both in \[fm\]) satisfy \[rule\] R\_0 (3 \_0)\^[+1/3]{}. For initial times in the range 0.2 $\leq \tau_0\leq$ 0.5 the dependence on $\alpha$ is very weak and the coherence length varies in the range 0.7 $\leq R_0 \leq$ 1.2 (all in \[fm\]). These values are certainly within the limits of conventional assumptions. Interpreted in terms of a thermodynamic equilibrium ensemble, $R_0\sim 1\mbox{fm}\sim \Te^{-1}$ implies the standard estimate $\Te\sim 200$ MeV for the chiral phase transition. With our choice $a/b=\tau_0=R_0/R_{\\| 0}$ for the ratio of the spatial and rapidity lattice constants, the initial time $\tau_0 \approx 1/3$ fm resulting from (\[rule\]) for $R_0=1$ fm then means that in the initial ensemble the initial rapidity coherence distance $R_{\\| 0}$ extends over three units of rapidity. Conclusion ========== We have presented numerical simulations of the dynamical evolution which chiral field configurations undergo in a rapidly expanding spatial volume. Starting at an initial time $\tau_0$ from a random hadronic field ensemble with restored chiral symmetry, we follow its ordering process and roll-down into the global potential minimum with spontaneously broken chiral symmetry. In accordance with standard concepts of heavy-ion physics we have considered one-dimensional longitudinal expansion of an essentially baryon-free region of high energy density, as it may be realized in the aftermath of an ultra-relativistic collision of heavy ions for central rapidities. Performed on a space-rapidity lattice in proper time of comoving frames, such simulations are very powerful instruments which allow to investigate a multitude of interesting features related to the chiral phase transition. We have concentrated here on the topological aspects which are directly related to baryon-antibaryon multiplicity as a sensitive signal for the phase transition. Mesonic abundancies could be analysed as well, both for $\pi$ and $\sigma$ mesons (or any other elementary fluctuations included in the chiral field). Not only their spectra can be obtained, but from the instantaneous configurations the spectral power of their momentum distribution could be extracted at every point in time. The method is not restricted to thermally equilibrated initial ensembles with global or local temperature; inhomogenities and anisotropy in the correlation lengths could be implemented naturally. Surface effects could be investigated by suitable boundary conditions. This may be interesting with respect to the $A$-dependence of spectra and multiplicities. Here we have applied only standard periodic conditions. The one-dimensional expansion could be replaced by anisotropic or spherically symmetric expansion, which may be of specific interest in cosmological applications. We have used the sudden quench approximation, which could be replaced by any desired time-dependence of the chiral potential with arbitrary quench times. We have selected ensembles with conserved net-baryon number $B=0$ or very small $B$. Any other choice would be possible, and it appears as a peculiarly attractive feature to study evolutions in ensembles with high net-baryon density, either fixed or in the form of grandcanonical ensembles. The method is well suited to analyse distribution, growth and realigning of domains with disoriented chiral condensate as has been shown previously in purely dissipative dynamics [@HoKl02]. The generalization to SU(3)-fields appears most interesting, to learn about strangeness production in terms of baryonic and kaonic abundance ratios. Evidently, the method opens up a wide field of applications. Unfortunately, however, we know very little about the nature and characteristics of the initial ensemble which enters crucially into all physical results. So, in our present analysis of antibaryon and pion multiplicities, the experimental data do not allow to draw definite conclusions about the validity of the topological approach, because the results depend on two initial coherence lengths, the spatial $R_0$ and rapidity $R_{\\| 0}$, (which for an isotropic initial ensemble are related by $R_0=R_{\\| 0}\tau_0$). We can only conclude that conventional assumptions about these quantities lead to results which are compatible with experimentally detected multiplicities. So, luckily, the mechanism is not ruled out. On the other hand, an assumption like $\tau_0=R_0$ ( which would imply that the correlations have grown with the speed of light from a pointlike origin ) is ruled out: it would overestimate the abundance ratio in (\[ratio\]) by a factor of 5. Acknowledgement =============== The author appreciates helpful discussions with J. Klomfass, H. Walliser, and H. Weigel. [99]{} =-0.1cm T.H.R. 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--- abstract: 'We demonstrate single atom anisotropic magnetoresistance on the surface of a topological insulator, arising from the interplay between the helical spin-momentum-locked surface electronic structure and the hybridization of the magnetic adatom states. Our first-principles quantum transport calculations based on density functional theory for Mn on Bi$_2$Se$_3$ elucidate the underlying mechanism. We complement our findings with a two dimensional model valid for both single adatoms and magnetic clusters, which leads to a proposed device setup for experimental realization. Our results provide an explanation for the conflicting scattering experiments on magnetic adatoms on topological insulator surfaces, and reveal the real space spin texture around the magnetic impurity.' author: - 'Awadhesh Narayan, Ivan Rungger, and Stefano Sanvito' title: Single atom anisotropic magnetoresistance on a topological insulator surface --- Introduction ============ Topological insulators (TIs) are a materials class holding great promises for new avenues in both fundamental and applied condensed matter physics. In TIs a spin-orbit-driven bulk band inversion results in time-reversal symmetry-protected surface states. The spin-momentum locked surface states are robust against non-magnetic disorder and perfect backscattering is forbidden as a consequence of counter-propagating electrons possessing opposite spins, which cannot be flipped by time reversal symmetry obeying impurities [@kane-rev; @zhang-rev]. Conventional magnetoresistance (MR) devices, spin valves, utilize two magnetic electrodes (polarizer and analyzer) separated by a spacer. Recently, Burkov and Hawthorn found a new kind of MR on TI surfaces, which requires only one ferromagnetic electrode [@hawthorn-mr]. TI surface-based spin valves, showing anomalous MR, have also been studied by model calculations [@nagaosa-spinvalve]. In this paper we report an extremely large anisotropic single-atom MR on the surface of a TI, stemming from the interplay between the helical surface states and the spin anisotropy of the magnetic adatom. Based on this huge anisotropic MR (of several hundred percents, compared to a few percents usually obtained in conventional ferromagnets), we propose a new device concept, which has a number of advantages over previous proposals for magnetic sensors: (i) it does not need any magnetic electrode, but requires only a magnetic adatom, (ii) it does not rely on opening a band gap in the surface states, and (iii) it does not require injecting a spin polarized current into the topological insulator. Our idea is based on the magnetic anisotropy of atoms on TI surfaces, which theory predicts to be large [@fazzio-adatoms] and magnetic circular dichroism confirms [@wiesendanger-febise1; @wiesendanger-febite], demonstrating controllable magnetic doping [@wiesendanger-febise2]. In presence of magnetic impurities spin-mixing on the surface of a TI is possible and scattering may be allowed. However, experiments have been conflicting and at the moment it is not clear whether or not scattering is observed [@yazdani-dopant; @madhavan-dopant]. Our results provide a possible way of reconciling these observations. We show that magnetic impurities open new backscattering channels, but these are found only at those energies where the impurity presents a large density of states and hybridizes with the underlying TI surface states. Then, the conductance depends strongly on the orientation of the local moment of the magnetic adatom, which implies a large MR. Away from these energies the transmission is close to the unperturbed value and no signature of the magnetic dopant is seen. Our large-scale density functional theory (DFT) calculations allow us to probe the real space spin texture around the magnetic adatom, where the inclusion of atomistic details reveals significant differences from previous model-based calculations [@zhang-magimp; @balatsky-magimp]. ![(Color online) Transport setup with a Mn atom adsorbed on a three quintuple layer Bi$_2$Se$_3$ slab, (a) viewed in the plane perpendicular to and (b) along the transport direction ($z$). The scattering region supercell consists of 8 Bi$_2$Se$_3$ primitive unit cells in the $xy$ plane and 16 unit cells along $z$, giving a concentration of 1 Mn atom in 1920 Bi/Se atoms ($\approx$ 0.05%).[]{data-label="setup"}](fig1.pdf) Computational Methods ===================== First-principles transport calculations are performed using the [smeagol]{} code [@sanvito-smeagol1; @sanvito-smeagol2; @sanvito-smeagol3], which interfaces the DFT [siesta]{} package [@soler-siesta] to a non-equilibrium Green’s function approach. Spin-orbit interaction is described by the on-site approximation [@sanvito-soc]. An order-$N$ implementation allows us to study large systems with a few thousand atoms, while maintaining a good basis set quality [@sanvito-sbqwell]. We use a double-$\zeta$ polarized basis, with a real space mesh cutoff of 300 Ry. The generalized gradient approximation for the exchange-correlation functional is used. The valence comprises Bi ($6s$, $6p$), Se ($4s$, $4p$) and Mn ($3d$, $4s$), while norm-conserving Troullier-Martins pseudopotentials describe the core electrons. Mn is studied in the Bi on-top geometry, which is the most stable [@fazzio-adatoms]. All the atoms in the top quintuple layer (QL) are allowed to move and the structures are relaxed until the forces are less that 0.001 eV/Å. For transport calculations, semi-infinite electrodes comprising 3 QL Bi$_2$Se$_3$ slabs are attached to the scattering region (shown in Fig. \[setup\]), and a minimum of 25 Å of vacuum is included along the slab thickness ($y$-direction). We use a $3\times 1\times 1$ $k$-point grid for converging the charge density, while a much denser grid of at least 80 $k_{x}$-points is employed to evaluate the transmission, reflection amplitudes and densities of states. Results and Discussion ====================== ![(Color online) Transmission and density of states projected on Mn for different Mn spin directions. In (a) and (b) results are for $k_{x}=0$, while in (c) and (d) they are averaged over all the incidence angles. For Mn spin along $x$, the transmission is unperturbed, while reduced transmission occurs for other directions, resulting in a single-atom anisotropic MR.[]{data-label="trms"}](fig2.pdf) The transmission coefficient for different orientations of the Mn magnetic moment is shown in Fig. \[trms\]. For $k_{x}=0$ \[normal incidence, Fig. \[trms\](a)\] and the Mn spin aligned along the $x$-axis, which is the spin direction of the incoming electron’s traveling along the positive $z$-axis, the transmission is close to two in the energy window of the topological state (approximately -0.1 eV to 0.3 eV), i.e. there is a unity contribution from each surface. In contrast, for the two orthogonal Mn spin directions, a dip in transmission occurs in the energy range, where a peak in the Mn projected density of states (PDOS) is found. When the Mn spin is along $x$, there is no reduction in transmission, even though there is a peak in Mn PDOS with a height comparable to the case of the other two directions. In all the three cases we find the Mn adatom having a moment close to 4.5 $\mu_\mathrm{B}$, in agreement with previous reports [@fazzio-adatoms], and a substantial in-plane magnetic anisotropy of 6 meV [@anisotropy]. After integrating over the entire Brillouin zone for all $k_{x}$ values a similar picture is obtained \[see Figs. \[trms\](c) and \[trms\](d)\]. Thus, we find that at the energies of the Mn states there emerges an anisotropic MR, depending upon the spin orientation of the magnetic adatom. We find a MR, $\mathrm{MR}=(T_{x,s}-T_{\alpha,s})/T_{\alpha,s}$, of 670% (here $T_{x,s}$ is the transmission at the top surface with the Mn spin along $x$ and $T_{\alpha,s}$ is the surface transmission for the other two Mn spin directions $\alpha=y, z$). We emphasize that this mechanism for MR does not involve opening a band gap in the surface state spectrum. Our findings can be compared to the results in Ref. [@rauch-dual], where, in the presence of a magnetic field, the gap in the surface state spectrum does not open as long as the mirror symmetry is preserved. This is a consequence of the dual topological character of chalcogenides, which are strong topological insulators as well as topological crystalline insulators. This mirror symmetry protects the Dirac crossing, however, the Dirac crossing can be shifted away from the time reversal invariant momenta. Applying a magnetic field perpendicular to a mirror plane of the crystal lattice breaks time-reversal symmetry and destroys the ${Z}_{2}$ topological phase, while the topological crystalline phase is still present, although with the Dirac point shifted away from $\Gamma$. In this case since we have a mismatch between the states from the electrodes (where the Dirac point has not been shifted off $\Gamma$) and the Dirac states close to the magnetic impurity, we find a high resistance state. In this particular setup the transport is along the $z$ direction and, for normal incidence, the spin-momentum relation locks the spin of the surface state along $x$. If the Mn impurity spin points along this direction, then electrons suffer minimal scattering and the resistance is low, while for other Mn spin directions we find a high resistance state. In contrast, if the electrodes are positioned in the orthogonal configuration, such that transport is along $x$, then the propagating electron spin will be along $z$. In this case the low resistance state will be obtained for the Mn spin parallel to the direction of propagation, $z$, while the other two directions will yield a high resistance state \[see Fig. \[trms-km\](e)\]. Since the resistance is given by the orientation of the local magnetic moment with respect to the transport direction, this MR is also anisotropic. In case of strong hexagonal warping, for instance in Bi$_2$Te$_3$, there is also a finite out-of-plane spin component [@henk-warping]. For transport along the $x$-direction, the impurity magnetization along $z$ would not be parallel to the spin of the propagating electron and this would result in a high resisting state. ![Scattering vectors, $q$, as a function of the incident wave vector, $k_{x}$, for Mn spin along (a) $x$, (b) $y$, and (c) $z$. The size of the circles is proportional to the reflection amplitude. The curves are plotted at energies corresponding to peaks in Mn density of states, $E-E_\mathrm{F}$= 0.08 eV, 0.10 eV and 0.08 eV, for Mn spin along $x$, $y$ and $z$, respectively. Here $a_{x}$ and $a_{z}$ are lengths of the electrode unit cell along $x$ and $z$ directions.[]{data-label="scatvec"}](fig3.pdf) From the previous results it is not possible to unequivocally distinguish whether the scattering occurs due to spin-flip between states on one surface or if the MR is an artifact of inter-surface scattering caused by the finite Bi$_2$Se$_3$ slab thickness. We clarify this issue by calculating the full scattering matrix and evaluating the transmission and reflection amplitudes for the individual scattering states on the top and bottom surfaces [@sanvito-bisestep2]. We obtain inter-surface reflection and transmission amplitudes always smaller than 0.008. For intra-surface scattering, in contrast, these quantities reach values up to 1, which confirms that the slab is thick enough to prevent significant coupling between opposite surfaces. A deeper analysis is provided by studying the scattering wave vectors, $q$, and the reflection amplitudes, $r$, on the top surface of the TI slab at the peak energy in Mn PDOS, as a function of the wave vector $k_{x}$ along the direction perpendicular to transport. Here $q=k_{z,\mathrm{out}}-k_{z,\mathrm{in}}$ is the difference between the outgoing, $k_{z,\mathrm{out}}$ and incoming, $k_{z,\mathrm{in}}$, $z$-components of the scattering wave vectors. Since in the bulk gap both $k_{z,\mathrm{in}}$ and $k_{z,\mathrm{out}}$ for the topological surface states are functions of $k_x$, we can evaluate $q$ as function of $k_x$. The result is in Fig. \[scatvec\], with the size of the circles denoting the reflection amplitude $r(k_{x})$. Since the constant energy surface in the energy range of the topological states is approximately circular, we also find the corresponding $q$-$k_x$ plot having a circular shape. For the Mn spin along $x$ and for small $k_x$ the reflection amplitude is vanishingly small, while it becomes larger when $k_x$ increases. This is because the overlap between the two counter-propagating surface states get larger when $k_{x}$ increases. Thus, when the Mn spin is along $x$ the impurity behaves as a non-magnetic scattering center [@sanvito-bisestep1]. In contrast, for the other two directions a large reflection is present even for $k_{x}=0$, which persists at larger $k_{x}$. The total reflection is obtained by integrating this function over all $k_x$, so that the underlying difference in reflection amplitude for small $k_x$ is what yields the anisotropic MR. ![(Color online) (a) Transmission and (b) adatom PDOS for the two-dimensional model, with the adatom spin pointing parallel and perpendicular to the electron spin. (c) Transmission and (d) average PDOS for a magnetic cluster in the two spin configurations. The insets are schematic of the two setups and the dashed lines indicate the transmission of one from the unperturbed edge. Here the adatom onsite energy is 0.1, the hopping integrals to the ribbon 0.3, the hopping between magnetic atoms 0.5 (in units of the nearest neighbor hopping) and the other parameters are the same as in Ref. \[\]. (e) Schematic of the four-probe geometry proposed to measure the anisotropic MR.[]{data-label="trms-km"}](fig4.pdf) Recent scanning tunneling microscopy studies of magnetic adatoms on TI surfaces have observed either new scattering channels, to be ascribed to magnetic scattering [@madhavan-dopant], or found the scattering independent of the magnetic nature of the adatom [@yazdani-dopant]. Our calculations provide a possible explanation for these conflicting observations. In the case of non-magnetic impurities there is scattering, but only for $k_{x}\neq 0$. Additional scattering, which can also occur at $k_{x}=0$, is found for certain directions of the moments of the magnetic impurities. However, this happens only at the energies where the impurity atoms present a large density of states. The transmission coefficients show that a new backscattering channel is created only at the energy of the adatom PDOS, while at all other energies where the topological state exists, no fingerprint of the magnetic adatom is visible. Thus, a likely explanation to reconcile experiments is that the adatom should not only hybridize with the TI surface, but also present peaks in density of states at relevant energies for being detected in the transmission spectra. These depend on the specific magnetic atom and the adsorption site and therefore can differ in different experiments. Away from these energies the transmission spectrum resembles the case of non-magnetic impurities. ![image](fig5.pdf) The anisotropic MR can be understood by considering the impurity as the source of an effective local magnetic field. If the spin of the adatom is parallel to the spin of the propagating electron such an effective field provides a collinear scattering potential, thus precluding spin mixing and backscattering. However, if the local spin forms an angle with that of the itinerant electrons, opposite spin electrons will couple and thus backscattering between helical states will become possible. A minimal two-dimensional model can be used to verify the generality of the MR. We use the Kane-Mele model [@kanemele] for a ribbon with a magnetic adatom or a magnetic cluster placed at the ribbon edge and an exchange coupling between the electron spin and the impurity [@sanvito-andreev]. The edge electrons in this model capture the essential physics of the $k_{x}=0$ case of three-dimensional TIs, which is responsible for the anisotropic MR. The results are shown in Fig. \[trms-km\]. The transmission is high for the adatom spin parallel to the electron spin, while it is low for other angles, thus that the model calculations confirm our first-principles results. Furthermore, for the magnetic cluster the MR is obtained over an energy range larger than that of the single adatom. The fact that the anisotropic MR is independent of the details of adatom means that one can select other magnetic ions to tailor the anisotropy direction. For instance, Cr and Co on Bi$_2$Se$_3$ exhibit an out of plane easy axis, while Mn and Fe an in plane one [@fazzio-adatoms]. In addition to a two-terminal device the anisotropic MR can be measured in a four-probe setup \[Fig. \[trms-km\](e)\]. When the impurity spin points in the direction shown (e.g. due to the magnetic shape anisotropy), then a measurement of the resistance between the electrodes $1$ and $2$ yields a low resistance state, while high resistance is measured between $3$ and $4$. If a thin film with in-plane magnetization is used, then an MR will be obtained depending on the in plane orientation of the magnetization. Out-of-plane magnetization, in contrast, always yields a high resistance state. In general, when the impurity spin points parallel to the helical electron spin the resistance is low, while other angles between the two spins will result in a higher resistance. A large magnetic anisotropy also implies the likely absence of Kondo-type features, which occur with degenerate ground states. Furthermore we expect the spin-flip of the impurity to be negligible as long as the bias is smaller than the magnetic anisotropy [@sanvito-iets; @sanvito-iets2]. Going a step further, we have found the same anisotropic MR for magnetic clusters, in which the aforementioned effects will be even smaller and the magnetic anisotropy may be engineered to be large. Since our ab initio calculations employ extremely large supercells, we are in the position to probe the real-space spin texture around the isolated magnetic impurity. This has been previously studied with Dirac-like effective Hamiltonians [@zhang-magimp; @balatsky-magimp], but here the full details of the electronic structure are included. A combination of atom projected DOS and local DOS is shown in Fig. \[spintex\] for the three different orientations of the Mn spin at the energy corresponding to the peak in Mn PDOS of any given orientation. The induced spins on the atoms around Mn are predominantly along the direction of the Mn spin. For Mn spin pointing along $y$, we find a hedgehog-like in-plane spin texture, with the spins pointing outwards from the impurity site. This contrasts continuum models, which yield a vortex-like in-plane structure [@zhang-magimp; @balatsky-magimp]. The out-of-plane spin points along the positive $y$ direction, in agreement with the model results. This spin is induced over the first QL. For Mn spin along $y$, the spin texture exhibits a three-fold rotational symmetry of the underlying lattice, which is not captured by the continuum low-energy model. For the other two directions, this lattice symmetry is broken by the Mn spin and the neighboring atoms exhibit a spin along the impurity spin direction. We have also investigated the spin texture at other energies and found similar directions as those presented in Fig. \[spintex\], although the magnitude of the induced spin decreases at energies away from Mn PDOS peak. Our spin texture predictions naturally call for an experimental corroboration via spin-polarized scanning tunneling microscopy [@heinrich-stm1; @heinrich-stm2]. Summary and Conclusions ======================= In conclusion we have discovered single-atom anisotropic magnetoresistance on topological insulator surfaces decorated with magnetic adatoms. This effect is a consequence of the spin-momentum locking of TI surface states interacting with the adatom spin. The MR does not originate from the opening of a gap in the surface band structure, nor from spin injection. Furthermore, our results provide a possible explanation for the conflicting observations concerning scattering from magnetic atoms on TI surfaces. Our order-$N$ code allowed us to study the real space spin texture around the adatom, which has differences from previous model calculations. Based on these findings we propose magnetoresistive devices with potentially large MR, utilizing either single magnetic atoms or thin film nanodots incorporated between non-magnetic electrodes, using an in plane rotation of the thin film magnetic moment. 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--- abstract: 'Negation scope has been annotated in several English and Chinese corpora, and highly accurate models for this task in these languages have been learned from these annotations. Unfortunately, annotations are not available in other languages. Could a model that detects negation scope be applied to a language that it hasn’t been trained on? We develop neural models that learn from cross-lingual word embeddings or universal dependencies in English, and test them on Chinese, showing that they work surprisingly well. We find that modeling syntax is helpful even in monolingual settings and that cross-lingual word embeddings help relatively little, and we analyze cases that are still difficult for this task.' author: - \| Federico Fancellu Adam Lopez Bonnie Webber\ ILCC, School of Informatics\ University of Edinburgh\ [f.fancellu@ed.ac.uk]{} bibliography: - 'emnlp2018.bib' title: 'Neural networks for cross-lingual negation scope detection' --- Introduction ============ Negation scope is the set of words whose meaning is affected by a word or morpheme expressing negation. For example, in (\[introex\]), the words ‘you’ and ‘drive’ are in the scope of the negation cue ‘not’. Detecting negation scope is important for many applications, including biomedical information retrieval [e.g. @morante2009metalearning], sentiment analysis [e.g. @councill2010what], and machine translation [e.g. @fancellu2014applying]. Its importance has prompted the development of several annotated corpora and classifiers that detect negation scope with high accuracy. Most of this work is confined to English. Supervised machine learning systems require annotated data, and annotating data for negation scope requires substantial effort, both to adapt annotation guidelines to new languages [@altuna2017scope], and for the annotation itself. As a consequence, there are only a handful of annotated datasets for languages other than English, such as the Chinese Negation and Speculation corpus [CNeSp, @zou2016research]. We ask: Can we learn a model that detects negation scope in English and use it in a language where annotations are not available? [UTF8]{}[gbsn]{} & must & & & because & it & is & dangerous\ & 因为 & 很 & 危险 & & & 能 &\ (.south)–(.north); (.south)–(.north); (.south)–(.north); (.south)–(.north); (.south)–(.north); (.south)–(.north); To answer this question, we develop models on English using language agnostic features only and apply them to Chinese; though annotations are available for Chinese we use them only for testing to simulate our zero-resource setting. Our initial model is the state-of-the-art bidirectional LSTM (BiLSTM) of @fancellu2016neural, initialized with cross-lingual word embeddings and universal part-of-speech (PoS) tags. But BiLSTMs are sensitive to word order, so we also experiment with a cross-lingual input representation that abstracts from word order—syntax in the form of universal dependencies [UD, @demarneffe2014universal]—since we expect that for examples like that in Fig. \[depex\], this will give our model a more consistent view of the input across languages. To condition our model on UD syntax, we consider two different encodings: a Bidirectional DependencyLSTM [D-LSTM below, modeled after the treeLSTM of @tai2015tree] and a Graph Convolutional Network [GCN below, @marcheggiani2017encoding] Our results show it is indeed possible to build models for cross-lingual negation scope detection with performance approaching that of a monolingual oracle. Modeling syntax in addition to surface word order is helpful, as shown by an ensemble of BiLSTM and D-LSTM models outperforming either model alone. Our results also show that cross-lingual word embeddings are not really necessary, suggesting that the model mainly relies on PoS, syntax, and punctuation boundaries—with the latter result reinforcing previous findings [@fancellu2017detecting]. Finally, error analysis show that our best model performs better when the cue is in the same dependency substructure as its scope (as it is in Fig \[depex\]) and fails to capture phenomena related to negation scope, such as neg-raising, where lexical information is required. The task ======== Our input is a sentence with a negation cue, which can be a word (e.g. ‘not’) or a multi-word unit (e.g. ‘by no means’) inherently expressing negation. Our task is to identify the set of words in the scope of the cue; we use gold cues and do not perform automatic cue detection. For example, and : Detecting negation scope is challenging because it often interacts with other semantic phenomena. To resolve (\[exinit\].i), the system needs to know that ‘must’ scopes over negation but other modals (e.g. ‘should’) do not. Likewise for neg-raising, as in (\[exinit\].ii), the presence of certain verbs like ‘think’ or ‘believe’ requires the negation scope to span the object clause (i.e. ‘I think he should not come’). Finally, in (\[exinit\].iii), the causal clause is in the scope despite the marker directly preceding the verb ‘miss’. Similar interactions are attested in Chinese. However, the lack of markers for certain syntactic environments may pose different challenges, since scope boundaries are not defined explicitly. This is the case of clausal complements and descriptive clauses in the following examples which lack explicit markers (‘to’ and ‘that’ in English). [UTF8]{}[gbsn]{} Related work {#prevwork} ============ Automatically detecting negation scope at the string level has been tackled by a variety of classifiers [e.g. @lapponi2012uio; @packard2014simple] exclusively in English or Chinese monolingual settings using language-specific heuristics or resources [e.g. DeepBank @flickinger2012deepbank]. Corpora also reflect this limitation, with only one available in a language other than English. Recently, @fancellu2016neural proposed a BiLSTM model that can be easily repurposed to a new dataset without feature engineering, since it requires only word and universal PoS tags embeddings. Its performance is state-of-the-art in both English and Chinese, but to train it on another language we would still need annotations in that language. In the absence of annotated data in a target language, many work have underlined the usefulness of Universal Dependencies, a cross-lingually consistent syntactic annotation framework. @tiedemann2015cross and @ammar2016many explore the problem of parsing across the languages annotated for UD, while @reddy2017universal have converted UD annotation to logical form for universal semantic parsing and @prazak2017cross have used UD for cross-lingual SRL. The models ========== BiLSTM ------ Our BiLSTM model follows @fancellu2016neural, to which we refer the reader for further detail. Given a sentence $w = w_1$...$w_n$, we encode $w_i$ as d-dimensional embedding vector, [w]{}$_i$ $\in {\rm I\!R^{d_w}}$. Alongside [w]{}$_i$, we also encode information 1) about whether a word $w_i$ is a cue or not, encoded in a cue-embedding vector [c]{}$_i$ $\in$ ${\rm I\!R^{d_c}}$ and 2) about the universal PoS tag of $w_i$, represented as a PoS embedding vector [p]{}$_i$ $\in$ ${\rm I\!R{^d_p}}$. We then concatenate these vectors to yield the input [x]{}$_i$ as follows: $$\begin{aligned} {\bf x}_i=[{\bf w}_i;{\bf c}_i;{\bf p}_i] \label{lstminput} \end{aligned}$$ Our goal is to predict the negation scope $s \in \{1,0\}^{\mid w\mid}$, where $s_i=1$ if a token is part of the scope and 0 otherwise. Bidirectional Dependency LSTM (D-LSTM) -------------------------------------- We now turn to the encoding of a dependency tree, considering the example in Figure \[depex\]. We can traverse the tree bottom-up, from leaves to root, or top-down, from root to leaves. A top-down pass seems insufficient, since negation since cues are usually leaves as in the example. On the other hand, a bottom-up pass would fully encode the subtree rooted at the parent of the cue (in Fig. \[depex\], ‘drive’) but would not be able to encode information about the subordinate being out of scope. Hence we need a bi-directional model that can encode the tree bottom-up and top-down. But this is still insufficient unless the passes communicate: that is, if the bottom-up pass first collects information about the children of ‘drive’, then the top-down pass can pick up that information and pass it downward, hence communicating information about ‘not’ to its sibling nodes in scope. The model accepts as input dependency trees. A dependency tree $g$ is a tuple (V,E), where $V_g$ is the set of word-nodes and $E_g$ the set of dependency edges. Each $e \in E$ is assigned a dependency label $l$. We define as $p(v)$ the parent of node $v$ and $C(v)$ the set of its children. $r$ is the root node. We represent each word-node $v \in V$ as shown in Eq.\[initbtmup\]. The input vector differs from the one used in the BiLSTM model in that we add an an extra embedding [l]{} representing the dependency label of the word in $v$ and a linear transformation to allow multiple layers to be stacked together, as ${\bf x}_v$ can be replaced with the hidden state from a previous layer. $$\begin{aligned} {\bf x}_v &= {\bf W}[{\bf x}_v;{\bf c}_v;{\bf p}_v;{\bf l}_v] + {\bf b} \label{initbtmup} \end{aligned}$$ The computation of the bottom-up pass is the same as in @tai2015tree. This pass returns the state $s_v^\uparrow$ = $\langle $ [h]{}$_v^\uparrow$, [c]{}$_v^\uparrow$ $\rangle$, where [h]{}$_v^\uparrow$ and [c]{}$_v^\uparrow$ are the hidden state and the memory cell of node $v$. To address the lack of bi-directionality in the original child-sum TreeLSTM of @tai2015tree, we add a second top-down pass where we feed the states computed during the bottom-up pass; in this our model is very similar to the one of @chen2017improved. The top-down pass is similar to the bottom-up one but traverses the vertices in a topological order. To create a dependency between passes, we made the states computed during the bottom-up pass, ${\bf s_v^\uparrow}$, available in the form of additional weighted feature during the top-down pass. We start by computing the representation of the root node $r$ as follows: $$\begin{aligned} {\bf s}_r^\downarrow &=LSTM({\bf x}_r,s_r^\uparrow) \end{aligned}$$ When computing the state of a node top-down, we use the parent state the same we did for the children states in the bottom-up pass. The hidden representation of the remaining nodes $v$, [$s_v^\downarrow$]{}, is computed as follows: ![The D-LSTM architecture. Each word is represented by the concatenation of word, universal PoS tags, dependency label and cue features. The latter is a binary feature which is 1 if the word is a cue (like ‘not’) and 0 otherwise. The bottom-up pass builds from the leaves (‘you’, ‘must’ and ‘not’) to the root (‘drive’) and the top-down in the opposite direction. The states built during both passes are exemplified by the $\uparrow$ and the $\downarrow$ respectively.[]{data-label="treelstmfigure"}](treelstm) $$\begin{aligned} s_v^\downarrow &= LSTM({\bf x}_v, s_v^\uparrow, s_{p(v)}^\downarrow) \end{aligned}$$ After both passes are computed, we pass the hidden states obtained at the end of the top-down pass to the softmax layer to compute the probability of a given node to be inside or outside the scope of negation.[^1] $$\label{softmax} \begin{aligned} \hat p(y\|{\bf h}_v) & = softmax({\bf W{\bf h}_v^\downarrow + b}) \end{aligned}$$ A summary of the architecture is shown in Fig. \[treelstmfigure\]. Graph convolutional networks ---------------------------- Our GCN is based on [@marcheggiani2017encoding], to which we refer the reader for details. The intuition behind a GCN is that the hidden representation for each node in the tree is a function that aggregates information from its immediate neighbors. To communicate information between nodes that are not immediate neighbors, this process is iterated a fixed number of times, where each iteration corresponds to a neural network layer. GCNs do not assume that their input directed, so they have no notion of bottom-up or top-down traversal and do not distinguish between parent or child nodes; directionality is encoded explicitly into the neighborhood function. The input to the model is a vector $[{\bf w_n;c_n;p_n}] \in {\rm I\!R}^{d^3}$, which is passed through a non-linearity or through a bi-LSTM before being fed to the GCN. The computation for the hidden state of a given node $v$ takes into account: the hidden state of a neighbor node $n$; the directionality of the edge between $v$ and $n$ and the dependency label with its directionality specified. For each directionality a different weight matrix W$_{dir(u,v)}$ is used. Unlike the D-LSTM, information regarding the dependency label is not encoded in the input but in the bias vector b$^{l(u,v)}$. This yields the following equation: $$\label{gcneq} \begin{aligned} {\bf h^{(K+1)}_v} &= ReLU(\\ & \sum_{u \in \mathcal{N}(v)} g^{(K)}_{v,u}({\bf W^{(K)}}_{dir(u,v)}{\bf h_v} + {\bf b^{l(u,v)}})) \end{aligned}$$ where g$_{(v,u)}$ is an edge-wise scalar gate to help weighing the importance of an edge-node pair amongst several neighbors and $K$ the current layer. However, whereas the original formulation of the GCN encodes information about the dependency labels in the bias term, we weight it alongside other input features. In this way, our GCN resembles the input of the D-LSTM. Our modification results in Eq \[gcneq2\] $$\label{gcneq2} \begin{aligned} {\bf h^{(K+1)}_v} &= ReLU(\\ & \sum_{u \in \mathcal{N}(v)} g^{(K)}_{v,u}({\bf W^{(K)}}_{dir(u,v)}{\bf h_v} + {\bf W_l^{(K)} l}_{(u,v)}+ {\bf b})) \end{aligned}$$ ![The CGN architecture. Hidden representations are built by aggregating neighboring nodes in the dependency trees, as represented by the dashed lines. The node itself is also taken into consideration as shown by the straight lines. Information propagates by stacking up different layers.[]{data-label="gcnfigure"}](gcn) A summary of the architecture is shown in Fig. \[gcnfigure\]. Ensemble -------- Finally, we experiment with two different ensemble models, where we join together the BiLSTM with either the D-LSTM and GCN. We ensemble together our sequential classifier with each of the structured models, to see whether syntactic information can benefit from sequential information and viceversa. We experimented with three different ensemble techniques: a) jointly train the two systems and concatenate the output states of each word before softmax; b) feed the input through a BiLSTM layer (as shown in Eq. \[lstminput\]) before passing it through either the D-LSTM or the GCN (same to what @marcheggiani2017encoding have done to improve the performance of the GCN model) and c) voting. We found voting to achieve the best performance. We also experimented with different kind of voting and we opted for ‘confidence’ voting, where for each word we choose the system where the absolute difference between probability of token being inside and outside the scope is larger. The results in the next section will be based on this last ensemble model. Data and experiment settings ============================ We experiment with <span style="font-variant:small-caps;">NegPar</span> [@liu2018negpar], a parallel English-Chinese corpus of four Sherlock Holmes stories annotated for negation. Although the English side of <span style="font-variant:small-caps;">NegPar</span> leverages pre-existing annotations [<span style="font-variant:small-caps;">ConanDoyleNeg</span> @morante2012conan], most of it has been reannotated to better capture semantic phenomena related to negation scope like modality and neg-raising. Note that the Chinese translation often converts positive English statements to negative—for example, ‘This dress is cheap’ becomes [UTF8]{}[gbsn]{}这件衣服贵 (‘This dress is expensive’). Hence the Chinese contains more negation instances (Table \[negpar\_stats\]). English Chinese ------- --------- --------- train 981 1206 dev 174 230 test 263 341 : Number of negation instances in the train, dev and test set in the English and Chinese sides of <span style="font-variant:small-caps;">NegPar</span>[]{data-label="negpar_stats"} We obtain PoS tags and dependency parses using the Stanford Parser [@chen2014fast]. In preliminary experiments, we compared UD version 1 and version 2, which have an important difference: the negation-specific neg relation in version 1 is replaced by the more general advmod label in version 2. We observed that UD1 performs consistently better, so all experiments reported below are based on version 1. PoS tags are converted into universal PoS tags.[^2] The word segmentation the Chinese side of <span style="font-variant:small-caps;">NegPar</span> is based on also leverages Stanford toolkits [@chang2008optimizing]. When testing across language, we remove language-specific dependency tags (e.g.conj:and$\to$ conj). We experimented with three different cross-lingual word embeddings: a) embeddings pre-trained on Wikipedia data [^3] where a linear transformation has mapped Chinese and English embeddings into a common space [@smith2017offline]; b) average cross-lingual word-embeddings [@guo2016representation], where the embedding vector of a Chinese word is an average of the embedding vectors of its English translations and c) where we take as the embedding vector of a Chinese word the one of the English word with the highest translation probability. We found that c) consistently outperforms the other methods and that’s what we are going to use in our experiments. We observed that method a) in particular suffers from a coverage problem since the embeddings cover only 64% of the training vocabulary. We obtain translation probabilities from approximately 2 million sentences of the UN corpus [@rafalovitch2009united] using fast\_align [@dyer2013simple]. Hyperparameter tuning was performed separately for each system. Both the D-LSTM and the GCN are optimized using Adam [@kingma2014adam], with an initial learning rate of 0.005. We found 4 layers to yield the best performance for the GCN models. We use a dropout as regularizer; in the D-LSTM, dropout is performed on the output layer, whereas in the GCN we follow @marcheggiani2017encoding in performing dropout on the neighbors $N(v)$. We evaluate our models using precision, recall, and $F_1$ over the number of scope tokens; and using the percentage of full scopes spans we correctly detect (PCS below). We evaluate our model cross-lingually by training in English and testing in Chinese (English$\to$Chinese); and for comparison we test models that are trained and test monolingually, on only English or Chinese. Results and Discussion ====================== We summarize the results in Table \[ressummary\] as follows: 1. [Modeling syntax is useful, though not on its own]{}. The ensembles that incorporate syntax outperform other models on both F$_1$ and PCS in both the monolingual and cross-lingual settings, showing that syntax is indeed beneficial—note that they outpeform the state-of-the-art BiLSTM of @fancellu2016neural [@fancellu2017detecting].[^4] The D-LSTM outperforms the GCN in the monolingual settings but the latter performs better in terms of full scope spans detected in the when training in English and testing in Chinese. 2. [The BiLSTM model on its own outperforms either syntactic model on its own by a large margin]{}. Perhaps surprisingly, the BiLSTM performs on par with the D-LSTM in the cross-lingual setting as well, despite relying solely on surface word order. We investigate this in more detail below. 3. [It is indeed possible to build a cross-lingual model of negation]{}, with performance that approaches that of a monolingual Chinese system. We also address the following questions:\ Do all features contribute in the same way? We perform feature ablation on our BiLSTM+D-LSTM ensemble by either removing the cross-lingual word embedding feature (-w) or the universal PoS embedding feature (-p) from either or both model in the ensemble (Table \[ensemble\_abl\]). Results for the BiLSTM+GCN ensemble are similar. Both ensembles show the same trend in that removing the cross-lingual word embedding or the universal PoS embedding feature from the structured models helps with both recall and $F_1$. This shows that both the D-LSTM and the GCN [leverage the dependency structure as main feature for cross-lingual negation scope detection, with little impact from the other two features]{}. Results also show that for both ensembles, results are worse when removing the PoS embedding feature from the BiLSTM model, suggesting that the BiLSTM relies on PoS to model word order. -- -- ------- --------------- --------------- ------- 75.09 [78.27]{} 74.32 P 62.59 62.36 64.38 R 68.27 69.55 69.17 F$_1$ 76.98 74.26 74.93 P 65.55 [69.12]{} 69.05 R 70.81 71.60 [71.87]{} F$_1$ 75.38 73.2 71.86 P 56.68 56.62 59.79 R 64.71 64.04 65.28 F$_1$ -- -- ------- --------------- --------------- ------- : (P)recision, (R)ecall, and F$_1$ for the feature ablation experiments, where models trained on all the features (all) are compared with models where cross-lingual word embeddings (-w) and universal PoS tags (-p) are removed. The row represents the three different ablated BiLSTM models, the columns the ablated D-LSTM models. Each cell represents an ensemble of these ablated models; for instance, the one with the highest F$_1$ (2$^{nd}$ row, 3$^{rd}$ column) is the ensemble of a BiLSTM where cross-lingual word embeddings are removed and a D-LSTM where universal PoS are removed.[]{data-label="ensemble_abl"} Why are BiLSTM useful in cross-lingual settings? Results in Table 2 might be surprising considering that the sequential nature of the BiLSTM does not adapt well with difference in word ordering that language can exhibit. This might be an artifact of the two languages used in the experiment, since English and Chinese have similar word order. However, this also could be explained by a striking observation by @fancellu2017detecting, who showed that recurrent classifiers are very accurate when negation scope is delimited by punctuation and sentence boundaries but inaccurate otherwise. For example, they would correctly predict the scope in Ex. (\[punct\].i), which we refer to as an easy case, but not the one in Ex. (\[punct\].ii), a hard case. To assess whether BiLSTM learns that punctuation is informative also in a cross-linguistic setting, we carry out two additional experiments. First, we replicate the experiments @fancellu2017detecting and divide the development instances into two groups, the easy instances, predictable by punctuation alone and the hard instances where scope cannot be predicted by punctuation alone. If the predictions of the BiLSTM are guided by punctuation we would expect easy instances to be predicted correctly more often than hard ones. Results in Table \[punctanalysis\] seems to confirm our prediction where the sequential model learns to use punctuation to detect negation scope. As for the hard cases we also noticed that in 47.6% of the cases prediction begins or ends at a punctuation token. condition easy hard ----------------------------- ------ ------ Chinese 48 22 English$\rightarrow$Chinese 27 5 : PCS for easy and hard instances using the BiLSTM model on the development set.[]{data-label="punctanalysis"} Given these results, we expect performance to worsen when punctuation is removed from the training and test data, so that the model cannot rely on it. We tested this in a second experiment where we compare the performance of the BiLSTM model with the same model where punctuation in the input has been removed. Results in Table \[punctnopunct\] confirm our hypothesis: the model with no punctuation performs significantly worse. condition P R F$_1$ PCS ------------------ ------ ------ ------- ------ with punctuation 66.2 71.0 68.5 13.8 without 57.7 59.8 58.4 8.6 : Comparison between two BiLSTM models in the cross-lingual task on the development set, one with (punct) and one without (no punct.) punctuation tokens.[]{data-label="punctnopunct"} Error Analysis ============== What is our model learning? To analyze the performance of our best ensemble model, the BiLSTM+D-LSTM, we look at the syntactic environment scope appears in. We approximate this by looking at the least common ancestor for all the nodes in the scope and by taking the label its parent edge; if the scope is discontinuous, we take into consideration the labels on top of all spans. For each of the most frequent dependency labels, we report token-level F$_1$, as well as the percentage of correct scope spans we recover (PCS). ----------- ------- ------ ------- ------ label F$_1$ PCS F$_1$ PCS root 76.4 41.1 70.5 22.9 conj 81.6 32 81.7 25.8 ccomp 77.5 43 66.5 10.7 nsubj 77.3 25 65.4 3.2 dep 79.8 44 78.0 30.2 dobj 70.8 6 63.2 9.3 nmod:prep 68.9 0 – – nmod – – 55.9 5.2 advmod 52.9 0 61 0 ----------- ------- ------ ------- ------ : Analysis of the syntactic environment around the scope where the dependency label represents the parent of the least common ancestor of all the nodes in the scope. Labels are ordered from most to least frequent.[]{data-label="depanalysis1"} Results are shown in Table \[depanalysis1\] for both English$\rightarrow$Chinese and Chinese settings. In the former, we notice that there is usually a substantial loss in performance in terms of PCS but not in terms of F$_1$, meaning that although the scope is not exactly captured the model is still able to correctly detect approximately the same proportion of tokens. [UTF8]{}[gbsn]{} In general, high performance is related to whether the cue is in the same dependency substructure as its scope. This happens when negation scope spans the entire sentence (‘root’) or when it spans complement (‘ccomp’) or coordinate clauses (‘conj’), where the root is usually a verb which is the cue’s parent. On the other hand, we found that the system do not detect full scope as well when the cue is in a different substructure than the scope. One of these cases involves neg-raising, which also requires lexical information and that the system always predict incorrectly. This is exemplified in (\[negraiseerr\].a) and (\[negraiseerr\].b) for the verbs 想到(‘to think’) and 认为(‘to believe’), where the cue appears in the matrix clause but the scope spans the complement clause. On the other hand, we found that in 8 out of 10 instances the universal quantifier was correctly predicted with respect to the scope of negation, as shown in the example below. Finally we found 8 cases where the systems does not distinguish the homographs 没有 (‘have not’), where both characters are part of the cue and 没有 (‘does not exist’), where only the first character is the cue and the second is the existential verb ‘there is’ which is part of the scope. In these cases, the systems always include 没有 as part of the scope as shown in (\[meiyouerr\]). Conclusion ========== Let us go back to our initial research question: when detecting negation scope in a language other than English, can we train a system to detect negation scope in English using language agnostic features and apply it to a language where no annotations are available? Although not quite as accurate as an oracle monolingual model, we show that this is indeed possible by an ensemble of neural networks, where syntactic and surface word order complement each other. More interestingly, we show that the contribution of other cross-lingual features, such as bilingual word embeddings is minor compared to the information extracted from syntax. We also found that this applies to recurrent models as well, where structural information is extracted in the form of punctuation boundaries around a negated scope. However, some phenomena related to negation scope, especially those requiring lexical information, are still missed by our system fail. We also suggest that future work could apply this method to languages where negation is realized in divergent ways from that of English, like those displaying double and morphological negation. [^1]: We also experimented with concatenating the two passes together but saw no difference in performance [^2]: Mapping available at <https://github.com/slavpetrov/universal-pos-tags> [^3]: Available at <https://github.com/Babylonpartners/fastText_multilingual> [^4]: Our results are not directly comparable to those of @fancellu2016neural [@fancellu2017detecting] since the annotation of the English data is different.
--- abstract: 'The problem of impulsive heating of dust grains in cold, dense interstellar clouds is revisited theoretically, with the aim to better understand leading mechanisms of the explosive desorption of icy mantles. It is rigorously shown that if the heating of a reactive medium occurs within a sufficiently localized spot (e.g., heating of mantles by cosmic rays), then the subsequent thermal evolution is characterized by a single dimensionless number $\lambda$. This number identifies a bifurcation between two distinct regimes: When $\lambda$ exceeds a critical value (threshold), the heat equation exhibits the explosive solution, i.e., the thermal (chemical) explosion is triggered. Otherwise, thermal diffusion causes the deposited heat to spread over the entire grain – this regime is commonly known as the whole-grain heating. The theory allows us to find a critical combination of the physical parameters that govern the explosion of icy mantles due to impulsive spot heating. In particular, the calculations suggest that heavy cosmic ray species (e.g., iron ions) colliding with dust are able to trigger the explosion. Based on the recently calculated local cosmic-ray spectra, the expected rate of the explosive desorption is estimated. The efficiency of the desorption, which affects all solid species independent of their binding energy, is shown to be comparable with other cosmic-ray desorption mechanisms typically considered in the literature. Also, the theory allows us to estimate maximum abundances of reactive species that may be stored in the mantles, which provides important constraints on available astrochemical models.' author: - 'A. V. Ivlev$^1$, T. B. Röcker$^1$, A. Vasyunin$^{1,2}$, and P. Caselli$^1$' bibliography: - 'refs.bib' title: Impulsive spot heating and thermal explosion of interstellar grains revisited --- Introduction ============ The earliest stages of star formation occur in cold ($T\sim10$ K), dense ($n(H)\gtrsim10^4$ cm$^{-3}$), and dark ($A_V\gtrsim10$ mag) molecular cloud cores [e.g., @Myers1987]. Under such physical conditions, rapid freeze-out of molecular species from the gas phase on interstellar grains should occur on a timescale of $\sim10^9/n(H)$ yrs. However, while infrared observations confirm the existence of thick icy mantles on interstellar grains [e.g., @Gibb_ea2004], molecular species are also observed in the dark cold gas [@Tafalla_ea2002; @Caselli_ea2002; @Tafalla_ea2004; @Caselli2012a]. As the lifetime of cold molecular cores is at least $\sim10^6$ yr [@Brunken_ea2014], a non-thermal desorption mechanism is required to maintain the observed gas-phase abundances of species. The recent discovery of complex organic molecules [@Oeberg_ea2010; @Bacmann_ea2012; @Cernicharo_ea2012] and deuterated methanol [@Bizzocchi2014] in the cold gas is further evidence for non-thermal processing and evaporation of cold icy mantles [@VasyuninHerbst2013a]. Interactions of interstellar grains with cosmic ray (CR) particles, X-ray and UV photons, and even their mutual collisions cause the grain heating and hence stimulate sublimation of ice [@dHendecourt1982; @Leger1985; @Hartquist1990; @Schutte1991; @Hasegawa1993; @Shalabiea1994; @Shen2004; @Bringa2004; @Cuppen2006; @Herbst2006; @Roberts2007]. Depending on the mechanism of energy deposition, the heated region may be localized or it may extend over the entire grain – these two scenarios are usually referred to as “spot heating” and “whole-grain heating”, respectively [@Leger1985; @Schutte1991; @Shen2004; @Bringa2004]. Also, some exothermic reactions occurring on the grain surface (e.g., the formation of molecular hydrogen) may result in the local heating and lead to the chemical desorption of weakly bound species [@Duley1993; @Garrod2007; @Cecchi-Pestellini2012; @Rawlings2013]. One can identify two distinct regimes of desorption occurring in response to the impulsive grain heating: The classical thermal evaporation, and the so-called “explosive desorption” triggered by the exothermic chemical reaction(s) between free radicals frozen in the bulk of ice [@dHendecourt1982; @Leger1985; @Schutte1991; @Shalabiea1994]. The essential difference between the two regimes is that the evaporation of the ice mantle (typically limited to the most volatile species) is accompanied by the grain cooling, whereas the chemical reactions (activated by the deposited energy) can lead to the runaway temperature growth. As a result, the explosive desorption may cause the ejection of the entire mantle off the grain surface. Since the 1980’s, there have been various mechanisms proposed to trigger the thermal explosion of icy mantles. In particular, these include inelastic collisions between the grains, when a certain fraction of their kinetic energy is converted into heat in the mantle [@dHendecourt1982; @Schutte1991; @Shalabiea1994], and the impact of energetic particles, such as CR and X rays [@Leger1985; @Shen2004]. The analysis, however, has been [almost completely]{} focused on the whole-grain-heating scenario, neglecting the initial thermal spikes emerging in a grain (e.g., along the CR paths). To the best of our knowledge, the possibility of thermal explosion due to CR spot heating was only discussed by [@Leger1985], who concluded that such process is not feasible.[^1] In this article we revisit the problem of spot heating of interstellar grains. We introduce a concept of the [localized]{} ignition spot and show that the evolution of the initial kinetic energy deposited in a reactive medium in this case is uniquely described by a [single]{} dimensionless number $\lambda$. This concept allows us to calculate a critical value of $\lambda$ above which the thermal explosion is triggered and, hence, to find a critical combination of the physical parameters that govern the explosion of icy mantles due to spot heating. We show that the energy deposited by iron CR are sufficient to cause such explosions. Furthermore, we demonstrate that the chemical explosion due to whole-grain heating is inhibited by efficient sublimation cooling. Based on the recent calculations of local CR energy spectra, we obtain the minimum expected rate of mantle disruption due to impacts of iron CR. Finally, the presented theory allows us to estimate the maximum abundances of reactive species that may be stored in the mantles and, thus, to impose important constraints on available astrochemical models. Theory ====== Consider the situation when a certain amount of kinetic energy is “instantaneously” deposited in a reactive medium. It is intuitive to expect that the exact form of the initial energy distribution must be unimportant for its subsequent evolution, provided this distribution is sufficiently localized [and]{} the energy is rapidly thermalized. Mathematically, the possibility of the thermal explosion in this case can be investigated by assuming the initial temperature distribution in the form of the delta function. Limits of applicability for such approximation of the ignition spot are determined from the numerical analysis, as discussed below. Let us consider the cases where the initial energy is concentrated on a plane, along an axis, or in a point. For such ignition spots, the problem is characterized by the symmetry indices $D=1,2,$ and 3, respectively, and the heat equation describing the temperature distribution $T(r,t)$ in a reactive medium has the following form [@LandauFluid]: $$\rho c\frac{\partial T}{\partial t}= Q_{\rm r}e^{-E_{\rm a}/k_{\rm B}T}+\kappa\left(\frac{\partial^2T}{\partial r^2} +\frac{D-1}r \frac{\partial T}{\partial r}\right),\label{heat_eq}$$ with the initial condition $$T(r,0)=\frac{q_D}{\rho c}\delta_D(r).\label{IC}$$ Here, $\delta_D(r)$ is the delta function in $D$ dimensions, $q_D$ is the initial energy density in the ignition spot, $\kappa$, $\rho$, and $c$ are, respectively, the thermal conductivity, mass density, and specific heat of the medium (treated as incompressible, so $c$ should be taken at constant pressure), $Q_{\rm r}$ is the heat of reaction per unit volume and time, and $E_{\rm a}$ is the relevant activation energy in the Arrhenius factor (definition of $Q_{\rm r}$ and proper choice of $E_{\rm a}$ are discussed in Sec. \[properties\]). We first assume the properties of the medium to be independent of the temperature – the cases when $c$ or/and $\kappa$ are functions of $T$ are considered later. For the analysis of Eqs. (\[heat\_eq\]) and (\[IC\]) we normalize the temperature by the activation energy, $\theta=k_{\rm B}T/E_{\rm a}$. For the dimensionless distance $\xi=r/r_$ we choose the scale $r_$ which provides unity normalization of Eq. (\[IC\]), while the dimensionless time $\tau=t/t_$ is determined by the timescale $t_$ of thermal diffusion at the distance $r_$. This yields $$\label{scales} r_=\left(\frac{q_D}{\rho cE_{\rm a}}\right)^{1/D},\quad t_=\frac{\rho c}{\kappa}r_^2,$$ so the heat equation is reduced to $$\label{heat_eq_norm} \frac{\partial \theta}{\partial \tau}= \lambda e^{-1/\theta}+\frac{\partial^2\theta}{\partial \xi^2} +\frac{D-1}{\xi}\frac{\partial \theta}{\partial \xi},$$ and the initial condition – to $$\label{IC_norm} \theta(\xi,0)=\delta_D(\xi).$$ Thus, in the dimensionless form the problem is characterized by a [single]{} number, $$\label{lambda} \lambda=\frac{Q_{\rm r}}{\kappa E_{\rm a}}\left(\frac{q_D}{\rho cE_{\rm a}}\right)^{2/D}.$$ The role of $\lambda$ is similar to that of the Frank-Kamenetskii number $\lambda_{\rm FK}$ \[see Eq. (\[lambda\_FK\])\] which governs the thermal stability of a steady state [@FrankKamenetskii; @LandauFluid]. The relation between $\lambda$ and $\lambda_{\rm FK}$ is discussed in Appendix \[App\_relation\]. The thermal explosion is triggered when $\lambda$ exceeds a certain critical value $\lambda_{\rm cr}$ – the explosion threshold. From the numerical solution of Eqs. (\[heat\_eq\_norm\]) and (\[IC\_norm\]) we obtain the following thresholds: $$\begin{aligned} D=1: &\quad& \lambda_{\rm cr}=1.45;\nonumber\\ D=2: &\quad& \lambda_{\rm cr}=9.94;\nonumber\\ D=3: &\quad& \lambda_{\rm cr}=22.1.\nonumber\end{aligned}$$ The bifurcation between the decaying and explosive evolutions is illustrated for $D=2$ in Fig. \[fig1\], where the temperature at the center of the ignition spot, $\theta(0,\tau)$, is plotted. For $\lambda<\lambda_{\rm cr}$ the integral effect of thermal diffusion is stronger than that of reaction heating, so the asymptotic temperature decay is described by the fundamental solution of the heat equation in free space [@LandauFluid], which yields $\theta(0,\tau)\propto\tau^{-D/2}$. When $\lambda>\lambda_{\rm cr}$, thermal diffusion becomes asymptotically negligible and the temperature approaches linear growth, since the Arrhenius term in Eq. (\[heat\_eq\_norm\]) tends to a constant ($\lambda$) for large $\theta$. In Fig. \[fig1\], the bifurcation occurs at $\tau\sim3$ (while for $D=1$ and 3 it is at $\tau\sim10$ and $\sim1$, respectively). We conclude that the explosion develops within the physical time of a few $t_$. For the numerical solution, we approximate the initial energy distribution by a rectangular function with width $w$ (the delta function formally corresponds to the limit $w\to0$). The obtained dependence $\lambda_{\rm cr}(w)$ is plotted in Fig. \[fig2\], showing that the explosion thresholds remain practically constant for $w\lesssim1$. Thus, the problem does not (practically) depend on the physical size of the ignition spot as long as it is smaller than $\sim r_$, i.e., the initial energy distribution for such [localized]{} spots is well represented by the delta function. Once the explosion is triggered, the hot reactive zone starts expanding away from the ignition spot. As discussed in Appendix \[App\_front\], the flame front propagates with a constant speed $U$ determined by Eq. (\[U\]). In Sec. \[imp1\] we demonstrate that the magnitude of $U$ is much smaller that the typical sound speed in solids. The above results can be generalized for the case where properties of the medium depend on the temperature. In Appendix \[App\_T\_dependence\] we show that for a power-law temperature dependence of the specific heat, $c(T)\propto T^{\alpha}$ (typical for solids, see Sec. \[properties\]), the explosion threshold rapidly decreases with the exponent $\alpha$. Figure \[fig3\], illustrating the case $D=2$, demonstrates that for the linear temperature dependence the value of $\lambda_{\rm cr}$ decreases by one order of magnitude, and for the quadratic – by two. Note that the number $\lambda$ as well as the front speed $U$ in this case are given by Eqs. (\[lambda1\]) and (\[U1\]). We also demonstrate that the temperature dependence of the thermal diffusivity $\chi=\kappa/\rho c$ has relatively weak effect on the results. Implication for interstellar dust grains {#implications} ======================================== In this section, the theory presented in Sec. \[theory\] is applied to interstellar dust grains, to obtain conditions when impulsive heating by energetic particles is expected to cause the thermal explosion of icy mantles. The impulsive heating by CR particles, sketched in Fig. \[fig4\], has the axial symmetry and is described by the solution for $D=2$. The initial energy density $q_2$, which enters the dimensionless number $\lambda$ in this case, is equal to the stopping power of a CR particle. The stopping power depends on the particle kinetic energy $\varepsilon$ (per nucleon) and exhibits a broad maximum at $\varepsilon=\varepsilon^{\rm max}$ [@ZieglerBook]: for protons, $\varepsilon_{\rm H}^{\rm max}\sim0.1$ MeV and $q_2(\varepsilon_{\rm H}^{\rm max}) \sim 10^{-10}$ J cm$^{-1}$; for iron ions, $\varepsilon_{\rm Fe}^{\rm max}\sim1$ MeV/nucleon and $q_2(\varepsilon_{\rm Fe}^{\rm max}) \sim10^{-8}$ J cm$^{-1}$. The heating by X rays is better described by the spherically-symmetric solution, $D=3$ (see discussion in Sec. \[imp1\]). Properties of icy mantles {#properties} ------------------------- Let us summarize typical physical properties of mantles which determine the magnitude of $\lambda$. For many amorphous solids (including ice) the specific heat $c$ increases approximately as $\propto T^2$ at lower temperatures, with $10^{-2}$ J cm$^{-3}$K$^{-1}$ $\lesssim\rho c\lesssim0.3$ J cm$^{-3}$K$^{-1}$ for $10$ K $\leq T\leq 50$ K; the growth becomes slower at higher temperatures, $\rho c\sim3$ J cm$^{-3}$K$^{-1}$ at $T\sim10^3$ K [@Zeller1971; @Leger1985]. We employ this generic dependence for the estimates below. For the thermal conductivity $\kappa=\rho c\chi$ we use the diffusivity $\chi\sim10^{-2}$ cm$^2$s$^{-1}$ [@dHendecourt1982; @Leger1985; @Schutte1991]; the latter is approximately constant for many amorphous solids at $T\gtrsim30$ K [@Zeller1971]. Note that $\chi$ may decrease with $T$ for amorphous water ice [@Andersson2002], but this should only have a minor effect on results (see Appendix \[App\_T\_dependence\]). For the sake of clarity we suppose that among the variety of reactive species (radicals) stored in the mantle, there is a pair (A and B) whose exothermic reaction dominates the heat release (the approach can be straightforwardly generalized to multiple reactions). The heat rate is then given by [@Leger1985] $Q_{\rm r}\simeq E_{\rm r}\varphi_{\rm A}\varphi_{\rm B}N\nu$, where $E_{\rm r}\sim3$ eV is the typical energy release per reaction, $\varphi_{\rm A,B}=N_{\rm A,B}/N$ are the fractional abundances of the species, $\nu\simeq2\times10^{12}$ s$^{-1}$ is their characteristic vibration frequency, and $N\simeq3\times 10^{22}$ cm$^{-3}$ is the total number density of molecules in ice [@Leger1985; @Schutte1991]. Thus, to estimate the magnitude of $Q_{\rm r}$ we need to know the abundance of reactive species. Let us elaborate on this point. Direct infrared observations of interstellar ices can only supply us with the abundances of major ice constituents which are in general not reactive under cold ISM conditions, with the exception of CO ice. As such, we have to rely on astrochemical modeling when estimating abundances of reactive species in a typical interstellar ice. Early astrochemical models did not have a distinction between reactive surface and more inert bulk of a thick icy mantle [@Hasegawa_ea1992; @Hasegawa1993]. Therefore, in these models all species adsorbed on a grain surface participate in efficient “surface” chemistry and the resulting fraction of radicals stored in the mantle is very low. However, in a number of more recent studies, several important effects were recognized that favor larger amounts of radicals to be stored in the interstellar ice. First, icy mantles in dark clouds are likely to be thick and consisting of several hundreds of monolayers [see, e.g., Sec. 4.2 of @Caselli2012b]. For this reason, reactive species in the inner layers of the ice may be quickly covered by new accreting species during the ice formation, and become excluded from the rapid surface chemistry. Reactive species become frozen into water ice and thus survive and accumulate [e.g., @Taquet2012]. Moreover, it is likely that icy mantles are exposed to the UV photons even in dark clouds. Photons can penetrate the entire mantle and dissociate stable molecules in the ice [@Cruz-Diaz2014; @Chang2014], thus producing radicals [@Garrod2013]. Finally, the amount of radicals in the ice may be affected by the internal ice structure: theoretical studies show that the porous structure of ice favors accumulation of the radicals [@Taquet2012]. However, some authors show that interstellar ices are rather compact than porous [@Garrod2013a]. To obtain quantitative estimates of the fraction of reactive species stored in the ice in a dark cloud, we simulate the formation of the icy mantle during the contraction of a diffuse cloud into a dense core using our MONACO code and a simple evolutionary model presented in @VasyuninHerbst2013b. Briefly, in the evolutionary model, the temperature linearly decreases with time from 20 K to 10 K, and the gas density increases from $10^3$ cm$^{-3}$ in the beginning to $10^5$ cm$^{-3}$ at the end of the contraction. Visual extinction $A_V$ increases self-consistently with the density, from $A_V=3$ to $A_V\geq10$. The MONACO code has been updated in comparison to @VasyuninHerbst2013b, and now it includes chemistry in the bulk, due to ice photoprocessing and intramantle diffusion of species (details will be described in a future paper). Mobility of species in the bulk of ice is likely to be significantly lower than on the surface, due to the larger number of neighboring species that bonded to each other [@Garrod2013]. In our microscopic formalism, this means higher diffusion energy for a species in the bulk than on the surface. Following @Garrod2013, we set the diffusion energy of species in the bulk to be two times the respective surface diffusion energy. The latter, in turn, is usually taken as a fraction of the sublimation enthalpy [typically, their ratio varies from 0.3 to 0.8, see, e.g., @Hasegawa_ea1992; @Ruffle2000], here we chose the value of 0.5 in agreement with the best-fit model by [@VasyuninHerbst2013b]. As such, the diffusion energy is $\simeq1150$ K for CO molecules, so CO as well as other abundant reactive species (with higher diffusion energies) that are produced during ice photoprocessing and entrapment of accreting material can effectively accumulate in the bulk of ice. Thus, we shall consider the CO diffusion energy as the relevant activation energy for the Arrhenius factor in Eq. (\[heat\_eq\]), i.e., $E_{\rm a}/k_{\rm B}=1150$ K. In Fig. \[fig5\], the fractional abundances $\varphi$ of the most abundant reactive species in the ice are plotted versus time. CO is mainly accreted from the gas phase. Some of it undergoes hydrogenation and ultimately converts to methanol and other saturated species, but significant fraction of CO molecules get buried in the icy mantle in a pristine form. The next most abundant species is OH, which is mainly produced via dissociation of water by photons and CR protons (according to @Andersson2008, only a fraction of the dissociation products recombine back to H$_2$O). Note that the abundance of OH in our model is about two orders of magnitude lower than in other models of multilayer ice [e.g., @Taquet2012]. Presumably, this is due to the fact that we take into account efficient recombination of OH with free H atoms that are generated in the bulk of ice and perform a random walk before reaching the ice surface. Finally, a certain fraction of HCO is produced in the bulk, mainly in dissociation of methanol by cosmic ray protons. We see that the abundances of CO and OH reach the values of $\varphi_{\rm CO}\sim10^{-1}$ and $\varphi_{\rm OH}\sim3\times10^{-3}$ at later stages of the contraction. We employ these characteristic values for the estimates below. Explosion due to spot heating {#imp1} ----------------------------- In Sec. \[theory\] we pointed out that the presented theory can be used as long as the physical size of the ignition spot does not exceed $\sim r_$. By substituting typical parameters (listed above, with $\rho c=0.3$ J cm$^{-3}$K$^{-1}$) in Eq. (\[scales\]) for $D=2$, we obtain $r_\sim 3\times10^{-6}$ cm for the heating by iron CR. This value is substantially larger than the diameter of the cylindrical volume where CR deposit their energy [$\lesssim100$ [Å]{}, @Leger1985] and, at the same time, is smaller than the size of large grains dominating the interstellar dust mass ($\sim10^{-5}$ cm). Furthermore, the cylindrical explosion develops during the time of the order of $3t_\sim3\times10^{-9}$ s, which is much longer than the time during which the deposited CR energy is thermalized [$\lesssim10^{-11}$ s, @Leger1985]. Thus, the theory is indeed applicable to study the reaction of large grains on impulsive heating by heavy CR species (assuming dust properties that are typically used in astrochemical modeling, see Sec. \[properties\]). Let us estimate the magnitude of $\lambda$ for individual collisions with iron CR. First, we assume a constant $c$ (and $\kappa$) for icy mantles. By substituting in Eq. (\[lambda\]) $q_2\sim10^{-8}$ J cm$^{-1}$ and $\rho c=0.3$ J cm$^{-3}$K$^{-1}$, and setting $E_{\rm r}=3$ eV and $\varphi_{\rm CO}\varphi_{\rm OH}=3\times10^{-4}$ for the reaction between CO and OH, we obtain $\lambda\sim30$, which exceeds $\lambda_{\rm cr}\simeq10$ for $D=2$. Hence, iron ions with the energy corresponding to the maximum of the stopping power are able to trigger the explosion,[^2] whereas CR protons with $q_2\sim10^{-10}$ J cm$^{-1}$ remain under-critical, since $\lambda\propto q_2$. Remarkably, when the temperature dependence of the specific heat is taken into account, the resulting ratio $\lambda/\lambda_{\rm cr}$ becomes even larger, i.e., the explosion condition is relaxed in comparison with the constant-$c$ case: For $c(T)\propto T^{\alpha}$, we employ the results of Appendix \[App\_T\_dependence\] and calculate the enthalpy scale $H_E$ with $\rho c(E)\sim3$ J cm$^{-3}$K$^{-1}$ and $1\leq\alpha\leq2$; using the dependence $\lambda_{\rm cr}(\alpha)$ plotted in Fig. \[fig3\], and substituting $H_E$ in Eq. (\[lambda1\]) we obtain $\lambda/\lambda_{\rm cr}$ in the range between $\sim3$ and $\sim30$ for iron CR. Thus, even if some numbers used above for estimating $\lambda$ would be somewhat less favorable (e.g., if $\varphi_{\rm A}\varphi_{\rm B}\sim3\times10^{-5}$), iron CR should still lead to the explosion. The flame front, generated in the mantle by the explosion, propagates with the speed $U$ given by Eqs. (\[U\]) or (\[U1\]). From this we obtain $U\sim10^4$ cm s$^{-1}$, which is more than an order of magnitude lower than the typical sound speed in ice [see, e.g., @Vogt2008]. The crossing time in a grain of the radius $a$ is $\sim a/U\sim10^{-9}$ s for $a\sim10^{-5}$ cm, so one would expect a practically instant evaporation of the whole mantle. Yet one should keep in mind that the flame front exerts enormous stress – the thermal pressure $\sim NT$ substantially exceeds GPa-level, while the tensile strength of ice is less than one MPa [e.g., @Petrovic2003]. This may lead to mechanical disruption of the mantle before it is completely evaporated. One can estimate the rate of mantle disruption due to the thermal explosions, $1/t_{\rm dis}$, which is determined by the local energy spectrum of iron CR. We assume a constant abundance of iron ions of $\phi_{\rm Fe}\sim10^{-4}$ [relative to protons, see, e.g., @Leger1985; @Shen2004] and employ the local proton spectrum $J_{\rm H}(\varepsilon)$ from [@Padovani2009], where $\varepsilon$ is the energy per nucleon. The disruption rate is equal to the product of the grain cross section and the CR flux contributing to the explosion. The minimum value of the latter can be roughly estimated as[^3] $\sim4\pi\phi_{\rm Fe}\varepsilon_{\rm Fe}^{\rm max}J_{\rm H}(\varepsilon_{\rm Fe}^{\rm max})$, where $\varepsilon_{\rm Fe}^{\rm max}\sim1$ MeV/nucleon corresponds to the maximum of stopping power for iron ions [@ZieglerBook]. Even for dense clouds [with the column density of molecular hydrogen of $\sim3\times10^{22}$ cm$^{-3}$, where the spectrum is strongly attenuated, @Padovani2009], we obtain that the disruption rate for large grains ($a\sim10^{-5}$ cm) is not lower than $$\begin{aligned} \nonumber 1/t_{\rm dis}\sim(2\pi a)^2\phi_{\rm Fe}\varepsilon_{\rm Fe}^{\rm max}J_{\rm H}(\varepsilon_{\rm Fe}^{\rm max}) \sim10^{-6}~{\rm yr}^{-1}.\end{aligned}$$ Furthermore, supposing the entire mantle evaporated upon disruption, we can also estimate the minimum desorption rate of molecules into the gas phase. For a mantle with thickness $\Delta a$, the desorption rate of species A is $\sim4\pi a^2\Delta aN\varphi_{\rm A}/t_{\rm dis}\propto a^5$ (assuming $\Delta a\propto a$). We see that the explosive desorption is heavily dominated by large grains, with the desorption rate of the order of $3\times10^{-7}$ molecules grain$^{-1}$s$^{-1}$ for CO molecules. This value is comparable to the desorption rates due to a combination of other mechanisms [@Shen2004; @Herbst2006], such as explosion due to whole-grain heating (see next section for its critical discussion), and evaporation due to whole-grain and spot heating. We note that the calculations of the desorption rate reported earlier [@Leger1985; @Hartquist1990; @Hasegawa1993; @Shen2004; @Bringa2004; @Herbst2006] do not take into account attenuation of the local CR spectrum, which is included in our analysis. It is noteworthy that we can practically exclude other sorts of energetic particles (e.g., X rays or UV photons) as possible causes of explosion due to spot heating. To demonstrate this, let us consider X rays as the most energetic species among such particles: The maximum energy which can be deposited in a grain by an X-ray photon is limited by the condition that the stopping range of electrons produced by the photon is smaller than the grain size; for $a\sim10^{-5}$ cm we get the upper energy limit of the order of a few keV [see, e.g., @Leger1985]. Since energetic electrons lose most of the energy at the end of their paths, the spherically-symmetric solution is more appropriate to describe the problem in this case. For $D=3$ (and otherwise the same parameters as above), from Eq. (\[lambda\]) we obtain that the minimum ignition energy to satisfy the condition $\lambda>\lambda_{\rm cr}\simeq22$ is $q_3\sim10^5$ eV, which exceeds the maximum deposited energy by about two orders of magnitude. Finally, the presented theory allows us to impose important constraints on the fractional abundance of reactive species in icy mantles, and thus to discriminate between different astrochemical models. In particular, one can estimate the [upper limit]{} of the abundance of radicals which can be stored in a mantle: For example, some models predict that at later stages of the cloud evolution, the product of the relative abundances of such radicals may be as high as $\varphi_{\rm A}\varphi_{\rm B}\sim3\times10^{-3}$ [or even higher, see, e.g., @Schutte1991; @Shalabiea1994; @Taquet2012; @Chang2014]. Since $\lambda\propto q_2\varphi_{\rm A}\varphi_{\rm B}$, its value for iron CR would then be about two orders of magnitude larger than $\lambda_{\rm cr}$, so the obtained abundances could already be marginally sufficient to satisfy the explosion condition for CR protons. However, the latter are $\sim10^{4}$ more abundant than iron CR, so the very possibility of mantle explosion due to impacts of CR protons would imply unrealistically high disruption rates, of $\sim10^{-3}~{\rm yr}^{-1}$ or even larger (these exceed the freeze-out rates at typical molecular cloud densities, i.e., icy mantles simply would not have time to grow). Hence, such high abundances of radicals can be ruled out based on the explosion theory. On the whole-grain heating {#imp2} -------------------------- If the stopping power of an energetic particle colliding with a grain is too low (under-critical), the deposited energy is rapidly redistributed over the whole grain. Even though the overall temperature increase in this case could be only a few tens of degrees, this leads to an exponential amplification of chemical heating in the entire reactive volume of the grain, with important consequences for surface chemistry and the chemical composition of icy mantles. The thermal stability in this regime is determined by the global balance between the volume heating and the surface cooling due to thermal radiation and sublimation [@Leger1985; @Schutte1991; @Cuppen2006]. Therefore, it has been usually argued that there is a critical temperature of the whole-grain heating, above which the explosion must be triggered [@dHendecourt1982; @Leger1985; @Schutte1991; @Shalabiea1994; @Shen2004]. As we pointed out in the introduction, other mechanisms of the whole-grain heating, e.g., due to inelastic grain-grain collisions, have also been suggested as a possible cause of the explosion. Let us consider the global thermal balance for a reactive spherical grain. The steady-state temperature distribution inside the grain is almost homogeneous, so the heating power $P_{\rm heat}$ is approximately the product of $Q_{\rm r}e^{-E_{\rm a}/k_{\rm B}T}$ and the reactive (mantle) volume $4\pi a^2\Delta a$. The surface cooling at temperatures above $\simeq25$ K is dominated by sublimation [@Leger1985; @Schutte1991]. The resulting cooling power $P_{\rm cool}$ is the product of the area $4\pi a^2$ and the cooling rate $\Delta H_{\rm sub}(2\pi mk_{\rm B}T)^{-1/2}p_0e^{-\Delta H_{\rm sub}/k_{\rm B}T}$, where $\Delta H_{\rm sub}$ and $m$ are the sublimation enthalpy and the mass of evaporating molecules, respectively, and $p_0$ is the pre-factor for the saturated vapor pressure [@Leger1983; @Leger1985]. By substituting the heat rate $Q_{\rm r}= E_{\rm r}\varphi_{\rm A}\varphi_{\rm B}N\nu$ for reactive species A and B, we obtain the heating-to-cooling power ratio, $$\begin{aligned} \nonumber \frac{P_{\rm heat}}{P_{\rm cool}}\sim\varphi_{\rm A}\varphi_{\rm B}\frac{E_{\rm r}N\Delta a\nu\sqrt{2\pi mk_{\rm B}T}}{\Delta H_{\rm sub}p_0}e^{(\Delta H_{\rm sub}-E_{\rm a})/k_{\rm B}T},\end{aligned}$$ which must exceed unity for the temperature to increase with time. For estimates we set $T=30$ K which ensures that, irrespective of poorly known emission efficiency of grains, the radiative cooling is negligible [@Leger1985; @Schutte1991; @Shen2004]. By adopting $E_{\rm a}/k_{\rm B}\simeq\Delta H_{\rm sub}/k_{\rm B}\simeq1150$ K for CO molecules, $\Delta a=2\times10^{-6}$ cm for the mantle thickness, and $p_0\simeq10^{12}$ dyne cm$^{-2}$ for the saturated CO-vapor pressure [@Leger1985], we get $P_{\rm heat}/P_{\rm cool}\sim10^{-4}$ (for the reaction between CO and OH). We see that the global grain cooling is much more efficient than the heating. The temperature could only increase with time if the sublimation enthalpy $\Delta H_{\rm sub}$ would be substantially larger than the activation energy $E_{\rm a}$, say by several hundreds of K. However, as was pointed out in Sec. \[properties\], these two values are estimated to be about the same,[^4] and therefore it is rather unlikely that the whole-grain heating could trigger the thermal explosion. Under-critical energetic particles colliding with a grain may nevertheless stimulate reactions between radicals stored in the mantle [e.g., @Reboussin2014]. To estimate this effect (assuming the whole-grain heating), we compare the characteristic time scales of the sublimation cooling and the chemical reactions. The time to burn the characteristic fraction $\bar{\varphi}= \sqrt{\varphi_{\rm A}\varphi_{\rm B}}$ of the major reactive species at a given temperature is $t_{\rm chem}\sim (\bar{\varphi}\nu)^{-1} e^{E_{\rm a}/k_{\rm B}T}$, while the cooling time is $t_{\rm cool}\sim E_{\rm dep}/P_{\rm cool}$, where $E_{\rm dep}$ is the total energy deposited in a grain. By substituting $E_{\rm dep}\simeq 2q_2a<10^4$ eV for CR protons and using otherwise the same parameters as above (and also taking into account the crossover to the radiative cooling at lower temperatures), we obtain $t_{\rm cool}/t_{\rm chem}<3\times10^{-5}$. We conclude that the chemical reactions stimulated by the whole-grain heating, due to collisions with under-critical particles, are many orders of magnitude slower than the cooling. Even though the collisions with (under-critical) CR protons are $\sim\phi_{\rm Fe}^{-1}\sim10^4$ more frequent than with (over-critical) iron ions, such reactions are not expected to noticeably affect the chemical composition of the ice mantle at a timescale of the explosive disruption [although the abundances of trace species, such as complex organic molecules, may be changed, e.g., @Reboussin2014]. It must be stressed, however, that the above estimates completely neglect effects of the local thermal spikes generated in the mantle by CR protons, at a timescale of thermal diffusion. The question of whether the integral effect of the heterogeneous chemistry stimulated by such heating is more profound than that due to whole-grain heating requires a separate careful study. Conclusions =========== The main result of this article is that we have identified the regime of [localized]{} spot heating of a reactive medium, and developed a rigorous theory describing the thermal evolution in this case. The problem is characterized by a single dimensionless number $\lambda$ which depends on the deposited energy and properties of the medium. The theory allows us to determine the universal explosion threshold and accurately describe impulsive heating of icy mantles by energetic particles. A collision with an over-critical energetic particle (when $\lambda$ exceeds the threshold) leads to the thermal explosion which, in turn, generates the flame front propagating in the mantle and leading to its disruption. We showed that heavy CR species, such as iron ions, are able to trigger the explosion, while the stopping power of the most abundant CR protons is insufficient for that (since the stopping power is roughly proportional to the squared atomic number of CR ions). Also, we practically ruled out other energetic species, e.g., X rays, as possible causes of explosion due to impulsive heating. It is important to stress that the question of how exactly the disruption occurs – whether the mantle is completely evaporated due to thermal explosion, or a part of it is ejected off the grain in a form of tiny ice pieces – remains unclear. Thus, the possibility of partial mechanical disruption of the mantle leads to a conclusion that interstellar medium may contain solid nanoparticles of predominantly water ice. Interestingly, the existence of a well-defined explosion threshold allows us to estimate also the upper limit of the abundance of radicals that may be stored in the mantle: For the assumed dust and CR properties, the product of the fractional abundances of two major radicals cannot exceed the value of $\sim3\times10^{-3}$, to avoid unrealistically large desorption rates. Thus, the presented theory enables us to put constraints on astrochemical models. When $\lambda$ is below the threshold, the deposited heat is quickly redistributed over the entire grain volume, i.e., the whole-grain heating scenario is realized. The rates of reactions between radicals frozen in the mantle exponentially depend on the temperature, so even a slight temperature increase can dramatically accelerate the release of chemical energy in the reactive volume – for this reason, the whole-grain heating has been considered so far as the prime possible cause of thermal explosion. However, we have demonstrated that the explosion is unlikely in this case, since the cooling from the grain surface (due to sublimation of volatile species) turns out to be very efficient. The [non-explosive]{} chemical processes, induced in the mantle by under-critical impulsive heating, represent another very important phenomenon which needs to be further investigated. We considered the whole-grain heating due to CR protons, and demonstrated that in this case the abundance of the major reactive species is not expected to noticeably change at a timescale of the explosive disruption (caused by heavy CR species). However, chemical reactions depend on the [local]{} temperature and therefore evolve much faster during short transient heating events, within small volumes where under-critical particles deposit their energy. Careful analysis of such heterogeneous chemistry (as opposed to the chemistry due to whole-grain heating), and the evaluation of its integral effect will be reported in a future paper. Appendix A\ Relation between unsteady and steady problems {#App_relation} ============================================= Consider a reactive medium which has a characteristic size $r_0$ and temperature $T_0$ at the boundary, and assume that there is a thermal equilibrium. The stability of such steady state is determined by the Frank-Kamenetskii number [@FrankKamenetskii; @LandauFluid], $$\label{lambda_FK} \lambda_{\rm FK}=\frac{Q_{\rm r}E_{\rm a}r_0^2e^{-E_{\rm a}/k_{\rm B}T_0}}{\kappa k_{\rm B}T_0^2},$$ which is the ratio of the time scales of thermal diffusion to chemical reaction: When $\lambda_{\rm FK}$ exceeds a certain threshold, the diffusive loss cannot compensate for temperature increase due to ongoing reaction and the steady state becomes unstable, i.e., the thermal explosion is triggered. The thresholds for $D=1,2,$ and 3 are $\lambda_{\rm FK,cr}=0.88,2,$ and 3.32, respectively [@FrankKamenetskii]. For the unsteady problem studied in this paper, the number $\lambda$ plays a role of $\lambda_{\rm FK}$. To understand their relation, let us calculate the “momentary” value of $\lambda_{\rm FK}$ for the unsteady process: The relevant scale for $T_0$ would be the temperature $T(0,t)$ at the center of the ignition spot, while for $r_0^2$ one should substitute the squared diffusion length $(q_D/\rho cT_0)^{2/D}$. Then, by employing Eq. (\[lambda\]) we get $$\begin{aligned} \lambda_{\rm FK}/\lambda\sim e^{-1/\theta_0}\theta_0^{-2(1+1/D)},\nonumber\end{aligned}$$ where $\theta_0(t)=k_{\rm B}T(0,t)/E_{\rm a}$. We see that $\lambda_{\rm FK}/\lambda$ is the sole function of $\theta_0$ and, thus, of $t$. It attains maximum at $\theta_0=2(1+1/D)$, where $\lambda_{\rm FK}/\lambda\sim1$, which identifies the “optimum moment” to trigger the explosion (provided $\lambda>\lambda_{\rm cr}$). Physically, the optimum comes out because the size of the reactive zone is too small in the beginning (i.e., the time scale of thermal diffusion is short), while at later times the reaction becomes exponentially slow. The derived relation allows us to obtain the dependence of $\lambda_{\rm cr}$ on the size of the ignition spot $w$. Using the relation $\theta_0w^D\sim1$ (where $w$ is in units of $r_$), we get the scaling $\lambda_{\rm cr}(w)\sim \exp(w^D) w^{-2(1+D)}$, which provides excellent fit to the curves in Fig. \[fig2\] at $w\geq2$. Thus, unlike the case of localized ignition spot, the unsteady problem for $w\gtrsim1$ is no longer characterized by a single dimensionless number. A similar problem of thermal explosion of large “hot spots” has been studied numerically in the 1960’s by [@Merzhanov1963] and [@Merzhanov1966] who showed that, for a given initial size $r_0$ and temperature $T_0$ of the spot, the explosion threshold $\lambda_{\rm FK,cr}$ has a logarithmic dependence on $T_0$. Appendix B\ Flame front {#App_front} =========== The explosion generates the flame front propagating away from the ignition spot. At sufficiently large times, when the front coordinate $\xi$ is much larger than the front thickness, the last term on the rhs of Eq. (\[heat\_eq\_norm\]) becomes asymptotically negligible (i.e., the front curvature is no longer important). Then, by separating the reactive ($\theta>\theta_{\rm tr}$) and inert ($\theta<\theta_{\rm tr}$) zones of the front [@FrankKamenetskii; @LandauFluid], we can approximately describe the temperature profile by the following equation: $$\begin{aligned} &\theta>\theta_{\rm tr}:&\quad\frac{\partial \theta}{\partial \tau}= \lambda +\frac{\partial^2\theta}{\partial \xi^2}, \nonumber\\ &\theta<\theta_{\rm tr}:&\quad\frac{\partial \theta}{\partial \tau}= \frac{\partial^2\theta}{\partial \xi^2},\nonumber\end{aligned}$$ where $\theta_{\rm tr}\sim1$ is the fitting parameter \[to be determined from numerical solution of Eqs. (\[heat\_eq\_norm\]) and (\[IC\_norm\])\]. We search the solution in the form $\theta(\xi,\tau)=\theta(s)$ with $s=\xi-u\tau$, which yields $\theta(s)=A_1e^{-us}-(\lambda/u)s+A_2$ for $\theta>\theta_{\rm tr}$ and $\theta(s)=A_3e^{-us}$ for $\theta<\theta_{\rm tr}$. By setting $\theta(0)=\theta_{\rm tr}$ and taking into account that in the reactive zone $\theta(s)$ cannot grow faster than linearly, we obtain $A_1=0$; constants $A_2$ and $A_3$ are determined from continuity of $\theta$ and $\partial\theta/\partial\xi$ at $s=0$. We get $u=\sqrt{\lambda/\theta_{\rm tr}}$ and $$\begin{aligned} &s<0:&\quad\theta(s)=-\sqrt{\lambda\theta_{\rm tr}}\:s+\theta_{\rm tr},\nonumber\\ &s>0:&\quad\theta(s)=\theta_{\rm tr}\exp\left(-\sqrt{\lambda/\theta_{\rm tr}}\:s\right),\nonumber\end{aligned}$$ the numerical fit yields $\theta_{\rm tr}\simeq1.3$. In physical units, the front speed, $$\label{U} U=\sqrt{\frac{Q_{\rm r}\kappa}{\theta_{\rm tr}(\rho c)^2E_{\rm a}}},$$ is determined by the reactive and transport properties of the medium. Appendix C\ Thermal explosion when $c$ or $\chi$ are functions of $T$ {#App_T_dependence} ========================================================= Let us consider the case when the specific heat is a function of temperature, $c(T)$, while the thermal diffusivity $\chi=\kappa/\rho c$ is first assumed to be constant. It is convenient [@LandauFluid] to introduce the enthalpy $H=\rho\int c\:dT\equiv F(T)$, noting that $F(T)$ is a single-valued (monotonously increasing) function. Then Eqs. (\[heat\_eq\]) and (\[IC\]) can be rewritten in the following identical form for $H$: $$\begin{aligned} \frac{\partial H}{\partial t}= Q_{\rm r}e^{-E_{\rm a}/k_{\rm B}T}+\chi\left(\frac{\partial^2H}{\partial r^2}+\frac{D-1}r \frac{\partial H}{\partial r}\right),\nonumber\\ H(r,0)=q_D\delta_D(r),\nonumber\end{aligned}$$ where $T=F^{-1}(H)$ is the inverse function. We introduce the enthalpy scale, $H_E=F(E_{\rm a})$, and conclude that the problem can be reduced to the dimensionless form of Eqs. (\[heat\_eq\_norm\]) and (\[IC\_norm\]), where (apart from the Arrhenius term) $H/H_E$ should be substituted for $\theta$; in the Arrhenius term, $\theta$ should be replaced with $F^{-1}(H/H_E)$, and $$\label{lambda1} \lambda=\frac{Q_{\rm r}}{\chi H_E}\left(\frac{q_D}{H_E}\right)^{2/D}.$$ Correspondingly, the speed of the flame front is given by $$\label{U1} U=\sqrt{\frac{Q_{\rm r}\chi}{\theta_{\rm tr}H_E}}.$$ For example, for $c\propto T^{\alpha}$ with the exponent $\alpha\geq0$, we get $H/H_E=\theta^{1+\alpha}$, where $H_E=\rho c(E_{\rm a})E_{\rm a}/(1+\alpha)$; for a constant specific heat ($\alpha=0$) Eqs. (\[lambda1\]) and (\[U1\]) are reduced to Eqs. (\[lambda\]) and (\[U\]), respectively. Figure \[fig3\] shows that the explosion threshold decreases dramatically with $\alpha$. Also, we analyzed the effect of the temperature-dependent thermal diffusivity $\chi$. In this case the diffusion term in the heat equation becomes nonlinear. From the numerical solution with $\chi\propto T^{\beta}$ we obtained the dependencies $\lambda_{\rm cr}(-\beta)$ that are qualitatively similar to $\lambda_{\rm cr}(\alpha)$ shown in Fig. \[fig3\] (i.e., $\lambda_{\rm cr}$ monotonically increases with $\beta$). However, the relative variation of $\lambda_{\rm cr}$ with $\beta$ turns out to be several times smaller than with $\alpha$, i.e., the effect of $\chi(T)$ on the explosion threshold is substantially weaker than that of $c(T)$. [^1]: We note that non-explosive desorption due to spot heating has been extensively studied [e.g., @Leger1985; @Shen2004; @Bringa2004]. [^2]: The explosion threshold could also be evaluated by directly comparing the rates of chemical reaction and thermal diffusion in Eq. (\[heat\_eq\]). However, as shown in Appendix \[App\_relation\], this comparison should be performed at the “optimum moment” (short before the bifurcation in Fig. \[fig1\]). Presumably, this latter point was not taken into account by [@Leger1985], who concluded that the thermal explosion due to spot heating is unlikely. [^3]: To obtain the CR flux contributing to the explosion, one should integrate $J_{\rm H}(\varepsilon)$ over the range of energies (around $\varepsilon_{\rm Fe}^{\rm max}$) where $q_2(\varepsilon)$ exceeds the critical value, i.e., where $\lambda(q_2)\gtrsim10$. Given the uncertainties in the local spectrum in this range [@Padovani2009], only the lower bound of the flux can be reasonably estimated. [^4]: Recent studies [@Theule2015] suggest that the diffusion energy of CO molecules in the bulk ice (which determines the magnitude of $E_{\rm a}$) might significantly [exceed]{} the value of $\Delta H_{\rm sub}$. If so, the ratio $P_{\rm heat}/P_{\rm cool}$ would be even smaller than estimated above.
--- abstract: 'We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.' author: - Charles Curry - 'Kurusch Ebrahimi-Fard' - 'Hans Munthe-Kaas' title: 'What is a post-Lie algebra and why is it useful in geometric integration' --- Introduction {#sect:intro} ============ In recent years classical numerical integration methods have been extended beyond applications in Euclidean space onto manifolds. In particular, the theory of Lie group methods [@Iserles00] has been developed rapidly. In this respect Butcher’s $B$-series [@HWL] have been generalized to Lie–Butcher series [@MK1; @MK2]. Brouder’s work [@Brouder] initiated the unfolding of rich algebro-geometric aspects of the former, where Hopf and pre-Lie algebras on non-planar rooted trees play a central role [@CHV; @MMMKV]. Lie–Butcher series underwent similar developments replacing non-planar trees by planar ones [@LMK; @MKW]. Correspondingly, pre-Lie algebras are to $B$-series what post-Lie algebras are to Lie-Butcher series [@EFLMK; @FMK]. In this note we explore the notion of a post-Lie algebra and outline its importance to integration methods. Post-Lie algebra and examples {#sect:postlie} ============================= We begin by giving the definition of a post-Lie algebra followed by a proposition describing the central result. The three subsequent examples illustrate the value of such algebras, in particular to Lie group integration methods. \[def:postLie\] A post-Lie algebra $(\mathfrak{g}, [\cdot,\cdot], \triangleright)$ consists of a Lie algebra $(\mathfrak g, [\cdot,\cdot])$ and a binary product $\triangleright : {\mathfrak g} \otimes {\mathfrak g} \rightarrow \mathfrak g$ such that, for all elements $x,y,z \in \mathfrak g$ the following relations hold $$\begin{aligned} \label{postLie1} x \triangleright [y,z] &= [x\triangleright y , z] + [y , x \triangleright z],\\ \label{postLie2} [x,y] \triangleright z &= {\rm{a}}_{\triangleright }(x,y,z) - {\rm{a}}_{\triangleright}(y,x,z),\end{aligned}$$ where the associator ${\rm{a}}_{\triangleright}(x,y,z):=x \triangleright (y \triangleright z) - (x \triangleright y) \triangleright z$. Post-Lie algebras first appear in the work of Vallette [@Vallette] and were independently described in [@MKW]. Comparing these references the reader will quickly see how different they are in terms of aim and scope, which hints at the broad mathematical importance of this structure. \[prop:Liebracket\] Let $(\mathfrak g, [\cdot, \cdot], \triangleright )$ be a post-Lie algebra. For $x, y \in \mathfrak g$ the bracket $$\label{postLie3} \llbracket x,y \rrbracket := x {\triangleright}y - y {\triangleright}x + [x,y]$$ satisfies the Jacobi identity. The resulting Lie algebra is denoted $(\overline{\mathfrak g},\llbracket \cdot,\cdot \rrbracket)$. \[cor:preLie\] A post-Lie algebra with an abelian Lie algebra $(\mathfrak g, [\cdot, \cdot]=0, \triangleright )$ reduces to a left pre-Lie algebra, i.e., for all elements $x,y,z \in \mathfrak g$ we have $$\begin{aligned} \label{preLie} {\rm{a}}_{\triangleright }(x,y,z) = {\rm{a}}_{\triangleright}(y,x,z).\end{aligned}$$ \[exm:connection\] Let $\mathcal{X}(M)$ be the space of vector fields on a manifold $M$, equipped with a linear connection. The covariant derivative $\nabla_X Y$ of $Y$ in the direction of $X$ defines an $\mathbb{R}$-linear, non-associative binary product $X\triangleright Y$ on $\mathcal{X}(M)$. The torsion $T$, a skew-symmetric tensor field of type $(1,2)$, is [defined by]{} $$\begin{aligned} \label{torsion} T(X,Y) := X\triangleright Y - Y\triangleright X - \llbracket X,Y \rrbracket,\end{aligned}$$ where the bracket $\llbracket \cdot,\cdot \rrbracket$ on the right is the Jacobi bracket of vector fields. The torsion admits a covariant differential $\nabla T$, a tensor field of type $(1,3)$. Recall the definition of the curvature tensor $R$, a tensor field of type $(1,3)$ given by $$R(X,Y)Z = X\triangleright(Y\triangleright Z) - Y\triangleright(X\triangleright Z) - \llbracket X,Y \rrbracket \triangleright Z.$$ In the case that the connection is flat and has constant torsion, i.e., $R=0=\nabla T$, we have that $(\mathcal{X}(M),-T(\cdot,\cdot),\triangleright)$ defines a post-Lie algebra. Indeed, the first Bianchi identity shows that $-T(\cdot,\cdot)$ obeys the Jacobi identity; as $T$ is skew-symmetric it therefore defines a Lie bracket. Moreover, flatness is equivalent to $(\ref{postLie2})$ as can be seen by inserting into the statement $R=0$, whilst $(\ref{postLie1})$ follows from the definition of the covariant differential of $T$: $$0 = \nabla T (Y,Z;X) = X \triangleright T(Y,Z) - T(Y,X \triangleright Z) - T(X \triangleright Y,Z).$$ The formalism of post-Lie algebras assists greatly in understanding the interplay between covariant derivatives and integral curves of vector fields, which is central to the study of numerical analysis on manifolds. \[exm:rootedtrees\] We now consider planar rooted trees with left grafting. Recall that a rooted tree is made out of vertices and non-intersecting oriented edges, such that all but one vertex have exactly one outgoing line and an arbitrary number of incoming lines. The root is the only vertex with no outgoing line and is drawn on bottom of the tree, whereas the leaves are the only vertices without any incoming lines. A planar rooted tree is a rooted tree with an embedding in the plane, that is, the order of the branches is fixed. We denote the set of planar rooted trees by ${\operatorname{OT}}$. $${\operatorname{OT}}= \Big\{\begin{array}{c} \scalebox{0.6}{\ab}, \scalebox{0.6}{\aabb},\scalebox{0.6}{\aaabbb}, \scalebox{0.6}{\aababb}, \scalebox{0.6}{\aaaabbbb}, \scalebox{0.6}{\aaababbb},\scalebox{0.6}{\aaabbabb}, \scalebox{0.6}{\aabaabbb}, \scalebox{0.6}{\aabababb},\ldots \end{array} \Big\}.$$ The left grafting of two trees $\tau_1 \triangleright \tau_2$ is the sum of all trees resulting from attaching the root of $\tau_1$ via a new edge successively to all the nodes of the tree $\tau_2$ from the left. $$\begin{aligned} \scalebox{0.6}{\AabB}\,\triangleright \scalebox{0.6}{\aababb} =\scalebox{0.6}{\aAabBababb} + \scalebox{0.6}{\aaAabBbabb} + \scalebox{0.6}{\aabaAabBbb}.\end{aligned}$$ Left grafting means that the tree $\tau_1$, when grafted to a vertex of $\tau_2$ becomes the leftmost branch of this vertex. We consider now the free Lie algebra $\mathcal{L}({\operatorname{OT}})$ generated by planar rooted trees. In [@LMK] is was shown that $\mathcal{L}({\operatorname{OT}})$ together with left grafting defines a post-Lie algebra. In fact, it is the free post-Lie algebra $\mathrm{PostLie}(\scalebox{0.6}{\ab})$ on one generator [@LMK]. Ignoring planarity, that is, considering non-planar rooted trees, turns left grafting into grafting, which is a pre-Lie product on rooted tree satisfying [@Manchon]. The space spanned by non-planar rooted trees together with grafting defines the free pre-Lie algebra $\mathrm{PreLie}(\scalebox{0.6}{\ab})$ on one generator [@CL]. \[exm:rmatrices\] Another rather different example of post-Lie algebra comes from projections on the algebra $\mathcal{M}_n(\mathbb{K})$ of $n \times n$ matrices with entries in the base field $\mathbb{K}$. More precisely, we consider linear projections involved in classical matrix factorization schemes, such as $LU$, $QR$ and Cholesky [@CN; @DLT]. Let $\pi^\ast_+$ be such a projection on $\mathcal{M}_n(\mathbb{K})$, where $\ast=LU$, $QR$, $Ch$. It turns out that both $\pi^\ast_+$ and $\pi^\ast_-:={{\mathrm{id}}}-\pi^\ast_+$ satisfy the Lie algebra identity $$[\pi^\ast_\pm M,\pi^\ast_\pm N]+\pi^\ast_\pm [M,N] =\pi^\ast_\pm([\pi^\ast_\pm M,N] + [M,\pi^\ast_\pm N]),$$ for all $M,N \in \mathcal{M}_n(\mathbb{K})$. In [@BGN] it was shown that $M {\triangleright}N := -[\pi^\ast_- M,N] $ defines a post-Lie algebra with respect to the Lie algebra defined on $\mathcal{M}_n(\mathbb{K})$. Corollary \[cor:preLie\] is more subtle in this context as it reflects upon the difference between classical and modified classical Yang–Baxter equation [@DLT; @EFMMK; @EFM]. Post-Lie algebras and Lie group integration {#sect:integration} =========================================== We now consider post-Lie algebras as they appear in numerical Lie group integration. Recall the standard formulation of Lie group integrators [@Iserles00], where differential equations on a homogeneous space $M$ are formulated using a left action $\cdot\colon G\times M\rightarrow M$ of a Lie group $G$ of isometries on $M$, with Lie algebra $\mathfrak{g}$. An infinitesimal action $\cdot\colon \g\times M\rightarrow TM$ arises from differentiation, $$V\cdot p := \left.\frac{\partial}{\partial t}\right\|_{t=0}\exp(tV)\cdot p .$$ In this setting any ordinary differential equation on $M$ can be written as $$y'(t) = f(y(t))\cdot y(t),\label{eq:ode}$$ where $f\colon M\rightarrow \g$. For instance ODEs on the 2-sphere $S^2\simeq SO(3)/SO(2)$ can be expressed using the infinitesimal action of $\so(3)$. Embedding $S^2\subset \mathbb{R}^3$ realizes the action and infinitesimal action as matrix-vector multiplications, where $SO(3)$ is the space of orthogonal matrices, and $\so(3)$ the skew-symmetric matrices. To obtain a description of the solution of (\[eq:ode\]), we begin by giving a post-Lie algebra structure to $\g^M$, the set of (smooth) functions from $M$ to $\g$. For $f,g\in \g^M$, we let $[f,g](p) := [f(p),g(p)]_\g$ and $$(f{\triangleright}g)(p) := \left.\frac{\partial}{\partial t}\right\|_{t=0} g(\exp(tf(p))\cdot p).$$ The flow map of (\[eq:ode\]) admits a Lie series expansion, where the terms are differential operators of arbitrary order, which live in the enveloping algebra of the Lie algebra generated by the infinitesimal action of $f$. Recall that for a Lie algebra $(\g,[\cdot,\cdot])$, the enveloping algebra is an associative algebra $(U(\g),\cdot)$ such that $\g\subset U(\g)$ and $[x,y] = x\cdot y-y\cdot x$ in $U(\g)$. As a Lie algebra $\g$ with product ${\triangleright}$, the enveloping algebra of $(\mathfrak{\g}, [\cdot,\cdot], \triangleright)$ is $U(\g)$ together with an extension of ${\triangleright}$ onto $U(\g)$ defined such that for all $x \in \g$ and $y,z\in U(\g)$ $$\begin{aligned} x{\triangleright}(y\cdot z) &= (x{\triangleright}y)\cdot z + y\cdot (x{\triangleright}z)\\ (x\cdot y){\triangleright}z &= a_{\triangleright}(x,y,z).\end{aligned}$$ Many of these operations are readily computable in practice. Recall that $\g$ has a second Lie algebra structure $\bar{\g}$ associated to the bracket $\llbracket\cdot,\cdot\rrbracket$, reflecting the Jacobi bracket of the vector fields on $M$ generated by the infinitesimal action of $\g$. As a vector space, its enveloping algebra $\left(U(\bar{\g}),\ast\right)$ is isomorphic to $U(\g)$. The Lie series solution of (\[eq:ode\]) is essentially the exponential in $U(\bar{\g})$, which in contrast to the operations of $U(\g)$ is in general difficult to compute. We are lead to the following:[[Basic aim:]{}]{} The fundamental problem of numerical Lie group integration is the approximation of the exponential $\exp^{\ast}$ in $\left(U(\bar{\g}),\ast\right)$ in terms of the operations of $\left(U(\g),\cdot,{\triangleright}\right)$, where $(\mathfrak{g}, [\cdot,\cdot], \triangleright)$ is the free Post-Lie algebra over a single generator. One may wonder why we use post-Lie algebras, which require flatness and constant torsion, and not structures corresponding to constant curvature and zero torsion such as the Levi-Civita connection on a Riemannian symmetric space. The key is that the extension of a ${\triangleright}$ onto $U(\g)$ allows for a nice algebraic representation of parallel transport, requiring flatness of the connection ${\triangleright}$. Indeed, the basic assumption is that ${\triangleright}$ extends to the enveloping algebra such that $x{\triangleright}(y{\triangleright}z)= (x\ast y){\triangleright}z$. From this follows $$\llbracket x,y\rrbracket{\triangleright}z = (x\ast y-y\ast x){\triangleright}z = x{\triangleright}(y{\triangleright}z)-y{\triangleright}(x{\triangleright}z),$$ and hence $R(x,y,z) = 0$. For any connection ${\triangleright}$, the corresponding parallel transport of $g$ is $$g + t f{\triangleright}g + \frac{t^2}{2}f{\triangleright}(f{\triangleright}g) + \frac{t^3}{3!}f{\triangleright}(f{\triangleright}(f{\triangleright}g)) +\cdots.$$ If the basic assumption above holds, this reduces to the formula $\exp^(tf){\triangleright}g$. Recall that the free Post-Lie algebra over a single generator is the post-Lie algebra of planar rooted trees $\postLie(\{\scalebox{0.6}{\ab}\})$ given in Example \[exm:rootedtrees\]. Freeness essentially means that it is a universal model for any post-Lie algebra generated by a single element, and in particular the post-Lie algebra generated by the infinitesimal action of a function $f\in\g^M$ on $M$. For instance, if we decide that $\scalebox{0.6}{\ab}$ represents the element $f \in \g^M$, then there is a unique post-Lie morphism $\F \colon \postLie(\{\scalebox{0.6}{\ab}\})\rightarrow \g^M$ such that $\F(\scalebox{0.6}{\ab}) = f$. Moreover, we then have, e.g., $$\F(\scalebox{0.6}{\aabb})=\F(\scalebox{0.6}{\ab}{\triangleright}\scalebox{0.6}{\ab}) = \F(\scalebox{0.6}{\ab}){\triangleright}\F(\scalebox{0.6}{\ab}) = f {\triangleright}f,$$ or $\F([\scalebox{0.6}{\ab},\scalebox{0.6}{\aabb}])= [f,f{\triangleright}f]$, and so on. This $\F$ is called the elementary differential map, associating planar rooted trees and commutators of these with vector fields on $M$. Hence, all concrete computations in $\g^M$ involving the operations ${\triangleright}$ and $[\cdot,\cdot]$ can be lifted to symbolic computations in the free post-Lie algebra $\postLie(\{\scalebox{0.6}{\ab}\})$. Revisiting our basic aim, we require a description of $U(\postLie(\{\scalebox{0.6}{\ab}\}))$, which is given as the linear combination of all ordered forests (OF) over the alphabet of planar rooted trees, including the empty forest $\one$, $${\text{OF}}=\Big\{\begin{array}{c} \one,\scalebox{0.6}{\ab},\scalebox{0.6}{\ab\ab},\scalebox{0.6}{\aabb}, \scalebox{0.6}{\ab\ab\ab},\scalebox{0.6}{\aabb\ab},\scalebox{0.6}{\ab\aabb}, \scalebox{0.6}{\aababb},\scalebox{0.6}{\aaabbb},\scalebox{0.6}{\ab\ab\ab\ab}, \ldots,\scalebox{0.6}{\aabaabbb},\scalebox{0.6}{\aaabbabb},\ldots \end{array}\Big\}.$$ So an element $a \in U(\postLie(\{\scalebox{0.6}{\ab}\}))$ could be, for instance, of the following form $$a = 3\one + 4.5\scalebox{0.6}{\ab} -2 \scalebox{0.6}{\ab\aabb} + 3 \scalebox{0.6}{\aabb\ab} + 6 \scalebox{0.6}{\aabaabbb\ab} + 7 \scalebox{0.6}{\ab\aaabbabb} - 2 \scalebox{0.6}{\ab\aaabbabb\ab} \cdots .$$ To be more precise, $U(\postLie(\{\scalebox{0.6}{\ab}\}))$ consists of all finite linear combinations of this kind, while infinite combinations such as the exponential live in $\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$ and are obtained by an inverse limit construction [@FMK]. Elements in the space $\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$ we call Lie–Butcher (LB) series. Note that all computations on such infinite series are done by evaluating the series on something finite in $U(\postLie(\{\scalebox{0.6}{\ab}\}))$. Indeed, we consider $\widehat{U}:=\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$ as the (linear) dual space of $U:=U\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$, with a bilinear pairing $\langle\cdot,\cdot\rangle\colon \widehat{U}\times U\rightarrow \RR$ defined such that ${\text{OF}}$ is an orthonormal basis, i.e. for $\omega,\omega'\in {\text{OF}}$ we have $\langle \omega,\omega'\rangle = 1$ if $\omega = \omega'$, and zero if $\omega \neq \omega'$. Two important subclasses of LB-series are $$\begin{aligned} \glb &:= \left\{\alpha\in \widehat{U}(\postLie(\{\ab\}) \ \colon\ \langle \alpha,\one\rangle = 0, \langle\alpha,\omega\shuffle\omega'\rangle = 0\ \forall \omega,\omega' \in {\text{OF}}\backslash\{\one\} \right\}\\ \Glb &:= \left\{\alpha\in \widehat{U}(\postLie(\{\ab\}) \ \colon\ \langle\alpha, \omega\shuffle\omega'\rangle = \langle\alpha,\omega\rangle \langle\alpha,\omega'\rangle\ \forall \omega,\omega'\in {\text{OF}}\right\},\end{aligned}$$ where $\shuffle$ denotes the usual shuffle product of words, e.g., $a \shuffle \one = \one \shuffle a=a$, $$ab \shuffle cd = a(b\shuffle cd) + c(ab \shuffle d).$$ Here elements in $\glb$ are called infinitesimal characters, representing vector fields on $M$ and elements in $\Glb$ are characters, representing flows (diffeomorphisms) on $M$. $\Glb$ forms a group under composition called the Lie–Butcher group. A natural question is how does an element $\gamma \in \Glb$ represents a flow on $M$? The elementary differential map sends $\gamma$ to the (formal[^1]) differential operator, i.e., $$\F(\gamma) = \sum_{\omega\in{\text{OF}}} \langle\gamma,\omega\rangle\F(\omega)\in \widehat{U}(\g)^M.$$ The flow map $\Psi_\gamma\colon M\rightarrow M$ is such that the differential operator $\F(\gamma)$ computes the Taylor expansion of a function $\phi \in C^\infty(M,\RR)$ along the flow $\Psi_\gamma$: $$\F(\gamma)[\phi] = \phi\circ \Psi_\gamma .$$ Recall from Proposition \[prop:Liebracket\] that any post-Lie algebra comes with two* Lie algebras $\g$ and $\overline{\g}$. Hence there are two enveloping algebras $U(\g)$ and $U(\overline{\g})$, with two different associative products. It turns out that $U(\g)$ and $U(\overline{\g})$ are isomorphic as Hopf algebras [@EFLMK; @EFM], such that the product of the latter can be represented in $U({\g})$. For LB-series, the resulting two associative products in $U({\g})$ are called the concatenation product and Grossman–Larson (GL) product [@GL]. Indeed, we have $\omega \cdot \omega' = \omega\omega'$ (sticking words together). The GL product is somewhat more involved, i.e., for $\alpha,\beta \in \glb$ we have $\alpha\ast \beta = \alpha\cdot\beta + \alpha{\triangleright}\beta$, see [@EFLMK] for the general formula. Interpreted as operations on vector fields on $M$, the GL product represents the standard composition (Lie product) of vector fields as differential operators, while the concatenation represents frozen composition, for $\alpha,\beta\colon M\rightarrow U(\g)$ we have $\alpha\cdot\beta(p) = \alpha(p)\cdot\beta(p)$. The two associative products on $U({\g})$ yield two exponential mappings $\exp^\cdot,\exp^\ast$ between $\glb$ and $\Glb$ obtained from these, $$\exp^\cdot(\alpha) = \one + \alpha+ \frac12 \alpha\cdot\alpha +\frac 16\alpha\cdot\alpha\cdot\alpha+\cdots, \; \exp^\ast(\alpha) = \one + \alpha+ \frac12 \alpha\ast\alpha +\frac 16\alpha\ast\alpha\ast\alpha+\cdots .$$ Both send vector fields on $M$ to flows on $M$. However, it turns out that the Grossman–Larson exponential $\exp^\ast$ sends a vector field to its exact solution flow, while the concatenation exponential $\exp^{\cdot}$ sends a vector field to the exponential Euler flow, $$y_1 = \exp(hf(y_0))\cdot y_0.\label{eq:expeuler}$$ All the basic Lie group integration methods can be formulated and analysed directly in the space of LB-series $\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$ with its two associative products and the lifted post-Lie operation. The Lie-Euler method which moves in successive timesteps along the exponential Euler flow is one such example. A slightly more intricate example is On the manifold $M$ a step of the Lie midpoint rule with time step $h$ for (\[eq:ode\]), is given as $$\begin{aligned} K &= hf(\exp(K/2)\cdot y_0)\\ y_1 &= \exp(K)\cdot y_0. \end{aligned}$$ In $\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$, the same integrator $\scalebox{0.6}{\ab}\mapsto \Phi\colon \glb\rightarrow \Glb$ is given as: $$\begin{aligned} K &= \exp^\cdot(K/2){\triangleright}(h\scalebox{0.6}{\ab})\in \glb\\ \Phi &= \exp^\cdot(K)\in \Glb. \end{aligned}$$ We conclude by commenting that the two exponentials are related exactly by a map $\chi: {\g} \to {\g}$, called post-Lie Magnus expansion [@EFLMMK; @EFMMK; @EFM], such that $\exp^\cdot(f)=\exp^\ast(\chi(f))$, $f \in {\g}$. The series $\chi(f)$ corresponds to the backward error analysis related to the Lie–Euler method. In $\widehat{U}\left(\postLie(\{\scalebox{0.6}{\ab}\})\right)$ we find $$\chi(\scalebox{0.6}{\ab}) = \scalebox{0.6}{\ab} - \frac{1}{2} \scalebox{0.6}{\aabb} + \frac{1}{12}[\scalebox{0.6}{\aabb},\scalebox{0.6}{\ab}] + \frac{1}{3} \scalebox{0.6}{\aaabbb} + \frac{1}{12} \scalebox{0.6}{\aababb} - \frac{1}{12} \scalebox{0.6}{\aaabbabb} +\cdots$$ This should be compared with the expansion $\beta$ on page 184 in [@LMK]. Chapoton and Patras studied the equality between these exponentials in the context of the free pre-Lie algebra [@CP]. Acknowledgments {#acknowledgments .unnumbered} =============== The research on this paper was partially supported by the Norwegian Research Council (project 231632). [99]{} 1.0ex , [[Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and PostLie algebras]{}]{}, Commun. Math. Phys. [[297]{}]{} number [2]{} (2010), 553–596. , [[Runge-Kutta methods and renormalization]{}]{}, Europ. Phys. J. 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(2011). , [[Butcher series: A story of rooted trees and numerical methods for evolution equations]{}]{}, accepted in Asia Pacific Mathematics Newsletter, 2017. , [[Lie–Butcher theory for Runge–Kutta methods]{}]{}, BIT Numer. Math. [[35]{}]{} (1995), 572–587. , [Runge–Kutta methods on Lie groups]{}, BIT Numer. Math. [[38]{}]{}:1 (1998), 92–111. , [[On the Hopf Algebraic Structure of Lie Group Integrators]{}]{}, Found. Comput. Math. [[8]{}]{} no. 2 (2008), 227–257. , [[Homology of generalized partition posets]{}]{}, J. Pure Appl. Algebra [[208]{}]{}(2): (2007), 699–725. [^1]: Neglecting convergence of infinite series at this point.
--- abstract: 'The main problem studied here is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety $X_0$ (or of an algebraic variety $X_0$ over a field of characteristic zero). Given $X_0$, we ask whether there is a global resolution of singularities ${{\sigma}}: X \to X_0$ such that the pulled-back cotangent sheaf of $X_0$ is generated by differential monomials in suitable coordinates at every point of $X$ (“Hsiang-Pati coordinates”). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of ${{\sigma}}$. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety $X_0$; there have been important applications of the latter to $L_2$-cohomology.' address: - 'AB,EB,PM: University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON, Canada M5S 2E4' - 'VG: Universidade Federal do Ceará, Departamento de Matemática, Campus do Pici, Bloco 914, Cep.60455-760 Fortaleza-Ce, Brasil' author: - André Belotto da Silva - Edward Bierstone - Vincent Grandjean - 'Pierre D. Milman' title: Resolution of singularities of the cotangent sheaf of a singular variety --- [^1] Introduction {#sec:intro} ============ The subject of this article is resolution of singularities or monomialization of differential forms on an algebraic or analytic variety. Let $X_0$ denote either an algebraic variety over a field of characteristic zero, or a complex- or real-analytic variety. We assume that $X_0$ is reduced; i.e., that its structure sheaf has no nilpotents. Let ${\mathrm{Sing}\,}X_0$ denote the singular subset of $X_0$. Our main goal is to prove the following conjecture. \[conj:main\] There is a resolution of singularities of $X_0$ (i.e., a proper birational or bimeromorphic morphism ${{\sigma}}: X \to X_0$ such that $X$ is smooth, ${{\sigma}}$ is an isomorphism over $X_0\backslash {\mathrm{Sing}\,}X_0$, and ${{\sigma}}^{-1}({\mathrm{Sing}\,}X_0)$ is the support of a simple normal crossings divisor $E$ on $X$), such that the pulled-back cotangent sheaf of $X_0$ is locally generated by differential monomials $$\label{eq:main} d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i}), \, i = 1,\ldots,s, \,\, \text{and } \, d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j), \, j=1,\ldots,n-s,$$ where $n=\dim X_0$, $({\pmb{u}},{\pmb{v}}) = (u_1,\ldots,u_s,v_1,\ldots,v_{n-s})$ are local (analytic or étale) coordinates on $X$, and 1. ${\mathrm{supp}\,}E = (u_1\cdots u_s = 0)$, 2. the multiindices ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_s \in {{\mathbb N}}^s$ are linearly independent over ${{\mathbb Q}}$, 3. $\{{\pmb{{{\alpha}}}}_i,{\pmb{{{\beta}}}}_j\}$ is totally ordered (with respect to the componentwise partial ordering of ${{\mathbb N}}^s$). An important consequence, for example in the case that $X_0$ is a complex projective variety, is that the pull-back to $X$ of the induced Fubini-Study metric on $X_0\backslash {\mathrm{Sing}\,}X_0$ is locally quasi-isometric to $$\sum_{i=1}^s d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i}) \otimes \overline{d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i})}\, +\, \sum_{j=1}^{n-s} d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j) \otimes \overline{d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)}.$$ We will show that the problem of desingularization of the cotangent sheaf can be reformulated in terms of principalization of logarithmic Fitting ideal sheaves (an approach suggested already by [@PS]); see Section \[sec:fitHP\] below. Given a resolution of singularities ${{\sigma}}: X \to X_0$, the logarithmic Fitting ideal ${{\mathcal F}}_k({{\sigma}})$ denotes the sheaf of ideals of ${{\mathcal O}}_X$ generated locally by the minors of order $n-k$ of the Jacobian matrix of ${{\sigma}}$ with respect to a logarithmic basis of 1-forms on $X$; see §\[subsec:logdiff\]. \[thm:fitHP\] Let ${{\sigma}}: (X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$ be a resolution of singularities of $X_0$. Then the following conditions are equivalent. 1. The logarithmic Fitting ideals ${{\mathcal F}}_k({{\sigma}})$, $k=0,\ldots,n-1$, are all principal monomial ideals (generated locally by monomials in components of the exceptional divisor). 2. The morphism ${{\sigma}}$ is a resolution of singularities of the cotangent sheaf of $X_0$, as in Conjecture \[conj:main\]. Conjecture \[conj:main\] can be strengthened by asking that ${{\sigma}}$ be a composite of blowings-up with smooth admissible centres (admissible means that each centre of blowing-up has only normal crossings with the exceptional divisor; also see §\[subsec:res\]). The following is our main result. \[thm:dim3\] Conjecture \[conj:main\] (in the preceding stronger form) holds for varieties of dimension $\leq 3$. The result was previously proved (at least locally) in the case of surfaces (2-dimensional varieties) with isolated singularities by W.-C. Hsiang and V. Pati [@HP 1985], and a more conceptual proof in this case was given by W. Pardon and M. Stern [@PS 2001]. Our formulation of Conjecture \[conj:main\] is due to B. Youssin [@Y 1998]. A system of coordinates as in Conjecture \[conj:main\] will be called Hsiang-Pati coordinates. One of the main interests of the Hsiang-Pati problem has been for applications to the $L_2$-cohomology of the smooth part of a singular variety, going back to the original ideas of Cheeger [@C1; @C2]. Hsiang and Pati used their result to prove that the intersection cohomology (with the middle perversity) of a complex surface $X_0$ equals the $L_2$-cohomology of $X_0 \backslash {\mathrm{Sing}\,}X_0$ (Cheeger-Goresky-Macpherson conjecture [@CGM]). Several controversial articles on both the Hsiang-Pati problem and the $L_2$-cohomology of singular varieties have perhaps discouraged work on these questions; we hope that our results will lead to a renewal of interest. Our main conjecture \[conj:main\] is closely related to the problem of monomialization or toroidalization of a morphism (see §\[subsec:mon\]), and our proof of Theorem \[thm:dim3\] is strongly influenced by Cutkosky [@Cut2]; in particular, the invariant $\rho$ of Section \[sec:inv\] below is introduced in the latter (but our article does not depend on [@Cut2]). Problems involving techniques similar to those developed in this article are treated in [@Be1; @Be2]. We are grateful to Franklin Vera Pacheco for several very helpful comments. Outline of the paper -------------------- The logarithmic Fitting ideals ${{\mathcal F}}_k({{\sigma}})$ cannot be principalized by a simple application of resolution of singularities because ${{\mathcal F}}_k({{\sigma}})$ does not, in general, commute with pull-back (even up to multiplication by a principal monomial ideal). We show, nevertheless, that standard desingularization techniques can be used to principalize the Fitting ideal ${{\mathcal F}}_0({{\sigma}})$ of highest order minors, as well as the Fitting ideals of lower order minors. More precisely, we can reduce to the case that, if the logarithmic Jacobian matrix has rank $r$ at $a \in X$, then ${{\mathcal F}}_0({{\sigma}})$ as well as ${{\mathcal F}}_{n-1}({{\sigma}})_a,\ldots, {{\mathcal F}}_{n-r-1}({{\sigma}})_a$ are principal, and the first $r+1$ components of ${{\sigma}}$ at $a$ (with respect to suitable local coordinates) are given by Hsiang-Pati monomials $${{\sigma}}_1 = v_1,\quad \ldots,\quad {{\sigma}}_r = v_r,\quad {{\sigma}}_{r+1} = {\pmb{u}}^{{\pmb{{{\alpha}}}}_1},$$ (where $({\pmb{u}},{\pmb{v}})$ are coordinates at $a$ in which ${\mathrm{supp}\,}E = (u_1\cdots u_s = 0)$; see §\[sec:start\]). An immediate consequence is that Conjecture \[conj:main\] holds in the case $\dim X_0 \leq 2$. Moreover, to prove Theorem \[thm:dim3\] (when $\dim X_0 = 3$), it remains only to principalize ${{\mathcal F}}_1({{\sigma}})$ at points of log rank $0$; the image of such points in $X_0$ forms a discrete subset. Principalization of ${{\mathcal F}}_1({{\sigma}})$ is technically the most difficult part of the paper. We argue by induction on the maximal value of an upper-semicontinuous local invariant $\rho$ of ${{\mathcal F}}_1({{\sigma}})$. The invariant $\rho$ has possible values $0,1,\ldots,\infty$, and $\rho(a) = 0$ if and only if ${{\mathcal F}}_1({{\sigma}})_a$ is a principal monomial ideal (Section \[sec:inv\]). Our proof of principalization of ${{\mathcal F}}_1({{\sigma}})$ has three main steps (cf. Section \[sec:outline\]): Step 1. Reduction to the case that $\rho(a) < \infty$, for all $a$. In this case, at a point $a$ of log rank $0$, we can write the components of ${{\sigma}}$ (with respect to suitable local coordinates as in Conjecture \[conj:main\]) as ${{\sigma}}_1 = {\pmb{u}}^{{\pmb{{{\alpha}}}}}$ and ${{\sigma}}_i = g_i({\pmb{u}}) + {\pmb{u}}^{{\pmb{{{\delta}}}}}T_i$, $i > 1$, where ${\pmb{u}}^{{\pmb{{{\alpha}}}}}$ divides each ${{\sigma}}_i$, every $dg_i$ is in the submodule generated by $d({\pmb{u}}^{{\pmb{{{\alpha}}}}})$, and (in the case that $a$ is a 1- or 2-point) $T_2$ can be written essentially in Weierstrass polynomial form with respect to a distinguished variable $v$ (Lemma \[lem:rho\]. We say that $a$ is an $s$-point when ${\mathrm{supp}\,}E = (u_1\cdots u_s = 0)$ at $a$.) Step 2. Reduction to prepared normal form (Lemma \[lem:prepnormal\] and Section \[sec:prepnorm\]). By further blowings-up, we reduce to the case that the coefficients of the $T_i$ as expansions in $v$ are monomials (times units) in local coordinates as above, and the zeroth coefficients (i.e., the coefficients of $v^0$) are essentially in Hsiang-Pati form (as components of a morphism in dimension two). Step 3. Decrease of $\rho$, by further blowings-up (Lemma \[lem:decrho\] and Section \[sec:decrho\]). Examples {#subsec:ex} -------- The following examples illustrate some of the challenges in the Hsiang-Pati problem. \[ex:1\] Let ${{\sigma}}= ({{\sigma}}_1,{{\sigma}}_2,{{\sigma}}_3)$ be the morphism given by $${{\sigma}}_1 = u^2,\quad {{\sigma}}_2 = u^3(v^2 + uw),\quad {{\sigma}}_3 = u^4v,$$ where $E = (u=0)$. Then the log Fitting ideals ${{\mathcal F}}_0({{\sigma}}),\, {{\mathcal F}}_2({{\sigma}})$ are principal, and ${{\mathcal F}}_1({{\sigma}}) = u^5\cdot(u,v)$ at the 1-point $a=0$. This is essentially the simplest example of a morphism written in prepared normal form (Lemma \[lem:prepnormal\]), that does not yet satisfy the conditions of Conjecture \[conj:main\]. In this example, ${{\sigma}}$ can be reduced to Hsiang-Pati form (in fact, ${{\sigma}}$ can be monomialized; cf. Section \[subsec:mon\] below) by two blowings-up. Let $\tau$ denote the blowing-up with centre $C = (u=v=0)$ ($C$ is the locus of points where the invariant $\rho =1$). Then $\tau$ can be covered by two coordinate charts. Let $b \in \tau^{-1}(a)$. There are two possibilities: 1. $b$ belongs to the “$u$-chart”, with coordinates $(x,{{\tilde v}},{{\tilde w}})$ in which $\tau$ is given by $(u,v,w) = (x,x{{\tilde v}},{{\tilde w}})$. Then $${{\sigma}}_1\circ\tau = x^2,\quad {{\sigma}}_2\circ\tau = x^4(x{{\tilde v}}+ {{\tilde w}}),\quad {{\sigma}}_3\circ\tau = x^5{{\tilde v}},$$ and ${{\sigma}}\circ\tau$ is in Hsiang-Pati form in this chart; in fact, ${{\sigma}}_2\circ\tau = x^4w'$ after a coordinate change $w' = {{\tilde w}}+ x{{\tilde v}}$, at any $b\in \tau^{-1}(a)$ in the chart. 2. $b$ is the origin of the $v$-chart, with coordinates $(x,y,{{\tilde w}})$ in which $\tau$ is given by $(u,v,w) = (xy,y,{{\tilde w}})$. In this chart, ${{\sigma}}_3\circ\tau = y(y+x{{\tilde w}})$; $b$ is a 2-point (with $\rho(b) = \infty$) and one more blowing-up, with centre the 2-curve $(x=y=0)$, is needed to reduce to Hsiang-Pati form. Note that this centre is globally defined in the source of $\tau$. We remark that Pati [@Pati] and Taalman [@Laura] claim that, in dimension three, ${{\mathcal F}}_1({{\sigma}})$ is necessarily principal at a 1-point, because “since $u$ does not divide $R$ \[where $R = T_2 = v^2 + uw$ in the example above\], the 2-form $du \wedge dR$ is nowhere-vanishing \[on $(u=0)$\], and thus $R$ is a coordinate independent of $u$” [@Laura p.258] (cf. [@Pati p.443]). The example above shows this is not true. \[ex:2\] Let ${{\sigma}}= ({{\sigma}}_1,{{\sigma}}_2,{{\sigma}}_3,{{\sigma}}_4,{{\sigma}}_5)$ be the morphism given by $${{\sigma}}_1 = u^2,\quad {{\sigma}}_2 = u^3(v^3 + (y^2 + ux^2)uv + u^3w),\quad {{\sigma}}_3 = u^4v,\quad {{\sigma}}_4 = u^4y,\quad {{\sigma}}_5 = u^4x,$$where $E = (u=0)$. Then ${{\mathcal F}}_0({{\sigma}})$ and ${{\mathcal F}}_4({{\sigma}})$ are principal at $0$. It might appear reasonable to blow up $(x=y=0)$ to principalize ${{\mathcal F}}_3({{\sigma}})$, but this centre is not in ${\mathrm{supp}\,}E$. Monomialization of a morphism {#subsec:mon} ----------------------------- Given $X_0$, we can ask whether there exists a resolution of singularities as in Conjecture \[conj:main\] such that the components of ${{\sigma}}$ (with respect to suitable local coordinates of a smooth local embedding variety of $X_0$) are themselves monomials (rather than only their differentials). This is not true, in general, because it would imply that $X_0$ locally has a toric structure. It is reasonable to ask, on the other hand, whether a morphism ${{\sigma}}: X \to Y$ can be monomialized by blowings-up in both the source and the target — this is the problem of monomialization. In its simplest formulation, we can ask whether, after suitable global blowings-up in the source and target, a proper birational or bimeromorphic morphism ${{\sigma}}$ can be transformed to a morphism that can be expressed locally as $$x_i = {\pmb{u}}^{{\pmb{{{\alpha}}}}_i},\quad y_j = {\pmb{u}}^{{\pmb{{{\beta}}}}_j}(c_j + v_j),\quad z_k = w_k,$$ with respect to coordinates $({\pmb{u}},{\pmb{v}},{\pmb{w}}) = (u_1,\ldots,u_p, v_1,\ldots,v_q, w_1,\ldots,w_r)$ and $({\pmb{x}},{\pmb{y}},{\pmb{z}}) \allowbreak = (x_1,\ldots,x_p, y_1,\ldots,y_q, z_1,\ldots,z_r)$ in the source and target (respectively), where ${\pmb{u}}$ and $({\pmb{x}},{\pmb{y}})$ represent the exceptional divisors, the exponents ${\pmb{{{\alpha}}}}_i$ are ${{\mathbb Q}}$-linearly independent, and the $c_j$ are nonzero constants. Our approach to Conjecture \[conj:main\] or Theorem \[thm:dim3\] is a step in this direction. By blowings-up in the source, we aim to express the successive components of ${{\sigma}}$ as the Hsiang-Pati monomials $({\pmb{u}}^{{\pmb{{{\alpha}}}}_i}, {\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)$, in the ordering of Conjecture \[conj:main\](3), plus additional terms whose differentials are in the submodule generated by ; see §\[subsec:fitHP\] and Lemma \[lem:rho\]. The passage from such a statement to monomialization of ${{\sigma}}$ by blowings-up in the source and target is an interesting problem that we plan to treat in a future article (cf. Cutkosky [@Cut1 Sections18,19] for morphisms from dimension three to two). The problem of monomialization or toroidalization of morphisms has an extensive literature (see, for example, [@AKMW; @Cut3] and references therein), though the only general results either are of a local nature or involve generically finite rather than birational or bimeromorphic modifications. Cutkosky has proved global monomialization for algebraic morphisms in dimension up to three [@Cut3; @Cut2]. The normal forms of [@Cut2 Section3] cannot be obtained, however, as claimed in the proof of [@Cut2 Lemma3.6] (see [@Cut4]); we use different normal forms in our Lemma \[lem:prepnormal\]. Cutkosky’s arguments do not extend to analytic morphisms in an evident way because they involve local blowings-up that are globalized by algebraic techniques (e.g., Zariski closure, Bertini’s theorem) that are not available in an analytic category. One of the main differences of our approach from that of [@Cut3; @Cut2] is that we chose as centres of blowing up only subspaces that a priori have global meaning (see Section \[sec:decrho\] below). Logarithmic Fitting ideals {#sec:logfit} ========================== Let $X_0$ denote either an algebraic variety over a field of characteristic zero, or a complex- or real-analytic variety. Assume that $X_0$ is reduced. Blowing up and resolution of singularities {#subsec:res} ------------------------------------------ A resolution of singularities of $X_0$ is a proper birational or bimeromorphic morphism ${{\sigma}}: X \to X_0$ such that $X$ is smooth, ${{\sigma}}$ is an isomorphism over $X_0\backslash {\mathrm{Sing}\,}X_0$, and ${{\sigma}}^{-1}({\mathrm{Sing}\,}X_0)$ is the support of a simple normal crossings divisor $E$ on $X$ (the exceptional divisor). See [@BMinv; @BMfunct]. We write ${{\sigma}}: (X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$. Given a smooth variety $X$ with a simple normal crossings divisor $E$, a blowing-up ${{\sigma}}: X' \to X$ is called admissible (for $E$) if the centre of ${{\sigma}}$ is smooth and has only normal crossings with $E$. An admissible blowing-up is called combinatorial if its centre is an intersection of components of $E$. A singular variety $X_0$ locally admits an embedding $X_0\|_U \hookrightarrow M_0$ in a smooth variety $M_0$ ($U$ denotes an open subset of $X_0$). A divisor $E_0$ on $X_0$ has only normal crossings if $E$ is the restriction of ambient normal crossings divisors, for a suitable covering of $X_0$ by embeddings in smooth varieties. The notions of simple normal crossings divisor, admissible and combinatorial blowing-up all make sense in the same way. If $X_0$ is an algebraic or compact analytic variety (with a simple normal crossings divisor $E_0$, perhaps empty), then a resolution of singularities $(X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$ can be obtained as a composite of finitely many smooth admissible blowings-up. In the case of a general analytic variety $X_0$, resolution of singularities can be obtained by a morphism which can be realized as a composite of finitely many smooth admissible blowings-up over any relatively compact open subset of $X_0$ (we will still say, somewhat loosely, that ${{\sigma}}$ is a “composite of blowings-up”). Logarithmic differential forms {#subsec:logdiff} ------------------------------ Let $X$ denote a smooth variety with simple normal crossings divisor $E$, and let ${{\Omega}}_X^1$ denote the ${{\mathcal O}}_X$-module of differential $1$-forms on $X$; ${{\Omega}}_X^1$ is locally free of rank $n = \dim X$. Let ${{\Omega}}_X^1(\log E)$ denote the ${{\mathcal O}}_X$-module of logarithmic 1-forms on $X$: If $U$ is an étale or analytic local coordinate chart of $X$ at a point $a$, with coordinates $({\pmb{u}},{\pmb{v}}) = (u_1,\ldots,u_s,v_1,\ldots,v_{n-s})$ such that $a=0$ and $(u_i=0)$, $i=1,\ldots,s$, are the components of $E$ in $U$ (we say the coordinates $({\pmb{u}},{\pmb{v}})$ are adapted to $E$), then ${{\Omega}}_X^1(\log E)$ is locally free at $a$ with basis given by $$\label{eq:logbasis} \frac{du_i}{u_i}, \, i=1,\ldots,s, \,\, \text{and } \, dv_j, \, j=1,\ldots,n-s.$$ There is a natural inclusion ${{\Omega}}_X^1 \hookrightarrow {{\Omega}}_X^1(\log E)$ (given by writing any 1-form in terms of a “logarithmic basis” ). Given a singular variety $X_0$, we also write ${{\Omega}}_{X_0}^1$ for the cotangent sheaf of $X_0$. (${{\Omega}}_{X_0}^1$ has stalk ${\underline{m}}_{X_0,a}/{\underline{m}}_{X_0,a}^2$ at $a$, where ${\underline{m}}_{X_0,a}$ denotes the maximal ideal of ${{\mathcal O}}_{X_0,a}$.) Suppose that ${{\sigma}}: (X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$ is a resolution of singularities (in particular, ${\mathrm{supp}\,}E = {{\sigma}}^{-1}({\mathrm{Sing}\,}X_0)$). Let ${{\sigma}}^{{\Omega}}_{X_0}^1$ denote the submodule of ${{\Omega}}_X^1$ generated by the pull-back of ${{\Omega}}_{X_0}^1$, and consider the quotient ${{\mathcal O}}_X$-module $$\label{eq:phi} \Phi := {{\Omega}}_X^1(\log E)/{{\sigma}}^{{\Omega}}_{X_0}^1.$$ (If $X_0 \hookrightarrow Z_0$, where $Z_0$ is smooth, then ${{\Omega}}_{X_0}^1$ is induced by the restriction to $X_0$ of ${{\Omega}}_{Z_0}^1$, and ${{\sigma}}^{{\Omega}}_{X_0}^1 = {{\sigma}}^{{\Omega}}_{Z_0}^1$.) \[rem:lindep\] Let ${\pmb{x}}:= (x_1,\ldots,x_n)$ and let ${\pmb{{{\beta}}}}_1,\ldots, {\pmb{{{\beta}}}}_k \in {{\mathbb N}}^n$. If ${\pmb{{{\gamma}}}}$ is linearly dependent on ${\pmb{{{\beta}}}}_1,\ldots, {\pmb{{{\beta}}}}_k$ over ${{\mathbb Q}}$, say ${\pmb{{{\gamma}}}}= \sum_{i=1}^k q_i {\pmb{{{\beta}}}}_i$, where each $q_i \in {{\mathbb Q}}$, then $d {\pmb{x}}^{{\pmb{{{\gamma}}}}} = \sum_i q_i x^{{\pmb{{{\gamma}}}}-{\pmb{{{\beta}}}}_i}d {\pmb{x}}^{{\pmb{{{\beta}}}}_i}$. In particular, if ${\pmb{{{\gamma}}}}\geq {\pmb{{{\beta}}}}_i$, for all $i$, then $d {\pmb{x}}^{{\pmb{{{\gamma}}}}}$ is in the submodule generated by the $d {\pmb{x}}^{{\pmb{{{\beta}}}}_i}$. If $X_0\|_V \hookrightarrow M_0$ is a local embedding over a neighbourhood $V$ of $b = {{\sigma}}(a)$, then ${{\sigma}}^({{\Omega}}_{X_0}^1\|_V) = {{\sigma}}^{{\Omega}}_{M_0}^1$. If $\dim M_0 = N$ and ${{\sigma}}$ is expressed in components ${{\sigma}}= ({{\sigma}}_1,\ldots,{{\sigma}}_N)$ with respect to local coordinates of $M_0$ at $b$, then $\Phi$ has a presentation at $a$ given by (the transpose of) the logarithmic Jacobian matrix of ${{\sigma}}$, $$\label{eq:logjac} \log {\mathrm{Jac}\,}{{\sigma}}= \begin{pmatrix} \displaystyle{u_1\frac{{{\partial}}{{\sigma}}_1}{{{\partial}}u_1}} & \cdots & \displaystyle{u_s \frac{{{\partial}}{{\sigma}}_1}{{{\partial}}u_s}} & \displaystyle{\frac{{{\partial}}{{\sigma}}_1}{{{\partial}}v_1}} & \cdots & \displaystyle{\frac{{{\partial}}{{\sigma}}_1}{{{\partial}}v_{n-s}}}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ \displaystyle{u_1\frac{{{\partial}}{{\sigma}}_N}{{{\partial}}u_1}} & \cdots & \displaystyle{u_s \frac{{{\partial}}{{\sigma}}_N}{{{\partial}}u_s}} & \displaystyle{\frac{{{\partial}}{{\sigma}}_N}{{{\partial}}v_1}} & \cdots & \displaystyle{\frac{{{\partial}}{{\sigma}}_N}{{{\partial}}v_{n-s}}} \end{pmatrix}.$$ Note that every minor of order $n$ of $\log {\mathrm{Jac}\,}{{\sigma}}$ equals $u_1\cdots u_s$ times the corresponding minor of order $n$ of the standard Jacobian matrix ${\mathrm{Jac}\,}{{\sigma}}$. The rank at $a$ of $\log {\mathrm{Jac}\,}{{\sigma}}$ will be called the logarithmic rank $\log {\mathrm{rk}\,}_a {{\sigma}}$ of ${{\sigma}}$ at $a$. It is clear from that, if $a \in {\mathrm{supp}\,}E$, then $\log {\mathrm{rk}\,}_a {{\sigma}}= {\mathrm{rk}\,}_a ({{\sigma}}\|_{E(a)})$, where $E(a)$ denotes the intersection of the components of $E$ containing $a$ (We call $E(a)$ the stratum of $E$ at $a$. If $a \notin {\mathrm{supp}\,}E$, then $\log {\mathrm{rk}\,}_a {{\sigma}}:= {\mathrm{rk}\,}_a {{\sigma}}$.) Logarithmic Fitting ideals {#subsec:fit} -------------------------- For each $k=0,\ldots,n-1$, the logarithmic Fitting ideal ${{\mathcal F}}_k = {{\mathcal F}}_k({{\sigma}})$ denotes the ideal (i.e., sheaf of ideals) of ${{\mathcal O}}_X$ generated by the minors of order $n-k$ of $\log {\mathrm{Jac}\,}{{\sigma}}$. The Fitting ideals ${{\mathcal F}}_k$ depend only on the quotient module $\Phi$ ; in particular, they are independent of the choices of adapted local coordinates of $(X,E)$, the local embedding $X_0\|_V \hookrightarrow M_0$ and the local coordinates of $M_0$ (cf. [@E §20.2]). The relevance of logarithmic Fitting ideals to the main conjecture was recognized by Pardon and Stern [@PS], and Theorem \[thm:fitHP\] is suggested by their work. Transformation of logarithmic Fitting ideals by blowing up {#subset:blup} ---------------------------------------------------------- Let ${{\beta}}$ denote an admissible blowing-up with centre $C \subset {\mathrm{supp}\,}E$, and let $E'$ denote the transform of $E$ by ${{\beta}}$ (by definition, the components of $E'$ are the strict transforms by ${{\beta}}$ of the components of $E$, together with the exceptional divisor of ${{\beta}}$). In this case, ${\mathrm{supp}\,}E' = {{\beta}}^{-1}({\mathrm{supp}\,}E)$; we will write ${{\beta}}: (X',E') \to (X,E)$. With the respect to adapted local coordinates $({\pmb{u}},{\pmb{v}})$ at a point $a \in C$, as above, we can assume that $C$ is given by $$u_1 = \cdots = u_k = 0, \ \text{for some} \ k \geq 1, \ \text{and}\ \, v_1 = \cdots = v_l = 0, \ \text{for some} \ l \geq 0.$$ The blowing-up ${{\beta}}$ is combinatorial (§\[subsec:res\]) precisely when $l=0$. Over the $({\pmb{u}},{\pmb{v}})$ coordinate chart, $X'$ can be covered by $k+l$ coordinate charts — a “$u_i$-chart”, for each $i=1,\ldots,k$, and a “$v_j$-chart”, for each $j=1,\ldots,l$. For example, the $u_1$-chart has coordinates $({\pmb{u}}',{\pmb{v}}') = (u_1',\ldots,u_s',v_1',\ldots,v_{n-s}')$ given by $$u_1' = u_1,\qquad u_i' = \left\{ \begin{array}{l l} u_i/u_1, & 2\leq i\leq k\\ u_i , & i > k \end{array}\right.,\qquad v_j' = \left\{ \begin{array}{l l} v_j/u_1 & j\leq l\\ v_j & j > l \end{array}\right.,$$ and the $v_1$-chart has coordinates $({\pmb{u}}',{\pmb{v}}') = (u_1',\ldots,u_{s+1}',v_2',\ldots,v_{n-s}')$ given by $$u_i' = \left\{ \begin{array}{l l} u_i/v_1, & 1\leq i\leq k\\ u_i , & k < i\leq s \end{array}\right.,\qquad u_{s+1}' = v_1,\qquad v_j' = \left\{ \begin{array}{l l} v_j/v_1 & 2\leq j\leq l\\ v_j & j > l \end{array}\right..$$ We compute $$\log {\mathrm{Jac}\,}({{\sigma}}\circ{{\beta}}) = ((\log {\mathrm{Jac}\,}{{\sigma}})\circ{{\beta}})\cdot B,$$ where $B$ denotes the $n \times n$ matrix $$\begin{pmatrix} A & 0\\ 0 & I \end{pmatrix} \cdot {\mathrm{Jac}\,}{{\beta}}\cdot \begin{pmatrix} C & 0\\ 0 & D \end{pmatrix},$$ with $A$ (respectively, $C$) the $s \times s$ diagonal matrix with diagonal entries $1/u_i$ (respectively, $u_i'$), $I$ the identity matrix of order $n-s$, and $D$ a diagonal matrix of order $n-s$ with first entry $1$ in the $u_1$-chart or $u_{s+1}'$ in the $v_1$-chart, and remaining diagonal entries $1$. We make the following simple but important observations. \[rem:fit\] (1) $\det B = {\mathrm{exc}}^l$, where ${\mathrm{exc}}$ denotes the exceptional divisor of ${{\sigma}}$; i.e., ${\mathrm{exc}}= u_1'$ in the $u_1$-chart or $u_{s+1}'$ in the $v_1$-chart. \(2) If ${{\beta}}$ is a combinatorial blowing-up ($l=0$), then $B$ is invertible, so that every minor of $\log {\mathrm{Jac}\,}({{\sigma}}\circ{{\beta}})$ is a linear combination of minors of the same order of $(\log {\mathrm{Jac}\,}{{\sigma}})\circ{{\beta}}$, and vice-versa. Therefore, in the notation above, we have the following. \[lem:fit\] 1. ${{\mathcal F}}_0({{\sigma}}\circ{{\beta}}) = exc^l \cdot {{\beta}}^{{\mathcal F}}_0({{\sigma}})$. 2. If ${{\beta}}$ is a combinatorial blowing-up, then ${{\mathcal F}}_k({{\sigma}}\circ{{\beta}}) = {{\beta}}^{{\mathcal F}}_k({{\sigma}})$, $k=0,\ldots$. \[thm:fit0\] Given a reduced variety $X_0$, there is a resolution of singularities ${{\sigma}}: (X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$ such that ${{\sigma}}$ is a composite of admissible blowings-up and ${{\mathcal F}}_0({{\sigma}})$ is a principal ideal generated locally by a monomial in generators of the components of $E$ (we will say that ${{\sigma}}$ is a principal $E$-monomial ideal, or a principal monomial ideal if $E$ is clear from the context). Moreover, if ${{\sigma}}$ is a resolution of singularities such that ${{\mathcal F}}_0({{\sigma}})$ is a principal monomial ideal, and ${{\beta}}: (X',E') \to (X,E)$ is an admissible blowing-up, then ${{\mathcal F}}_0({{\sigma}}\circ{{\beta}})$ is a principal monomial ideal. The second assertion is immediate from Lemma \[lem:fit\](1). Suppose that ${{\sigma}}$ is a resolution of singularities of the variety $X_0$ by admissible blowings-up. We can then apply resolution of singularities of an ideal to ${{\mathcal F}}_0({{\sigma}})$, and the first assertion again follows from Lemma \[lem:fit\](1). Regularization of the Gauss mapping {#subset:Gauss} ----------------------------------- Given a smooth variety $M_0$ and $n \leq \dim M_0$, let $G(n,M_0)$ denote the Grassmann bundle of $n$-dimensional linear subspaces of the tangent spaces to $M_0$ at every point. If $X_0 \hookrightarrow M_0$ and $Y_0 = {\mathrm{Sing}\,}X_0$, then there is a natural Gauss mapping $G_{X_0}: X_0\backslash Y_0 \to G(n,M_0)$, where $n = \dim X_0$, given by $a \mapsto$ tangent space of $X_0$ at $a$. Theorem \[thm:fit0\] also provides a regularization of the Gauss mapping: \[thm:Gauss\] Let ${{\sigma}}: (X,E) \to (X_0,Y_0)$ denote a resolution of singularities. Then the following are equivalent: 1. the pull-back ${{\sigma}}^G_{X_0}$ extends to a regular (or analytic) morphism on $X$; 2. ${{\sigma}}^{{\Omega}}_{X_0}^1$ is a locally free ${{\mathcal O}}_X$-module of rank $n$; 3. the Fitting ideal ${{\mathcal F}}_0({{\sigma}})$ is a principal ideal (not necessarily monomial). The equivalence of (1) and (2) is a consequence of the fact that a sheaf of ${{\mathcal O}}_X$-modules is locally free if and only if it defines a vector bundle (see [@Ha Exercise II.5.18]). To see that (1)$\iff$(3), note that the Grassmannian $\text{Grass}(n,N)$ of $n$-planes in ${{\mathbb C}}^N$; i.e., the space of linear injections ${\pmb{y}}= {{\lambda}}({\pmb{x}})$ from ${{\mathbb C}}^n \to {{\mathbb C}}^N$, is the complex projective space of dimension $\binom{N}{n} -1$ with homogeneous coordinates given by the minors of order $n$, $$\frac{{{\partial}}(y_{i_1},\ldots,y_{i_n})}{{{\partial}}(x_1,\ldots,x_n)},\quad i_1 < \dots < i_n,$$ of the Jacobian matrix of ${{\lambda}}$ with respect to coordinate systems ${\pmb{x}}= (x_1,\ldots,x_n)$, ${\pmb{y}}= (y_1,\ldots,y_N)$ of ${{\mathbb C}}^n$, ${{\mathbb C}}^N$ (respectively). \[rem:Gauss\] Given $a \in X$, condition (1) of Theorem \[thm:Gauss\] can be used to choose coordinates for $M_0$ at ${{\sigma}}(a)$ such that each $d{{\sigma}}_i$, $i>n$, is in the submodule generated by $d{{\sigma}}_1,\ldots,d{{\sigma}}_n$ at $a$. We will not use this result, but have included Theorem \[thm:Gauss\] for historical reasons. Logarithmic Fitting ideals and desingularization of the cotangent sheaf {#sec:fitHP} ======================================================================= We continue to use the notation of Section \[sec:logfit\]. Equivalence of the main conjecture and principalization of logarithmic Fitting ideals {#subsec:fitHP} ------------------------------------------------------------------------------------- In this subsection, we will prove Theorem \[thm:fitHP\]. According to Theorem \[thm:fit0\], there is a resolution of singularities ${{\sigma}}$ such that ${{\mathcal F}}_0({{\sigma}})$ is a principal monomial ideal, and the latter condition is stable by admissible blowings-up. Although Theorem \[thm:fitHP\] may seem attractive because our main conjecture \[conj:main\] is obtained “in one shot” from the condition (1), the Fitting ideals ${{\mathcal F}}_k({{\sigma}})$, $k > 0$, do not enjoy the stability property of ${{\mathcal F}}_0({{\sigma}})$, so in practice it may be difficult to obtain condition (1) step-by-step. (Example. Consider the morphism ${{\sigma}}(u,v,w) = (u^2,u^3v,u^4w)$ and the admissible blowing-up with centre $(u = v+w^2 = 0)$.) We will show, nevertheless, that, given a resolution of singularities ${{\sigma}}$ by admissible blowings-up, then, after further admissible blowings-up, ${{\mathcal F}}_{n-1}({{\sigma}})$ is a principal monomial ideal (Theorem \[thm:start\]). In fact, we will show that, after further admissible blowings-up, ${{\mathcal F}}_{n-1}({{\sigma}}),\ldots,{{\mathcal F}}_{n-(r+1)}({{\sigma}})$ are principal monomial ideals at every point of log rank $r$. This seems useful as a way to begin an inductive proof of our main conjecture. In particular, Theorems \[thm:fit0\], \[thm:fitHP\] and \[thm:start\] immediately establish Conjecture \[conj:main\] in the case that $\dim X_0 \leq 2$ (see Corollary \[cor:HP2\]). We will use Theorem \[thm:start\] to begin an inductive proof in the $3$-dimensional case, in Section \[sec:prepnorm\]. It will be useful to have the more precise local statement of the following lemma, which immediately implies Theorem \[thm:fitHP\]. The proof of Lemma \[lem:fitHP\] will include the precise relationship between the exponents of monomials generating the log Fitting ideals, and the exponents appearing in the differential monomials involved in Hsiang-Pati coordinates. \[lem:fitHP\] Let ${{\sigma}}: X \to M_0$ be a morphism between smooth varieties. Say $n=\dim X$, $N=\dim M_0$. Suppose that the critical locus of ${{\sigma}}$ (i.e., $\{a \in X: {\mathrm{rk}\,}_a {{\sigma}}< n\}$) is the support of a simple normal crossings divisor $E$ on $X$. Let $a \in X$. Then, for each $k = 1,\ldots,n$ the following are equivalent: 1. The logarithmic Fitting ideals ${{\mathcal F}}_{n-1}({{\sigma}}), \ldots, {{\mathcal F}}_{n-k}({{\sigma}})$ are all principal $E$-monomial ideals at $a$. 2. There are (analytic or étale) coordinates $({\pmb{u}},{\pmb{v}})$ of $X$ at $a$ adapted to $E$, and coordinates ${\pmb{z}}= (z_1,\ldots,z_N)$ of $M_0$ at ${{\sigma}}(a)$, such that, writing ${{\sigma}}= ({{\sigma}}_1,\ldots,{{\sigma}}_N)$ with respect to the coordinates ${\pmb{z}}$, 1. the submodule ${{\mathcal M}}_k$ of ${{\Omega}}_{X,a}^1$ generated by the pull-backs ${{\sigma}}^dz_m = d{{\sigma}}_m$, $m=1,\ldots,k$, is also generated by differential monomials $d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i})$, $i = 1,\ldots,l_k$, and $d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)$, $j=1,\ldots,k-l_k$, for some $l_k \leq k$, satisfying conditions as in Conjecture \[conj:main\]; 2. for each $m > k$, ${{\sigma}}_m = g_m + S_m$, where $d g_m \in {{\mathcal M}}_k$ and $S_m$ is divisible by ${\pmb{u}}^{\max\{{\pmb{{{\alpha}}}}_{l_k}, {\pmb{{{\beta}}}}_{k-l_k}\}}$. (2)$\implies$(1). Given “partial” Hsiang-Pati coordinates at a point $a$ of $X$, as in (2), each log Fitting ideal ${{\mathcal F}}_{n-m}({{\sigma}})$, $m=1,\ldots,k$, at $a$ is generated by a minor of order $m$ of the matrix with rows given by the coefficients of the differential monomials $d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i})$ and $d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)$ with respect to the logarithmic basis . Each row is a vector $({\pmb{\xi}},{\pmb{\eta}})$, where the components of ${\pmb{\xi}}$ (respectively, ${\pmb{\eta}}$) are log derivatives with respect to the coordinates $u_i$ (respectively, derivatives with respect to the $v_j$). The row vector corresponding to ${\pmb{u}}^{{\pmb{{{\alpha}}}}_i}$ (respectively, to ${\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j$) is ${\pmb{u}}^{{\pmb{{{\alpha}}}}_i}({\pmb{{{\alpha}}}}_i,0)$ (respectively, ${\pmb{u}}^{{\pmb{{{\beta}}}}_j}(v_j{\pmb{{{\beta}}}}_j,(j))$, where ${\pmb{\eta}}= (j)$ denotes the vector with $1$ in the $j$’th place and $0$ elsewhere). It is easy to see that each ${{\mathcal F}}_{n-m}$ is generated by ${\pmb{u}}^{{\pmb{{{\gamma}}}}_m}$, where ${\pmb{{{\gamma}}}}_m$ is the sum of the first $m$ elements of $\{{\pmb{{{\alpha}}}}_i,{\pmb{{{\beta}}}}_j\}$ (as an ordered set). (1)$\implies$(2). Assume that ${{\mathcal F}}_{n-m} = ({\pmb{u}}^{{\pmb{{{\gamma}}}}_m})$, $m=1,\ldots,k$, where ${\pmb{{{\gamma}}}}_1 \leq \cdots \leq {\pmb{{{\gamma}}}}_k$. Write ${{\sigma}}$ as ${{\sigma}}= ({{\sigma}}_1,\ldots,{{\sigma}}_N)$ with respect to local coordinates ${\pmb{z}}= (z_1,\ldots,z_N)$ of $M_0$ at ${{\sigma}}(a) = 0$. We will prove (2) (in fact, a stronger statement) by induction on $k$. Given $k\geq 0$, assume there are local coordinates $({\pmb{u}},{\pmb{v}})$ at $a=0$ such that: 1. The ${{\mathcal O}}_{X,a}$-module generated by $d{{\sigma}}_1,\ldots,d{{\sigma}}_k$ is generated by $d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i})$, $i=1,\ldots,l_k$, and $d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)$, $j=1,\ldots,k-l_k$, where $\{{\pmb{{{\alpha}}}}_i,{\pmb{{{\beta}}}}_j\}$ is totally ordered and ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_{l_k}$ are linearly independent over ${{\mathbb Q}}$. 2. For all $m=1,\ldots,k$, $${{\sigma}}_m = g_m + S_m$$ (sum of analytic functions, or regular functions in the étale chart), where, 1. for every monomial ${\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}}$ appearing (i.e., with nonzero coefficient) in the formal expansion of $g_m$ at $a=0$, $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}})$ is linearly dependent on the $({\pmb{{{\alpha}}}}_i,0)$, $i=1,\ldots,l_{m-1}$, and $({\pmb{{{\beta}}}}_j,(j))$, $j=1,\ldots,m-1-l_{m-1}$, over ${{\mathbb Q}}$, and ${\pmb{{{\beta}}}}\geq {\pmb{{{\alpha}}}}_i,\, {\pmb{{{\beta}}}}_j$, for all such $i,j$; 2. $S_m = {\pmb{u}}^{{\pmb{{{\alpha}}}}_{l_{m-1}+1}}$ or $S_m = {\pmb{u}}^{{\pmb{{{\beta}}}}_{m-1-l_{m-1} +1}} v_{m-1-l_{m-1} +1}$. Note that, if $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}}) = \sum_{i=1}^{l_{m-1}} q_i ({\pmb{{{\alpha}}}}_i,0) + \sum_{j=1}^{m-1-l_{m-1}} r_j ({\pmb{{{\beta}}}}_j,(j))$, where the $q_i, r_j \in {{\mathbb Q}}$ and $r_j \neq 0$ for some $j$, then ${\pmb{{{\beta}}}}\geq {\pmb{{{\beta}}}}_j \implies ({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}}) \geq ({\pmb{{{\beta}}}}_j,(j))$, so (by Remark \[rem:lindep\]), $dg_m$ is in the ${{\mathcal O}}_{X,a}$-submodule generated by the $d{\pmb{u}}^{{\pmb{{{\alpha}}}}_i}$ and $d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j)$. 3. For each $m=k+1,\ldots, N$, $${{\sigma}}_m = g_{mk} + {\pmb{u}}^{{\pmb{{{\delta}}}}_{mk}}S_{mk},$$ where $g_{mk}$ and $S_{mk}$ are analytic or regular functions such that 1. for every monomial ${\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}}$ appearing in the formal expansion of $g_{mk}$ at $a=0$, $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}})$ is linearly dependent on the $({\pmb{{{\alpha}}}}_i,0)$, $i=1,\ldots,l_k$, and $({\pmb{{{\beta}}}}_j,(j))$, $j=1,\ldots,k-l_k$, over ${{\mathbb Q}}$, and ${\pmb{{{\beta}}}}\geq {\pmb{{{\alpha}}}}_i,\, {\pmb{{{\beta}}}}_j$, for all such $i,j$ (so $dg_{mk}$ is in the submodule generated by $d{{\sigma}}_1,\ldots,d{{\sigma}}_k$); 2. ${\pmb{{{\delta}}}}_{mk} \geq \max\{{\pmb{{{\alpha}}}}_{l_k}, {\pmb{{{\beta}}}}_{k-l_k}\}$ and, if $S_{mk}$ is a unit, then ${\pmb{{{\delta}}}}_{mk}$ is linearly independent of ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_{l_k}$; 3. $S_{mk}$ is divisible by no $u_i$ (unless $S_{mk} = 0$). 4. $\sum_{i=1}^{l_k} {\pmb{{{\alpha}}}}_i + \sum_{j=1}^{k-l_k} {\pmb{{{\beta}}}}_j = {\pmb{{{\gamma}}}}_k$. The assumptions above (except for (3)$_k$(c)) are all empty if $k=0$. Note that the generator ${\pmb{u}}^{{\pmb{{{\gamma}}}}_k}$ of ${{\mathcal F}}_{n-k}$ is given by a coefficient of $d{{\sigma}}_1\wedge \cdots \wedge d{{\sigma}}_k$ (with respect to the log basis). It is easy to see that the generator ${\pmb{u}}^{{\pmb{{{\gamma}}}}_{k+1}}$ of ${{\mathcal F}}_{n-(k+1)}$ is given by a coefficient of $$d{{\sigma}}_1 \wedge \cdots \wedge d{{\sigma}}_k \wedge d{{\sigma}}_{m_0},$$ for some $m_0 \geq k+1$ (e.g., using elementary row and column operations on $\log {\mathrm{Jac}\,}{{\sigma}}$). Set $${{\Omega}}_k := \bigwedge_{i=1}^{l_k} d({\pmb{u}}^{{\pmb{{{\alpha}}}}_i}) \wedge \bigwedge_{j=1}^{k-l_k} d({\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j).$$ Then $${{\Omega}}_k = {\pmb{u}}^{{\pmb{{{\gamma}}}}_k}\left\{ \sum_I c_I \bigwedge_I\hspace{-.2em}\frac{du_i}{u_i} \wedge dv_1 \wedge \cdots \wedge dv_{k-l_k} + \eta_k\right\},$$ where 1. $I$ runs over all sets $\{i_1,\ldots,i_{l_k}\}$ such that $1 \leq i_1 < \cdots < i_{l_k} \leq s$, 2. $\displaystyle{ \bigwedge_I\hspace{-.1em}\frac{du_i}{u_i} := \frac{du_{i_1}}{u_{i_1}} \wedge \cdots \wedge \frac{du_{i_k}}{u_{i_k}} }$, 3. $c_I$ is a unit, for some $I$, 4. $\eta_k \in (v_1,\dots, v_{k-l_k})\cdot{{\Omega}}_{X,a}$. Let us compute ${{\Omega}}_k \wedge d({\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}})$, where ${\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}}$ is a monomial with ${\pmb{{{\beta}}}}\geq \max\{{\pmb{{{\alpha}}}}_{l_k}, {\pmb{{{\beta}}}}_{k-l_k}\}$ (for example, a monomial in the formal expansion of ${\pmb{u}}^{{\pmb{{{\delta}}}}_{mk}}S_{mk}$, where $m \geq k+1$). Case (i) ${\pmb{{{\gamma}}}}= 0$. Now, $d{\pmb{u}}^{{\pmb{{{\beta}}}}} = {\pmb{u}}^{{\pmb{{{\beta}}}}} \sum {{\beta}}_i du_i/u_i$. If ${\pmb{{{\beta}}}}$ is ${{\mathbb Q}}$-linearly dependent on ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_{l_k}$, then ${{\Omega}}_k \wedge d{\pmb{u}}^{{\pmb{{{\beta}}}}} = 0$. If ${\pmb{{{\beta}}}}$ is ${{\mathbb Q}}$-linearly independent of the ${\pmb{{{\alpha}}}}_i$, then some coefficient of ${{\Omega}}_k \wedge d{\pmb{u}}^{{\pmb{{{\beta}}}}}$ (with respect to the log basis) is $u^{{\pmb{{{\gamma}}}}_k + {\pmb{{{\beta}}}}}$ times a unit, and all other coefficients are divisible by $u^{{\pmb{{{\gamma}}}}_k + {\pmb{{{\beta}}}}}$. Case (ii) $\|{\pmb{{{\gamma}}}}\| \geq 1$. Then $$d({\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}}) = {\pmb{u}}^{{\pmb{{{\beta}}}}} {\pmb{v}}^{{\pmb{{{\gamma}}}}} \sum {{\beta}}_i \frac{du_i}{u_i} + {\pmb{u}}^{{\pmb{{{\beta}}}}} \sum {{\gamma}}_j {\pmb{v}}^{{\pmb{{{\gamma}}}}- (j)} dv_j.$$ If $\|{\pmb{{{\gamma}}}}\| > 1$, then all coefficients of ${{\Omega}}_k \wedge d({\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}})$ are divisible by some $v_j$. Consider $\|{\pmb{{{\gamma}}}}\| = 1$. If ${\pmb{{{\gamma}}}}= (j)$, for some $j \leq k-l_k$, then ${\pmb{u}}^{{\pmb{{{\beta}}}}} {\pmb{v}}^{(j)} = u^{{\pmb{{{\beta}}}}- {\pmb{{{\beta}}}}_j}u^{{\pmb{{{\beta}}}}_j}v_j$ and ${{\Omega}}_k \wedge d({\pmb{u}}^{{\pmb{{{\beta}}}}} {\pmb{v}}^{(j)})$ is in the submodule of logarithmic $(k+1)$-forms divisible by $v_j$. If ${\pmb{{{\gamma}}}}= (q)$, for some $q > k-l_k$, then some coefficient of ${{\Omega}}_k \wedge d({\pmb{u}}^{{\pmb{{{\beta}}}}} {\pmb{v}}^{(q)})$ is ${\pmb{u}}^{{\pmb{{{\gamma}}}}_k + {\pmb{{{\beta}}}}}$ times a unit, and all other coefficients are divisible by ${\pmb{u}}^{{\pmb{{{\gamma}}}}_k + {\pmb{{{\beta}}}}}$. We can assume that the monomial ${\pmb{u}}^{{\pmb{{{\gamma}}}}_{k+1}}$ generating ${{\mathcal F}}_{n-(k+1)}$ is given (up to a unit) by a coefficient of $$d{{\sigma}}_1 \wedge \cdots \wedge d{{\sigma}}_k \wedge d{{\sigma}}_{k+1}.$$ Then $${\pmb{{{\gamma}}}}_{k+1} = {\pmb{{{\gamma}}}}_k + {\pmb{{{\delta}}}}_{k+1,k}\quad \text{and }\quad {\pmb{{{\delta}}}}_{mk} \geq {\pmb{{{\delta}}}}_{k+1,k},\,\, m \geq k+1.$$ Moreover, by the computation above, either there is a monomial $P = {\pmb{u}}^{{\pmb{{{\alpha}}}}}$ appearing in the formal expansion of ${\pmb{u}}^{{\pmb{{{\delta}}}}_{k+1,k}}S_{k+1,k}$ such that ${\pmb{{{\delta}}}}_{k+1,k} = {\pmb{{{\alpha}}}}$, or there is a monomial $P = {\pmb{u}}^{{\pmb{{{\beta}}}}}v_q$, where $q> k - l_k$, appearing in the formal expansion of ${\pmb{u}}^{{\pmb{{{\delta}}}}_{k+1,k}}S_{k+1,k}$ such that ${\pmb{{{\delta}}}}_{k+1,k} = {\pmb{{{\beta}}}}$. We can now obtain (1)$_{k+1}$. Set $S := S_{k+1,k}$. First suppose $P = {\pmb{u}}^{{\pmb{{{\alpha}}}}}$. Then $S$ is a unit and ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_{l_k},{\pmb{{{\alpha}}}}$ are linearly independent over ${{\mathbb Q}}$. Let ${\pmb{{{\epsilon}}}}= ({{\epsilon}}_1,\ldots,{{\epsilon}}_s) \in {{\mathbb Q}}^s$ denote a shortest vector such that $$\begin{aligned} \langle {\pmb{{{\alpha}}}}_i, {\pmb{{{\epsilon}}}}\rangle &= 0,\quad i=1,\ldots,l_k,\\ \langle {\pmb{{{\alpha}}}},{\pmb{{{\epsilon}}}}\rangle &= 1,\end{aligned}$$ and consider the coordinate change $$\begin{aligned} {3} {\overline{u}}_h &= S^{{{\epsilon}}_h} u_h, &&h = 1,\ldots,s,\\ {\overline{v}}_j &= S^{-\langle{\pmb{{{\beta}}}}_j,{\pmb{{{\epsilon}}}}\rangle}v_j,\quad &&j = 1,\ldots,k-l_k,\\ {\overline{v}}_j &= v_j, &&j > k-l_k.\end{aligned}$$ Then ${\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}} = S^{\langle{\pmb{{{\beta}}}},{\pmb{{{\epsilon}}}}\rangle}{\pmb{u}}^{{\pmb{{{\beta}}}}}$, for any ${\pmb{{{\beta}}}}$, so $$\begin{aligned} {3} {\overline{\pmb{u}}}^{{\pmb{{{\alpha}}}}_i} &= {\pmb{u}}^{{\pmb{{{\alpha}}}}_i}, &&i=1,\ldots,l_k,\\ {\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}_j}{\overline{v}}_j &= {\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j,\quad &&j=i,\ldots, k-l_k,\\ {\overline{\pmb{u}}}^{{\pmb{{{\alpha}}}}} &= S {\pmb{u}}^{{\pmb{{{\alpha}}}}}. &&\end{aligned}$$ Set $$\begin{aligned} {\pmb{{{\alpha}}}}_{l_k + 1} &:= {\pmb{{{\alpha}}}}= {\pmb{{{\gamma}}}}_{k+1} - {\pmb{{{\gamma}}}}_k,\\ l_{k+1} &:= l_k + 1.\end{aligned}$$ Then the ${{\mathcal O}}_X$-module generated by $d{{\sigma}}_1,\ldots,d{{\sigma}}_{k+1}$ is also generated by $$d{\overline{\pmb{u}}}^{{\pmb{{{\alpha}}}}_i},\,\, i=1,\ldots,l_{k+1},\quad \text{and }\quad d({\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}_j}{\overline{v}}_j),\,\, j=1,\ldots,k+1-l_{k+1}.$$ Secondly, suppose $P = {\pmb{u}}^{{\pmb{{{\beta}}}}}v_q$, where $q > k - l_k$. Then $S(0)=0$ and $({{\partial}}S/{{\partial}}v_q)(0) \neq 0$. We can assume that $q = k-l_k+1$. Consider the coordinate change $$\begin{aligned} {3} {\overline{\pmb{u}}}&= {\pmb{u}}, &&\\ {\overline{v}}_j &= v_j,\quad &&j\neq k-l_k+1,\\ {\overline{v}}_{k-l_k+1} &= S. &&\end{aligned}$$ Set $l_{k+1} := l_k$. Then the ${{\mathcal O}}_X$-module generated by $d{{\sigma}}_1,\ldots,d{{\sigma}}_{k+1}$ is also generated by $$d{\overline{\pmb{u}}}^{{\pmb{{{\alpha}}}}_i},\,\, i=1,\ldots,l_{k+1},\quad \text{and }\quad d({\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}_j}{\overline{v}}_j),\,\, j=1,\ldots,k+1-l_{k+1}.$$ Properties (2)$_{k+1}$ and (4)$_{k+1}$ are clear from the construction. It remains to verify (3)$_{k+1}$. Clearly, $g_{mk}({\overline{\pmb{u}}},{\overline{\pmb{v}}}) = g_{mk}({\pmb{u}},{\pmb{v}})$, for all $m \geq k+1$; i.e., ${\overline{\pmb{u}}}^{{\pmb{{{\alpha}}}}_i} = {\pmb{u}}^{{\pmb{{{\alpha}}}}_i}$ and ${\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}_j}{\overline{v}}_j = {\pmb{u}}^{{\pmb{{{\beta}}}}_j}v_j$, for all monomials involved. In the analytic case, for each $m\geq k+2$, we can define $g_{m,k+1}$ by adding to $g_{mk}({\overline{\pmb{u}}},{\overline{\pmb{v}}})$ all monomials ${\overline{\pmb{u}}}^{{\pmb{{{\beta}}}}}{\overline{\pmb{v}}}^{{\pmb{{{\gamma}}}}}$ (times nonzero constants) appearing in ${{\sigma}}_m - g_{mk}$ such that $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}})$ is a ${{\mathbb Q}}$-linear combination of $({\pmb{{{\alpha}}}}_i,0),\, ({\pmb{{{\beta}}}}_j,(j))$, $i=1,\ldots,l_{k+1},\, j=1,\ldots,k+1-l_{k+1}$. Note that such $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}})$ is $\geq$ all the $({\pmb{{{\alpha}}}}_i,0),\, ({\pmb{{{\beta}}}}_j,(j))$. In the algebraic case, the preceding construction provides formal expansions $\hat{g}_{m,k+1}$ and ${\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}\widehat{S}_{m,k+1}$ that are not a priori* algebraic. In this case, for each $m\geq k+2$, we can define $g_{m,k+1} := \hat{g}_{m,k+1} - \hat{h}_{m,k+1}$ and $S_{m,k+1} := \widehat{S}_{m,k+1} + \hat{h}_{m,k+1}/{\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}$, where $\hat{h}_{m,k+1}$ denotes the sum of all terms $c_{{\pmb{{{\beta}}}}{\pmb{{{\gamma}}}}}{\pmb{u}}^{{\pmb{{{\beta}}}}}{\pmb{v}}^{{\pmb{{{\gamma}}}}}$ ($c_{{\pmb{{{\beta}}}}{\pmb{{{\gamma}}}}} \neq 0$) of $\hat{g}_{m,k+1}$ with $({\pmb{{{\beta}}}},{\pmb{{{\gamma}}}}) > ({\pmb{{{\delta}}}}_{m,k+1},0)$. Then we still have (3)$_{k+1}$ (as well as (1)$_{k+1}$, (2)$_{k+1}$ and (4)$_{k+1}$). Moreover, $g_{m,k+1}$ and $S_{m,k+1}$ are algebraic; we explain this in Remark \[rem:fitHP\](1) following because the remark will be needed also in the proof of Lemma \[lem:rho\]. This completes the proof of Lemma \[lem:fitHP\]. \[rem:fitHP\] (1) Given ${{\sigma}}_m = \hat{g}_{m,k+1} + {\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}\widehat{S}_{m,k+1}$ as well as $g_{m,k+1}$ and $S_{m,k+1}$, as above, let $Q$ and $R$ denote the quotient and remainder of ${{\sigma}}_m$ (respectively) after division by the monomial ${\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}$. (This means that, formally, $R$ is the sum of all terms of $\hat{g}_{m,k+1}$ which are not divisible by ${\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}$.) Then $Q$ and $R$ are algebraic. Moreover, $g_{m,k+1} = R$ and $S_{m,k+1} = Q$ unless $\hat{g}_{m,k+1}$ includes a term $c{\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}$ (with nonzero coefficient $c$). In the latter case, ${\pmb{{{\delta}}}}_{m,k+1}$ is linearly dependent on ${\pmb{{{\alpha}}}}_1,\ldots,{\pmb{{{\alpha}}}}_{l_k}$, and $g_{m,k+1} = R + c{\pmb{u}}^{{\pmb{{{\delta}}}}_{m,k+1}}$, $S_{m,k+1} = R - c$. \(2) Theorem \[thm:fitHP\] says that, under the assumption that all log Fitting ideals ${{\mathcal F}}_k({{\sigma}})$ are principal, we get Hsiang-Pati coordinates at every point of $X$, as in Conjecture \[conj:main\]; according to the proof of Theorem \[thm:fitHP\], it may happen that the first differential monomial according to the order in condition (3) of the latter is $d({\pmb{u}}^{{\pmb{{{\beta}}}}_1}v_1)$. In practice, we will principalize ${{\mathcal F}}_0$ using Theorem \[thm:fit0\] and then try to principalize ${{\mathcal F}}_{n-1}, {{\mathcal F}}_{n-2},\ldots$ inductively. The resulting ordered list of differential monomials will always begin with either ${\pmb{u}}^{{\pmb{{{\alpha}}}}_1}$ or $v_1$ (see Theorem \[thm:start\] below). One can add to Conjecture \[conj:main\] the condition that each ${\pmb{{{\beta}}}}_j$ be linearly dependent on all preceding ${\pmb{{{\alpha}}}}_i$ in the ordered list. See also [@Y]. Given Hsiang-Pati coordinates as in Conjecture \[conj:main\], one can obtain the additional condition at least locally, by further admissible blowings-up over $E$ if necessary. We do not know of a situation where the stronger condition is needed, but it may simplify proofs. \(3) Lemma \[lem:fitHP\] includes, in particular, the statement of Theorem \[thm:fitHP\] locally at a point $a \in X$. Since Lemma \[lem:fitHP\](1) is an open condition, the lemma implies that Hsiang-Pati coordinates at a point of $X$ induce Hsiang-Pati coordinates at nearby points. Logarithmic rank and principalization of Fitting ideals of low-order minors {#sec:start} --------------------------------------------------------------------------- In this section, we assume that $X_0$ is a complex-analytic variety or an algebraic variety over an algebraically closed field of characteristic zero. The results also hold, however, for a real-analytic variety or an algebraic variety over an arbitrary field of characteristic zero (see Remark \[rem:real\]). Let ${{\sigma}}: (X,E) \to (X_0, Y_0 :={\mathrm{Sing}\,}X_0)$ be a resolution of singularities of $X_0$. In particular, ${{\sigma}}^{{\mathcal I}}_{Y_0}$ is a principal $E$-monomial ideal. Set $$p := \max_E \log{\mathrm{rk}\,}{{\sigma}}= \dim Y_0,$$ and, for each $k=0,\ldots, p$, set $$\begin{aligned} \Sigma_k &:= \{a \in E: \log{\mathrm{rk}\,}_a {{\sigma}}\leq p-k\},\\ Y_k &:= {{\sigma}}(\Sigma_k).\end{aligned}$$ Since ${{\sigma}}$ is proper, each $Y_k$ is a closed subvariety of $X_0$, and $${\mathrm{Sing}\,}X_0 = Y_0 \supset Y_1 \supset \cdots \supset Y_p.$$ Define $${{\mathcal I}}_{Y_k} := \text{ideal of $Y_k$ in ${{\mathcal O}}_{X_0}$},\quad k=0,\ldots,p.$$ \[lem:start\] After further admissible blowings-up over $Y_0$ if necessary, we can assume that, for each $k$, $\Sigma_k = {{\sigma}}^{-1}(Y_k)$, $Y_k\backslash Y_{k+1}$ is smooth, and ${{\sigma}}^{{\mathcal I}}_{Y_k}$ is a principal monomial ideal. \[rem:start\] (1) It follows from the lemma that, for each $k$, $\log {\mathrm{rk}\,}{{\sigma}}= p-k$ on $\Sigma_k\backslash \Sigma_{k+1}$. \(2) The condition that ${{\sigma}}^{{\mathcal I}}_{Y_k}$ be a principal monomial ideal is not stable under admissible blowing-up. $Y_0 = {\mathrm{Sing}\,}X_0$, so that $\Sigma_0 = {\mathrm{supp}\,}E = {{\sigma}}^{-1}(Y_0)$ and ${{\sigma}}^{{\mathcal I}}_{Y_0}$ is a principal monomial ideal with support $\Sigma_0$. Of course, ${{\Sigma}}_1 \subset {{\Sigma}}_0$. Set $$Y'_1 := {{\sigma}}({{\Sigma}}_1) \cup {\mathrm{Sing}\,}Y_0.$$ Then $\dim Y'_1 \leq p-1$. By resolution of singularities, after further admissible blowings-up with centres over $Y'_1$ (i.e., in the inverse image of $Y'_1$), we can assume that ${{\sigma}}^{{\mathcal I}}_{Y'_1}$ is a principal monomial ideal. Then ${{\Sigma}}_1 = {\mathrm{supp}\,}{{\sigma}}^{{\mathcal I}}_{Y'_1} = {{\sigma}}^{-1}(Y'_1)$, $Y_1 = Y'_1$ and $Y_0\backslash Y_1$ is smooth. Again, ${{\Sigma}}_2 \subset {{\Sigma}}_1$. Let $$Y'_2 := {{\sigma}}({{\Sigma}}_2) \cup {\mathrm{Sing}\,}Y_1.$$ Then $\dim Y'_2 \leq p-2$. After further admissible blowings-up with centres over $Y'_2$, we can assume that ${{\sigma}}^{{\mathcal I}}_{Y'_2}$ is a principal monomial ideal. Then ${{\Sigma}}_2 = {\mathrm{supp}\,}{{\sigma}}^{{\mathcal I}}_{Y'_2} = {{\sigma}}^{-1}(Y'_2)$, $Y_2 = Y'_2$ and $Y_1\backslash Y_2$ is smooth. Moreover, it is still true that ${{\sigma}}^{{\mathcal I}}_{Y_1}$ is a principal monomial ideal, ${{\Sigma}}_1 = {\mathrm{supp}\,}{{\sigma}}^{{\mathcal I}}_{Y_1} = {{\sigma}}^{-1}(Y_1)$, and $Y_0\backslash Y_1$ is smooth. We can continue in the same way to prove the lemma. \[rem:real\] If $X_0$ is a real-analytic variety, or an algebraic variety over a field that is not algebraically closed, then the ${{\sigma}}({{\Sigma}}_k)$ need not be closed varieties. The proof of Lemma \[lem:start\] goes through, however, provided that each ${{\sigma}}({{\Sigma}}_k)$ lies in a closed variety of dimension $= d_k :=\max \{\log{\mathrm{rk}\,}_a {{\sigma}}: a \in {{\Sigma}}_k\}$; in this case, we can simply replace each $Y'_k$ in the proof by the smallest closed subvariety of $Y_{k-1}$ containing ${{\sigma}}({{\Sigma}}_k) \cup {\mathrm{Sing}\,}Y_{k-1}$. The preceding condition holds in the algebraic case, in general (cf. [@Rond]). It holds in the real-analytic case because $X_0$ has a complexification $X_0^{{\mathbb C}}$ [@Tognoli], and ${{\sigma}}$ is induced by a resolution of singularities ${{\sigma}}^{{\mathbb C}}: (X^{{\mathbb C}}, E^{{\mathbb C}}) \to (X_0^{{\mathbb C}}, {\mathrm{Sing}\,}X_0^{{\mathbb C}})$ of $X_0^{{\mathbb C}}$ (in particular, ${{\sigma}}^{{\mathbb C}}$ is proper). Lemma \[lem:start\] applies to ${{\sigma}}^{{\mathbb C}}$. In the proof of Lemma \[lem:start\] in the real case, we can take $Y'_k$ to be the real part (i.e., the invariance space with respect to the canonical autoconjugation) of a union of components of $(Y_k^{{\mathbb C}})'$; cf. [@Hiro Sect.2]. \[thm:start\] Let ${{\sigma}}: (X,E) \to (X_0, Y_0 :={\mathrm{Sing}\,}X_0)$ denote a resolution of singularities of $X_0$ satisfying the conclusion of Lemma \[lem:start\]. (We use the notation at the beginning of the subsection.) Let $a \in E$, and let $r := \log {\mathrm{rk}\,}_a {{\sigma}}$. Then, for any local embedding $X_0 \hookrightarrow M_0$ (at $a$) in a smooth variety $M_0$, we can choose coordinates $({\pmb{u}},{\pmb{v}}) = (u_1,\ldots,u_s,v_1,\ldots,v_{n-s})$ adapted to $E$ for $X$ at $a=0$, and coordinates $z=(z_1,\ldots,z_N)$ for $M_0$ at ${{\sigma}}(a)$, with respect to which, if ${{\sigma}}= ({{\sigma}}_1,\ldots,{{\sigma}}_N)$, then $${{\sigma}}_1 = v_1,\,\, \ldots,\,\, {{\sigma}}_r = v_r,\,\, {{\sigma}}_{r+1} = {\pmb{u}}^{{\pmb{{{\alpha}}}}_1},$$ where ${\pmb{{{\alpha}}}}_1 \in {{\mathbb N}}^s$, ${{\mathcal I}}_{Y_{p-r}}$ is generated by $z_{r+1},\ldots,z_N$ at ${{\sigma}}(a)$, and ${{\sigma}}^{{\mathcal I}}_{Y_{p-r}}$ is generated by ${{\sigma}}_{r+1} = {{\sigma}}^(z_{r+1})$ at $a$. Let $E_i$, $i=1,\ldots,s$, denote the components of $E$ at $a$. We call $\cap E_i$ the stratum $E(a)$ of $a$ (cf. \[subsec:logdiff\]). We can assume that $${\mathrm{supp}\,}{{\sigma}}^{{\mathcal I}}_{Y_{p-r}} = \bigcup_{i=1}^t E_i,$$ at $a$, where $t \leq s$. Then $\log {\mathrm{rk}\,}_b {{\sigma}}= r$, for all $b \in \cup_{i=1}^t E_i$ near $a$. Let $z=(z_1,\ldots,z_N)$ denote coordinates for $M_0$ at ${{\sigma}}(a)$. It follows from the implicit function theorem that, after permuting the $z_j$ if necessary, we can choose coordinates $({\pmb{u}},{\pmb{v}}) = (u_1,\ldots,u_s,v_1,\ldots,v_{n-s})$ for $X$ adapted to $E$ at $a=0$, such that $(v_1,\ldots,v_r)$ forms part of a system of coordinates for $E(a)$ at $a$, and 1. ${{\sigma}}_1 = v_1,\, \ldots,\, {{\sigma}}_r = v_r$ at $a$; 2. for each $j > r$, ${{\sigma}}_j = {{\sigma}}_j(v_1,\ldots,v_r)$ on $E_i$ at $a$, $i=1,\ldots,t$. Since $Y_{p-r}$ is smooth at ${{\sigma}}(a)$, then $z_j - {{\sigma}}_j(z_1,\ldots,z_r)$, $j > r$, generate the ideal of $Y_{p-r}$ at $a$. After a coordinate change $${\overline{z}}_j := z_j - {{\sigma}}_j(z_1,\ldots,z_r),\quad j> r,$$ we can therefore assume that $z_{r+1},\ldots,z_N$ generate ${{\mathcal I}}_{Y_{p-r}}$ at ${{\sigma}}(a)$ (so that ${{\sigma}}_{r+1},\ldots, {{\sigma}}_N$ generate ${{\sigma}}^{{\mathcal I}}_{Y_{p-r}}$ at $a$). Since the latter is a principal monomial ideal, we can also assume that ${{\sigma}}_{r+1} = {\pmb{u}}^{{\pmb{{{\alpha}}}}_1}$, as required. The following is an immediate consequence of Theorems \[thm:fit0\], \[thm:fitHP\] and \[thm:start\]. \[cor:HP2\] Conjecture \[conj:main\] holds in the case that $\dim X_0 \leq 2$. Invariant of a logarithmic Fitting ideal {#sec:inv} ======================================== We use the notation of Section \[sec:fitHP\]. Let ${{\sigma}}: (X,E)\to (X_0,{\mathrm{Sing}\,}X_0)$ denote a resolution of singularities of $X_0$, $n = \dim X_0$. \[def:inv\] Given $k=0,\ldots,n-1$ and $a \in X$, let ${{\mathcal R}}_{k,a}$ denote the residual ideal of ${{\mathcal F}}_k({{\sigma}})_a$ in ${{\mathcal O}}_{X,a}$; i.e., ${{\mathcal F}}_k({{\sigma}})_a = \prod_q {{\mathcal I}}_{E_q,a}^{\mu_q} \cdot {{\mathcal R}}_{k,a}$, with the $\mu_q \in {{\mathbb N}}$ as large as possible, where $\{E_q\}$ denotes the set of components of $E$ and ${{\mathcal I}}_{E_q}$ is the ideal sheaf of $E_q$. Let $\rho_k(a)$ denote the order of ${{\mathcal R}}_{k,a}$ in the local ring ${{\mathcal O}}_{X,a}/\sum_{a\in E_q} {{\mathcal I}}_{E_q,a}$ (cf. [@BMfunct Section5], [@Cut2 Section2]; $\rho_k(a) := \infty$ if and only if ${{\mathcal R}}_{k,a} = 0$ in the preceding local ring). The following lemma lists several properties of the basic invariant $\rho_k(a)$ that are all either clear or easy to prove. \[lem:invprops\] 1. $0 \leq \rho_k(a) \leq \infty$. 2. $\rho_k(a) = 0$ if and only if ${{\mathcal F}}_k({{\sigma}})_a$ is a principal monomial ideal. 3. $\rho_k$ is upper-semicontinuous in the Zariski topology of $X$. 4. If $a$ is an $n$-point, then $\rho_k(a) =0$ or $\infty$. 5. If ${{\beta}}: (X',E') \to (X,E)$ is a combinatorial blowing-up and $a' \in {{\sigma}}^{-1}(a)$, then $\rho_k(a') \leq \rho_k(a)$. We will only need $\rho_k$ in the case that $k=n-2$ in this article; i.e., for the log Fitting ideal of $2\times 2$ minors. Write $\rho := \rho_{n-2}$. Lemma \[lem:rho\] below extends in a straightforward way to $\rho_k(a)$, for any $k$, with the assumption that the Fitting ideals ${{\mathcal F}}_{n-1}({{\sigma}})_a, {{\mathcal F}}_{n-2}({{\sigma}})_a,\ldots,{{\mathcal F}}_{k+1}({{\sigma}})_a$ are all principal monomial ideals. We present the lemma only in the case needed for the remainder of the paper, in part so that we can fix notation that will be used in the following sections. \[lem:rho\] Let $a \in {\mathrm{supp}\,}E$ be an $s$-point ($1 \leq s \leq n$). Suppose that ${{\mathcal F}}_{n-1}({{\sigma}})_a$ is a principal monomial ideal and that $\log{\mathrm{rk}\,}_a {{\sigma}}= 0$. Then: 1. Let $M_0$ denote a local embedding variety for $X_0$ at ${{\sigma}}(a)$. Then there are adapted local coordinates $({\pmb{u}},{\pmb{v}}) = (u_1,\ldots,u_s,v_1,\ldots v_{n-s})$ for $X$ at $a$ (where the $(u_k=0)$ are the components of $E$ at $a$), and local coordinates ${\pmb{z}}=(z_1,\ldots,z_N)$ for $M_0$ at ${{\sigma}}(a)$ with respect to which the components ${{\sigma}}_i$ of ${{\sigma}}$ can be written $$\label{eq:prelim} \begin{aligned} {{\sigma}}_1 &= {\pmb{u}}^{{\pmb{{{\alpha}}}}}, \quad {\pmb{{{\alpha}}}}\in {{\mathbb N}}^s,\\ {{\sigma}}_i &= g_i({\pmb{u}}) + {\pmb{u}}^{{\pmb{{{\delta}}}}} T_i, \quad i=2,\ldots,N, \end{aligned}$$ where ${\pmb{u}}^{{\pmb{{{\alpha}}}}}$ divides all ${{\sigma}}_i$, each $g_i$ and $T_i$ is analytic (or regular), each $dg_i$ is in the submodule generated by $d{\pmb{u}}^{{\pmb{{{\alpha}}}}}$, the $T_i$ are not simultaneously divisible by any $u_k$, and ${\pmb{{{\delta}}}}$ is linearly independent of ${\pmb{{{\alpha}}}}$ if some $T_i$ is a unit. 2. Given coordinates as above in which satisfied, let $$d = d(a) := \min\{\|{\pmb{{{\gamma}}}}\|: {\pmb{{{\gamma}}}}\in {{\mathbb N}}^{n-s} \text{ and } {{\partial}}_{{\pmb{v}}}^{{\pmb{{{\gamma}}}}} T_i \text{ is a unit, for some } i\}.$$ Then 1. $\rho(a) < \infty$ if and only if $d(a) < \infty$; 2. $\rho(a) = 0$ if and only if $d(a) = 0$ or $1$; 3. if $0 < \rho(a) < \infty$, then $d(a) = \rho(a) +1$. 3. Suppose that $0 < \rho(a) < \infty$. Then there are adapted coordinates $({\pmb{u}},v,{\pmb{w}}) = (u_1,\ldots,u_s,v,w_1,\ldots w_{n-s-1})$ for $X$ at $a$ and coordinates ${\pmb{z}}=(z_1,\ldots,z_N)$ for $M_0$ at ${{\sigma}}(a)$ with respect to which the components ${{\sigma}}_i$ of ${{\sigma}}$ can be written as in with $$\label{eq:weier} T_i({\pmb{u}},v,{\pmb{w}}) = {{\widetilde T}}_i({\pmb{u}},v,{\pmb{w}})v^d + \sum^{d-1}_{j=0} a_{ij}({\pmb{u}},{\pmb{w}}) v^j,\quad i=2,\ldots,N, $$ where ${{\widetilde T}}_2$ is a unit that we will also denote $U$, $a_{2,d-1} = 0$, and all monomials of order $\leq d + \|{\pmb{{{\delta}}}}\|$ in the formal expansions of the ${\pmb{u}}^{{\pmb{{{\delta}}}}} T_i$ at $a$, $i=2,\ldots,N$, are “linearly independent” of ${\pmb{u}}^{{\pmb{{{\alpha}}}}}$ (i.e., have exponents with respect to $({\pmb{u}},v,{\pmb{w}})$ that are linearly independent of $({\pmb{{{\alpha}}}},0,{\pmb{0}})$. It is easy to obtain (1), where the formal expansion of each $g_i$ at $a$ is a sum of monomials ${\pmb{u}}^{{\pmb{{{\beta}}}}}$ where each ${\pmb{{{\beta}}}}$ is a rational multiple $q{\pmb{{{\alpha}}}}$, $q\geq 1$ (see Remarks \[rem:lindep\] and \[rem:fitHP\](1)). In (2), if $d=0$, then ${\pmb{{{\delta}}}}$ is linearly independent of ${\pmb{{{\alpha}}}}$ and $T_i$ is a unit, for some $i$; say $i=2$. Then, after a coordinate change, we can assume $T_2 = 1$, so that ${{\mathcal F}}_{n-2}({{\sigma}})_a$ is generated by ${\pmb{u}}^{{\pmb{{{\alpha}}}}+{\pmb{{{\delta}}}}}$. If $d=1$, then, after a coordinate change, we can assume $T_2 = v_1$, and again ${{\mathcal F}}_{n-2}({{\sigma}})_a$ is generated by ${\pmb{u}}^{{\pmb{{{\alpha}}}}+{\pmb{{{\delta}}}}}$. On the other hand, (a) is clear and it is easy to see directly from the log Jacobian matrix of ${{\sigma}}$ that the residual ideal ${{\mathcal R}}_a$ of ${{\mathcal F}}_{n-2}({{\sigma}})_a$ is generated modulo $\sum_{a\in E_k} {{\mathcal I}}_{E_k,a}$ by the partial derivatives ${{\partial}}_{v_l}T_i$, $i=2,\ldots,N$, $l =1,\ldots, n-s$, together with $({{\alpha}}_p{{\delta}}_q - {{\alpha}}_q{{\delta}}_p)T_i$, $i=2,\ldots,N$, $p,q=1,\ldots,s$. It follows that, if $d \geq 1$, then $\rho(a) = d-1$. Given , after a permutation of the coordinates $(z_1,\ldots,z_N)$ and a generic linear coordinate change in ${\pmb{v}}= (v_1,\ldots, v_{n-s})$, we can write the $T_i$ in the form , where ${{\widetilde T}}_2=U$ is a unit, if we allow the sum in $T_2$ also to go from $j=0$ to $d-1$. Then we can eliminate $a_{2,d-1}$ by completing the $d$th power with respect to $v$. The final condition in (3) can be obtained by “moving” the lower order monomials of (the formal expansions of) the ${\pmb{u}}^{{\pmb{{{\delta}}}}} T_i$ that are rational powers of ${\pmb{u}}^{{\pmb{{{\alpha}}}}}$ to the $g_i$. Three-dimensional case: outline of the proof {#sec:outline} ============================================ In this section, we outline the proof Theorem \[thm:dim3\], which will be completed in Sections 6, 7 following. Assume that $\dim X_0 =3$. By Theorems \[thm:fit0\], \[thm:fitHP\] and \[thm:start\], there is a resolution of singularities ${{\sigma}}: (X,E) \to (X_0, {\mathrm{Sing}\,}X_0)$ such that ${{\mathcal F}}_0({{\sigma}}),\, {{\mathcal F}}_2({{\sigma}})$ are principal monomial ideal sheaves and, moreover, if $a \in {\mathrm{supp}\,}E$ and $\log {\mathrm{rk}\,}_a {{\sigma}}> 0$, then ${{\mathcal F}}_1({{\sigma}})_a$ is also a principal monomial ideal. We will make ${{\mathcal F}}_1({{\sigma}})$ a principal monomial ideal sheaf by admissible blowings-up that preserve the preceding conditions on ${{\sigma}}$. Recall that combinatorial blowings-up, in particular, preserve these conditions (Lemma \[lem:fit\](2)). The blowings-up that we use to principalize ${{\mathcal F}}_1({{\sigma}})$ will have the additional property that $\log {\mathrm{rk}\,}{{\sigma}}$ is identically zero on the image in $X$ of every centre (or, equivalently, that every centre of blowing up lies over ${\mathrm{supp}\,}{{\mathcal F}}_2({{\sigma}})$). Thus we will blow up only over a discrete subset of $X_0$. We will say that $a$ is resolved if ${{\mathcal F}}_1({{\sigma}})_a$ is a principal monomial ideal; i.e., $\rho(a) = 0$. Our proof of Theorem \[thm:dim3\] has three main steps: [Step 1.]{} Reduction to the case that $\rho(a) < \infty$, for all $a$. If $0 < \rho(a) < \infty$ and $\log {\mathrm{rk}\,}_a {{\sigma}}= 0$, then the conclusions of Lemma \[lem:rho\](3) hold; in this case, we will say that ${{\sigma}}$ is in Weierstrass form at $a$. Note that the set of 2-points forms a collection of curves (“2-curves”) each given by (a connected component of) the intersection of precisely two components of $E$. A 2-curve either is closed or has limiting 3-points. By Lemma \[lem:start\], we can assume that, throughout each 2-curve, either $\log {\mathrm{rk}\,}{{\sigma}}= 0$ or $\log {\mathrm{rk}\,}{{\sigma}}= 1$. A 2-curve on which $\log {\mathrm{rk}\,}{{\sigma}}= 0$ is relatively compact. By Theorem \[thm:start\], $\rho = 0$ on every 2-curve where $\log {\mathrm{rk}\,}{{\sigma}}= 1$. \[lem:rhofinite\] By combinatorial blowings-up (more precisely, by composing ${{\sigma}}$ with a morphism $\tau: ({{\widetilde X}},{{\widetilde E}}) \to (X,E)$ that restricts to a finite sequence of combinatorial blowings-up over any relatively compact open subset of $X_0$), we can reduce to the case that 1. $\rho(a) < \infty$ at every point $a$ (and, therefore, by Lemma \[lem:rho\], we can choose coordinates at every nonresolved point $a$ and its image ${{\sigma}}(a)$ in which ${{\sigma}}$ has Weierstrass form); 2. $\rho$ is generically zero on every component of the set of 1-points, and on every 2-curve. In particular, in this case, the set of nonresolved points comprises isolated 2-points, isolated 1-points, and closed curves that are generically 1-points. Moreover, $\tau$ can be chosen so that no centre of blowing up includes points over a 2-curve in $X$ where $\rho = 0$ (in particular, every centre lies over the locus $(\log {\mathrm{rk}\,}{{\sigma}}= 0)$). The proof of Lemma \[lem:rhofinite\] following involves repeated blowings-up of 2-curves and 3-points. The last assertion of the lemma is important in the case of (not necessarily compact) analytic varieties because it means that only relatively compact 2-curves will be blown up, and this will imply that only finitely many blowings-up will be needed over a given 2-curve in $X$. In fact, Lemma \[lem:rhofinite\] involves only finitely many blowings-up over each point of the discrete subset $\Gamma$ of $X_0$ given by the image of $(\log {\mathrm{rk}\,}{{\sigma}}= 0)$. It is clear that $\rho(a) < \infty$ at every 1-point $a$. It follows from Lemma \[lem:rho\] that $\rho$ is generically zero on every component of the set of 1-points. Recall we can assume that $\log {\mathrm{rk}\,}{{\sigma}}$ is constant (either $0$ or $1$) on every 2-curve in $X$, and $\rho =0$ on every 2-curve in $X$ where $\log {\mathrm{rk}\,}{{\sigma}}= 1$. Three-points are isolated. It is also clear that, by combinatorial blowings-up, we can reduce to the case that $\rho(a) = 0$ at every 3-point $a$. (The blowings-up involved have centres that are 3-points or closures of 2-curves, but it is unnecessary to blow up 2-curves that are already resolved; i.e., on which $\rho =0$.) It follows that $\rho$ is generically zero on every 2-curve with a 3-point $a$ as a limiting point. Suppose that $a$ is a 2-point. If $\rho(a) < \infty$ at a 2-point $a$, then $\rho$ is generically zero on the 2-curve containing $a$, again by Lemma \[lem:rho\]. A blowing-up with centre given by the intersection of two components of $E$ is combinatorial. If $\rho(a) = \infty$, then we can reduce to the case that $\rho < \infty$ over $a$ by finitely many such blowings-up (since the effect of such blowings-up is to principalize the ideal generated by the coefficients $a_{ij}(u_1,u_2)$ of formal expansions $T_i = \sum_{j=0}^\infty a_{ij}(u_1,u_2)v^j$ at $a$; cf. Lemma \[lem:rho\]). It therefore follows that we can reduce to $\rho < \infty$ on every 2-curve, by a locally-finite sequence of combinatorial blowings-up. [Step 2.]{} Reduction to prepared normal form. The following lemma \[lem:prepnormal\] will be proved in Section \[sec:prepnorm\]. \[lem:prepnormal\] Suppose that $\rho(a) < \infty$, for all $a\in X$ (and that $\rho = 0$ on every 2-curve with non-compact closure; cf. Step 1 above). Assume that $\rho$ takes a maximum value $\rho_{\max} > 0$, and let $\Sigma \subset X$ denote the closed subset on which $\rho$ takes the value $\rho_{\max}$ (so that $\Sigma \subset {\mathrm{supp}\,}E$). Then, by a (locally) finite sequence of admissible blowings-up over $(\log {\mathrm{rk}\,}{{\sigma}}= 0)$, we can reduce to the case that, for every point $a\in \Sigma$, ${{\sigma}}$ has the Weierstrass form of Lemma \[lem:rho\], where the coefficients $a_{ij}$ satisfy the following additional conditions. 1. At a 2-point $a$, with adapted coordinates $({\pmb{u}},v)=(u_1,u_2,v)$, where ${\mathrm{supp}\,}E = (u_1u_2=0)$, $$\label{eq:norm2} \begin{alignedat}{2} a_{ij} &= {\pmb{u}}^{{\pmb{r}}_{ij}}{{\tilde a}}_{ij}({\pmb{u}}), \quad &&i = 2,\ldots,N,\ \ j=1,\ldots,d-1,\\ a_{i_0,0} &= {\pmb{u}}^{{\pmb{{{\beta}}}}}, &&\text{for some } i_0, \end{alignedat}$$ where each ${{\tilde a}}_{ij}$ is either zero or a unit, ${\pmb{{{\delta}}}}+ {\pmb{{{\beta}}}}$ is linearly independent of ${\pmb{{{\alpha}}}}$, and ${\pmb{u}}^{{\pmb{{{\beta}}}}}$ divides $a_{i0}$, for all $i$. 2. At a 1-point $a$, with adapted coordinates $(u,v,w)$, where ${\mathrm{supp}\,}E = (u=0)$, $$\label{eq:norm1} \begin{alignedat}{2} a_{ij} &= u^{r_{ij}}w^{s_{ij}}{{\tilde a}}_{ij}(u,w) \quad &&i = 2,\ldots,N,\ \ j=1,\ldots,d-1,\\ a_{i_0,0} &= u^{{{\beta}}}w, &&\text{for some } i_0, \end{alignedat}$$ where each ${{\tilde a}}_{ij}$ is either zero or a unit, and $u^{{\beta}}$ divides $a_{i0}$, for all $i$. The blowings up involved do not increase the value of $\rho$ over any point. We will say that $a$ is a prepared 2-point (resp., a prepared 1-point) if ${{\sigma}}$ has Weierstrass form at $a$, where the coefficients are given by (resp., ), with respect to suitable adapted coordinates at $a$. In either case, we will also say that ${{\sigma}}$ has prepared normal form at $a$. At a generic prepared 1-point, all $s_{ij} = 0$ in . Points where not all $s_{ij} = 0$ will be called non-generic. \[rem:transverse\] Suppose that $\rho_{\max} > 0$. Then all points of the maximum locus $\Sigma$ of $\rho$ are unresolved and, if ${{\sigma}}$ has prepared normal form at every point of $\Sigma$, then (the closure of) any curve of 1-points in $\Sigma$ has only normal crossings with respect to 2-curves. [Step 3.]{} Further admissible blowings-up to decrease the maximal value of the invariant $\rho$. Lemma \[lem:decrho\] following is the subject of Section \[sec:decrho\]. Theorem \[thm:dim3\] then follows by induction on the maximal value of $\rho$. \[lem:decrho\] Assume that $\rho(a)<\infty$, for all $a \in X$, and that $\rho$ takes a maximum value $\rho_{\max} > 0$ on $X$. Let $\Sigma \subset X$ denote the maximum locus of $\rho$. Suppose that ${{\sigma}}$ has prepared normal form at every point $a \in \Sigma$ (see Lemma \[lem:prepnormal\]). Let $A$ denote the discrete set of all non-generic points of $\Sigma$ (i.e., all 2-points and non-generic 1-points of $\Sigma$). Then there is a morphism $\tau: (\widetilde{X},\widetilde{E}) \to (X,E)$ given by a locally finite sequence of admissible blowings-up over $\Sigma$ such that $\rho < \rho_{\max}$ throughout ${{\widetilde X}}$. Moreover, $\tau$ can be realized as a composite $\tau = \tau_3\circ \tau_2\circ \tau_1$, where 1. $\tau_1: (X_1,E_1) \to (X,E)$ is a single blowing-up with centre $A$; 2. $\tau_2:(X',E') \to (X_1,E_1)$ is the composite of a locally finite sequence of admissible blowings-up $(X_{i+1},E_{i+1}) \to (X_i,E_i)$, $i\geq 1$, with centres $\overline{\Sigma_i \setminus A_i}$, where $\Sigma_i$ is the maximum locus of $\rho$ and $A_i$ the preimage of $A$ in $X_i$; 3. $\tau_3: (\widetilde{X},\widetilde{E}) \to (X',E')$ is the composite of a locally finite sequence of blowings-up with centres over $A$. The theorem follows from Lemmas \[lem:rhofinite\], \[lem:prepnormal\] and \[lem:decrho\] by induction on the maximal value of $\rho$, at least for $X$ on which $\rho$ assumes a maximum value (e.g., algebraic varieties or the restrictions of analytic varieties to relatively compact open sets). Theorem \[thm:dim3\] follows for analytic varieties, in general, because the preceding lemmas show that, if we start with a resolution of singularities ${{\sigma}}$ as at the beginning of this section, then $\rho$ can be everywhere decreased to zero by finitely many blowings-up over each point of the discrete subset $\Gamma$ of $X_0$ given by the image of $(\log {\mathrm{rk}\,}{{\sigma}}= 0)$. Prepared normal form {#sec:prepnorm} ==================== In this section, we prove Lemma \[lem:prepnormal\]. The proof is by induction on pairs $(\rho(a),{{\iota}}(a))$ (ordered lexicographically), where ${{\iota}}(a)$ is a secondary invariant with values in ${{\mathbb N}}$, introduced in the following subsection. Secondary invariant {#subsec:io} ------------------- Suppose that ${{\sigma}}$ has Weierstrass form in adapted coordinates $({\pmb{u}},v)$ at a 2-point $a$; respectively, in adapted coordinates $(u,v,w)$ at a 1-point $a$. For each $i=2,\ldots,N$, write $$\begin{aligned} d{{\sigma}}_1\|_{(v=0)}\wedge d{{\sigma}}_i\|_{(v=0)} &= d({\pmb{u}}^{{\pmb{{{\alpha}}}}})\wedge d({\pmb{u}}^{{\pmb{{{\delta}}}}}a_{i0}({\pmb{u}}))\\ &= H_i({\pmb{u}}) \frac{du_1}{u_1} \wedge \frac{du_2}{u_2};\end{aligned}$$ respectively, $$\begin{aligned} d{{\sigma}}_1\|_{(v=0)}\wedge d{{\sigma}}_i\|_{(v=0)} &= d(u^{{{\alpha}}})\wedge d(u^{{{\delta}}}a_{i0}(u,w))\\ &= H_i(u,w) \frac{du}{u} \wedge dw .\end{aligned}$$ In either case, let ${{\mathcal H}}_a$ denote the ideal generated by $H_i$, $i=2,\ldots,N$, in the local ring of ${(v=0)}$ at $a$. \[rem:H\] ${{\mathcal H}}_a$ is the log Fitting ideal of $2\times 2$ minors of the morphism ${{\sigma}}\|_{(v=0)}$ at $a$. Blowing up of the point $a$ in ${(v=0)}$ is admissible for $E\|_{(v=0)}$. If ${{\mathcal H}}_a$ is a principal monomial ideal (i.e., generated by a monomial in components of $E\|_{(v=0)}$), then, by Theorem \[thm:fit0\] and Lemma \[lem:fitHP\], ${{\sigma}}\|_{(v=0)}$ can be written in Hsiang-Pati form $d{{\sigma}}_1\|_{(v=0)} = d({\pmb{u}}^{{\pmb{{{\alpha}}}}})$ (resp., $d(u^{{{\alpha}}})$), and $a_{i_0,0} = {\pmb{u}}^{{\pmb{{{\beta}}}}}$ (resp., $u^{{{\beta}}}w$), for some $i_0$, where ${\pmb{u}}^{{\pmb{{{\beta}}}}}$ (resp., $u^{{{\beta}}}$) satisfies the additional conditions given in Lemma \[lem:prepnormal\]. Let ${{\mathcal G}}_a$ denote the ideal $${{\mathcal G}}_a := \Bigg(\prod_{(i,j)\in J}a_{ij}\Bigg)\cdot {{\mathcal H}}_a,$$ where $J:= \{(i,j): a_{ij}\neq 0,\, i=2,\ldots,N,\, j=1,\ldots,d-1\}$. \[rem:G\] (1) Since ${{\mathcal G}}_a$ is an ideal of functions in two variables, it follows from resolution of singularities that, after a finite number ${{\iota}}(a; {\pmb{u}})$ (respectively, ${{\iota}}(a;(u,w))$) of blowings-up of discrete sets (beginning with $a$ and) lying over $a$, the pull-back of ${{\mathcal G}}_a$ is a principal ideal generated by a monomial ${\tilde{\pmb{u}}}^{{\pmb{{{\gamma}}}}}$ (resp., ${{\tilde u}}^p {{\tilde w}}^q$) with respect to adapted coordinates ${\tilde{\pmb{u}}}$ (resp., $({{\tilde u}},{{\tilde w}})$) at any point over $a$. (At each step, the centre of blowing-up is the finite set of points over $a$ at which the pull-back of ${{\mathcal G}}_a$ is not already a principal ideal generated by such a monomial.) Note that the blowing-up of $(v=0)$ with centre $a$ corresponds to the blowing-up of $X$ with centre $({\pmb{u}}= {\pmb{0}})$ (resp., $(u=w=0)$). \(2) In particular, multiplication of ${{\mathcal G}}_a$ by a monomial in components of the exceptional divisor does not change the value of ${{\iota}}(a; {\pmb{u}})$ (resp., ${{\iota}}(a;(u,w))$). \[def:io\] Let ${{\iota}}(a)$ denote the minimum of ${{\iota}}(a; {\pmb{u}})$ (respectively, ${{\iota}}(a;(u,w))$) over all adapted coordinate systems $({\pmb{u}},v)$ (resp., $(u,v,w)$) at $a$ in which ${{\sigma}}$ has Weierstrass form. Of course, ${{\mathcal H}}_a$ and ${{\mathcal G}}_a$ themselves depend on the coordinates in Weierstrass form. \[lem:io=0\] If ${{\iota}}(a) = 0$, then ${{\sigma}}$ has prepared normal form at $a$. Assume that ${{\iota}}(a) = 0$. Then we can choose adapted local coordinates in which each coefficient $a_{ij}$, $j>0$, in is a monomial times a unit as in (resp., ), and ${{\mathcal H}}_a$ is a principal ideal generated by a monomial ${\pmb{u}}^{{\pmb{{{\gamma}}}}}$ (resp., $u^p w^q$) as in Remarks \[rem:G\]. In the 1-point case, necessarily $q=0$ since ${{\sigma}}\|(v=0)$ has rank $2$ outside ${\mathrm{supp}\,}E$ (i.e., since the Fitting ideal of $3 \times 3$ minors of $\log {\mathrm{Jac}\,}{{\sigma}}$ is supported in $E$). By Lemma \[lem:fitHP\], we can choose coordinates also in which the coefficients $a_{i0}$ satisfy the conditions of Lemma \[lem:prepnormal\]. \[lem:semicontin\] If ${{\iota}}(a) \neq 0$, then there is a neighbourhood of $a$ in which $(\rho(b), {{\iota}}(b)) < (\rho(a), {{\iota}}(a))$, $b\neq a$. This is clear. Proof of Lemma \[lem:prepnormal\] {#subset:proofprepnormal} --------------------------------- The lemma will be given by an algorithm presented in three distinct cases, beginning with ${{\sigma}}$ in Weierstrass form as in Lemma \[lem:rho\]: - $a$ is a 2-point; - $a$ is a 1-point with ${\mathrm{ord}}_a (T_i) = d(a)$ (where $(T_i)$ denotes the ideal generated by $T_2,\ldots,T_N$ and ${\mathrm{ord}}_a$ means the order at $a$); - $a$ is a 1-point where ${\mathrm{ord}}_a (T_i) < d(a)$. The third case is the most delicate. Case that $a$ is a 2-point {#subsec:2pt} -------------------------- \[lem:2pt\] Let $a\in X$ be a 2-point. Suppose that $\rho(a) > 0$, ${{\iota}}(a)>0$ and ${{\sigma}}$ is in Weierstrass form at $a$ (Lemma \[lem:rho\]). Let $C$ denote the $2$-curve through $a$. Then $C \subset (\log {\mathrm{rk}\,}{{\sigma}}= 0)$. Let $\tau: (\widetilde{X},\widetilde{E}) \to (X,E)$ denote the combinatorial blowing-up with centre $C$. Then $(\rho(b),{{\iota}}(b)) < (\rho(a),{{\iota}}(a))$, for all $b \in \tau^{-1}(a)$. Consider ${{\sigma}}$ in the Weierstrass form of Lemma \[lem:rho\] in adapted coordinates $({\pmb{u}},v)$, where $d=d(a)=\rho(a)+1$ and ${{\iota}}(a) = {{\iota}}(a;{\pmb{u}})$. Since $\rho(a) > 0$, $\log {\mathrm{rk}\,}_a {{\sigma}}= 0$. Then $\log {\mathrm{rk}\,}{{\sigma}}= 0$ in a neighbourhood of $a$ in $C$ (by ), and therefore on $C$. Let $b\in \tau^{-1}(a)$. If $b$ is a 2-point, then, without loss of generality, there are adapted coordinates $({\pmb{x}},z)=(x_1,x_2,z)$ at $b$, with ${{\widetilde E}}=(x_1x_2=0)$, in which $\tau$ is given by $$u_1=x_1,\quad u_2 = x_1x_2,\quad v=z.$$ If $b$ is a 1-point, then we can assume that ${{\alpha}}_2 \neq 0$, and there are adapted coordinates $(x,y,z)$ such that $$u_1 = x (\eta+y),\quad u_2 = x (\eta+y)^{-{{\alpha}}_1/{{\alpha}}_2},\quad v = z,$$ where $\eta\neq 0$. Thus, if $b$ is a 2-point (resp., 1-point), we can write $$\begin{aligned} \tau^\sigma_1 &= {\pmb{x}}^{{{\tilde {\pmb{{{\alpha}}}}}}},\\ \tau^\sigma_i &= {{\tilde g}}_i({\pmb{x}}) + {\pmb{x}}^{{{\tilde {\pmb{{{\delta}}}}}}} \tau^T_i, \end{aligned} \qquad\quad \text{resp.,}\qquad\quad \begin{aligned} \tau^\sigma_1 &= x^{{{\tilde {{\alpha}}}}},\\ \tau^\sigma_i &= {{\tilde g}}_i(x,y) + x^{{{\tilde {{\delta}}}}} \widetilde{U} \tau^T_i, \end{aligned}$$ $i=2,\ldots,N$, where ${{\tilde {\pmb{{{\alpha}}}}}}= ({{\alpha}}_1+{{\alpha}}_2,{{\alpha}}_2)$, ${{\tilde {\pmb{{{\delta}}}}}}=({{\delta}}_1+{{\delta}}_2,{{\delta}}_2)$ (resp., ${{\tilde {{\alpha}}}}= {{\alpha}}_1+{{\alpha}}_2$, ${{\tilde {{\delta}}}}={{\delta}}_1+{{\delta}}_2$, and $\widetilde{U} = (\eta+y)^{{{\delta}}_1 - {{\delta}}_2 {{\alpha}}_1 / {{\alpha}}_2}$ is a unit). In either case, $$\tau^T_i = z^d \tau^{{\widetilde T}}_i + \sum^{d-1}_{j=0} b_{ij} z^j,\quad i=2,\ldots,N,$$ where $b_{ij}= \tau^a_{ij}$ and $b_{ij} = b_{ij}({\pmb{x}})$ (resp., $b_{ij}= b_{ij}(x,y)$). Clearly, in either case, $\rho(b)\leq \rho(a)$. Moreover, in either case it follows from Lemma \[lem:fit\](1) that ${{\mathcal H}}_b = \tau^{{\mathcal H}}_a$, so that ${{\mathcal G}}_b = \tau^{{\mathcal G}}_a$; therefore, ${{\iota}}(b) < {{\iota}}(a)$ and then $(\rho(b), {{\iota}}(b)) < (\rho(a), {{\iota}}(a))$, by the definition of ${{\iota}}$ (see Remarks \[rem:G\]). \[rem:2pt\] Lemma \[lem:prepnormal\] in the case that $a$ is a 2-point thus follows directly from resolution of singularities of the ideal ${{\mathcal G}}_a$. If $a$ is a 1-point, then blowing-up $(u=w=0)$ (with respect to adapted coordinates as in Lemma \[lem:rho\]) likewise gives $(\rho(b), {{\iota}}(b)) < (\rho(a), {{\iota}}(a))$, for $b\in \tau^{-1}(a)$. This can be used to prove a local version of Lemma \[lem:prepnormal\], but the centre $(u=w=0)$ need not have a global meaning in $X$. The challenge in §§\[subsec:1ptd\],\[subsec:1pt<d\] is to decrease the value of the invariant $(\rho,{{\iota}})$ by global* blowings-up only. Case that $a$ is a 1-point with ${\mathrm{ord}}_a (T_i) = d(a)$ {#subsec:1ptd} --------------------------------------------------------------- \[lem:1pt1\] Let $a\in X$ be a 1-point. Suppose that $\rho(a) > 0$, ${{\iota}}(a)>0$, and ${{\sigma}}$ is in Weierstrass form at $a$ (Lemma \[lem:rho\]), in adapted coordinates $(u,v,w)$, where ${\mathrm{ord}}_a(T_i)=d(a)=\rho(a)+1$ and ${{\iota}}(a) = {{\iota}}(a;(u,w))$. Let $\tau: (\widetilde{X},\widetilde{E}) \to (X,E)$ denote the blowing-up with centre $a$. Then $(\rho(b),{{\iota}}(b)) < (\rho(a),{{\iota}}(a))$, for all $b \in \tau^{-1}(a)$. We again write the components of ${{\sigma}}$ using the notation of Lemma \[lem:rho\]. We consider three cases, depending on the coordinate chart of $X$ containing $b$. Case I. The point $b$ belongs to the $u$-chart. This chart has adapted coordinates $(x,{{\tilde v}},{{\tilde w}})$ in which $\tau$ is given by $$u=x,\quad v = x{{\tilde v}},\quad w=x{{\tilde w}},$$ and $b \in (x=0)$; say, $b=(0,\nu,{{\omega}})$. Since ${\mathrm{ord}}_a (T_i) = d$, $$\begin{aligned} \tau^{{\sigma}}_1 &= x^{{\alpha}},\\ \tau^{{\sigma}}_i &= g_i(x) + x^{{{\delta}}+d}\cdot\frac{\tau^T_i}{x^d},\quad i=2,\ldots,N,\end{aligned}$$ and each $$\label{eq:u} \frac{\tau^T_i}{x^d} = {{\tilde v}}^d \tau^{{\widetilde T}}_i + \sum^{d-1}_{j=0} b_{ij}(x,{{\tilde w}}) {{\tilde v}}^j,$$ where each $b_{ij} = \tau^a_{ij}/x^{d-j}$. It is clear from that, if $\nu\neq 0$, then $d(b) \leq d(a)-1 < d(a)$ (recall that ${{\widetilde T}}_2 = U$ is a unit). Assume that $\nu =0$. Then $d(b)\leq d(a)$, by . Moreover, it follows from Lemma \[lem:fit\](1) that ${{\mathcal H}}_b = x\cdot\tau^{{\mathcal H}}_a$, so that ${{\mathcal G}}_b = x^q\cdot\tau^{{\mathcal G}}_a$, where $q = 1 - \sum_{(i,j)\in J}(d-j)$; therefore, ${{\iota}}(b) < {{\iota}}(a)$ and $(\rho(b), {{\iota}}(b)) < (\rho(a), {{\iota}}(a))$. Case II. The point $b$ belongs to the $w$-chart, but not to the $u$-chart. The $w$-chart has adapted coordinates $({{\tilde u}},x,{{\tilde v}})$, where ${{\widetilde E}}= ({{\tilde u}}=x=0)$, in which $$u = x{{\tilde u}},\quad v= x{{\tilde v}},\quad w=x,$$ and $b=(0,0,\nu)$. Similarly to Case I, $$\begin{aligned} \tau^{{\sigma}}_1 &= x^{{\alpha}}{{\tilde u}}^{{\alpha}},\\ \tau^{{\sigma}}_i &= g_i(x{{\tilde u}}) + {{\tilde u}}^{{\delta}}x^{{{\delta}}+d} \cdot\frac{\tau^T_i}{x^d},\quad i=2,\ldots,N,\end{aligned}$$ and $$\frac{\tau^T_i}{x^d} = {{\tilde v}}^d \tau^{{\widetilde T}}_i + \sum^{d-1}_{j=0} b_{ij}({{\tilde u}},x) {{\tilde v}}^j,$$ where $b_{ij} = \tau^a_{ij}/x^{d-j}$. If $\nu\neq 0$, then $d(b) \leq d(a)-1 < d(a)$. Assume $\nu=0$. Then $d(b)\leq d(a)$, and ${{\mathcal H}}_b = x\cdot\tau^{{\mathcal H}}_a$, by Lemma \[lem:fit\](1). Again, ${{\mathcal G}}_b = x^q\cdot\tau^{{\mathcal G}}_a$, with $q$ as in Case I, and $(\rho(b), {{\iota}}(b)) < (\rho(a), {{\iota}}(a))$. Case III. The point $b$ belongs to the $v$-chart, but not to the $u$- or $w$-charts. The $v$-chart has adapted coordinates $({{\tilde u}},x,{{\tilde w}})$, where ${{\widetilde E}}= ({{\tilde u}}=x=0)$, in which $$u = x{{\tilde u}},\quad v= x,\quad w=x{{\tilde w}},$$ and $b=(0,0,0)$. In the same way as before, $$\begin{aligned} \tau^{{\sigma}}_1 &= x^{{\alpha}}{{\tilde u}}^{{\alpha}}\\ \tau^{{\sigma}}_i &= g_i(x{{\tilde u}}) + {{\tilde u}}^{{\delta}}x^{{{\delta}}+d} \cdot\frac{\tau^T_i}{x^d},\quad i=2,\ldots,N,\end{aligned}$$ and $$\frac{\tau^T_i}{x^d} = \tau^{{\widetilde T}}_i + \sum^{d-1}_{j=0} b_{ij}({{\tilde u}},x,{{\tilde w}}),$$ where each $b_{ij}({{\tilde u}},x,{{\tilde w}}) = a_{ij}(x{{\tilde u}},x{{\tilde w}})/x^{d-j}$; in particular, all $b_{ij}(0,0,0) = 0$. Hence $\tau^T_2/x^d$ is a unit at $b$ and, moreover, $({{\delta}},{{\delta}}+d)$ is linearly independent of $({{\alpha}},{{\alpha}})$ (since $d=d(a)\neq 0$). Therefore, $d(b) = 0$, so that $\rho(b) = 0$. Case that $a$ is a 1-point with ${\mathrm{ord}}_a (T_i) < d(a)$ {#subsec:1pt<d} --------------------------------------------------------------- Let $a\in X$ be a 1-point. Suppose that $\rho(a) > 0$, ${{\iota}}(a)>0$, and ${{\sigma}}$ is in Weierstrass form at $a$ (Lemma \[lem:rho\]), in adapted coordinates $(u,v,w)$, where ${\mathrm{ord}}_a(T_i) < d(a)=\rho(a)+1$ and ${{\iota}}(a) = {{\iota}}(a;(u,w))$. Set $\mu:= {\mathrm{ord}}_a(T_i)$. We rewrite as $$\label{eq:weiermu} T_i = \sum_{k=\mu}^{d-1} u^{{{\alpha}}_{ik}} P_{ik}(u,v,w) + \left({{\widetilde T}}_i v^d + \sum^{d-1}_{j=0} c_{ij}(u,w) v^j\right), \quad i=2,\ldots,N,$$ where $c_{2,d-1}=0$, ${\mathrm{ord}}_a c_{ij}\geq d-j$, for all $i,\,j$, and $u^{{{\alpha}}_{ik}} P_{ik}$ is a homogeneous polynomial of degree $k$ such that either $P_{ik}=0$ or $P_{ik}(0,v,w) \neq 0$ (i.e., $P_{ik}$ is not divisible by $u$), for all $i,\,k$. \[rem:alpha\] For all $i,\,k$ such that $P_{ik}\neq 0$, we have $0 < {{\alpha}}_{ik} < k$. The left-hand inequality is clear from the definition of $\rho(a)$. On the other hand, if ${{\alpha}}_{ik} = k$, then $u^{{{\alpha}}_{ik}} P_{ik}(u,v,w) = u^k P_{ik}(0,0,0)$ is a monomial of degree $< d$; since ${{\alpha}}_{ik}$ is trivially linearly dependent on ${{\alpha}}$, this would contradict the definition of Weierstrass form. Let ${{\mathfrak m}}_{X,a}$ denote the maximal ideal of ${{\mathcal O}}_{X,a}$ and let $I := \{(i,k): P_{ik}\neq 0\}$. Let ${{\mathcal J}}$ denote the ideal $${{\mathcal J}}:= {{\mathfrak m}}_{X,a}^{d} + \sum_{(i,k) \in I} u^{\alpha_{ik}} {{\mathfrak m}}_{X,a}^{k-\alpha_{ik}}.$$ Clearly, $V({{\mathcal J}}) = \{a\}$. \[lem:1pt2\] Let $a\in X$ be a 1-point. Suppose that $\rho(a) > 0$, ${{\iota}}(a)>0$, and ${{\sigma}}$ is in Weierstrass form at $a$ (Lemma \[lem:rho\]), in adapted coordinates $(u,v,w)$, where $\mu = {\mathrm{ord}}_a(T_i)<d(a)=\rho(a)+1$ and ${{\iota}}(a) = {{\iota}}(a;(u,w))$. Then there is a morphism $\tau: (\widetilde{X},\widetilde{E}) \to (X,E)$ given by a sequence of admissible blowings-up that principalizes ${{\mathcal J}}$, such that 1. $\tau$ is an isomorphism over $X\setminus \{a\}$; 2. $(\rho(b),{{\iota}}(b)) < (\rho(a),{{\iota}}(a))$, for all $b \in \tau^{-1}(a)$. Let $\tau_1: (X_1,E_1)\to (X,E)$ denote the blowing-up with centre the point $a$, and let $\tau_2: (\widetilde{X},\widetilde{E}) \to (X_1,E_1)$ denote a morphism given by admissible blowings-up that principalizes $\tau_1^{{\mathcal J}}$. Set $\tau=\tau_1\circ\tau_2$. Clearly, $\tau$ is an isomorphism over $X\setminus \{a\}$. Let $b\in \tau^{-1}(a)$. We write the components of ${{\sigma}}$ using the notation of , and again consider three cases, depending on the coordinate chart containing $\tau_2(b)$. Case I. The point $\tau_2(b)$ belongs to the $u$-chart of $\tau_1$.* This chart has adapted coordinates $(x,{{\tilde v}},{{\tilde w}})$ in which $\tau_1$ is given by $$u=x,\quad v = x{{\tilde v}},\quad w=x{{\tilde w}},$$ and $\tau_2(b) \in (x=0)$. It follows that $\tau_1^{{\mathcal J}}$ is a principal monomial ideal in this chart, so that $\tau_2$ is an isomorphism over the chart. Since ${\mathrm{ord}}_a (T_i)\geq \mu$, $$\begin{aligned} \tau_1^{{\sigma}}_1 &= x^{{\alpha}},\\ \tau_1^{{\sigma}}_i &= g_i(x) + x^{{{\delta}}+\mu}\cdot\frac{\tau_1^T_i}{x^\mu},\quad i=2,\ldots,N,\end{aligned}$$ and each $$\label{eq:u2} \frac{\tau_1^T_i}{x^\mu} = \sum_{k=\mu}^{d-1} x^{k-\mu}Q_{ik}(1,{{\tilde v}},{{\tilde w}}) + x^{d-\mu} \left({{\tilde v}}^d \tau_1^{{\widetilde T}}_i + \sum^{d-1}_{j=0} d_{ij}(x,{{\tilde w}}) {{\tilde v}}^j\right),$$ where each $Q_{ik} = \tau_1^P_{ik}/x^{k-{{\alpha}}_{ik}}$ and each $d_{ij} = \tau_1^c_{ij}/x^{d-j}$. Since $\mu < d$, there exists $i_0$ such that $Q_{i_0,\mu}\neq 0$, and $$\frac{\tau_1^T_{i_0}}{x^\mu} = Q_{i_0,\mu}(1,{{\tilde v}},{{\tilde w}}) + x R(x,{{\tilde v}},{{\tilde w}}),$$ for some $R$. By Remark \[rem:alpha\], $Q_{i_0,\mu}$ is a non constant polynomial of degree $< \mu$. Therefore, $d(\tau_2(b)) \leq \deg\, Q_{i_0,\mu} < d(a)$. Case II. The point $\tau_2(b)$ belongs to the $w$-chart, but not to the $u$-chart.* The $w$-chart has adapted coordinates $({{\tilde u}},{{\tilde w}},{{\tilde v}})$, where ${{\widetilde E}}= ({{\tilde u}}={{\tilde w}}=0)$, in which $$u = {{\tilde u}}{{\tilde w}},\quad v= {{\tilde v}}{{\tilde w}},\quad w={{\tilde w}},$$ and $\tau_2(b)\in V({{\tilde u}},{{\tilde w}})$. Similarly to Case I, $$\label{eq:w2} \begin{aligned} \tau_1^{{\sigma}}_1 &= {{\tilde u}}^{{\alpha}}{{\tilde w}}^{{\alpha}},\\ \tau_1^{{\sigma}}_i &= g_i({{\tilde u}}{{\tilde w}}) + {{\tilde u}}^{{\delta}}{{\tilde w}}^{{{\delta}}+\mu} \cdot\frac{\tau_1^T_i}{{{\tilde w}}^\mu},\quad i=2,\ldots,N, \end{aligned}$$ and each $$\label{eq:w2a} \frac{\tau_1^T_i}{{{\tilde w}}^\mu} = \sum_{k=\mu}^{d-1} {{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde w}}^{k-\mu}Q_{ik}({{\tilde u}},{{\tilde v}}) + {{\tilde w}}^{d-\mu} \left({{\tilde v}}^d \tau_1^{{\widetilde T}}_i + \sum^{d-1}_{j=0} d_{ij}({{\tilde u}},{{\tilde w}}) {{\tilde v}}^j\right),$$ where each $Q_{ik} = \tau_1^P_{ik}/{{\tilde w}}^{k-{{\alpha}}_{ik}}$ and each $d_{ij} = \tau_1^c_{ij}/{{\tilde w}}^{d-j}$. Then, for each $i,k$, either $Q_{ik}=0$ or there exists $j_{ik} < k$ such that: $$\label{eq:w2expansion} Q_{ik}({{\tilde u}},{{\tilde v}}) = {{\tilde v}}^{j_{ik}}Q_{i,j_{ik},k} + \sum_{j=0}^{j_{ik}-1}{{\tilde v}}^jQ_{ijk} +{{\tilde u}}R_{ik}({{\tilde u}},{{\tilde v}})$$ where $Q_{ij_{ik}k}$ is a nonzero constant. Moreover, $\tau_1^{{\mathcal J}}$ is the ideal $$\tau_1^{{\mathcal J}}= {{\tilde w}}^{\mu}\cdot\left({{\tilde w}}^{d-\mu};\, {{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde w}}^{k-\mu},\, (i,k) \in I\right),$$ and principalization of $\tau_1^{{\mathcal J}}$ is equivalent to principalization of ${{\mathcal K}}:= {{\tilde w}}^{-\mu}\cdot \tau_1^{{\mathcal J}}$. Since $\tau_1^{{\mathcal J}}$ is generated by finitely many exceptional monomials in two variables, $\tau_2: ({{\widetilde X}},{{\widetilde E}}) \to (X_1,E_1)$ is a composite of combinatorial blowings-up, over the $w$-chart. We consider two subcases, depending on whether $b$ is a 2-point or a 1-point; each of these subcases will be divided into further subcases, depending on which generator of ${{\mathcal K}}$ pulls back to a generator of the principal ideal $\tau_2^{{\mathcal K}}$. Subcase II.1. The point $b$ is a 2-point.* There are adapted coordinates $({\pmb{x}},z) = (x_1,x_2,z)$ centred at $b$ such that ${{\widetilde E}}= (x_1x_2=0)$ and $${{\tilde u}}= {\pmb{x}}^{{\pmb{{{\lambda}}}}_{1}},\quad {{\tilde w}}= {\pmb{x}}^{{\pmb{{{\lambda}}}}_{2}},\quad {{\tilde v}}= \zeta + z,$$ where ${\pmb{{{\lambda}}}}_{1}, {\pmb{{{\lambda}}}}_{2}$ are ${{\mathbb Q}}$-linearly independent. By , $$\label{eq:w2.1} \begin{aligned} \tau^{{\sigma}}_1 &= {\pmb{x}}^{{{\alpha}}({\pmb{{{\lambda}}}}_{1}+{\pmb{{{\lambda}}}}_{2})},\\ \tau^{{\sigma}}_i &= {{\tilde g}}_i({\pmb{x}}) + {\pmb{x}}^{{{\delta}}{\pmb{{{\lambda}}}}_{1} + ({{\delta}}+\mu){\pmb{{{\lambda}}}}_{2}} \cdot\frac{\tau^T_i}{{\pmb{x}}^{\mu{\pmb{{{\lambda}}}}_{2}}},\quad i=2,\ldots,N. \end{aligned}$$ Subcase II.1.1. The ideal $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde w}}^{d-\mu}) = {\pmb{x}}^{{\pmb{{{\beta}}}}}$, where ${\pmb{{{\beta}}}}= (d-\mu){\pmb{{{\lambda}}}}_{2}$.* Then $$\frac{\tau^T_i}{{\pmb{x}}^{d{\pmb{{{\lambda}}}}_{2}}} = (\zeta + z)^d\tau^{{\widetilde T}}_i + \sum_{j=0}^{d-1}b_{ij}({\pmb{x}})(\zeta+z)^j, \quad i=2,\ldots,N,$$ where $b_{ij} = \tau^a_{ij}/{\pmb{x}}^{(d-j){\pmb{{{\lambda}}}}_{2}}$. Clearly, if $\zeta\neq0$, then $d(b) \leq d-1 < d(a)$. Suppose that $\zeta = 0$. Then $d(b) \leq d(a)$. Moreover, by Lemma \[lem:fit\](1),(2), ${{\mathcal H}}_b = {\pmb{x}}^{{\pmb{{{\lambda}}}}_{2}}\cdot\tau^{{\mathcal H}}_a$, so that $${{\mathcal G}}_b = {\pmb{x}}^{{\pmb{{{\lambda}}}}_{2}}\prod_{(i,j)\in J}\frac{1}{{\pmb{x}}^{{\pmb{{{\beta}}}}+(\mu-j){\pmb{{{\lambda}}}}_{2}}}\cdot\tau^{{\mathcal G}}_a;$$ therefore, ${{\iota}}(b) < {{\iota}}(a)$ and $(\rho(b),{{\iota}}(b))<(\rho(a),{{\iota}}(a))$. Subcase II.1.2. There exists $i_0$ and $k_0$ (where $\mu \leq k_0 < d$) such that $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde u}}^{{{\alpha}}_{i_0, k_0}}{{\tilde w}}^{k_0-\mu}) = {\pmb{x}}^{{\pmb{{{\beta}}}}}$, where ${\pmb{{{\beta}}}}= {{\alpha}}_{i_0, k_0}{\pmb{{{\lambda}}}}_{1}+(k_0-\mu){\pmb{{{\lambda}}}}_{2}$.* By , $$\frac{\tau^T_{i_0}}{{\pmb{x}}^{{\pmb{{{\beta}}}}+\mu{\pmb{{{\lambda}}}}_{2}}} = {{\widetilde Q}}_{i_0,k_0}({\pmb{x}},z) + {{\widetilde R}}({\pmb{x}},z),$$ where ${{\widetilde Q}}_{i_0,k_0} = \tau_2^Q_{i_0,k_0}$ and ${{\widetilde R}}(0,z) = 0$. By , $${{\widetilde Q}}_{i_0,k_0} = z^{j_0}Q_{i_0, j_0, k_0} + \sum_{j=0}^{j_0-1} z^j Q_{i_0, j, k_0} + {\pmb{x}}^{{\pmb{{{\lambda}}}}_{1}}R_{i_0,k_0}({\pmb{x}},z),$$ where $j_0 = j_{i_0, k_0}$. If $j_0 > 0$, then $d(b) \leq j_0 < k_0 < d(a)$. On the other hand, if $j_0 = 0$, then, by , $$\tau^{{\sigma}}_{i_0} = {{\tilde g}}_{i_0}({\pmb{x}}) + {\pmb{x}}^{({{\delta}}+{{\alpha}}_{i_0,k_0}){\pmb{{{\lambda}}}}_{1} + ({{\delta}}+k_0){\pmb{{{\lambda}}}}_{2}} \cdot \frac{\tau^T_{i_0}}{{\pmb{x}}^{{\pmb{{{\beta}}}}+\mu{\pmb{{{\lambda}}}}_{2}}};$$ since ${{\alpha}}_{i_0,k_0} < k_0$ and ${\pmb{{{\lambda}}}}_{1},\, {\pmb{{{\lambda}}}}_{2}$ are linearly independent, we conclude that $d(b) = 0 < d(a)$. This completes Subcase II.1. Subcase II.2. The point $b$ is a 1-point. There are adapted coordinates $(x,y,z)$ centred at $b$ such that ${{\widetilde E}}=(x=0)$ and $${{\tilde u}}= x^{{{\lambda}}_1}(\eta+y)^{-1},\quad {{\tilde w}}= x^{{{\lambda}}_2}(\eta+y),\quad {{\tilde v}}= \zeta+ z,$$ where $\eta\neq 0$. By , $$\label{eq:w2.2} \begin{aligned} \tau^{{\sigma}}_1 &= x^{{{\alpha}}({{\lambda}}_1+{{\lambda}}_2)},\\ \tau^{{\sigma}}_i &= {{\tilde g}}_i(x,y) + x^{{{\delta}}{{\lambda}}_1 + ({{\delta}}+\mu){{\lambda}}_2} \cdot\frac{\tau^T_i}{x^{\mu{{\lambda}}_2}},\quad i=2,\ldots,N. \end{aligned}$$ Subcase II.2.1. The ideal $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde w}}^{d-\mu}) = x^{(d-\mu){{\lambda}}_2}(\eta+y)^{d-\mu}$.* Then $$\frac{\tau^T_i}{x^{d{{\lambda}}_2}} = (\zeta + z)^d(\eta + y)^d\tau^{{\widetilde T}}_i + \sum_{j=0}^{d-1}b_{ij}(x,y)(\zeta+z)^j, \quad i=2,\ldots,N,$$ where $b_{ij} = \tau_2^({{\tilde w}}^ja_{ij})/x^{d{{\lambda}}_2}$. Since $\eta\neq 0$, it is clear that, if $\zeta\neq 0$, then $d(b) \leq d-1 < d(a)$. Suppose that $\zeta = 0$. Then $d(b) \leq d(a)$. As above, ${{\mathcal H}}_b = x^{{{\lambda}}_2}\cdot\tau^{{\mathcal H}}_a$, and $${{\mathcal G}}_b = x^{{{\lambda}}_2}\prod_{(i,j)\in J}\frac{1}{x^{{{\beta}}+(\mu-j){{\lambda}}_2}}\cdot\tau^{{\mathcal G}}_a,$$ so that $(\rho(b),{{\iota}}(b))<(\rho(a),{{\iota}}(a))$. Subcase II.2.2. The ideal $\tau_2^{{\mathcal K}}$ is not generated by $\tau_2^({{\tilde w}}^{d-\mu})$.* Then there exists $(i,k) \in I$ such that $\tau_2^({{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde w}}^{k-\mu})$ generates $\tau_2^{{\mathcal K}}$. Set $${{\Lambda}}:= \{(i,k)\in I: \tau_2^({{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde w}}^{k-\mu}) \text{ generates } \tau_2^{{\mathcal K}}\}.$$ Note that, if $(i,k_1),(i,k_2) \in {{\Lambda}}$, then ${{\lambda}}_1{{\alpha}}_{i,k_1} + {{\lambda}}_2(k_1-\mu) = {{\lambda}}_1{{\alpha}}_{i,k_2} + {{\lambda}}_2(k_2-\mu) = {{\beta}}$, say, so that $$\label{eq:pairs} {{\alpha}}_{i,k_1} = {{\alpha}}_{i,k_2} + (k_2 - k_1)\frac{{{\lambda}}_2}{{{\lambda}}_1}.$$ From , we rewrite $$\tau^{{\sigma}}_i = {{\tilde g}}_i(x,y) + x^{{{\delta}}{{\lambda}}_1 + ({{\delta}}+\mu){{\lambda}}_2 + {{\beta}}} \cdot\frac{\tau^T_i}{x^{\mu{{\lambda}}_2+{{\beta}}}}.$$ For each $i$, let ${{\Lambda}}_i := \{k: (i,k)\in {{\Lambda}}\}$. By , $$\label{eq:w2avar} \frac{\tau^T_i}{x^{\mu{{\lambda}}_2+{{\beta}}}} = \sum_{k\in {{\Lambda}}_i} (\eta+y)^{k-{{\alpha}}_{ik}}(\tau_2^Q_{ik})(x,z) + xR_i(x,y,z).$$ We claim that, for each $i$, the exponents $k-{{\alpha}}_{ik}$ in are all distinct. Indeed, if $k_1-{{\alpha}}_{i,k_1} = k_2-{{\alpha}}_{i,k_2}$, where $(i,k_1),(i,k_2) \in {{\Lambda}}$, $k_1\neq k_2$, then ${{\lambda}}_1=-{{\lambda}}_2$, by ; a contradiction since ${{\lambda}}_1,{{\lambda}}_2>0$. By , for each $(i,k) \in {{\Lambda}}$, $$(\tau_2^Q_{ik})(x,z) = (\zeta+z)^{j_{ik}}Q_{i,j_{ik},k} + \sum_{j=0}^{j_{ik}-1} (\zeta + z)^j Q_{ijk} + xR_{ik}(x,y),$$ where $Q_{i,j_{ik},k} \neq 0$. Set $j_0 := \max\{j_{ik}: (i,k) \in {{\Lambda}}\}$ and $${{\Gamma}}:= \{(i,k) \in {{\Lambda}}: j_{ik} = j_0\}.$$ We can rewrite as $$\label{eq:w2avar1} \frac{\tau^T_i}{x^{\mu{{\lambda}}_2+{{\beta}}}} = \sum_{k\in {{\Gamma}}_i} z^{j_0} (\eta+y)^{k-{{\alpha}}_{ik}} Q_{i,j_0,k} + \sum_{j=0}^{j_0-1}z^jR_{ij}(x,y) + xR_i(x,y,z),$$ where ${{\Gamma}}_i := \{k: (i,k)\in {{\Gamma}}\}$. Recall that the $k-{{\alpha}}_{ik}$, $k \in {{\Gamma}}_i$ are distinct. Choose $i_0$ such that ${{\Gamma}}_{i_0} \neq \emptyset$. We consider three subcases of Subcase II.2.2, depending on $j_0$. First suppose that $j_0 = 0$. Choose $k_0$ such that $k_0-{{\alpha}}_{i_0,k_0} = \max\{k-{{\alpha}}_{ik}: k \in {{\Gamma}}_{i_0}\}$. Then $0<k_0-{{\alpha}}_{i_0,k_0}<d$ and ${{\partial}}_y^{k_0-{{\alpha}}_{i_0,k_0}}\left(\tau^T_{i_0}/x^{\mu{{\lambda}}_2+{{\beta}}}\right)$ does not vanish at $b$; therefore, $d(b) \leq k_0-{{\alpha}}_{i_0,k_0}<d(a)$. Secondly, suppose that $0 < j_0 \leq \mu$.* By , $$\label{eq:sum} {{\partial}}_z^{j_0}\left(\frac{\tau^T_{i_0}}{x^{\mu{{\lambda}}_2+{{\beta}}}}\right) = j_0! \sum_{k\in {{\Gamma}}_{i_0}} (\eta+y)^{k-{{\alpha}}_{i_0,k}} Q_{i_0,j_0,k} + xR_{i_0}(x,y,z).$$ Since $\mu\leq k < d$ in , there are at most $d-\mu$ terms in the sum, with distinct exponents. Hence there exists $l_0 < d-\mu$ such that ${{\partial}}_y^{l_0}$ applied to the sum in is a unit; therefore, ${{\partial}}_z^{j_0}{{\partial}}_y^{l_0}\left(\tau^T_{i_0}/x^{\mu{{\lambda}}_2+{{\beta}}}\right)$ is a unit. It follows that $d(b) \leq j_0 + l_0 \leq \mu + l_0 < \mu + d(a) - \mu = d(a)$. Finally, suppose that $\mu < j_0$. If $k \in {{\Gamma}}_{i_0}$, then $j_0 < k$. So the sum in has at most $d-j_0$ terms, with distinct exponents. As above, there exists $l_0 < d-j_0$ such that ${{\partial}}_y^{l_0}$ applied to this sum is a unit; therefore, ${{\partial}}_z^{j_0}{{\partial}}_y^{l_0}\left(\tau^T_{i_0}/x^{\mu{{\lambda}}_2+{{\beta}}}\right)$ is a unit, and again $d(b) \leq j_0 + l_0 < j_0 + d(a) - j_0 = d(a)$. This completes Case II of the proof of Lemma \[lem:1pt2\]. Case III following has a similar pattern. Case III. The point $\tau_2(b)$ belongs to the $v$-chart, but not to the $u$- or $w$-charts.* The $v$-chart has adapted coordinates $({{\tilde u}},{{\tilde v}},{{\tilde w}})$, where ${{\widetilde E}}= ({{\tilde u}}={{\tilde v}}=0)$, in which $$u = {{\tilde u}}{{\tilde v}},\quad v= {{\tilde v}},\quad w={{\tilde v}}{{\tilde w}},$$ and $\tau_2(b)=0$. Similarly to Case II, $$\label{eq:w3} \begin{aligned} \tau_1^{{\sigma}}_1 &= {{\tilde u}}^{{\alpha}}{{\tilde v}}^{{\alpha}},\\ \tau_1^{{\sigma}}_i &= g_i({{\tilde u}}{{\tilde v}}) + {{\tilde u}}^{{\delta}}{{\tilde v}}^{{{\delta}}+\mu} \cdot\frac{\tau_1^T_i}{{{\tilde v}}^\mu},\quad i=2,\ldots,N, \end{aligned}$$ and each $$\frac{\tau_1^T_i}{{{\tilde v}}^\mu} = \sum_{k=\mu}^{d-1} {{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde v}}^{k-\mu}Q_{ik}({{\tilde u}},{{\tilde w}}) + {{\tilde v}}^{d-\mu} \left(\tau_1^{{\widetilde T}}_i + \sum^{d-1}_{j=0} d_{ij}({{\tilde u}},{{\tilde v}},{{\tilde w}}) \right),$$ where each $Q_{ik} = \tau_1^P_{ik}/{{\tilde v}}^{k-{{\alpha}}_{ik}}$ and each $d_{ij} = \tau_1^c_{ij}/{{\tilde v}}^{d-j}$. Then, for each $i,k$, either $Q_{ik}=0$ or there exists $j_{ik} < k$ such that: $$Q_{ik}({{\tilde u}},{{\tilde w}}) = {{\tilde w}}^{j_{ik}}Q_{i,j_{ik},k} + \sum_{j=0}^{j_{ik}-1}{{\tilde w}}^jQ_{ijk} +{{\tilde u}}R_{ik}({{\tilde u}},{{\tilde w}})$$ where $Q_{i,j_{ik},k}$ is a nonzero constant. Moreover, $\tau_1^{{\mathcal J}}$ is the ideal $$\tau_1^{{\mathcal J}}= {{\tilde v}}^{\mu}\cdot\left({{\tilde v}}^{d-\mu};\, {{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde v}}^{k-\mu},\, (i,k) \in I\right),$$ and principalization of $\tau_1^{{\mathcal J}}$ is equivalent to principalization of ${{\mathcal K}}:= {{\tilde v}}^{-\mu}\cdot \tau_1^{{\mathcal J}}$. Since $\tau_1^{{\mathcal J}}$ is generated by finitely many exceptional monomials in two variables, $\tau_2: ({{\widetilde X}},{{\widetilde E}}) \to (X_1,E_1)$ is a composite of combinatorial blowings-up, over the $v$-chart. As in Case II, we consider two subcases, depending on whether $b$ is a 2-point or a 1-point; each of these subcases will be divided into further subcases, depending on which generator of ${{\mathcal K}}$ pulls back to a generator of the principal ideal $\tau_2^{{\mathcal K}}$. Subcase III.1. The point $b$ is a 2-point.* There are adapted coordinates $({\pmb{x}},z) = (x_1,x_2,z)$ centred at $b$ such that ${{\widetilde E}}= (x_1x_2=0)$ and $${{\tilde u}}= {\pmb{x}}^{{\pmb{{{\lambda}}}}_{1}},\quad {{\tilde v}}= {\pmb{x}}^{{\pmb{{{\lambda}}}}_{2}},\quad {{\tilde w}}= z,$$ where ${\pmb{{{\lambda}}}}_{1}, {\pmb{{{\lambda}}}}_{2}$ are ${{\mathbb Q}}$-linearly independent. By , $$\begin{aligned} \tau^{{\sigma}}_1 &= {\pmb{x}}^{{{\alpha}}({\pmb{{{\lambda}}}}_{1}+{\pmb{{{\lambda}}}}_{2})},\\ \tau^{{\sigma}}_i &= {{\tilde g}}_i({\pmb{x}}) + {\pmb{x}}^{{{\delta}}{\pmb{{{\lambda}}}}_{1} + ({{\delta}}+\mu){\pmb{{{\lambda}}}}_{2}} \cdot\frac{\tau^T_i}{{\pmb{x}}^{\mu{\pmb{{{\lambda}}}}_{2}}},\quad i=2,\ldots,N.\end{aligned}$$ Subcase III.1.1. The ideal $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde v}}^{d-\mu}) = {\pmb{x}}^{(d-\mu){\pmb{{{\lambda}}}}_{2}}$.* Then $$\tau^{{\sigma}}_2 = {{\tilde g}}_2({\pmb{x}}) + {\pmb{x}}^{{{\delta}}{\pmb{{{\lambda}}}}_{1} + ({{\delta}}+d){\pmb{{{\lambda}}}}_{2}} \cdot \left(\tau^T_2 / {\pmb{x}}^{\mu{\pmb{{{\lambda}}}}_{2}}\right)$$ and $$\frac{\tau^T_2}{{\pmb{x}}^{d{\pmb{{{\lambda}}}}_{2}}} = \tau^U + R({\pmb{x}},z),$$ where $R({\pmb{0}},z)=0$. Since ${\pmb{{{\lambda}}}}_{1}, {\pmb{{{\lambda}}}}_{2}$ are linearly independent and $d\neq 0$, it follows that $d(b)=0<d(a)$, so $\rho(b)=0<\rho(a)$. Subcase III.1.2. There exists $i_0$ and $k_0$ (where $\mu \leq k_0 < d$) such that $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde u}}^{{{\alpha}}_{i_0, k_0}}{{\tilde v}}^{k_0-\mu})$. In this subcase, we can show that $d(b)<d(a)$ exactly as in Subcase II.1.2 above. Subcase III.2. The point $b$ is a 1-point. There are adapted coordinates $(x,y,z)$ centred at $b$ such that ${{\widetilde E}}=(x=0)$ and $${{\tilde u}}= x^{{{\lambda}}_1}(\eta+y)^{-1},\quad {{\tilde v}}= x^{{{\lambda}}_2}(\eta+y),\quad {{\tilde w}}= z,$$ where $\eta\neq 0$. By , $$\begin{aligned} \tau^{{\sigma}}_1 &= x^{{{\alpha}}({{\lambda}}_1+{{\lambda}}_2)},\\ \tau^{{\sigma}}_i &= {{\tilde g}}_i(x,y) + x^{{{\delta}}{{\lambda}}_1 + ({{\delta}}+\mu){{\lambda}}_2} \cdot\frac{\tau^T_i}{x^{\mu{{\lambda}}_2}},\quad i=2,\ldots,N.\end{aligned}$$ Subcase III.2.1. The ideal $\tau_2^{{\mathcal K}}$ is generated by $\tau_2^({{\tilde v}}^{d-\mu}) = x^{(d-\mu){{\lambda}}_2}(\eta+y)^{d-\mu}$.* Then $$\frac{\tau^T_2}{x^{d{{\lambda}}_2}} = (\eta + y)^d\tau^U + \sum_{j=0}^{d-2}\tau^a_{2j}x^{(j-d){{\lambda}}_2}(\eta+y)^j,$$ and $\tau^a_{2j} = a_{2j}(x^{{{\lambda}}_1+{{\lambda}}_2},x^{{{\lambda}}_2}(\eta+y)z)$. Since $\eta\neq0$, it is clear that $d(b) \leq d(a)-1$. Subcase III.2.2. The ideal $\tau_2^{{\mathcal K}}$ is not generated by $\tau_2^({{\tilde v}}^{d-\mu})$. Then there exists $(i,k) \in I$ such that $\tau_2^({{\tilde u}}^{{{\alpha}}_{ik}}{{\tilde v}}^{k-\mu})$ generates $\tau_2^{{\mathcal K}}$. In this subcase, we can show that $d(b)<d(a)$ precisely as in Subcase II.2.2 above. This completes the proof of Lemma \[lem:prepnormal\]. Decreasing the main invariant {#sec:decrho} ============================= In this section, we prove Lemma \[lem:decrho\] and thus complete the proof of Theorem \[thm:dim3\]. We continue to use the notation of Sections \[sec:outline\] and \[sec:prepnorm\]. A sequence of blowings-up as in the conclusion of Lemma \[lem:decrho\] will be called permissible. To prove Lemma \[lem:decrho\], we will show that, in general, every prepared point $a \in {\mathrm{supp}\,}E$ admits a neighbourhood $U$ over which there is a morphism $\tau: (\widetilde{U},\widetilde{E}) \to (U,E\|_U)$ given by a permissible finite sequence of blowings-up over $(\rho = \rho(a))$, such that $\rho(b) < \rho(a)$, for all $b \in {{\widetilde U}}$ (see Lemmas \[lem:decrho1\], \[lem:decrho2\], \[lem:decrho3\], depending on the nature of $a$). Lemma \[lem:decrho\] clearly follows from this local statement, because the sequence of blowings-up in Lemma \[lem:decrho\](2) is uniquely determined by the maximum value of $\rho$. Note that, if $a$ is a generic prepared 1-point, then there is a neighbourhood $U$ of $a$ over which $A = \emptyset$, so that a permissible blowing-up sequence means a finite sequence as in Lemma \[lem:decrho\](2). Declared local exceptional divisor {#subsec:declared} ---------------------------------- Suppose that $\sigma$ has prepared normal form in adapted coordinates $(\pmb{u},v)$ at a $2$-point $a$ (respectively, prepared normal form in adapted coordinates $(u,v,w)$ at a $1$-point $a$). Although $v$ (and $w$) are not globally defined, there is a neighbourhood $U$ of $a$ in which $(v=0)$ (or $(v=0)$ and $(w=0)$) are smooth hypersurfaces that we can add to $E$ to obtain a divisor $D$ on $U$. We consider $D$ a “declared exceptional divisor”. The divisor $D$ and a corresponding monomial idea ${{\mathcal I}}= {{\mathcal I}}({{\sigma}},a)$ are defined according to the nature of $a$, as follows. \[def:declared\] We use the notation of Lemma \[lem:prepnormal\]. - If $a$ is a prepared 2 point , $$\begin{aligned} D &:= E\|_U +(v=0) = (u_1 u_2 v=0), \\ \mathcal{I} &:= (v^d,\, \pmb{u}^{\pmb{r}_{ij}}v^j, {\pmb{u}}^{{\pmb{{{\beta}}}}}),\end{aligned}$$ where ${\pmb{{{\beta}}}}$ will also be denoted ${\pmb{r}}_{i_0,0}$ (for reasons evident from ); - If $a$ is a generic prepared 1-point ( with all $s_{ij}=0$), $$\begin{aligned} D &:= E\|_U +(v=0) = (uv=0),\\ \mathcal{I} &:= (v^d,\, u^{r_{ij}}v^j,\,u^{\beta}),\end{aligned}$$ where ${{\beta}}$ will also be denoted $r_{i_0,0}$; - If $a$ is a non-generic prepared 1-point , $$\begin{aligned} D &:= E\|_U +(v=0) +(w=0) = (uvw=0),\\ \mathcal{I} &:= (v^d,\, u^{r_{ij}} w^{s_{ij}}v^j,\, u^{\beta}w),\end{aligned}$$ where again ${{\beta}}$ will also be denoted $r_{i_0,0}$. The notation for ${{\mathcal I}}$ in each case above is understood to mean that $(i,j)$ runs over the index set $J := \{(i,j): a_{ij}\neq 0,\, i=2,\ldots,N,\, j=1,\ldots,d-1\}$. \[rem:declared\] (1) $D$ and $\mathcal{I}({{\sigma}},a)$ depend on the adapted coordinates at $a$. Nevertheless, if $U$ is a small open neighbourhood of $a$ and $\rho(b) = \rho(a)$, where $b \in U$, then $D$ induces a declared exceptional divisor at $b$, and ${{\mathcal I}}({{\sigma}},a)$ induces ${{\mathcal I}}({{\sigma}},b)$. \(2) A blowing-up that is admissible for $D$ is also admissible for $E\|_U$. Suppose that $\tau: {{\widetilde U}}\to U$ is given by a finite sequence of blowings-up that are admissible for $D$. If ${{\widetilde D}},\,{{\widetilde E}}$ denote the transforms of the divisors $D,\, E\|_U$, respectively, then ${{\widetilde D}}= {{\widetilde E}}$ $+$ strict transform of $(v=0)$ (or ${{\widetilde D}}={{\widetilde E}}$ $+$ strict transforms of $(v=0)$ and $(w=0)$ in the non-generic prepared 1-point case). \[prop:declared\] Let $a\in X$ and suppose that $\sigma$ has prepared normal form at $a$. Take $U,\, D$ and ${{\mathcal I}}$ as in Definitions \[def:declared\]. Let $\tau: (\widetilde{U},\widetilde{E}) \to (U,E\|_U)$ be a sequence of blowings-up with centres over ${\mathrm{supp}\,}E\|_U$ that are combinatorial with respect to $D$, and let $b \in \tau^{-1}(a)$. If $\tau^(\mathcal{I})$ is a principal ${{\widetilde D}}$-monomial ideal, then $\tau^(\mathcal{I})$ is also ${{\widetilde E}}$-monomial, and $\rho(b) < \rho(a)$. Note that the hypotheses of Proposition \[prop:declared\] does not exclude $(v=w=0)$ as centre of blowing up (in the non-generic 1-point case). Proposition \[prop:declared\] is a purely local assertion; we will not claim to principalize ${{\mathcal I}}({{\sigma}},a)$ by blowings-up that are global over $X$. The proposition plays an important part in the proof of Lemma \[lem:decrho\] but, in the latter, we do not necessarily principalize ${{\mathcal I}}({{\sigma}},a)$ at every point $b$ over a given $a \in \Sigma$ because $\rho$ may decrease before ${{\mathcal I}}$ becomes principal. The proof of Proposition \[prop:declared\] is a case-by-case analysis which we leave to the end; we first complete the proof of Lemma \[lem:decrho\]. Permissible sequences of blowings-up ------------------------------------ As indicated above, in order to prove Lemma \[lem:decrho\], it is enough to show that every prepared point $a\in {\mathrm{supp}\,}E$ admits a neighbourhood $U$ over which the invariant $\rho$ can be decreased by a permissible finite sequence of blowings-up. This will be done separately in the case that $a$ is a generic 1-point, a 2-point or a non-generic 1-point, in the three lemmas \[lem:decrho1\], \[lem:decrho2\], \[lem:decrho3\] following. In each of these lemmas, $U$ denotes a (relatively compact) open neighbourhood of $a$ in which $\rho \leq \rho(a)$, and $\Sigma := \{x\in U: \rho(x) = \rho(a)\}$. We assume that $U$ is small enough that the declared exceptional divisor $D$ can be defined as in §\[subsec:declared\], and we use the notation of Lemma \[lem:prepnormal\] and Definitions \[def:declared\]. \[lem:decrho1\] Let $a$ be a generic 1-point of $\Sigma$. Then there is a finite sequence of permissible blowings-up $\tau: (\widetilde{U},\widetilde{E}) \to (U,E\|_U)$ which is combinatorial with respect to $D$, such that $\rho(b)<\rho(a)$ for all $b \in \widetilde{U}$. Moreover, the weak transform of ${{\mathcal I}}$ by $\tau$ is principal except perhaps on $\widetilde{E} \cap \widetilde{H}$, where $\widetilde{H}$ is the strict transform of $H := (v=0)$. (The weak transform means the residual ideal after factoring out the exceptional divisor as much as possible.) For brevity, we write $E$ instead of $E\|_U$. Note that the unique permissible centre of blowing up in $U$ is $V(u,v) = (u=v=0)$. Let $\mu(a) := \min\{{{\beta}}, r_{ij}+j\}$. We first show that we can reduce to the case $\mu(a) < d$. Suppose that $\mu(a) \geq d$. Consider the blowing-up $\tau_1: (U_1,E_1) \to (U,E)$ with centre $V(u,v)$, and let $b \in \tau_1^{-1}(a)$. There are two possibilities: 1. $b$ belongs to the $v$-chart, with coordinates $({{\tilde u}},{{\tilde v}},{{\tilde w}})$ in which $\tau_1$ is given by $u = {{\tilde u}}{{\tilde v}},\, v={{\tilde v}},\, w={{\tilde w}}$. (The strict transform of $H$ does not intersect this chart.) Then $\tau_1^{{\mathcal I}}$ is the principal ideal generated by ${{\tilde v}}^d$ at $b$; therefore, $\rho(b) < \rho(a)$, by Proposition \[prop:declared\]. 2. $b$ is the origin of the $u$-chart, with coordinates $(x,{{\tilde v}},{{\tilde w}})$ in which $\tau_1$ is given by $u=x,\, v=x{{\tilde v}},\, w={{\tilde w}}$. Then $$\tau_1^{\ast}T_i = x^d\left(\tilde{v}^d + \sum_{i=1}^{d-1} x^{r_{ij}+j -d }\tilde{v}^j\tau_1^{\ast}\tilde{a}_{ij} + x^{\beta-d} \tau_1^{\ast}\tilde{a}_{i0} \right).$$ Clearly, $\rho(b) \leq \rho(a)$ and $\mu(b) < \mu(a)$ if $\rho(b) = \rho(a)$. We therefore assume that $\mu(a) < d$. Again let $\tau_1: (U_1,E_1) \to (U,E)$ denote the blowing-up with centre $V(u,v)$, let $b \in \tau_1^{-1}(a)$, and consider the two coordinate charts as above. 1. Suppose that $b$ belongs to the $u$-chart. If $b\neq 0$, then ${{\tilde v}}\neq 0$ at $b$, so that $\tau_1^{{\mathcal I}}$ is a principal ideal generated by a monomial in $x$, and $\rho(b) < \rho(a)$, by Proposition \[prop:declared\]. On the other hand, if $b=0$, then $$\tau_1^{\ast}T_i = x^{\mu(a)}\left(\tilde{v}^dx^{d-\mu(a)} + \sum_{i=1}^{d-1} x^{r_{ij}+j -\mu(a)}\tilde{v}^j\tau_1^{\ast}\tilde{a}_{ij} + x^{\beta-\mu(a)} \tau_1^{\ast}\tilde{a}_{i0}\right),$$ and clearly $\rho(b) < \rho(a)$. We remark that, in this case, the weak transform of ${{\mathcal I}}$ is supported in $\widetilde{E} \cap \widetilde{H}$. 2. Otherwise, $b$ is the origin of the $v$-chart, where $E_1 = (\tilde{u}\tilde{v}=0)$ and $$\label{eq:excideal} \tau_1^{\ast}\mathcal{I} = {{\tilde v}}^{\mu(a)}\left( \tilde{v}^{d-\mu(a)},\, \tilde{u}^{r_{ij}}\tilde{v}^{r_{ij}+j-\mu(a)},\, \tilde{u}^{\beta}\tilde{v}^{\beta-\mu(a)} \right).$$ Now, let ${{\mathcal J}}$ denote the monomial ideal on $U_1$ determined by the right hand side of , and let $\tau_2$ be a sequence of combinatorial blowings-up such that $\tau_2^\mathcal{J}$ is a principal monomial ideal. Clearly, $\tau = \tau_2 \circ \tau_1$ is a sequence of combinatorial blowings-up with respect to $F$, and the weak transform of $\mathcal{I}$ by $\tau$ is supported on $\widetilde{E} \cap \widetilde{H}$. Furthermore, if $c \in \tau^{-1}(a)$, then $\rho(c) < \rho(a)$, either by the preceding case (1) or by Proposition \[prop:declared\]. Consider a sequence of blowings-up $\overline{\tau}: (\overline{U},\overline{E}) \to (U_1,E_1)$ that are combinatorial with respect to $E$ (e.g., part of the sequence $\tau_2$). Let $b$ be the origin of the $v$-chart in the preceding case (2), and let $c \in \overline{\tau}^{-1}(b)$ be a point where $\overline{\tau}^\mathcal{J}$ is not principal. We will show that $\rho(c) = \infty$. This implies that we can principalize ${{\mathcal J}}$ by combinatorial blowings-up given at each step by the maximal locus of $\rho$; in fact, the locus $(\rho = \infty)$: If ${{\mathcal J}}$ is not principal at $b$, then $(\rho = \infty)$ is given by the 2-curve $({{\tilde u}}= {{\tilde v}}= 0)$ in . After each blowing-up with centre $(\rho = \infty)$, the new locus $(\rho = \infty)$ is a disjoint union of analogous 2-curves defined in the various coordinate charts. In other words, ${{\mathcal J}}$ can be principalized by a sequence $\tau_2$ of permissible blowings-up. To show that $\rho(c) = \infty$ above: Since $\overline{\tau}^{\ast}\mathcal{J}$ is not principal, $c$ must be a 2-point of $\overline{E}$. Therefore, there are coordinates $({\pmb{x}},y)= (x_1,x_2,y)$ at $c$ in which $\overline{\tau}$ us given by $${{\tilde u}}= x_1^{\kappa_1}x_2^{\kappa_2}, \quad {{\tilde v}}= x_1^{{{\lambda}}_1}x_2^{{{\lambda}}_2},\quad \tilde{w}=y,$$ so that $$\overline{\tau}^{\ast}\tilde{v}^d = \pmb{x}^{\pmb{\bar{r}}_d}, \quad \overline{\tau}^{\ast}\tilde{u}^{r_{ij}}\tilde{v}^{r_{ij}+j} = \pmb{x}^{\pmb{\bar{r}}_{ij}} , \quad \overline{\tau}^{\ast}\tilde{u}^{\beta} = \pmb{x}^{\pmb{\bar{\beta}}},$$ with suitable exponents. Since $(\kappa_1,\kappa_2)$ and $(\lambda_1,\lambda_2)$ are linearly independent, these exponents are distinct for fixed $i$, and all except $\pmb{\bar{\beta}}$ are linearly independent of $\pmb{\bar{\alpha}}$, where $\overline{\tau}^{\ast}\sigma_1 = \pmb{x}^{\pmb{\bar{\alpha}}}$. Let $\mathcal{K}$ denote the ideal generated by the coefficients of the formal expansions with respect to $y$ of $\overline{\tau}^{\ast}T_i$, $i=2, \ldots, N$. Note that $\mathcal{K} \subset \overline{\tau}^{\ast}\mathcal{J}$, and that each monomial $\overline{\tau}^{\ast}\tilde{v}^d$, $\overline{\tau}^{\ast}\tilde{u}^{r_{ij}}\tilde{v}^{r_{ij}+j}$ in $\overline{\tau}^{\ast}\mathcal{J}$ appears in the expansion of the constant term (i.e., the coefficient of $y^0$) of the formal expansion of $\overline{\tau}^{\ast}T_i$, for some $i$. Moreover, $\overline{\tau}^{\ast}\tilde{u}^{\beta}$ appears in the coefficient of $y$ in $\overline{\tau}^{\ast}T_{i_0}$. It follows that $\mathcal{K}$ is principal if and only if $\overline{\tau}^{\ast}\mathcal{J}$ is principal. Therefore, $\rho(c) = \infty$, as claimed. \[lem:decrho2\] Let $a$ be 2-point of $\Sigma$. Then there is a finite sequence of permissible blowings-up $\tau: (\widetilde{U},\widetilde{E}) \to (U,E)$ which is combinatorial with respect to $D$, such that $\rho(b)<\rho(a)$ for all $b \in \widetilde{U}$. Moreover, $\tau$ can be realized as a composite $\tau = \tau_3\circ \tau_2\circ \tau_1$, where $\tau_1$ is a single blowing-up with centre $a$, $\tau_2$ is the composite of a finite sequence of permissible blowings-up as in Lemma \[lem:decrho\](2), and $\tau_3$ is the composite of a finite sequence of blowings-up over $a$. The proof consists of four steps. Step 1. Decreasing $\rho$ outside the preimage of $a$. Let $W := U \setminus \{a\}$. Then every point of $\Sigma \cup W$ is a generic prepared 1-point. (If $a$ is an isolated point of $\Sigma$, then the proof of the lemma reduces essentially to Step 3 below.) By Lemma $\ref{lem:decrho1}$, there is a finite sequence $\tilde{\tau}$ of permissible blowings-up $\tilde{\tau} : (\widetilde{W},\widetilde{E}) \to (W, E\|_W)$ such that $\rho(b) < \rho(a)$ for all $b \in \widetilde{W}$. Moreover, all blowings-up involved are combinatorial with respect to $D\|_W$ and, after a first blowing-up with centre $a$ (to separate curves that will be blown up simultaneously according to Lemma \[lem:decrho1\]), we can take the closures of all the centres of blowings-up that comprise $\tilde{\tau}$, to get a sequence $${{\tilde \tau}}_1: (U_1,E_1) \to (U,E)$$ of permissible blowings-up over $U$, such that $\rho(b) < \rho(a)$ for all $b$ outside ${{\tilde \tau}}_1^{-1}(a)$. Each centre of blowing up is a union (necessarily disjoint) of closures of curves given by Lemma \[lem:decrho1\]). Moreover, all blowings-up are combinatorial with respect to $D$. In particular, ${{\tilde \tau}}_1$ also makes sense as a morphism ${{\tilde \tau}}_1 : (U_1,D_1) \to (U,D)$. Set $\mathcal{I}_1 := {{\tilde \tau}}_1^{\ast}\mathcal{I}$. Let $H_1$ denote the strict transform of $H := (v=0)$ by ${{\tilde \tau}}_1$, and $\mathcal{J}_1$ the residual ideal sheaf of ${{\mathcal I}}_1$ after factoring out the greatest possible monomial in local generators of components of the exceptional divisor $E_1$. By Lemma \[lem:decrho1\], $V(\mathcal{J}_1) \subset {{\tilde \tau}}_1^{-1}(a) \cup (E_1 \cap H_1)$. Let $\Gamma_1$ denote the closure of $V(\mathcal{J}_1)\setminus {{\tilde \tau}}^{-1}_1(a)$. Then $\Gamma_1$ is an analytic (or regular) curve (unless it is empty) with at most two connected components ${{\gamma}}_1^{(k)}$ ($k=1$ or $k=1,2$), each of which intersects ${{\tilde \tau}}^{-1}_1(a)$ in a point $p_1^{(k)}$. If there are two components, then they were separated by the first blowing-up (with centre $a$) of the sequence ${{\tilde \tau}}_1$, so that $p_1^{(1)},\, p_1^{(2)}$ are distinct. Step 2. Decreasing $\rho$ at the limit points of the 1-curve(s) ${{\gamma}}_1^{(k)}$. Let $p_1$ denote either of the points $p_1^{(k)}$, and ${{\gamma}}_1$ the corresponding curve ${{\gamma}}_1^{(k)}$. Assume that ${{\mathcal I}}_1$ is not principal at $p_1$. Then there is a coordinate system $({\pmb{x}},y) = (x_1,x_2,y)$ at $p_1$ in which ${{\tilde \tau}}_1$ is given by: $$u_1 = \pmb{x}^{\pmb{\lambda}_1} ,\quad u_2 = \pmb{x}^{\pmb{\lambda}_2} ,\quad v= \pmb{x}^{\pmb{\lambda}_3} y,$$ where $\pmb{\lambda}_1$, $\pmb{\lambda}_2$ are $\mathbb{Q}$-linearly independent, $E_1 = (x_1x_2=0)$ and $D_1 = (x_1x_2y=0)$. We can assume that $(x_1=0)$ is the component of $E_1$ that does not project to $a$, so that ${{\gamma}}_1 = V(x_1,y)$. In particular, in this coordinate neighbourhood of $p_1$, the ideal ${{\mathcal I}}_1$ has the form $$\label{eq:I1} {{\mathcal I}}_1 = ( \pmb{x}^{\pmb{\tilde{r}}_{2d}}y^d, \pmb{x}^{\pmb{\tilde{r}}_{ij}}y^j,\pmb{x}^{\pmb{\tilde{\beta}}})$$ for suitable $\pmb{\tilde{r}}_{2d}$, $\pmb{\tilde{r}}_{ij}$ and $\pmb{\tilde{\beta}}$. (Recall that the first monomial in ${{\mathcal I}}_1$ in comes from $T_2$, and the last comes from $T_{i_0}$.) Write $\pmb{\tilde{r}}_{i_0,0}= \pmb{\tilde{\beta}}$ (cf. Definitions \[def:declared\]). Each $\pmb{\tilde{r}}_{ij}$ is a pair $\pmb{\tilde{r}}_{ij} =(r_{ij1},r_{ij2})$. Let $m$ denote the minimum of $r_{ij1}$ over all $(i,j)$ corresponding to monomials in , and let $\Lambda := \{(i,j): r_{ij1}= m\}$. Take $(i_1,j_1) \in \Lambda$ with minimal $j_1$. Since $d(b) < d(a)=d$ for $b$ outside ${{\tilde \tau}}^{-1}(a)$ (cf. notation of Lemma \[lem:rho\]), we see that $j_1 < d(a)$. We can also assume that $r_{i_1,j_1,2}$ is minimal over all pairs $(i,j)$; indeed, we can blow up $p_1 = (0,0,0)$ once, and then: 1. After a further sequence of blowings-up of curves of the form $(x_2=y=0)$ (which are combinatorial with respect to $D_1$ and project to $a$), we can assume that $r_{i_1,j_1,2} \leq r_{ij2}$, for all $(i,j) \in \Lambda$. (This does not change the values of $r_{ij1}$.) 2. After a further sequence of blowings-up of curves of the form $(x_1 = x_2 =0)$ (which are combinatorial in respect to $D_1$ and project to $a$), we can assume that $r_{i_1,j_1,2} \leq r_{ij2}$, for all $(i,j) \notin \Lambda$. (Again this does not change the values of $r_{ij1}$.) The above construction applies to $p_1 = p_1^{(k)}$, $k=1$ or $k=1,2$, and provides a sequence of permissible blowings-up $${{\tilde \tau}}_2 : (U_2,E_2) \to (U_1,E_1)$$ (where the first blowing-up in the sequence has centre $p_1^{(1)} \cup p_1^{(2)}$ if $k=1,2$). Let $D_2$, $\mathcal{I}_2$, $\mathcal{J}_2$, $p_2^{(k)}$ and $\gamma_2^{(k)}$ denote the objects defined after ${{\tilde \tau}}_2$ that are analogous to $D_1$, $\mathcal{I}_1$, $\mathcal{J}_1$, $p_1^{(k)}$ and $\gamma_1^{(k)}$. Clearly, at $p_2 = p_2^{(k)}$, the ideal $\mathcal{I}_2$ has the form $$\label{eq:I2} {{\mathcal I}}_2 = \pmb{x}^{\pmb{\tilde{r}}_{i_1,j_1}} ( y^{j_1}, \pmb{x}^{\pmb{r}_{ij}}y^j, \pmb{x}^{\pmb{\beta}} ),$$ (where $\gamma_2 = V(x_1,y)$) for suitable $\pmb{r}_{ij}$, $\pmb{\beta}$, and it follows that $\rho(p_2) <\rho(a)$. In particular, if $j_1 = 0$, then $\mathcal{I}_2$ is principal at $p_2$ and $\rho(p_2)=0$. Step 3. Decreasing $\rho$ over $a$, outside the preimage(s) of the limit point(s) $p_2^{(k)}$. Let $W$ denote the complement of the curve(s) $\gamma_2^{(k)}$ in $U_2$. Then the ideal $\mathcal{J}_2\cdot \mathcal{O}_{W}$ has support in the preimage of $a$. There is a sequence ${{\tilde \tau}}: (\widetilde{W},\widetilde{E}) \to (W,E_2\|_W)$ of blowings-up that are combinatorial with respect to $D_2\|_W$, which principalizes $\mathcal{J}_2\cdot\mathcal{O}_{W}$. Then ${{\tilde \tau}}$ is permissible (all centres project to $a$), and $\rho < \rho(a)$ throughout $\widetilde{W}$, by Proposition \[prop:declared\]. Since all blowings-up involved are combinatorial with respect to $D_2\|_W$, we can take the closures of all centres to get a sequence of permissible blowings-up $${{\tilde \tau}}_3 : (U_3,E_3) \to (U_2,E_2),$$ where $\rho(b) < \rho(a)$ for every $b$ not in the preimage of $p_2^{(k)}$, and all blowings-up are combinatorial with respect to the divisor $D_2$. We define $D_3$, $\mathcal{I}_3$, $\mathcal{J}_3$, $p_3^{(k)}$ and $\gamma_3^{(k)}$ in the same way as before. The precise form of the ideal $\mathcal{I}_3$ at $p_3^{(k)}$ is important for the next step, so let us compute it. Again let $p_3$ denote either point $p_3^{(k)}$ and let ${{\gamma}}_3 = {{\gamma}}_3^{(k)}$. By construction (see ), the centres of all blowings-up in ${{\tilde \tau}}_3$ containing $p_2$ (or the analogous limit point after blowing up) are of the form $V(y,x_2)$. Therefore, there is a coordinate system $(u_1,u_2,v)$ at $p_3$ in which ${{\tilde \tau}}_3$ is given by $$x_1 = u_1 ,\quad x_2 = u_2 ,\quad y= u_2^{\lambda}v.$$ It follows that, in these coordinates, ${{\gamma}}_3 = V(u_1,v)$ and $\mathcal{I}_3$ has the form $${{\mathcal I}}_3 = \pmb{u}^{\pmb{\delta}} (u_2^{r_{j_1}}v^{j_1}, \pmb{u}^{\pmb{r}_{ij}}v^{j} , \pmb{u}^{\pmb{\beta}}),$$ for suitable exponents $\pmb{\delta}$, $r_{j_1}$, $\pmb{r}_{ij}$ and $\pmb{\beta}$ (where we use notation unchanged from before for simplicity). Since $\mathcal{I}_3$ is principal outside the preimage of $p_2$ (in the preimage of $a$), we have $\beta_2 =0$ and $$\label{eq:I3} {{\mathcal I}}_3 = \pmb{u}^{\pmb{\delta}} (u_2^{r_{j_1}}v^{j_1}, \pmb{u}^{\pmb{r}_{ij}}v^{j}, u_1^{\beta_1}).$$ Step 4. Decreasing $\rho$ in the preimage(s) of the point(s) $p_2^{(k)}$. Let $W$ denote the complement of the curve(s) $\gamma_3^{(k)}$ in $U_3$. Then the ideal $\mathcal{J}_3\cdot \mathcal{O}_{W}$ has support in the preimage of $a$. There is a permissible sequence ${{\tilde \tau}}: (\widetilde{W},\widetilde{E}) \to (W,E_3\|_W)$ of blowings-up that are combinatorial with respect $D_3\|_W$, which principalizes $\mathcal{J}_3\cdot \mathcal{O}_{W}$. By Proposition \[prop:declared\], $\rho < \rho(a)$ throughout $\widetilde{W}$. Since all blowings-up are combinatorial with respect to $D_3\|_W$, we can take the closures of all centres to get a sequence of permissible blowings-up $${{\tilde \tau}}_4 : (U_4,E_4) \to (U_3,E_3),$$ where $\rho(b) < \rho(a)$ for every $b$ not in the preimage of $p_3^{(k)}$, and all blowings-up are combinatorial with respect to $D_3$. We define $\mathcal{I}_4$ as before. We claim that $\rho < \rho(a)$ in the preimage of $p_3 = p_3^{(k)}$, as required to finish the proof. By , the centres of all blowings-up in ${{\tilde \tau}}_4$ containing $p_3$ (or the analogous limit point after blowing up) are of the form $V(u_1,u_2)$, and they must principalize the ideal $$\mathcal{K} = ( u_2^{r_{j_1}}, \pmb{u}^{\pmb{r}_{ij}}, u_1^{\beta_1}).$$ Let $b$ denote a point in the preimage of $p_3$. There are two possible cases. Case I. $b$ is a 2-point. Then there are coordinates $(x_1,x_2,y)$ at $b$ in which ${{\tilde \tau}}_4$ is given by $$u_1 = \pmb{x}^{\pmb{\lambda}_1} ,\quad u_2 = \pmb{x}^{\pmb{\lambda}_2} ,\quad v=y.$$ Since ${{\tilde \tau}}_4^{\ast}\mathcal{K}$ is principal, there exists $j_2 \leq j_1$ such that $${{\mathcal I}}_4 = \pmb{u}^{\pmb{\tilde{\delta}}} ( y^{j_2}, \pmb{u}^{\pmb{\tilde{r}}_{ij}}y^{j}, \pmb{u}^{\pmb{\tilde{\beta}}}),$$ for suitable $\pmb{\tilde{\delta}}$, $\pmb{\tilde{r}}_{ij}$, $\pmb{\tilde{\beta}}$. Therefore, $\rho(b)<\rho(a)$. In particular, if $j_2 = 0$, then $\mathcal{I}_4$ is principal. Case II. $b$ is a 1-point. Then there are coordinates $(x,y,z)$ at $b$ in which ${{\tilde \tau}}_4$ is given by $$u_1 = x^{\lambda_1} ,\quad u_2 = x^{\lambda_2}(\zeta + z),\quad v=y.$$ Since ${{\tilde \tau}}_4^{\ast}\mathcal{K}$ is principal, there exists $j_2 \leq j_1$ such that $${{\mathcal I}}_4 = x^{\tilde{\delta}} ( y^{j_2}, x^{\tilde{r}_{ij}}y^{j}, x^{\tilde{\beta}})$$ for suitable $\tilde{\delta}$, $\tilde{r}_{ij}$, $\tilde{\beta}$. Therefore, $\rho(b)<\rho(a)$. In particular, if $j_2 = 0$, then $\mathcal{I}_4$ is principal. \[lem:decrho3\] Let $a$ be a non-generic 1-point of $\Sigma$. Then there is a finite sequence of permissible blowings-up $\tau: (\widetilde{U},\widetilde{E}) \to (U,E)$ which is combinatorial with respect to $D$, such that $\rho(b)<\rho(a)$ for all $b$ in $\widetilde{U}$. Moreover, $\tau$ can be realized as a composite $\tau = \tau_3\circ \tau_2\circ \tau_1$, where $\tau_1$ is a single blowing-up with centre $a$, $\tau_2$ is the composite of a finite sequence of permissible blowings-up as in Lemma \[lem:decrho\](2), and $\tau_3$ is the composite of a finite sequence of blowings-up over $a$. The proof consists of four steps, as for Lemma \[lem:decrho2\]. Step 1. Decreasing $\rho$ outside the preimage of $a$. Let ${{\gamma}}$ and ${{\delta}}$ denote the curves $(v=u=0)$ and $(v=w=0)$, respectively. Then ${{\gamma}}$ coincides with $\Sigma$, since ${{\delta}}\setminus \{a\}$ lies outside ${\mathrm{supp}\,}E$. We first blow up with centre $a$ to separate ${{\gamma}}$ and ${{\delta}}$, and define the morphism ${{\tilde \tau}}_1$ as in the proof of Lemma \[lem:decrho2\] (so that ${{\tilde \tau}}_1$ consists of blowings-up over ${{\gamma}}$). Set $\mathcal{I}_1 := {{\tilde \tau}}_1^{\ast}\mathcal{I}$. Let $H_1$ and $K_1$ denote the strict transforms of $H := (v=0)$ and $K := (w=0)$ by ${{\tilde \tau}}_1$ (respectively), and let $\mathcal{J}_1$ be the residual ideal sheaf of ${{\mathcal I}}_1$ after factoring out the greatest possible monomial in local generators of components of the exceptional divisor $E_1$. By Lemma \[lem:decrho1\], $V(\mathcal{J}_1) \subset {{\tilde \tau}}_1^{-1}(a) \cup (E_1 \cap H_1) \cup (K_1 \cap H_1)$. The closure $\Gamma_1$ of $V(\mathcal{J}_1)\setminus {{\tilde \tau}}^{-1}_1(a)$ is a union of two curves ${{\gamma}}_1 = E_1 \cap H_1$ (analogous to ${{\gamma}}_1^{(1)}$ in the proof of Lemma \[lem:decrho2\]) and ${{\delta}}_1 = K_1 \cap H_1$, which intersect ${{\tilde \tau}}^{-1}_1(a)$ in distinct points $p_1$ and $q_1$. Step 2. Decreasing $\rho$ at the limit points $p_1,\, q_1$ of the curves ${{\gamma}}_1,\, {{\delta}}_1$. For $p_1$, we can repeat the argument of Lemma \[lem:decrho2\], Step 2. Consider $q_1$. First note that, if $s_{ij} = 0$ for some $i,j$ (i.e., if $w$ does not appear in the monomial part of the coefficient of $v^j$, for some $i$ and some $j< d$), then we can again repeat the argument of Lemma \[lem:decrho2\], Step 2, thinking of $(u,w)$ here as $(u_2, u_1)$ in Lemma \[lem:decrho2\]. In this case, we can finish the proof of Lemma \[lem:decrho3\] as in Lemma \[lem:decrho2\]. It may, however, happen that $s_{ij} > 0$ for all $i,j$. In this case, after finitely many blowings-up of $q_1$ and its preimages in successive liftings of ${{\delta}}_1$, we get ${{\delta}}_2$ and $q_2$ with the property that there is a coordinate system $(x,y,z)$ at $q_2$, such that $E_2 = (x=0)$, ${{\delta}}_2 = V(y,z)$ and $${{\mathcal J}}_2 = (x^{r_d}y^d,\, x^{\tilde{r}_{ij}}z^{s_{ij}}y^j,\, z) = (x^{r_d}y^d,\, z),$$ for suitable $r_d,\, \tilde{r}_{ij}$, and it follows that $\rho(q_2) = 0$. (We define $D_2$, ${{\mathcal I}}_2$ and ${{\mathcal J}}_2$ as in Lemma \[lem:decrho2\].) Step 3. Decreasing $\rho$ over $a$, outside the preimages of $p_2,\, q_2$. As in the proof of Lemma \[lem:decrho2\], we define ${{\tilde \tau}}_3 : (U_3,E_3) \to (U_2,E_2)$, where $\rho(b) < \rho(a)$ for every $b$ not in the preimages of $p_2, q_2$. We define $D_3$, $\mathcal{I}_3$, $\mathcal{J}_3$, $p_3,\, q_3$, $\gamma_3$ and ${{\delta}}_3$ as before. At $p_3$, we can compute ${{\mathcal I}}_3$ or ${{\mathcal J}}_3$ as in the proof of Lemma \[lem:decrho2\]. At $q_3$, there is a coordinate system $(u,v,w)$ in which ${{\tilde \tau}}_3$ is given by $$x=u,\quad y=v,\quad z = u^{r_d}w,$$ and in which ${{\delta}}_3 = V(v,w)$ and $$\label{eq:NG3} {{\mathcal J}}_3 = (v^d, w).$$ It follows that $\rho(q_3) =0$ and that, if $W$ denotes the complement of the curves $\gamma_3$, ${{\delta}}_3$ in $U_3$, then ${{\mathcal J}}_3\cdot{{\mathcal O}}_W$ has support disjoint from a neighbourhood $V$ of ${{\delta}}_3$. Step 4. Decreasing $\rho$ in the preimages of $p_2,\, q_2$. We define ${{\tilde \tau}}_4 : (U_4,E_4) \to (U_3,E_3)$ as in the proof of Lemma \[lem:decrho2\]. Then, as in the latter, $\rho < \rho(a)$ outside the preimages of $p_3,\ q_3$, and, moreover, $\rho < \rho(a)$ in the preimage of $p_3$. Since ${{\tilde \tau}}_4$ is an isomorphism over $V$, we have $\rho < \rho(a)$ on $U_4$. This completes the proof of Lemma \[lem:decrho\]. Proof of Proposition \[prop:declared\] -------------------------------------- Consider $\tau: (\widetilde{U},\widetilde{D}) \to (U,D)$. In each case in Definitions \[def:declared\], ${\mathrm{supp}\,}{{\mathcal I}}$ has codimension at least two; therefore, ${\mathrm{cosupp}\,}\tau^({{\mathcal I}}) \subset {\mathrm{supp}\,}{{\widetilde E}}$ and $\tau^({{\mathcal I}})$ is ${{\widetilde E}}$-principal. The proof of the proposition will be divided in three cases depending on the nature of $a$. Case I. $a$ is a generic 1-point. Then $D = (uv = 0)$. We consider two cases depending on whether $b$ is a 1- or 2-point with respect to $\widetilde{D}$: Subcase I.1. $b$ is a 1-point of $\widetilde{D}$. There are adapted coordinates $(x,y,z)$ at $b$, where $\widetilde{D} = (x=0)$ and $\tau$ is given by $$u =x^{\lambda_1}, \quad v = x^{\lambda_2}(\zeta + z),\quad w=y,$$ where $\zeta \neq 0$. Clearly, $\widetilde{E} = \widetilde{D}$ at $b$, since ${\mathrm{supp}\,}\widetilde{E} = V(\tau^(u))$. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, each $$\label{eq:I.1} \tau^{\ast}T_i = x^{\tilde{r}_d} (\zeta+z)^d \overline{T}_i + \sum_{j=1}^{d-1} x^{\tilde{r}_{ij}} (\zeta+z)^j {{\tilde b}}_{ij} + b_{i0},$$ for suitable $\tilde{r}_d$, $\tilde{r}_{ij}$, where ${{\tilde b}}_{ij} = {{\tilde b}}_{ij}(x,y) = \tau^{{\tilde a}}_{ij}$, $\overline{T}_i = \tau^\widetilde{T}_i$ and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = x^{\tilde{\beta}} y$, for some $\tilde{\beta}$. All the exponents can be computed explicitly from the ${{\lambda}}_i$ and the original $r_{ij}$, $\beta$, but we will not need the explicit formulas. Let $x^{\tilde{\gamma}}$ denote a generator of $\tau^\mathcal{I}$. Then becomes $$\label{eq:I.1sigma} \begin{aligned} \sigma_1 & = x^{\tilde{\alpha}},\\ \sigma_i & = g_i(x) + x^{\tilde{\delta} + \tilde{\gamma}} \frac{\tau^T_i}{x^{\tilde{\gamma}}}, \end{aligned}$$ for suitable $\tilde{\alpha}$, $\tilde{\delta}$. We now consider three further subcases depending on which monomial generator of $\mathcal{I}$ pulls back to a generator of $\tau^\mathcal{I}$. Subcase I.1.1. $\tau^(v^d)$ generates $\tau^\mathcal{I}$.* Then $$\frac{\tau^T_2}{x^{\tilde{\gamma}}} = (\zeta +z)^d \overline{T}_2 + \frac{1}{x^{{{\tilde {{\gamma}}}}}}\left(\sum_{j=1}^{d-2} x^{ \tilde{r}_{2j}} (\zeta +z)^j {{\tilde b}}_{2j} + b_{20}\right),$$ where $\zeta \neq 0$, $\overline{T}_2$ is a unit, and ${{\tilde b}}_{2j},\, b_{20}$ are independent of $z$. It follows that $d(b) < d(a)$, so that $\rho(b)<\rho(a)$. Subcase I.1.2. $\tau^(v^d)$ does not generate $\tau^\mathcal{I}$, but $\tau^(u^{r_{ij}}v^{j})$ generates $\tau^\mathcal{I}$, for some $(i,j)$.* Let $(i_1,j_1)$ denote such $(i,j)$ with maximal $j$. Then $$\frac{\tau^{\ast}T_{i_1}}{x^{\tilde{\gamma}}} = (\zeta +z)^{j_1} {{\tilde b}}_{i_1,j_1} + \frac{1}{x^{{{\tilde {{\gamma}}}}}}\left(\sum_{j=1}^{j_1-1} x^{\tilde{r}_{i_1,j}} (\zeta +z)^j {{\tilde b}}_{i_1,j} + b_{i_1,0} \right) + x R(x,y,z).$$ Since the ${{\tilde b}}_{i_1,j}$ and $b_{i_1,0}$ are independent of $z$, $d(b) \leq j_1$, so that $\rho(b)<\rho(a)$. Subcase I.1.3. Neither $\tau^(v^d)$ nor any $\tau^(u^{r_{ij}}v^{j})$ generates $\tau^\mathcal{I}$.* Then $\tau^(u^{\beta})$ generates $\tau^{\ast}\mathcal{I}$. In this case, $$\tau^T_{i_0} = x^{\tilde{\gamma}} \left( y + x R(x,y,z)\right),$$ so that $\rho(b)= 0 <\rho(a)$. Subcase I.2. $b$ is a 2-point of $\widetilde{D}$: There are adapted coordinates $(\pmb{x},y)=(x_1,x_2,y)$ at $b$ such that $\widetilde{D} = (x_1x_2=0)$ and $\tau$ is given by $$u = \pmb{x}^{\pmb{\lambda}_1}, \quad v = \pmb{x}^{\pmb{\lambda}_2}, \quad w=y,$$ where $\pmb{\lambda}_1,\pmb{\lambda}_2$ are $\mathbb{Q}$-linearly independent. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, $$\label{eq:I.2} \tau^{\ast}T_i = \pmb{x}^{\pmb{\tilde{r}}_d} \overline{T}_{i} + \sum_{j=1}^{d-1} \pmb{x}^{\pmb{\tilde{r}}_{ij}} {{\tilde b}}_{ij} + b_{i0},$$ for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{i,j}$, where ${{\tilde b}}_{ij} = \tau^{{\tilde a}}_{ij}$, $\overline{T}_i = \tau^\widetilde{T}_i$ and $b_{i0}= \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}} y$, for some $\pmb{\tilde{\beta}}$. Again all exponents can be computed explicitly from the $\pmb{\lambda}_i$ and the original exponents. In particular, since $\pmb{\lambda}_1$, $\pmb{\lambda}_2$ are linearly independent, the multi-indices $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{i,j}$ (for fixed $i$) and $\pmb{\tilde{\beta}}$ are all distinct. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^\mathcal{I}$. Then becomes $$\begin{aligned} \sigma_1 & = \pmb{x}^{\pmb{\tilde{\alpha}}}, \\ \sigma_i & = g_i(\pmb{x}) + \pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}} \frac{\tau^T_i}{\pmb{x}^{\pmb{\tilde{\gamma}}}}, \end{aligned} \label{eq:LocalI2prelim}$$ for some $\pmb{\tilde{\alpha}}$, $\pmb{\tilde{\delta}}$. Note that $\pmb{x}^{\pmb{\tilde{\alpha}}}$ and $\pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. We again consider three further subcases depending on the generator of $\tau^\mathcal{I}$. Subcase I.2.1. $\tau^(v^d)$ generates $\tau^\mathcal{I}$. Then $$\tau^T_2 = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( \overline{T}_2 + R(\pmb{x},y) \right),$$ where $R(0,y) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= \delta \pmb{\lambda}_1 + d \pmb{\lambda}_2$ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$; therefore, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$ and $\pmb{\tilde{\alpha}}$ are linearly independent. By , $\rho(b) = 0 < \rho(a)$. Subcase I.2.2. $\tau^(v^d)$ does not generate $\tau^\mathcal{I}$, but $\tau^(u^{r_{ij}}v^{j})$ generates $\tau^\mathcal{I}$, for some $(i,j)$.* Let $(i_1,j_1)$ denote such $(i,j)$ with maximal $j$. Then $$\tau^T_{i_1} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( b_{i_1,j_1} + R(\pmb{x},y) \right),$$ where $R(0,y) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= (\delta+r_{i_1,j_1}) \pmb{\lambda}_1 +j_1 \pmb{\lambda}_2$ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$; therefore, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$ and $\pmb{\tilde{\alpha}}$ are linearly independent. By , $\rho(b) = 0 < \rho(a)$. Subcase I.2.3. $\tau^(u^{\beta})$ is the only generator of $\tau^\mathcal{I}$.* Then $$\tau^T_{i_0} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( y + R(\pmb{x},y) \right),$$ where $R(0,y) = 0$, so that $d(b) = 1< d(a)$ and $\rho(b)<\rho(a)$. Case II. $a$ is a non-generic 1-point.* Then $D = (uvw = 0)$. We consider three cases depending on whether $b$ is a 1-, 2- or 3-point of $\widetilde{D}$. Subcase II.1. $b$ is a 1-point of $\widetilde{D}$. We follow the steps of Subcase I.1. There are adapted coordinates $(x,y,z)$ at $b$, where $\widetilde{D} = (x=0)$ and $\tau$ is given by $$u = x^{\lambda_1},\quad w = x^{\lambda_2}(\eta+y) ,\quad v=x^{\lambda_3}(\zeta+z).$$ where $\eta \neq 0$, $\zeta \neq 0$. Again $\widetilde{E} = \widetilde{D}$ at $b$, since ${\mathrm{supp}\,}\widetilde{E} = V(\tau^(u))$. By $\eqref{eq:weier}$ and $\eqref{eq:norm1}$, we again have formulas , for suitable $\tilde{r}_d$, $\tilde{r}_{ij}$, where now ${{\tilde b}}_{ij} = {{\tilde b}}_{ij}(x,y) = \tau^{{\tilde a}}_{ij}$ times a unit, $\overline{T}_i = \tau^\widetilde{T}_i$ and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = x^{\tilde{\beta}} (\eta +y)$, for some $\tilde{\beta}$. Let $x^{\tilde{\gamma}}$ be a generator of $\tau^\mathcal{I}$. Again ${{\sigma}}$ has the form , for suitable $\tilde{\alpha}$, $\tilde{\delta}$, and we consider three further subcases depending on the generator of $\tau^\mathcal{I}$. Subcase II.1.1. $\tau^(v^d)$ generates $\tau^\mathcal{I}$. In this case, we can repeat Subcase I.1.1 word-for-word. Subcase II.1.2. $\tau^(v^d)$ does not generate $\tau^\mathcal{I}$, but some $\tau^(u^{r_{ij}}w^{s_{ij}}v^{j})$ generates $\tau^\mathcal{I}$. Then we can repeat Subcase I.1.2 word-for-word. Subcase II.1.3. $\tau^(u^{\beta}w)$ is the only generator of $\tau^\mathcal{I}$. Then $$\tau^T_{i_0} = x^{\tilde{\gamma}} \left((\eta + y) + x R(x,y,z)\right),$$ so that $\rho(b)= 0 <\rho(a)$. Subcase II.2. $b$ is a $2$-point of $\widetilde{D}$.* Then there are two possibilities: 1. There are adapted coordinates $(\pmb{x},z) = (x_1,x_2,z)$ at $b$ such that $\widetilde{D} = (x_1x_2=0)$ and $\tau$ is given by $$u = \pmb{x}^{\pmb{\lambda}_1},\quad w= \pmb{x}^{\pmb{\lambda}_2},\quad v = \pmb{x}^{\pmb{\lambda}_3}(\zeta+z), $$ where $\pmb{\lambda}_1, \pmb{\lambda}_2$ are $\mathbb{Q}$-linearly independent and $\zeta\neq 0$. 2. There are adapted coordinates $(\pmb{x},y) = (x_1,x_2,y)$ such that $\widetilde{D} = (x_1x_2=0)$ and $\tau$ is given by $$u = \pmb{x}^{\pmb{\lambda}_1}, \quad w= \pmb{x}^{\pmb{\lambda}_2}(\eta+y),\quad v = \pmb{x}^{\pmb{\lambda}_3}, $$ where $\pmb{\lambda}_1, \pmb{\lambda}_3$ are linearly independent, $\pmb{\lambda}_1, \pmb{\lambda}_2$ are linearly dependent, and $\eta\neq 0$. We have to consider both (a) and (b). Subcase II.2(a). By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, $$\label{eq:II.2(a)} \tau^{\ast}T_i = \pmb{x}^{\pmb{\tilde{r}}_d} (\zeta+z)^d \overline{T}_i + \sum_{j=1}^{d-1} \pmb{x}^{\pmb{\tilde{r}}_{ij}} (\zeta+z)^j {{\tilde b}}_{ij} + b_{i0},$$ for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = {{\tilde b}}_{ij}({\pmb{x}}) = \tau^{\ast}{{\tilde a}}_{ij}$, $\overline{T}_2 = \tau^\widetilde{T}_i$, and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}}$, for some $\pmb{\tilde{\beta}}$. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^{\ast}\mathcal{I}$. Then ${{\sigma}}$ takes the form , for suitable $\pmb{\tilde{\alpha}},\, \pmb{\tilde{\delta}}$. Note that $\pmb{x}^{\pmb{\tilde{\alpha}}}, \pmb{x}^{\pmb{\tilde{\delta}}}$ and $\pmb{x}^{\pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. As before, we consider three further subcases depending on which monomial generator of $\mathcal{I}$ pulls back to a generator of $\tau^{\ast}\mathcal{I}$. Subcase II.2.1(a). $\tau^(v^d)$ generates $\tau^\mathcal{I}$. This subcase is similar to I.1.1 and II.1.1. We have $$\frac{\tau^{\ast}T_2}{\pmb{x}^{\pmb{\tilde{\gamma}}}} = (z+\zeta)^d \overline{T}_2 + \frac{1}{\pmb{x}^{\pmb{\tilde{\gamma}}}}\left(\sum_{j=1}^{d-2} \pmb{x}^{ \pmb{\widetilde{r}}_{2j}} (z+\zeta)^j {{\tilde b}}_{2j} + b_{20}\right),$$ where $\zeta \neq 0$, $\overline{T}_2$ is a unit and ${{\tilde b}}_{2j},\, b_{20}$ are independent of $z$. It follows that $d(b) < d(a)$, so $\rho(b)<\rho(a)$. Subcase II.2.2(a). $\tau^(v^d)$ does not generate $\tau^\mathcal{I}$, but $\tau^(u^{r_{i_1,j_1}}w^{s_{i_1,j_1}}v^{j_1})$ (with maximal $j_1$) generates $\tau^\mathcal{I}$. As in I.1.2 and II.1.2, $$\frac{\tau^{\ast}T_{i_1}}{\pmb{x}^{\pmb{\tilde{\gamma}}}} = (z+\zeta)^{j_1} {{\tilde b}}_{i_1,j_1} + \frac{1}{\pmb{x}^{\pmb{\tilde{\gamma}}}}\left(\sum_{j=1}^{j_1-1} \pmb{x}^{\pmb{\widetilde{r}}_{i_1,j}} (z+\zeta)^j {{\tilde b}}_{i_1,j} + b_{i_1,0} \right) + \pmb{x} R(\pmb{x},z),$$ where the ${{\tilde b}}_{i_1,j}$ and $b_{i_1,0}$ are independent of $z$. Therefore, $d(b) \leq j_1$, so that $\rho(b)<\rho(a)$. Subcase II.2.3(a). $\tau^(u^{\beta}w)$ is the only generator of $\tau^\mathcal{I}$. This is similar to I.2.1 or I.2.2. We have $$\tau^{\ast}T_{i_0} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( 1 + R(\pmb{x},z)\right),$$ where $R(0,z) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}} = (\delta+\beta) \pmb{\lambda}_1 + \pmb{\lambda}_2$ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$, so that $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$ and $\pmb{\tilde{\alpha}}$ are linearly independent. Since $\pmb{x}^{\pmb{\tilde{\gamma}}}$ is supported in $\widetilde{E}$, $\rho(b) = 0 < \rho(a)$. Subcase II.2(b). We follow the steps of Subcase I.2. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we have formulas , for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = \tau^{\ast}{{\tilde a}}_{ij}$ times a unit, $\overline{T}_i = \tau^{\ast}\widetilde{T}_i$ and $b_{i0}= \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}}(\eta + y)$ for suitable $\tilde{\beta}$. For fixed $i$, since $\pmb{\lambda}_1,\,\pmb{\lambda}_3$ are linearly independent, the exponents $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$ and $\pmb{\tilde{\beta}}$ are distinct. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ denote a generator of $\tau^\mathcal{I}$. Then takes the form , for suitable $\pmb{\tilde{\alpha}},\, \pmb{\tilde{\delta}}$. Note that $\pmb{x}^{\pmb{\tilde{\alpha}}}$ and $\pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. We consider three further subcases II.2.1(b), II.2.2(b) and II.2.3(b) analogous to I.2.1, I.2.2 and I.2.3, respectively, in each of which we can argue word-for-word as in the latter. Subcase II.3. $b$ is a $3$-point of $\widetilde{D}$. We follow the steps of Subcase I.2. There are adapted coordinates $\pmb{x}=(x_1,x_2,x_3)$ at $b$, such that $\widetilde{D} = (x_1x_2x_3=0)$ and $\tau$ is given by $$u = \pmb{x}^{\pmb{\lambda}_1},\quad w=\pmb{x}^{\pmb{\lambda}_2} ,\quad v = \pmb{x}^{\pmb{\lambda}_3},$$ where $\pmb{\lambda}_1,\pmb{\lambda}_2,\pmb{\lambda}_3$ are $\mathbb{Q}$-linearly independent. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we again have formulas (here of course $\pmb{x}=(x_1,x_2,x_3)$), for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = \tau^{\ast}{{\tilde a}}_{ij}$, $\overline{T}_i = \tau^{\ast}\widetilde{T}_i$ and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}}$ for suitable $\pmb{\tilde{\beta}}$. For fixed $i$, since $\pmb{\lambda}_1,\, \pmb{\lambda}_2,\, \pmb{\lambda}_3$ are $\mathbb{Q}$-linearly independent, the expoments $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$ and $\pmb{\tilde{\beta}}$ are distinct. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^\mathcal{I}$. Then equation takes the form , for suitable $\pmb{\tilde{\alpha}},\, \pmb{\tilde{\delta}}$, where $\pmb{x}^{\pmb{\tilde{\alpha}}}$, $\pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. We consider three subcases as before. Subcase II.3.1. $\tau^{\ast}(v^d) = \pmb{x}^{d \pmb{\lambda}_2}$ generates $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}T_2 = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( \overline{T}_2 + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= \delta \pmb{\lambda}_1 + d \pmb{\lambda}_3$ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$, so that $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$, $\pmb{\tilde{\alpha}}$ are linearly independent. By , $\rho(b) = 0 < \rho(a)$. Subcase II.3.2. $\tau^{\ast}(v^d)$ does not generate $\tau^{\ast}\mathcal{I}$, but $\tau^{\ast}(u^{r_{i_1,j_1}}w^{s_{i_1,j_1}}v^{j_1})$ (with maximal $j_1$) generates $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}T_{i_1} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( b_{i_1,j_1} + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= (\delta+r_{i_1,j_1}) \pmb{\lambda}_1 +s_{i_1,j_1} \pmb{\lambda}_2 + j_1 \pmb{\lambda}_3$ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$, so that $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$, $\pmb{\tilde{\alpha}}$ are linearly independent. By , $\rho(b) = 0 < \rho(a)$. Subcase II.3.3. $\tau^{\ast}(u^{\beta})$ is the only generator of $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}T_{i_0} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( 1 + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= (\delta+\beta) \pmb{\lambda}_1 + \pmb{\lambda}_2 $ and $\pmb{\tilde{\alpha}} = \alpha \pmb{\lambda}_1$, so that $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$, $\pmb{\tilde{\alpha}}$ are linearly independent. By , $\rho(b) = 0 < \rho(a)$. Case III. $a$ is a 2-point. Then $D:= (u_1u_1v = 0)$. Since $\tau$ is combinatorial in respect to $D$, we consider three cases depending on whether $b$ is a 1-, 2- or 3-point of $\widetilde{D}$: Subcase III.1. $b$ is a 1-point of $\widetilde{D}$. We follow the steps of Subcase I.1. There are adapted coordinates $(x,y,z)$ at $b$, where $\widetilde{D} = (x=0)$ and $\tau$ is given by $$u_1 = x^{\lambda_1}(\eta+y)^{\alpha_2},\quad u_2 = x^{\lambda_2}(\eta+y)^{-\alpha_1} ,\quad v=x^{\lambda_3}(\zeta+z),$$ where $\eta \neq 0$, $\zeta \neq 0$. Clearly, $\widetilde{E} = \widetilde{D}$ at $b$, since ${\mathrm{supp}\,}\widetilde{E} = V(\tau^{\ast}\pmb{u}^{\pmb{\alpha}})$. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we have formulas , for suitable $\tilde{r}_d$, $\tilde{r}_{ij}$, where ${{\tilde b}}_{ij} = {{\tilde b}}_{ij}(x,y) = \tau^{\ast}{{\tilde a}}_{ij}$ times a unit, $\overline{T}_i = \tau^{\ast}\widetilde{T}_i$, and $b_{i0} = \tau^a_{i0}$. Note that $\tau^{\ast}(\pmb{u}^{\pmb{\delta}}a_{i_0,0}) = x^{\tilde{\beta}+\tilde{\delta}} (\eta +y)^{\tilde{\epsilon}}$ for appropriate $\tilde{\delta}$, $\tilde{\beta}$ and $\tilde{{{\epsilon}}}$, where $\tilde{{{\epsilon}}}\neq 0$ since $\tilde{{{\epsilon}}} = (\delta_2+\beta_2) \alpha_1 - (\delta_1 + \beta_1) \alpha_2$ and $\pmb{\alpha}$, $\pmb{\delta}+\pmb{\beta}$ are $\mathbb{Q}$-linearly independent. Let $x^{\tilde{\gamma}}$ denote a generator of $\tau^\mathcal{I}$. Then ${{\sigma}}$ has the form , and we consider three further subcases III.1.1, III.1.2 and III.1.3 as before, depending on which monomial generator of ${{\mathcal I}}$ pulls back to a generator of $\tau^{\ast}\mathcal{I}$. Subcases III.1.1 and III.1.2 parallel I.1.1 and I.1.2 (respectively), word-for-word. Subcase III.1.3. $\tau^{\ast}\pmb{u}^{\pmb{\beta}}$ is the only generator of $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}(\pmb{u}^{\pmb{\delta}}T_{i_0}) = x^{\tilde{\gamma}+\tilde{\delta}} \left((\eta + y)^{\tilde{{{\epsilon}}}} + x R(x,y,z)\right);$$ therefore, $\rho(b)= 0 <\rho(a)$. Subcase III.2. $b$ is a 2-point of $\widetilde{D}$. Then there are two possibilities: 1. There are adapted coordinates $(\pmb{x},z) = (x_1,x_2,z)$ at $b$ such that $\widetilde{D} = (x_1x_2=0)$ and $\tau$ is given by $$u_1 = \pmb{x}^{\pmb{\lambda}_1},\quad u_2= \pmb{x}^{\pmb{\lambda}_2},\quad v = \pmb{x}^{\pmb{\lambda}_3}(\zeta+z),$$ where $\pmb{\lambda}_1, \pmb{\lambda}_2$ are linearly independent and $\zeta\neq 0$. In this case, $\widetilde{E} = \widetilde{D}$. 2. There are adapted coordinates $(\pmb{x},y) = (x_1,x_2,y)$ at $b$ such that $\widetilde{D} = (x_1x_2=0)$ and $\tau$ is given by $$u_1 = \pmb{x}^{\pmb{\lambda}_1}(\eta+y)^{\alpha_2}, \quad u_2= \pmb{x}^{\pmb{\lambda}_2}(\eta+y)^{-\alpha_1},\quad v = \pmb{x}^{\pmb{\lambda}_3},$$ where $\pmb{\lambda}_1,\, \pmb{\lambda}_3$ are linearly independent, $\pmb{\lambda}_1,\, \pmb{\lambda}_2$ are linearly dependent, and $\eta\neq 0$. We again consider both (a) and (b). Subcase III.2(a). By equations $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we have formulas , for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = {{\tilde b}}_{ij}({\pmb{x}}) = \tau^{\ast}{{\tilde a}}_{i,j}$, $\overline{T}_i = \tau^{\ast}\widetilde{T}_i$ and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}}$ for suitable $\pmb{\tilde{\beta}}$. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^{\ast}\mathcal{I}$. Then takes the form , for suitable $\pmb{\tilde{\alpha}}$, $\pmb{\tilde{\delta}}$. We again consider three further subcases III.2.1(a), III.2.2(a) and III.2.3(a), depending on which monomial generator of $\mathcal{I}$ pulls back to a generator of $\tau^{\ast}\mathcal{I}$. In each subcase, we can follow the corresponding subcase of II.2(a) essentially word-for-word. Subcase III.2(b).* We follow the steps of I.2. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we have formulas , for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = \tau^{\ast}{{\tilde a}}_{ij}$ times a unit, $\overline{T}_i = \tau^{\ast}\widetilde{T}_i$ and $b_{i0} = \tau^a_{i0}$. Note that $\tau^{\ast}(\pmb{u}^{\pmb{\delta}}a_{i_0,0}) = \pmb{x}^{\pmb{\tilde{\delta}}+\pmb{\tilde{\beta}}}(\eta + y)^{\tilde{{{\epsilon}}}}$, suitable $\pmb{\tilde{\delta}}$, $\pmb{\tilde{\beta}}$ and $\tilde{{{\epsilon}}}$, where $\tilde{{{\epsilon}}} \neq 0$ since $\tilde{{{\epsilon}}} = \alpha_2(\beta_1+\delta_1)-\alpha_1(\delta_2+\delta_1)$ and $\pmb{\alpha}$, $\pmb{\delta}+\pmb{\beta}$ are linearly independent. Moreover, for each fixed $i$, since $\pmb{\lambda}_1$, $\pmb{\lambda}_3$ are linearly independent, the exponents $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$ and $\pmb{\tilde{\beta}}$ are distinct. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^\mathcal{I}$. Then takes the form , for suitable $\pmb{\tilde{\alpha}}$, $\pmb{\tilde{\delta}}$, and $\pmb{x}^{\pmb{\tilde{\alpha}}}$, $\pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. As usual, we consider three subcases. The first two, III.2.1(b) and III.2.2(b), parallel I.2.1 and I.2.2 (respectively). Subcase III.2.3(b). $\tau^{\ast}(\pmb{u}^{\pmb{\beta}})$ is the only generator of $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}\pmb{u}^{\pmb{\delta}}T_{i_0} = \pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}} \left( (\eta+ y)^{\tilde{{{\epsilon}}}} + R(\pmb{x},y) \right),$$ where $R(0,y) = 0$. It follows that $\rho(b) = 0 <\rho(a)$. Subcase III.3. $b$ is a 3-point of $\widetilde{D}$. We can again follow the steps of Subcase I.2. There are adapted coordinates $\pmb{x}=(x_1,x_2,x_3)$ at $b$, such that $\widetilde{D} = (x_1x_2x_3=0)$ and $\tau$ is given by: $$u_1 = \pmb{x}^{\pmb{\lambda}_1},\quad u_2=\pmb{x}^{\pmb{\lambda}_2} ,\quad v = \pmb{x}^{\pmb{\lambda}_3},$$ where $\pmb{\lambda}_1,\, \pmb{\lambda}_2,\, \pmb{\lambda}_3$ are linearly independent. By $\eqref{eq:weier}$, $\eqref{eq:norm1}$, we have formulas , for suitable $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$, where ${{\tilde b}}_{ij} = \tau^{\ast}{{\tilde a}}_{ij}$, $\overline{T}_i = \tau^{\ast}\tilde{T}_i$ and $b_{i0} = \tau^a_{i0}$; in particular, $b_{i_0,0} = \pmb{x}^{\pmb{\tilde{\beta}}}$ for suitable $\pmb{\tilde{\beta}}$. For each fixed $i$, since $\pmb{\lambda}_1$, $\pmb{\lambda}_2$, $\pmb{\lambda}_3$ are linearly independent, the exponents $\pmb{\tilde{r}}_d$, $\pmb{\tilde{r}}_{ij}$ and $\pmb{\tilde{\beta}}$ are distinct. Let $\pmb{x}^{\pmb{\tilde{\gamma}}}$ be a generator of $\tau^\mathcal{I}$. Then takes the form , for suitable $\pmb{\tilde{\alpha}}$, $\pmb{\tilde{\delta}}$; moreover, $\pmb{x}^{\pmb{\tilde{\alpha}}}$, $\pmb{x}^{\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}}$ are supported in $\widetilde{E}$. We consider three subcases, as usual. Subcase III.3.1. $\tau^{\ast}(v^d)$ generates $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}T_2 = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( \bar{T}_2 + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= \delta_1 \pmb{\lambda}_1 +\delta_2 \pmb{\lambda}_2 + d \pmb{\lambda}_3$ and $\pmb{\tilde{\alpha}} = \alpha_1 \pmb{\lambda}_1+\alpha_2 \pmb{\lambda}_2$; therefore, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$ and $\pmb{\tilde{\alpha}}$ are linearly independent, and it follows that $\rho(b) = 0 < \rho(a)$. Subcase III.3.2. $\tau^{\ast}(v^d)$ does not generate $\tau^{\ast}\mathcal{I}$, but $\tau^{\ast}(\pmb{u}^{\pmb{r}_{i_1,j_1}}v^{j_1})$ (with maximal $j_1$) generates $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}T_{i_1} = \pmb{x}^{\pmb{\tilde{\gamma}}} \left( b_{i_1,j_1} + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= (\delta_1+r_{i_1,j_1,1}) \pmb{\lambda}_1 +(\delta_2+r_{i_1,j_1,2}) \pmb{\lambda}_2 + j_1 \pmb{\lambda}_3$ and $\pmb{\tilde{\alpha}} = \alpha_1 \pmb{\lambda}_1+\alpha_2 \pmb{\lambda}_2$; therefore, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}$ and $\pmb{\tilde{\alpha}}$ are linearly independent, and it follows that $\rho(b) = 0 < \rho(a)$. Subcase III.3.3. $\tau^{\ast}(\pmb{u}^{\pmb{\beta}})$ is the only generator of $\tau^{\ast}\mathcal{I}$. Then $$\tau^{\ast}\pmb{u}^{\pmb{\delta}}T_{i_0} = \pmb{x}^{\pmb{\tilde{\delta}}+\pmb{\tilde{\gamma}}} \left( 1 + R(\pmb{x}) \right),$$ where $R(0) = 0$. Moreover, $\pmb{\tilde{\delta}} + \pmb{\tilde{\gamma}}= (\delta_1+\beta_1) \pmb{\lambda}_1 + (\delta_2+\beta_2)\pmb{\lambda}_2 $ and $\pmb{\tilde{\alpha}} = \alpha_1 \pmb{\lambda}_1+\alpha_2 \pmb{\lambda}_2$. 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--- abstract: \| In analyzing the point-to-point wireless channel, insights about two qualitatively different operating regimes—bandwidth- and power-limited—have proven indispensable in the design of good communication schemes. In this paper, we propose a new scaling law formulation for wireless networks that allows us to develop a theory that is analogous to the point-to-point case. We identify fundamental operating regimes of wireless networks and derive architectural guidelines for the design of optimal schemes. Our analysis shows that in a given wireless network with arbitrary size, area, power, bandwidth, etc., there are three parameters of importance: the short-distance SNR, the long-distance SNR, and the power path loss exponent of the environment. Depending on these parameters we identify four qualitatively different regimes. One of these regimes is especially interesting since it is fundamentally a consequence of the heterogeneous nature of links in a network and does not occur in the point-to-point case; the network capacity is [both ]{} power and bandwidth limited. This regime has thus far remained hidden due to the limitations of the existing formulation. Existing schemes, either multihop transmission or hierarchical cooperation, fail to achieve capacity in this regime; we propose a new hybrid scheme that achieves capacity. author: - 'Ayfer [Ö]{}zg[ü]{}r, Ramesh Johari, , David Tse, and Olivier Lévêque, [^1] [^2] [^3] [^4]' title: Information Theoretic Operating Regimes of Large Wireless Networks --- Ad hoc Wireless Networks, Distributed MIMO, Hierarchical Cooperation, Multihopping, Operating Regimes, Scaling Laws. Introduction ============ The classic capacity formula $C = W \log_2 (1+ P_r/N_0W)$ bits/s of a point-to-point AWGN channel with bandwidth $W$ Hz, received power $P_r$ Watts, and white noise with power spectral density $N_0/2$ Watts/Hz plays a central role in communication system design. The formula not only quantifies exactly the performance limit of communication in terms of system parameters, but perhaps more importantly also identifies two fundamentally different operating regimes. In the power-limited (or low SNR) regime, where $\SNR:= P_r/N_0 W \ll 0$ dB, the capacity is approximately linear in the power and the performance depends critically on the power available but not so much on the bandwidth. In the bandwidth-limited (or high SNR) regime, where $\SNR \gg 0$ dB, the capacity is approximately linear in the bandwidth and the performance depends critically on the bandwidth but not so much on the power. The regime is determined by the interplay between the amount of power and degrees of freedom available. The design of good communication schemes is primarily driven by the parameter regime one is in. Can analogous operating regimes be identified for ad hoc wireless networks, with multiple source and destination pairs and nodes relaying information for each other? To address this question, we are confronted with several problems. First, we have no exact formula for the capacity of networks, even in the simplest case of a single source-destination pair plus one relay. Second, unlike in the point-to-point case, there is no single received SNR parameter in a network. The channels between nodes closer together can be in the high SNR regime while those between nodes farther away can be in the low SNR regime. One approach to get around the first problem is through the scaling law formulation. Pioneered by Gupta and Kumar [@GK00], this approach seeks not the exact capacity of the network but only how it [scales]{} with the number of nodes in the network and the number of source-destination pairs. The capacity scaling turns out to depend critically on how the area of the network scales with the number of nodes. Two network models have been considered in the literature. In [dense]{} networks [@GK00; @AS06; @OLT07], the area is fixed while the density of the nodes increases linearly with the number of nodes. In [extended]{} networks [@XK04; @JVK04; @LT05; @XK06; @AJV06; @OLP06; @OLT07], the area grows linearly with the number of nodes while the density is fixed. For a given path loss exponent, the area of the network determines the amount of power that can be transferred across the network and so these different scalings couple the power transferred and the number of nodes in different ways. There are two significant limitations in using the existing scaling law results to identify fundamental operating regimes of ad hoc networks. First, the degrees of freedom available in a network depend on the number of nodes in addition to the the amount of bandwidth available. By [a priori]{} coupling the power transferred in the network with the number of nodes in specific ways, the existing formulations may be missing out on much of the interesting parameter space. Second, neither dense nor extended networks allow us to model the common scenario where the channels between different node pairs can be in different SNR regimes. More concretely, let us interpret a channel to be in high SNR in a large network if the SNR goes to infinity with $n$, and in low SNR if the SNR goes to zero with $n$.[^5] Then it can be readily verified that in dense networks, the channels between all node pairs are in the high SNR regime, while in extended networks, the channels between all node pairs are in the low SNR regime. In this paper, we consider a generalization that allows us to overcome these two limitations of the existing formulation. Instead of considering a fixed area or a fixed density, we let the area of the network scale like $n^\nu$ where $\nu$ can take on any real value. Dense networks correspond to $\nu =0$ and extended networks correspond to $\nu =1$. By analyzing the problem for all possible values of $\nu$, we are now considering all possible interplay between power and degrees of freedom. Note that in networks where $\nu$ is strictly between $0$ and $1$, channels between nodes that are far away will be at low SNR while nodes that are closer by will be at high SNR. Indeed, the distance between nearest neighbors is of the order of $\sqrt{A/n} = n^{(\nu-1)/2}$ and, assuming a path loss exponent of $\alpha$, the received SNR of the transmitted signal from the nearest neighbor scales like $n^{\alpha(1-\nu)/2}$, growing with $n$. On the other hand, the received SNR of the transmitted signal from the [farthest]{} nodes scales like $(\sqrt{A})^{-\alpha} = n^{-\alpha \nu/2}$, going to zero with $n$. Note that scaling the area by $n^{\nu}$ is completely equivalent to scaling the nearest neighbor $\SNR$ as $n^\beta$, where $\beta := \alpha(1-\nu)/2$. Since $\SNR$ is a physically more relevant parameter in designing communication systems, we will formulate the problem as scaling directly the nearest neighbor $\SNR$. The main result of this paper is as follows. Consider $2n$ nodes randomly located in an area $2A$ such that the received SNR for a transmission over the typical nearest neighbor distance of $\sqrt{A/n}$ is $\SNR_s:= n^\beta$. The path loss exponent is $\alpha \ge 2$. Each transmission goes through an independent uniform phase rotation. There are $n$ source and destination pairs, randomly chosen, each demanding the same rate. Let $C_n(\alpha,\beta)$ denote the total capacity of the network, which is the highest achievable sum rate, in bits/s/Hz and its scaling exponent be defined as, $$\label{def:scexp} e(\alpha, \beta) := \lim_{n \rightarrow \infty} \frac{\log C_n(\alpha,\beta)}{\log n}.$$ The following theorem is the main result of this paper. \[thm1\] The scaling exponent $e(\alpha, \beta)$ of the total capacity $C_n(\alpha,\beta)$ is given by $$e(\alpha,\beta) = \left \{ \begin{array}{ll} 1 & \beta \geq \alpha/2 -1 \\ 2-\alpha/2 + \beta & \beta < \alpha/2 - 1 \mbox{ and } 2 \le \alpha \le 3 \\ 1/2 + \beta & \beta \le 0 \mbox{ and } \alpha > 3\\ 1/2 + \beta/(\alpha-2) & 0 < \beta < \alpha/2 - 1 \mbox{ and } \alpha > 3.\label{eq:main}\end{array} \right .$$ Note that dense networks correspond to $\beta = \alpha/2$, with an exponent $e(\alpha,\alpha/2) = 1$ (first case), and extended networks correspond to $\beta = 0$, with an exponent equal to: $$e(\alpha,0) = \left \{ \begin{array}{ll} 2 - \alpha/2 & 2 \le \alpha \le 3 \\ 1/2 & \alpha > 3 \end{array} \right .$$ (second and third cases respectively). These special cases are the main results of [@OLT07]. Observe that in the general case the scaling exponent $e(\alpha,\beta)$ depends on the path loss exponent $\alpha$ and the nearest neighbor $\SNR$ exponent $\beta$ [separately]{}, so the general result cannot be obtained by a simple re-scaling of distances in the dense or extended model. To interpret the general result (\[eq:main\]) and to compare it to the point-to-point scenario, let us re-express the result in terms of system quantities. Recall that $\SNR_s$ is the SNR over the smallest scale in the network, which is the typical nearest neighbor distance. Thus, $$\label{SNRs} \SNR_s = n^\beta = \frac{P_r}{N_0 W},$$ where $P_r$ is the received power from a node at the typical nearest neighbor distance $\sqrt{A/n}$ and $W$ Hz is the channel bandwidth. Let us also define the SNR over the largest scale in the network, the diameter $\sqrt{A}$, to be $$\label{SNRl} \SNR_l:=n\,\frac{n^{-\alpha/2}P_r}{N_0 W}=n^{1-\alpha/2+\beta},$$ where $n^{-\alpha/2}P_r$ is the received power from a node at distance diameter of the network. The result (\[eq:main\]) can be used to give the following approximation to the total capacity $C$, in bits/s: [^6] $$\label{eq:approx} C \approx \left \{ \hspace{-0.2cm}\begin{array}{ll} n W & \SNR_l \gg 0 \mbox { dB} \\ n^{2-\alpha/2} P_r/N_0 & \SNR_l \ll 0 \mbox { dB and } 2\le\alpha \le 3 \\ \sqrt{n} P_r/N_0 & \SNR_s \ll 0 \mbox{ dB and } \alpha > 3\\ \sqrt{n} W^{\frac{\alpha -3}{\alpha -2}} (P_r/N_0)^{\frac{1}{\alpha -2}} & \SNR_l \ll 0 \mbox{ dB},\, \SNR_s \gg 0 \mbox{ dB}\\ & \mbox {and } \alpha > 3. \end{array} \right .$$ Note two immediate observations in (\[eq:approx\]). First, there are two SNR parameters of interest in networks, the short and the long distance SNR’s, as opposed to the point-to-point case where there is a single SNR parameter. Second, the most natural way to measure the long-distance SNR in networks is not the SNR of a pair separated by a distance equal to the diameter of the network, but it is $n$ times this quantity as defined in (\[SNRl\]). Note that there are order $n$ nodes in total located at a diameter distance to any given node in the network, hence $n$ times the SNR between farthest nodes is the total SNR that can be transferred to this node across this large scale. On the other hand a node has only a constant number of nearest neighbors, and hence the short-distance SNR in (\[SNRs\]) is simply the SNR between a nearest neighbor pair. The four regimes in (\[eq:approx\]) are shown in Figure \[fig:phase\_diagram\]. In Regime-I, the performance is achieved by hierarchical cooperation and long range MIMO transmission, the scheme introduced in [@OLT07]. At the highest level of hierarchy, clusters of size almost order $n$ communicate via MIMO, at distance the diameter of the network. The quantity $\SNR_l$ corresponds to the total received SNR at a node during these MIMO transmissions. Since this quantity is larger than $0$ dB, the long range MIMO transmissions, and hence the performance of the network, are in the bandwidth limited regime, with performance roughly linear in the bandwidth $W$. The performance is linear in the number of nodes, implying that interference limitation is removed by cooperation, at least as far as scaling is concerned. Performance in this regime is qualitatively the same as that in dense networks. In all the other regimes, the total long-range received SNR is less than $0$ dB. Hence we are power-limited and the transfer of power becomes important in determining performance. In Regime-II, i.e., when $\alpha \le 3$, signal power decays slowly with distance, and the total power transfer is maximized by long-range MIMO transmission. This performance can be achieved by bursty hierarchical cooperation with long-range MIMO, much like in extended networks. When $\alpha > 3$, signal power decays fast with distance, and the transfer of power is maximized by short-range communications. If the nearest-neighbor $\SNR \ll 0$ dB (Regime-III), these transmissions are in the power-efficient regime and this power gain translates linearly into capacity, so nearest-neighbor multihop is optimal. This is indeed the case in extended networks, and hence nearest-neighbor multihop is optimal for extended networks when $\alpha > 3$. The most interesting case is the fourth regime, when $\alpha > 3$ and $0 < \beta < \alpha/2 -1$. This is the case when $\SNR_s \gg 0 $ dB, so nearest-neighbor transmissions are bandwidth-limited and not power-efficient in translating the power gain into capacity gain. There is the potential of increasing throughput by spatially multiplexing transmission via cooperation within clusters of nodes and performing distributed MIMO. Yet, the clusters cannot be as large as the size of the network since power attenuates rapidly for $\alpha > 3$. Indeed, it turns out that the optimal scheme in this regime is to cooperate hierarchically within clusters of an intermediate size, perform MIMO transmission between adjacent clusters and then multihop across several clusters to get to the final destination. (See Figure \[fig:mimo\_multihop\]). The optimal cluster size is chosen such that the received SNR in the MIMO transmission is at $0$ dB. Any smaller cluster size results in power inefficiency. Any larger cluster size reduces the amount of power transfer because of the attenuation. Note that the two extremes of this architecture are precisely the traditional multihop scheme, where the cluster size is $1$ and the number of hops is $\sqrt{n}$, and the long-range cooperative scheme, where the cluster size is of order $n$ and the number of hops is $1$. Note also that because short-range links are bandwidth-limited and long-range links are power-limited, the network capacity is [both]{} bandwidth and power-limited. Thus the capacity is sensitive to both the amount of bandwidth and the amount of power available. This regime is fundamentally a consequence of the heterogeneous nature of links in a network and does not occur in point-to-point links, nor in dense or extended networks. The organization of the paper is as follows. In the following section we present our model in more detail. Section \[sec:ub\] derives a tight upper bound on the scaling exponent in (\[def:scexp\]). Section \[sec:ach\] introduces schemes that achieve the upper bound presented in the previous section. The two sections together prove our main result in Theorem \[thm1\]. Section \[sec:conc\] contains our conclusions. Model ===== There are $2n$ nodes uniformly and independently distributed in a rectangle of area $2\sqrt{A}\times\sqrt{A}$. Half of the nodes are sources and the other half are destinations. The sources and destinations are randomly paired up one-to-one without any consideration on node locations. Each source has the same traffic rate $R$ in bits/s/Hz to send to its destination node and a common average transmit power budget of $P$ Watts. The total throughput of the system is $T = n R$. We assume that communication takes place over a flat channel of bandwidth $W$ Hz around a carrier frequency of $f_c$, $f_c \gg W$. The complex baseband-equivalent channel gain between node $i$ and node $k$ at time $m$ is given by: $$\label{eq:ch_model} H_{ik}[m] = \sqrt{G}\,r_{ik}^{-\alpha/2} \exp(j \theta_{ik}[m])$$ where $r_{ik}$ is the distance between the nodes, $\theta_{ik}[m]$ is the random phase at time $m$, uniformly distributed in $[0,2\pi]$ and $\{\theta_{ik}[m]; 1\leq i\leq 2n, 1\leq k\leq 2n\}$ is a collection of independent identically distributed random processes. The $\theta_{ik}[m]$’s and the $r_{ik}$’s are also assumed to be independent. The parameters $G$ and $\alpha \ge 2$ are assumed to be constants; $\alpha$ is called the power path loss exponent. The path-loss model is based on the standard far-field assumption: we assume that the distance $r_{ik}$ is much larger than the carrier wavelength $\lambda_c$. When the distances are comparable or shorter than the carrier wavelength, the simple path-loss model obviously does not hold anymore as path loss can potentially become path “gain”. Moreover, the phases $\theta_{ik}[m]$ depend on the distance between the nodes modulo the carrier wavelength and they can only be modeled as completely random and independent of the actual positions of the nodes if the nodes’ separation is large enough. Indeed, a recent result [@FMM08] showed that, without making an a priori assumption of i.i.d. phases, the degrees of freedom are limited by the diameter of the network (normalized by the carrier wavelength $\lambda_c$). This is a [spatial]{} limitation and holds regardless of how many communicating nodes there are in the network. This result suggests that as long as the number of nodes $n$ is smaller than this normalized diameter, the spatial degrees of freedom limitation does not kick in, the degrees of freedom are still limited by the number of nodes, and the i.i.d. phase model is still reasonable. For example, in a network with diameter $1$ km and carrier frequency $3$ GHz, the number of nodes should be of the order of $10^4$ or less for the i.i.d. phase model to be valid. While the present paper deals exclusively with the standard i.i.d. random phase model, it would be interesting to apply our new scaling law formulation to incorporate regimes where there is a spatial degrees of freedom limitation as well. Note that the channel is random, depending on the location of the users and the phases. The locations are assumed to be fixed over the duration of the communication. The phases are assumed to vary in a stationary ergodic manner (fast fading). We assume that the phases $\{\theta_{ik}[m]\}$ are known in a casual manner at all the nodes in the network. The signal received by node $i$ at time $m$ is given by $$Y_i[m]=\sum_{k\neq i}H_{ik}[m]X_k[m]+Z_i[m]$$ where $X_k[m]$ is the signal sent by node $k$ at time $m$ and $Z_i[m]$ is white circularly symmetric Gaussian noise of variance $N_0$ per symbol. Cutset Upper Bound {#sec:ub} ================== We consider a cut dividing the network area into two equal halves. We are interested in upper bounding the sum of the rates of communications $T_{L\rightarrow R}$ passing through the cut from left to right. These communications with source nodes located on the left and destination nodes located on the right half domain are depicted in bold lines in Fig. \[fig:cutset\]. Since the S-D pairs in the network are formed uniformly at random, $T_{L\rightarrow R}$ is equal to $1/4$’th of the total throughput $T$ w.h.p.[^7] The maximum achievable $T_{L\rightarrow R}$ in bits/s/Hz is bounded above by the capacity of the MIMO channel between all nodes $S$ located to the left of the cut and all nodes $D$ located to the right. Under the fast fading assumption, we have $$T_{L\rightarrow R} \leq \hspace{-0.3cm}\max_{\substack{Q(H) \geq 0 \\ \EE(Q_{kk}(H)) \leq P, \, \forall k\in S}} \hspace{-0.5cm}\EE \left(\log \det(I +\frac{1}{N_0W} H Q(H) H^) \right) \label{cutset}$$ where $$H_{ik} = \frac{\sqrt{G} \; e^{j \, \theta_{ik}}}{r_{ik}^{\alpha/2}}, \quad k\in S, i\in D.$$ The mapping $Q(\cdot)$ is from the set of possible channel realizations $H$ to the set of positive semi-definite transmit covariance matrices. The diagonal element $Q_{kk}(H)$ corresponds to the power allocated to the $k$th node for channel state $H$. Let us simplify notation by introducing $$\label{SNR_0} \SNR_s:=\frac{GP}{N_0W(A/n)^{\alpha/2}}$$ which can be interpreted as the average SNR between nearest neighbor nodes since $\sqrt{A/n}$ is the typical nearest neighbor distance in the network. Let us also rescale the distances in the network by this nearest neighbor distance, defining $$\hat{r}_{ik}:=\frac{1}{\sqrt{A/n}}\, r_{ik} \et \hat{H}_{ik} := \frac{ e^{j \, \theta_{ik}}}{\hat{r}_{ik}^{\alpha/2}}. \label{dis_ext}$$ Note that the first transformation rescales space and maps our original network of area $2\sqrt{A}\times \sqrt{A}$ to a network of area $2\sqrt{n}\times \sqrt{n}$, referred to as an extended network in the literature. Consequently, the matrix $\hat{H}$ defined in terms of the rescaled distances relates to such an extended network with area $2n$. We can rewrite (\[cutset\]) in terms of these new variables as $$T_{L\rightarrow R} \leq \hspace{-0.3cm}\max_{\substack{Q(\hat{H}) \geq 0 \\ \EE(Q_{kk}(\hat{H})) \leq 1, \, \forall k\in S}} \hspace{-0.5cm}\EE \left( \log \det(I + \SNR_{s}\,\hat{H} Q(\hat{H}) \hat{H}^) \right). \label{cutset_simp}$$ In order to upper bound (\[cutset\_simp\]), we will use an approach similar to the one developed in [@OLT07 Sec.V-B] for analyzing the capacity scaling of extended networks. Note that although due the rescaling in (\[dis\_ext\]), $\hat{H}$ in (\[cutset\_simp\]) governs an extended network, the problem in (\[cutset\_simp\]) is not equivalent to the classical extended setup since here we do not necessarily assume $\SNR_{s}=1$. Indeed, we want to keep full generality and avoid such arbitrary assumptions on $\SNR_{s}$ in the current paper. Formally, we are interested in characterizing the whole regime $\SNR_{s}=n^\beta$ where $\beta$ can be any real number. One way to upper bound (\[cutset\_simp\]) is through upper bounding the capacity by the total received SNR, formally using the relation $$\label{up:tr} \log \det(I + \SNR_{s}\,\hat{H} Q(\hat{H}) \hat{H}^)\leq \Tr\left(\SNR_{s}\,\hat{H} Q(\hat{H}) \hat{H}^* \right).$$ The upper bound is tight only if the SNR received by each right-hand side node (each diagonal entry of the matrix $\SNR_{s}\,\hat{H} Q(\hat{H}) \hat{H}^$) is small. (Note that the relation in (\[up:tr\]) relies on the inequality $\log(1+x)\leq x$ which is only tight if $x$ is small.) In the extended setup, where $\SNR_{s}=1$, the network is highly power-limited and the received SNR is small, that is decays to zero with increasing $n$, for every right-hand side node. Using (\[up:tr\]) yields a tight upper bound in that case. However, in the general case $\SNR_{s}$ can be arbitrarily large which can result in high received SNR for certain right-hand side nodes that are located close to the cut or even for all nodes, depending on how large exactly $\SNR_{s}$ is. Hence, before using (\[up:tr\]) we need to distinguish between those right-hand side nodes that receive high SNR and those that have poor power connections to the left-hand side. For the sake of simplicity in presentation, we assume in this section that there is a rectangular region located immediately to the right of the cut that is cleared of nodes. Formally, we assume that the set of nodes $E=\{i\in D: 0\leq \hat{x}_i\leq 1\}$ is empty, where $\hat{x}_i$ denotes the horizontal coordinate of the rescaled position $\hat{r}_i=(\hat{x}_i,\hat{y}_i)$ of node $i$. In fact, w.h.p this property does not hold in a random realization of the network. However, making this assumption allows us to exhibit the central ideas of the discussion in a simpler manner. The extension of the analysis to the general case (without this particular assumption) is given in Appendix \[sec:app2\]. Let $V_D$ denote the set of nodes located on a rectangular strip immediately to the right of the empty region $E$. Formally, $V_D=\{i\in D: 1\leq \hat{x}_i\leq \hat{w}\}$ where $1\leq \hat{w}\leq \sqrt{n}$ and $\hat{w}-1$ is the rescaled width of the rectangular strip $V_D$. See Fig. \[fig:cutset\]. We would like to tune $\hat{w}$ so that $V_D$ contains the right-hand side nodes with high received SNR; i.e., those with received SNR larger than a threshold, say $1$. Note however that we do not yet know the covariance matrix $Q$ of the transmissions from the left-hand side nodes, which is to be determined from the maximization problem in (\[cutset\_simp\]). Thus, we cannot compute the received SNR of a right-hand side node. For the purpose of determining $V_D$ however, let us arbitrarily look at the case when $Q$ is the identity matrix and define the received SNR of a right-hand side node $i\in D$ when left-hand side nodes are transmitting [independent]{} signals at full power to be $$\label{SNRi} \SNR_i := \frac{P}{N_0W}\,\sum_{k\in S} \|H_{ik}\|^2\,=\,\SNR_{s}\,\sum_{k\in S} \|\hat{H}_{ik}\|^2=\,\SNR_{s}\,\hat{d}_i.$$ where we have defined $$\label{di} \hat{d}_i:=\sum_{k\in S} \|\hat{H}_{ik}\|^2.$$ Later, we will see that this arbitrary choice of identity covariance matrix is indeed a reasonable one. A good approximation for $\hat{d}_i$ is $$\label{di_app} \hat{d}_i\approx \hat{x}_i^{2-\alpha}$$ where $\hat{x}_i$ denotes the rescaled horizontal coordinate of node $i$. (See [@OLT07 Lemma 5.4].) Using (\[SNRi\]) and (\[di\_app\]), we can identify three different regimes and specify $\hat{w}$ accordingly: - If $\SNR_{s}\,\geq\,n^{\alpha/2-1}$, then $\SNR_i\gtrsim 1,\,\forall i\in D$. Thus, let us choose $\hat{w}=\sqrt{n}$ or equivalently $V_D=D$. - If $\SNR_{s}\,<\,1$, then $\SNR_i\lesssim 1,\,\forall i\in D$. Thus, let us choose $\hat{w}=1$ or equivalently $V_D=\emptyset$.[^8] - If $1\leq\SNR_{s}\,<\,n^{\alpha/2-1}$, then let us choose $$\hat{w}=\left \{ \begin{array}{ll} \sqrt{n} & \text{if}\qquad \alpha =2 \\ \SNR_{s}^\frac{1}{\alpha-2} & \text{if}\qquad\alpha > 2 \end{array} \right .$$ so that we ensure $\SNR_i \gtrsim 1, \,\forall i\in V_D$. We now would like to break the information transfer from the left-half domain $S$ to the right-half domain $D$ in (\[cutset\_simp\]) into two terms. The first term governs the information transfer from $S$ to $V_D$. The second term governs the information transfer from $S$ to the remaining nodes on the right-half domain, i.e., $D\setminus V_D$. Recall that the characteristic of the nodes $V_D$ is that they have good power connections to the left-hand side, that is the information transfer from $S$ to $V_D$ is not limited in terms of power but can be limited in degrees of freedom. Thus, it is reasonable to bound the rate of this first information transfer by the cardinality of the set $V_D$ rather than the total received SNR. On the other hand, the remaining nodes $D\setminus V_D$ have poor power connections to the left-half domain and the information transfer to these nodes is limited in power, hence using (\[up:tr\]) is tight. Formally, we proceed by applying the generalized Hadamard’s inequality which yields $$\begin{aligned} \log \det(I &+ \SNR_s\hat{H} Q(\hat{H}) \hat{H}^)\\ & \leq \log\det(I + \SNR_s\hat{H}_1 Q(\hat{H}) \hat{H}_1^)\\ &\hspace{0.5cm}+\log\det(I + \SNR_s\hat{H}_2 Q(\hat{H}) \hat{H}_2^)\end{aligned}$$ where $\hat{H}_1$ and $\hat{H}_2$ are obtained by partitioning the original matrix $\hat{H}$: $\hat{H}_1$ is the rectangular matrix with entries $\hat{H}_{ik}, k\in S, i\in V_D$ and $\hat{H}_2$ is the rectangular matrix with entries $\hat{H}_{ik}, k\in S, i\in D\setminus V_D$. In turn, (\[cutset\_simp\]) is bounded above by $$\begin{aligned} \lefteqn{\hspace{-0.4cm}T_{L\rightarrow R} \leq \hspace{-0.7cm}\max_{\substack{Q(\hat{H}_1) \geq 0 \\ \EE(Q_{kk}(\hat{H}_1)) \leq 1, \, \forall k\in S}} \hspace{-0.7cm}\EE \left( \log \det(I + \SNR_s\hat{H}_1 Q(\hat{H}_1) \hat{H}_1^) \right) \nonumber}\\ & + & \hspace{-1.2cm} \max_{\substack{Q(\hat{H}_2) \geq 0 \\ \EE(Q_{kk}(\hat{H}_2)) \leq 1, \, \forall k\in S}} \hspace{-0.7cm}\EE \left( \log \det(I + \SNR_s\hat{H}_2 Q(\hat{H}_2) \hat{H}_2^) \right)\label{cutset3}\end{aligned}$$ The first term in (\[cutset3\]) can be bounded by considering the sum of the capacities of the individual MISO channels between nodes in $S$ and each node in $V_D$, $$\begin{aligned} \lefteqn{\hspace{-0.5cm}\max_{\substack{Q(\hat{H}_1) \geq 0 \\ \EE(Q_{kk}(\hat{H}_1)) \leq 1, \, \forall k\in S}} \hspace{-0.5cm}\EE \left( \log \det(I + \SNR_s\hat{H}_1 Q(\hat{H}_1) \hat{H}_1^) \right)}\nonumber\\ & \leq & \hspace{-0.3cm}\sum_{i\in V_D}\log(1+n\,\SNR_{s}\sum_{k\in S} \|\hat{H}_{ik}\|^2)\nonumber\\ & \leq & \hspace{-0.3cm}(\hat{w}-1)\,\sqrt{n}\log n\,\log(1+n^{1+\alpha(1/2+\delta)}\,\SNR_{s})\label{ub1}\end{aligned}$$ w.h.p. for any $\delta>0$, where we use the fact that for any covariance matrix $Q$ of the transmissions from the left-hand side, the SNR received by each node $i\in V_D$ is smaller than $n\,\SNR_{s} \hat{d}_i$ and $\hat{d}_i\leq n^{\alpha(1/2+\delta)}$ since the rescaled minimal separation between any two nodes in the network is larger than $\frac{1}{n^{1/2+\delta}}$ w.h.p. for any $\delta>0$. The number of nodes in $V_D$ is upper bounded by $(\hat{w}-1)\,\sqrt{n}\log n$ w.h.p. The second term in (\[cutset3\]) is the capacity of the MIMO channel between nodes in $S$ and nodes in $D\setminus V_D$. Using (\[up:tr\]), we get $$\begin{aligned} \max_{\substack{Q(\hat{H}_2) \geq 0 \\ \EE(Q_{kk}(\hat{H}_2)) \leq 1,\, \forall k\in S}} &\hspace{-0.5cm}\EE \left( \log \det(I + \SNR_s\hat{H}_2 Q(\hat{H}_2) \hat{H}_2^) \right)\nonumber\\ &\le \hspace{-0.7cm}\max_{\substack{Q(\hat{H}_2) \geq 0 \\ \EE(Q_{kk}(\hat{H}_2)) \leq 1,\, \forall k\in S}} \hspace{-0.7cm}\EE \left(\Tr\left(\SNR_s\hat{H}_2 Q(\hat{H}_2) \hat{H}_2^\right)\right)\nonumber\\ &\le n^\epsilon\,\SNR_{tot}\label{ub2}\end{aligned}$$ for any $\epsilon>0$ w.h.p, where $$\label{eq:snrtot} \SNR_{tot}=\sum_{i\in D\setminus V_D} \SNR_i= \SNR_s \sum_{i\in D\setminus V_D}\hat{d}_i.$$ Inequality (\[ub2\]) is proved in [@OLT07 Lemma 5.2] and is precisely showing that an identity covariance matrix is good enough for maximizing the power transfer from the left-hand side. Recall that $\SNR_i$ in (\[eq:snrtot\]) has already been defined in (\[SNRi\]) to be the received SNR of node $i$ under [independent]{} signalling from the left-hand side. Note that (\[eq:snrtot\]) is equal to zero when $D\setminus V_D=\emptyset$ or equivalently when $\hat{w}=\sqrt{n}$. If $D\setminus V_D\neq\emptyset$, the last summation in (\[eq:snrtot\]) can be approximated with an integral since nodes are uniformly distributed on the network area. Using also (\[di\_app\]), it is easy to derive the following approximation for the summation $$\sum_{i\in D\setminus V_D}\hat{d}_i\,\approx\,\int_0^{\sqrt{n}}\int_{\hat{w}}^{\sqrt{n}} \hat{x}^{(2-\alpha)} d\hat{x}\,d\hat{y}.$$ Here we state a precise result that can be found by straight forward modifications of the analysis in [@OLT07]. If $\hat{w}\neq \sqrt{n}$, we have $$\SNR_{tot}\leq \left \{ \begin{array}{ll} K_1\,\SNR_{s}\, n\, (\log n)^3 & \alpha =2 \vspace{0.1cm}\\ K_1\,\SNR_{s}\, n^{2-\alpha/2}(\log n)^2 &2 < \alpha <3 \vspace{0.1cm}\\ K_1\,\SNR_{s}\,\sqrt{n} \, (\log n)^3 & \alpha=3\vspace{0.1cm} \\ K_1\,\SNR_{s}\,\hat{w}^{3-\alpha}\,\sqrt{n} \,(\log n)^2 & \alpha > 3. \end{array} \right.\label{SNRtot}$$ where $K_1>0$ is a constant independent of $SNR_s$ and $n$. Combining the upper bounds (\[ub1\]) and (\[ub2\]) together with our choices for $\hat{w}$ specified earlier, one can get an upper bound on $T_{L\rightarrow R}$ in terms of $\SNR_s$ and $n$. Here, we state the final result in terms of scaling exponents: Let us define $$\beta:=\lim_{n\to\infty}\frac{\log \SNR_s}{\log n}$$ and $$e(\alpha,\beta):=\lim_{n\to\infty}\frac{\log T}{\log n}\,=\,\lim_{n\to\infty}\frac{\log T_{L\rightarrow R}}{\log n}.\label{expo}$$ We have, $$e(\alpha, \beta) \leq \left \{ \begin{array}{ll} 1 & \beta \ge \alpha/2-1 \\ 2-\alpha/2+\beta & \beta < \alpha/2-1\,\, \text{and}\,\, 2\leq \alpha <3\\ 1/2+\beta & \beta \leq 0 \,\,\text{and}\,\, \alpha \ge 3\\ 1/2+\beta/(\alpha-2) & 0 < \beta < \alpha/2-1 \,\,\text{and}\,\, \alpha \ge 3 \end{array} \right.\label{upp_expo}$$ where we identify four different operating regimes depending on $\alpha$ and $\beta$. Note that in the first regime the upper bound (\[ub1\]) is active with $\hat{w}=\sqrt{n}$ (or equivalently $V_D=D$) while (\[ub2\]) is zero. The capacity of the network is limited by the degrees of freedom in an $n\times n$ MIMO transmission between the left and the right hand side nodes. In the second regime, (\[ub2\]), with the corresponding upper bound being the second line in (\[SNRtot\]), yields a larger contribution than (\[ub1\]). The capacity is limited by the total received SNR in a MIMO transmission between the left-hand side nodes and $D\setminus V_D$. Note that this total received SNR is equal (in order) to the power transferred in a MIMO transmission between two groups of $n$ nodes separated by a distance of the order of the diameter of the network, i.e., $n^2\times (\sqrt{n})^{-\alpha}\times \SNR_s$. In the third regime, (\[ub2\]) is active with $\hat{w}=1$ (or equivalently $V_D=\emptyset$) while (\[ub1\]) is zero. The corresponding upper bound is the fourth line in (\[SNRtot\]). Note that this is where we make use of the assumption that there are no nodes located at rescaled distance smaller than $1$ to the cut. Due to this assumption, the choice $\hat{w}=1$ vanishes the upper bound (\[ub1\]) and simultaneously yields $K_1 \SNR_s \sqrt{n} (\log n)^2$ in the last line in (\[SNRtot\]). If there were nodes closer than rescaled distance $1$ to the cut, we would need to choose $\hat{w}<1$ to vanish the contribution from (\[ub1\]) which would yield a larger value for the term $K_1 \SNR_s \hat{w}^{3-\alpha}\sqrt{n} (\log n)^2$. The capacity in the third regime is still limited by the total SNR received by nodes in $D\setminus V_D$ ($=D$ now) but in this case the total is dominated by the SNR transferred between the nearest nodes to the cut, i.e., $\sqrt{n}$ pairs separated by the nearest neighbor distance, yielding $\sqrt{n}\times\SNR_s$. The most interesting regime is the fourth one. Both (\[ub1\]) and (\[ub2\]) with the choice $\hat{w}=\SNR_s^{\frac{1}{\alpha-2}}$ yield the same contribution. Note that (\[ub1\]) upper bounds the information transfer to $V_D$, the set of nodes that have bandwidth-limited connections to the left-hand side. This information transfer is limited in degrees of freedom. On the other hand, (\[ub2\]) upper bounds the information transfer to $D\setminus V_D$, the set of nodes that have power-limited connections to the left-hand side. This second information transfer is power-limited. Eventually in this regime, the network capacity is both limited in degrees of freedom and power, since increasing the bandwidth increases the first term (\[ub1\]) and increasing the power increases the second term (\[ub2\]). Order Optimal Communication Schemes {#sec:ach} =================================== In this section, we search for communication schemes whose performance meets the upper bound derived in the previous section. The derivation of the upper bound already provides hints on what these schemes can be: In the first two regimes, the capacity of the network is limited by the degrees of freedom and received SNR respectively, in a network wide MIMO transmission. The recently proposed hierarchical cooperation scheme in [@OLT07] is based on such MIMO transmissions so it is a natural candidate for optimality in these regimes. In the third regime, the information transfer between the two halves of the network is limited by the power transferred between the closest nodes to the cut. This observation suggests the following idea: if the objective is to transfer information from the left-half network to the right-half, then it is enough to employ only those pairs that are located closest to the cut and separated by the nearest neighbor distance. (The rest of the nodes in the network can undertake simultaneous transmissions suggesting the idea of spatial reuse.) In other words, the upper bound derivation suggests that efficient transmissions in this regime are the point-to-point transmissions between nearest neighbors. Indeed, this is how the well-known multihop scheme transfers power across the network so the multihop scheme arises as a natural candidate for optimality in the third regime. In the derivation of the upper bound for the fourth regime, we have seen that the two terms (\[ub1\]) and (\[ub2\]), governing the information transfer to $V_D$ and $D\setminus V_D$ respectively, yield the same contribution with the particular choice $\hat{w}=\SNR_s^{\frac{1}{\alpha-2}}$. Since the contributions of the two terms are equal (and since we are interested in order here) the derivation of the upper bound suggests the following idea: information can be transferred optimally from the left-half network to the right-half by performing MIMO transmission only between those nodes on both sides of the cut that are located up to $\hat{w}=\SNR_s^{\frac{1}{\alpha-2}}$ rescaled distance to the cut. Note that (\[ub1\]) corresponds to the degrees of freedom in such a MIMO transmission. As in the case of multihop, we can have spatial reuse and allow the rest of the nodes in the network to perform simultaneous transmissions. Thus, the derivation of the upper bound suggests that efficient transmissions in the fourth regime are MIMO transmissions at the scale $\hat{w}=\SNR_s^{\frac{1}{\alpha-2}}$. Combined with the idea of spatial reuse this understanding suggests to transfer information in the network by performing MIMO transmissions at the particular (local) scale of $\hat{w}=\SNR_s^{\frac{1}{\alpha-2}}$ and then multihopping at the global scale. This new scheme is introduced in Section \[sec:ach:new\].[^9] Known Schemes in the Literature {#sec:ach:old} ------------------------------- There are two fundamentally different communication schemes suggested for wireless networks in the literature: The multihop scheme and the hierarchical cooperation scheme. The multihop scheme is based on multihopping packets via nearest neighbor transmissions. Its aggregate throughput is well known to be $$T_{multihop}=\sqrt{n}\,\log\left(1+\frac{\SNR_s}{1+K_2\SNR_s}\right)$$ w.h.p where $\log(1+\frac{\SNR_s}{1+K_2\SNR_s})$ is the throughput achieved in the nearest neighbor transmissions. $K_2\SNR_s$ is the interference from simultaneous transmissions in the network to noise ratio where $K_2>0$ is a constant independent of $n$ and $\SNR_s$. The factor $\sqrt{n}$ is the number of nearest neighbor transmissions that can be parallelized over a given cut. The scaling exponent $e_{multihop}(\alpha, \beta)$ of the multihop scheme (defined analogously to (\[expo\])) is given by $$\label{expo_mh} e_{multihop}(\alpha, \beta)=\left \{ \begin{array}{ll} 1/2 & \beta > 0 \\ 1/2+\beta & \beta \le 0 \end{array} \right .$$ As can be expected, multihop only achieves the upper bound in (\[upp\_expo\]) in the third regime when $\beta\leq 0$ and $\alpha\geq 3$. In other words, when even the nearest neighbor transmissions in the network are power limited and signals attenuate sufficiently fast so that pairs located farther apart than the nearest neighbor distance cannot contribute to the power transfer effectively, the optimal strategy is to confine to nearest neighbor transmissions. The second scheme for wireless networks in [@OLT07] is based on a hierarchical cooperation architecture that performs distributed MIMO transmissions between clusters of nodes. The overhead introduced by the cooperation scheme is small so that the throughput achieved by the distributed MIMO transmissions is not that different (at least in scaling sense) from the throughput of a classical MIMO system where transmit and receive antennas are collocated and can cooperate for free. Indeed, the aggregate throughput achieved by the scheme is almost equal to the rate of a MIMO transmission between two clusters of the size of the network $n$ and separated by a distance equal to the diameter of the network $\sqrt{A}$. More precisely, $$\label{hc} T_{HC}\geq K_3\,n^{1-\epsilon}\,\log\left(1+ n\frac{GP}{N_0W\,(\sqrt{A})^\alpha}\right)$$ for any $\epsilon>0$ and a constant $K_3>0$ w.h.p, where $n^{-\epsilon}$ is the loss in performance due to cooperation overhead. The quantity $n\frac{GP}{N_0W\,(\sqrt{A})^\alpha}$ is the total power received by a node in the receive cluster, when nodes in the transmit cluster are signalling independently at full power. Expressing $T_{HC}$ in terms of $\SNR_s$ in (\[SNR\_0\]), we have $$T_{HC}\geq K_3\,n^{1-\epsilon}\,\log\left(1+ n^{1-\alpha/2}\,\SNR_s\right).$$ Thus, the scaling exponent of hierarchical cooperation is given by $$\label{expo_hier} e_{HC}(\alpha, \beta)=\left \{ \begin{array}{ll} 1 & \beta \geq \alpha/2-1 \\ 2-\alpha/2+\beta & \beta < \alpha/2-1. \end{array} \right .$$ The performance in the second line is achieved by using a bursty version of the hierarchical cooperation scheme, where nodes operate the original scheme only a fraction $\frac{1}{n^{\alpha/2-1}}$ of the total time and stay inactive in the rest to save power. See [@OLT07 Sec. V-A]. We see that hierarchical cooperation meets the upper bound in (\[upp\_expo\]) in the first regime when $\beta \geq \alpha/2-1$, i.e., when power is not a limitation. When power is limited but $2\leq \alpha\leq 3$, bursty hierarchical cooperation can be used to achieve the optimal power transfer. We see that neither multihop nor hierarchical cooperation is able to meet the upper bound in the fourth regime. A New Hybrid Scheme: Cooperate Locally, Multihop Globally {#sec:ach:new} --------------------------------------------------------- Let us divide our network of $2n$ nodes and area $2\sqrt{A}\times\sqrt{A}$ into square cells of area $A_c=\frac{2A}{2n}\,\SNR_s^{1/(\alpha/2-1)}$. Note that $A_c\leq A$, hence this is a valid choice, if $\beta\leq \alpha/2-1$. If also $\beta>0$, each cell contains of the order of $M=\SNR_s^{1/(\alpha/2-1)}$ nodes w.h.p. We transmit the traffic between the source-destination pairs in the network by multihopping from one cell to the next. More precisely let the S-D line associated to an S-D pair be the line connecting its source node to its destination node. Let the packets of this S-D pair be relayed along adjacent cells on its S-D line just like in standard multihop. See Fig \[fig:mimo\_multihop\]. The total traffic through each cell is that due to all S-D lines passing through the cell, which is $O(\sqrt{nM})$. Let us randomly associate each of these $O(\sqrt{nM})$ S-D lines passing through a cell with one of the $M$ nodes in the cell, so that each node is associated with $O(\sqrt{n/M})$ S-D lines. The only rule that we need to respect while doing this association is that if an S-D line starts or ends in a certain cell, then the node associated to the S-D line in this cell should naturally be its respective source or destination node. The nodes associated to an S-D line are those that will decode, temporarily store and forward the packets of this S-D pair during the multihop operation. The following lemma states a key result regarding the rate of transmission between neighboring cells. \[lem:multiHC\] There exists a strategy (based on hierarchical cooperation) that allows each node in the network to relay its packets to their respective destination nodes in the adjacent cells at a rate $$R_{relay}\geq K_4\, n^{-\epsilon}$$ for any $\epsilon>0$ and a constant $K_4>0$. In steady-state operation, the outbound rate of a relay node given in the lemma should be shared between the $O(\sqrt{n/M})$ S-D lines that the relay is responsible for. Hence, the rate per S-D pair is given by $$\label{rate:mhc} R\geq K_4 \sqrt{M}\, n^{-1/2-\epsilon}$$ or equivalently, the aggregate rate achieved by the scheme is $$T_{multihop+HC}\geq K_4\, n^{1/2-\epsilon}\, \SNR_s^{\frac{1}{\alpha-2}}.$$ In terms of the scaling exponent, we have $$e_{multihop+HC}(\alpha,\beta)= 1/2+\beta/(\alpha-2) \hspace{0.3cm}\textrm{if } 0<\beta\leq \alpha/2-1$$ which matches the upper bound (\[upp\_expo\]) in the third regime. Note that considering (\[rate:mhc\]), it is beneficial to choose $M$ as large as possible since it reduces the relaying burden. However, Lemma \[lem:multiHC\] does not hold for any arbitrary $M$. The proof of the lemma reveals a key property regarding our initial choice for $M$ (or $A_c$). [Proof of Lemma \[lem:multiHC\]:]{} Let us concentrate only on two neighboring cells in the network. (Consider for example the two cells highlighted in Fig. \[fig:mimo\_multihop\]): The two neighboring cells together form a network of $2M$ nodes randomly and uniformly distributed on a rectangular area $2\sqrt{A_c}\times\sqrt{A_c}$. Let the $M$ nodes in one of the cells be sources and the $M$ nodes in the other cell be destinations and let these source and destination nodes be paired up randomly to form $M$ S-D pairs. (This traffic will later be used to model the hop between two adjacent cells.) As we have already discussed in (\[hc\]), using hierarchical cooperation one can achieve an aggregate rate $$\begin{aligned} M\, R_{relay}\,&\geq K_3\,M^{1-\epsilon}\,\log\left(1+ M\frac{GP}{N_0W\,(\sqrt{A_c})^\alpha}\right)\\ &=K_3\,M^{1-\epsilon}\,\log\left(1+ M^{1-\alpha/2}\SNR_s\right)\end{aligned}$$ for these $M$ source destination pairs. The second equation is obtained by substituting $A_c=MA/n$ and $\SNR_s=\frac{GP}{N_0W(A/n)^{\alpha/2}}$. Note that if $M^{1-\alpha/2}\SNR_s\geq 1$, then $$\label{rate:mhc2} R_{relay}\,\geq\, K_3\,M^{-\epsilon}\,\geq\, K_3\,n^{-\epsilon}.$$ In other words, $M=\SNR_s^{1/(\alpha/2-1)}$ is the largest cell size one can choose while still maintaining almost constant transmission rate for each of the $M$ S-D pairs. Now let us turn back to our original problem concerning the steady-state operation of the multihop scheme. At each hop, each of the $M$ nodes in a cell needs to relay its packets to one of the four (left, right, up and down) adjacent cells. Since the S-D lines are randomly assigned to the nodes in the cell, there are $M/4$ nodes on the average that want to transmit in each direction. These transmissions can be realized successively using hierarchical cooperation and the relaying rate in (\[rate:mhc2\]) can be achieved in each transmission. On the other hand the TDMA between the four transmissions will reduce the overall relaying rate by a factor of $4$. Indeed, one should also consider a TDMA scheme between the cells allowing only those cells that are sufficiently separated in space to operate simultaneously so that the inter-cell-interference in the network does not degrade the quality of the transmissions significantly. The inter-cell-interference and TDMA will further reduce the rate in (\[rate:mhc2\]) by a constant factor however will not affect the scaling law. Such insights on scheduling and interference are standard by-now and are not central to our analysis on scaling laws. We refer the reader to [@OLT07 Lemma 4.2] for more details.$\square$ Note that the new scheme illustrated in Fig. \[fig:mimo\_multihop\] is a combination of multihop and hierarchical cooperation. Packets are transferred by multihopping on the network level and each hop is realized via distributed MIMO transmissions. Our analysis shows that multihopping and distributed MIMO are two fundamental strategies for wireless networks. However, optimality can only be achieved if these two strategies are combined together appropriately; the optimal combination depends on the SNR level in the network. When $\alpha>3$, we identify three different regimes in wireless networks: the high, low and hybrid SNR regimes. The high SNR regime ($\beta\ge \alpha/2-1$) is the extremal case when even the long-distance SNR in the network is large ($\SNR_l\gg 0$ dB). Distributed MIMO with hierarchical cooperation achieves capacity in this case. In the hybrid SNR regime ($0<\beta\le \alpha/2-1$), the long-distance SNR in the network is low ($\SNR_l\ll 0$ dB), and packets need to be transmitted by multihopping at this scale; while close by pairs are still in the high SNR regime ($\SNR_s\gg 0$ dB) and distributed MIMO provides the optimal information transfer at this smaller scale. The low SNR regime ($\beta\le 0$) is the other extreme when even the short distance SNR is low ($\SNR_s\ll0$ dB). The multihop MIMO scheme reduces to pure multihop in this last case. Conclusion {#sec:conc} ========== Suppose you are asked to design a communication scheme for a particular network with given size, area, power budget, path loss exponent, etc. What would be the efficient strategy to operate this wireless network? In this paper, we answer this question by connecting engineering quantities that can be directly measured in the network to the design of good communication schemes. In a given wireless network, we identify two SNR parameters of importance, the short-distance and the long-distance SNR’s. The short-distance SNR is the SNR between nearest neighbor pairs. The long-distance SNR is the SNR between farthest nodes times the size of the network. If the long-distance SNR is high, then the network is in the bandwidth limited regime. Long-distance communication is feasible and good communication schemes should exploit this feasibility. If the long-distance SNR is low, then the network is power-limited and good communication schemes need to maximize the power transfer across the network. When the power path loss exponent is small so that signals decay slowly, this power transfer is maximized by global cooperation. When the power path loss exponent is large and signals decay fast, the power transfer is maximized by cooperating in smaller scales. The cooperation scale is dictated by the power path loss exponent and the short-distance SNR in the network. The current results in the literature, in particular [@OLT07] that provides the complete picture for the dense and the extended scaling regimes, fail to answer this engineering question because they only address two specific cases that couple the degrees of freedom and power in the network in two very particular ways. The picture is much richer than what can be delineated by these two settings. In that sense, the current paper suggests the abandonment of the existing formulation of wireless networks in terms of dense and extended scaling regimes, a formulation that has been dominant in the literature over the last decade. A better delineation is obtained by treating the power and degrees of freedom available in the network as two independent parameters and studying the interplay between them. Removing the Assumption of an Empty Strip in Section \[sec:ub\] {#sec:app2} =============================================================== While proving the upper bound on network capacity in Section \[sec:ub\], we have considered a vertical cut of the network that divides the network area into two equal halves and assumed that there is an empty rectangular region to the right of this cut, of width equal to the nearest neighbor distance in the network (or of width equal to $1$ in the corresponding rescaled network). With high probability, this assumption does not hold in a random realization of the network. Indeed for any linear cut of the random network, w.h.p. there will be nodes on both sides of the cut that are located at a distance much smaller than the nearest neighbor distance to the cut. In order to prove the result in Section \[sec:ub\] rigorously for random networks, we need to consider a cut that is not necessarily linear but satisfies the property of having no nodes located closer than the nearest neighbor distance to it. Below, we show the existence of such a cut using methods from percolation theory. See [@FDT06] for a more general discussion of applications of percolation theory to wireless networks. \[lem:percol\] For any realization of the random network and a constant $0<c<1/7\sqrt{e}$ independent of $n$ and $A$, w.h.p. there exists a vertical cut of the network area that is not necessarily linear but is located in the middle of the network in a slab not wider than $L=c\sqrt{A/n}\log n$ and is such that there exists no nodes at distance smaller than $\frac{c}{2}\sqrt{A/n}$ to the cut on both sides. See Fig. \[fig:percolation\]. The assumption of an empty region $E$ in Section \[sec:ub\], allowed us to plug in $\hat{w}=1$ in the fourth line of (\[SNRtot\]) and conclude that when the left-hand side nodes $S$ are transmitting independent signals, the total SNR received by all nodes $D$ to the right of the linear cut is bounded above by $$\begin{aligned} \SNR_{tot}&=\sum_{i\in D}\SNR_i\nonumber\\ &\leq \left \{ \begin{array}{ll} K_1\,\SNR_{s}\, n\, (\log n)^3 & \alpha =2 \vspace{0.1cm}\\ K_1\,\SNR_{s}\, n^{2-\alpha/2}(\log n)^2 &2 < \alpha <3 \vspace{0.1cm}\\ K_1\,\SNR_{s}\,\sqrt{n} \, (\log n)^3 & \alpha=3\vspace{0.1cm} \\ K_1\,\SNR_{s}\,\sqrt{n} \,(\log n)^2 & \alpha > 3, \end{array} \right.\label{SNRtot_lcut}\end{aligned}$$ where $\SNR_i$ is defined in (\[SNRi\]). The same result can be proven for the cut given in Lemma \[lem:percol\] without requiring any special assumption. Let $B$ denote the set of nodes located to the right of the cut but inside the rectangular slab mentioned in the lemma. See Figure \[fig:percolation\]. Then $$\label{eq:sumB} \SNR_{tot}=\sum_{i\in B}\SNR_i\,+\,\sum_{i\in D\setminus B}\SNR_i.$$ For any node $i\in B$, an approximate upper bound for $\SNR_i$ is $$\SNR_i\lesssim\SNR_s\int_{0}^{\sqrt{2\pi}}\int_{c}^{\sqrt{n}}\frac{1}{\hat{r}^\alpha} \hat{r}d\hat{r}d\theta,$$ since Lemma \[lem:percol\] guarantees that there are no left-hand side nodes located at rescaled distance smaller than $c$ to a right-hand side node $i$. Moreover, nodes are uniformly distributed on the network area so the summation in (\[SNRi\]) over the left-hand side nodes $S$ can be approximated by an integral. A precise upper bound on $\SNR_i$ can be found using the binning argument in [@OLT07 Lemma 5.2] which yields $$\SNR_i \leq K_1\,\SNR_s\,\log n.$$ Since there are less than $\sqrt{n}\log n$ nodes in $B$ with high probability, the first summation in (\[eq:sumB\]) can be upperbounded by $$\sum_{i\in B}\SNR_i\leq K_1\,\SNR_s\,\sqrt{n}\,(\log n)^2.$$ Note that this contribution is smaller than any of the terms in (\[SNRtot\_lcut\]). The second summation $\sum_{i\in D\setminus B}\SNR_i$ in (\[eq:sumB\]) is equal or smaller in order to (\[SNRtot\_lcut\]) since when the nodes $B$ are removed there is a empty region of width at least $c$ between the nodes $S$ and remaining nodes $D\setminus B$. Hence for the second term in (\[eq:sumB\]), we are back in the situation discussed in Section \[sec:ub\], hence the upperbound (\[SNRtot\_lcut\]) applies. [Proof of Lemma \[lem:percol\]:]{} Let us divide our network of area $2\sqrt{A}\times\sqrt{A}$ into square cells of side length $c\sqrt{A/n}$ where $0<c<1$ is a constant independent of $A$ and $n$. We say that a cell is closed if it contains at least one node and open if it contains no nodes. Since the $2n$ nodes are uniformly and independently distributed on the network area $2A$, the probability that a given cell is closed is upper bounded by the union bound by $$\PP[\textrm{a cell is closed}]\leq c^2.$$ Similarly, the probability that a given set of $m$ cells $\{c_1,\dots,c_m\}$ are simultaneously closed is upper bounded by $$\begin{aligned} \PP[&\{c_1,\dots,c_m\}\textrm{ is closed}]\nonumber\\ &=\PP[c_1\textrm{ is closed}]\times\PP[c_2\textrm{ is closed}\|c_1\textrm{ is closed}]\times\dots\nonumber\\ &\leq c^2\times c^2\dots\times c^2\,=c^{2m}\label{eq:pn}\end{aligned}$$ since by the union bound we have, $$\begin{aligned} \PP[c_{k+1}\textrm{ is closed}&\|c_1,\dots, c_{k}\textrm{ is closed}]\\ &\leq \frac{(c^2A/n)}{A-k(c^2A/n)} (n-k)\\ &\leq\, c^2\end{aligned}$$ when $0<c<1$. Now let us consider a slab of width $c\sqrt{A/n}\log n$ in the middle of the network. Equivalently, this is a rectangle of $\log n\times \sqrt{n}/c$ cells. By choosing $c$ properly, we will show that this slab contains at least one [open path]{} that crosses the network from top to bottom. Such a path is called an open top-bottom crossing. A path is called open if it is composed of neighboring cells that are open, a neighboring cell being one of the four cells located immediately to the top, bottom, left and right of a cell. See Fig. \[fig:percolation\]. On the other hand, we define a closed path in a slightly different manner: A closed path is composed of neighboring cells that are closed but a neighboring cell can now be one of the $8$ cells located immediately at the top, top-left, left, bottom-left, bottom, bottom-right, right, top-right of a cell. See Fig. \[fig:percolation2\]. With these definitions of closed and open paths, we have $$\begin{aligned} \PP[&\textrm{the slab contains an open top-bottom crossing}]\\ &=1-\PP[\textrm{the slab contains a closed left-right crossing}]\end{aligned}$$ where a closed left-right crossing refers to a closed path that connects the left-boundary $\mathscr{L}$ of the slab to its right boundary $\mathscr{R}$. Let $\PP(i\leftrightarrow\mathscr{R})$ denote the probability that there exists a closed path starting from a particular cell $i\in\mathscr{L}$ and ending at the right-boundary. Note that such a path should be at least of length $\log n$ cells. Denoting by $N_i$ the number of closed paths of length $\log n$ that start from the cell $i$, we have $$\PP(i\leftrightarrow\mathscr{R})\leq\PP(N_i\geq 1).$$ By (\[eq:pn\]), a given path of length $\log n$ is closed with probability less than $c^{2\log n}$. By the union bound, we have $$\PP(N_i\geq 1)\leq c^{2\log n} \sigma_i(\log n),$$ where $\sigma_i(\log n)$ denotes the number of distinct, loop-free paths of length $\log n$ starting from $i$. This number is obviously not larger than $ \sigma_i(\log n)\leq 5\times 7^{(\log n-1)}. $ Combining the three inequalities, we have $$\begin{aligned} \PP[&\textrm{the slab contains a closed left-right crossing}]\\ &\leq\sum_{i=1}^{\sqrt{n}/c} \PP(i\leftrightarrow\mathscr{R}) \,\leq\frac{5}{7c}\sqrt{n} (7c^2)^{\log n}. \end{aligned}$$ Choosing $c^2<\frac{1}{7\sqrt{e}}$, the last probability decreases to $0$ as $n$ increases. This concludes the proof of the lemma. $\square$. [xx]{} P. Gupta and P. R. Kumar, The Capacity of Wireless Networks, IEEE Trans. on Information Theory 42 (2), pp.388-404, 2000. S. Aeron, V. Saligrama, Wireless Ad hoc Networks: Strategies and Scaling Laws for the Fixed SNR Regime, IEEE Trans. on Information Theory 53 (6), 2007, 2044 - 2059. L. -L. Xie and P. R. Kumar, A Network Information Theory for Wireless Communications: Scaling Laws and Optimal Operation, IEEE Trans. on Information Theory 50 (5), 2004, 748-767. A. Jovicic, P. Viswanath and S. R. Kulkarni, Upper Bounds to Transport Capacity of Wireless Networks, IEEE Trans. on Information Theory 50 (11), 2004, 2555-2565. O. L[é]{}v[ê]{}que and E. Telatar, Information Theoretic Upper Bounds on the Capacity of Large, Extended Ad-Hoc Wireless Networks, IEEE Trans. on Information Theory 51 (3), 2005, 858- 865. L. -L. Xie, P. R. Kumar, On the Path-Loss Attenuation Regime for Positive Cost and Linear Scaling of Transport Capacity in Wireless Networks, IEEE Trans. on Information Theory, 52 (6), 2006, 2313-2328. S. Ahmad, A. Jovicic and P. Viswanath, Outer Bounds to the Capacity Region of Wireless Networks, IEEE Trans. Inform. Theory, 52 (6), 2006, 2770-2776. A. [Ö]{}zg[ü]{}r, O. L[é]{}v[ê]{}que, E. Preissmann, Scaling laws for one and two-dimensional random wireless networks in the low attenuation regime, IEEE Trans. on Information Theory 53 (10), 2006, 3573-3585. A. [Ö]{}zg[ü]{}r, O. L[é]{}v[ê]{}que, D. Tse, Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad-Hoc Networks, IEEE Trans. on Information Theory 53 (10), 2007, 3549-3572. M. Franceschetti, M.D. Migliore, P. Minero, The capacity of wireless networks: information-theoretic and physical limits, preprint, 2007. U. Niesen, P. Gupta, D. Shah, On Capacity Scaling in Arbitrary Wireless Networks, eprint arXiv: 0711.2745, arxiv.org, 2007. D. Tse and P. Viswanath, [Fundamentals of Wireless Communication]{}, Cambridge University Press, 2005. M. Franceschetti, O. Dousse, D. Tse, P. Thiran, Closing the Gap in the Capacity of Wireless Networks Via Percolation Theory, IEEE Trans. on Information Theory, 53 (3), 2007, 1009 - 1018. [^1]: The work of Ayfer [Ö]{}zg[ü]{}r was supported by Swiss NSF grant Nr 200020-118076. The work of Ramesh Johari was supported by the Defense Advanced Research Projects Agency under the ITMANET program, and the U.S. National Science Foundation under grant 0644114. The work of David Tse was supported by the U.S. National Science Foundation via an ITR grant: “The 3R’s of Spectrum Management: Reuse, Reduce and Recycle”. The material in this paper was presented in part at the IEEE Symposium on Information Theory, Toronto, July 2008. [^2]: A. Özgür and O. Lévêque are with the Ecole Polytechnique Fédérale de Lausanne, Faculté Informatique et Communications, Building INR, Station 14, CH - 1015 Lausanne, Switzerland (e-mails: {ayfer.ozgur,olivier.leveque}@epfl.ch). [^3]: R. Johari is with the Stanford University, Department of Management Science and Engineering, Stanford, CA 94305, USA (e-mail: ramesh.johari@stanford.edu). [^4]: D. Tse is with the University of California at Berkeley, Department of EECS, Berkeley, CA 94720, USA, (e-mail: dtse@eecs.berkeley.edu). [^5]: We interpret a channel in both high and low SNR, if the SNR does not depend on $n$. [^6]: Note that $C=W\,C_n(\alpha,\beta)$. [^7]: with high probability, i.e., with probability going to $1$ as $n$ grows. [^8]: Note that this is when we use the earlier assumption of an empty strip $E$ of width $1$. Without the assumption, we would need to choose $\hat{w}<1$ in this part. [^9]: In a different context, a similar scheme has been suggested recently in an independent work [@NGS07].
--- abstract: 'We present a total of $\sim$45 ksec (3$\times$15 ksec) of XRT observations for three non-magnetic nova-like (NL) Cataclysmic Variables (CVs) (MV Lyr, BZ Cam, V592 Cas) in order to study characteristics of Boundary Layers (BL) in CVs. The nonmagnetic NLs are found mostly in a state of high mass accretion rate ($\ge$1$\times$10$^{-9}$ yr$^{-1}$) and some show occasional low states. Using the XRT data, we find optically thin multiple-temperature cooling flow type emission spectra with X-ray temperatures (kT$_{max}$) of 21-50 keV. These hard X-ray emitting boundary layers diverge from simple isobaric cooling flows indicating X-ray temperatures that are of virial values in the disk. In addition, we detect power law emission components from MV Lyr and BZ Cam and plausibly from V592 Cas which may be a result of the Compton scattering of the optically thin emission from the fast wind outflows in these systems and/or Compton up-scattering of the soft disk photons. The X-ray luminosities of the (multi-temperature) thermal plasma emission in the 0.1-50.0 keV range are (0.9-5.0)$\times$10$^{32}$ erg/sec. The ratio of the X-ray and disk luminosities (calculated from the UV-optical wavelengths) yield an efficiency (L$_{x}$/L$_{disk}$)$\sim$0.01-0.001. Given this non-radiative ratio for the X-ray emitting boundary layers with no significant optically thick blackbody emission in the soft X-rays (consistent with $ROSAT$ observations) together with the high/virial X-ray temperatures, we suggest that high state NL systems may have optically thin BLs merged with ADAF-like flows and/or X-ray coronae. In addition, we note that the axisymmetric bipolar and/or rotation dominated fast wind outflows detected in these three NLs (particularly BZ Cam and V592 Cas) or some other NL may also be explained in the context of ADAF-like BL regions.' author: - Şölen Balman - 'Patrick Godon, Edward M. Sion' title: ' XRT Observations of the Nova-like Cataclysmic Variables MV Lyr, BZ Cam and V592 Cas ' --- Introduction ============ Cataclysmic Variables (CVs) are short period (up to $\sim$2 day) close binary systems in which a white dwarf (WD) accretes matter from a late-type main sequence star filling its Roche lobe (Warner 1995). In non-(or weakly-) magnetic CVs the transferred material forms an accretion disk around the WD and reaches all the way to the stellar surface, as the magnetic field is not strong enough to disrupt the accretion disk. NL systems are a class of CVs usually found in a state of high mass accretion rate, though some NLs are sometimes found in a low state of reduced accretion. NLs have two main subclasses where VY Scl stars exibit high states and occasional low states of optical brightness and the UX UMa stars remain in the high state and low states are not seen (Warner 1995). RW Tri stars are NLs that are eclipsing UX UMa systems (so they’re all high inclination NLs). While all NL variables show emission lines, UX UMa sub-type of NLs also exhibit broad absorption lines in the optical and/or UV wavelengths. In the high states of NLs, the accretion rates are typically a few $\times$10$^{-8}$ yr$^{-1}$ to a few $\times$10$^{-9}$ yr$^{-1}$. Virtually all NLs reside above the period gap with a concentration of them between P$_{orb}$ of 3 and 4 hours (Ritter & Kolb 1998). There is only one NL below the period gap (BK Lyn), this system has been identified with a nova in ancient times from chinese records (see Patterson et al. 2013). The high accretion rates are inferred from luminous disks seen best and modeled in the FUV with steady-state disk models. In addition, bipolar outflows and/or rotationally dominated winds from NLs are detected primarily in the FUV, and P Cygni profiles of the resonance doublet of CIV are almost invariably seen from these collimated outflows in systems with disk inclinations less than 60-70 degrees (Guinan & Sion 1982, Sion 1985). The high accretion rates may be driven in part by irradiation of the donor star if the accreting WD remains hot (e.g., long after a nova event). Existence of hot WDs in NLs in turn leads to irradiation of the donor hence a driving of the high mass transfer by the donor, either by irradiation driving a wind off of the donor or by irradiation causing the Roche-Lobe filling donor star to be bloated. There is another subclass of nonmagnetic CVs called SW Sex stars often listed as part of NLs. These are defined by specific spectroscopic characteristics and have orbital periods between 3-4 hrs (see Rodriguez-Gil et al. 2007 for a review). Not all NLs show these characterstics. Our present paper does not deal with SW Sex stars. Observations of CV disk systems at low mass accretion rate (namely dwarf nova CVs in quiescence) have successfully led to the determination of the temperature and luminosity of the accreting white dwarfs from FUV observations (where it is the dominant emitting component at low $\dot{\rm M}$) and X-ray observations have yielded the temperature and luminosity of the optically thin BLs. The dwarf novae in quiescence (low $\dot{\rm M}$ systems) were observed with recent X-ray telescopes like , , and $Suzaku$(e.g. Szkody et al. 2002, Pandel et al. 2005, Rana et al. 2006, Okada et al. 2008, Ishida et al. 2009, Mukai et al. 2009, Balman et al. 2011) and the results seem in accordance with one-dimensional numerical simulations of the optically thin BLs in standard steady-state disks of DNe in quiescence with low accretion rates (10$^{-10}$-10$^{-12}$ yr$^{-1}$, (Narayan & Popham 1993, Pringle 1981). However, note that a recent study of X-ray variability and inner disk structure of DN in quiescence reveals optically thick disk truncation and plausible formation of hot (coronal) flows in the inner parts of the quiescent DN accretion disks (Balman & Revnivtsev 2012). One of the earliest comprehensive studies on hard X-ray emission from the boundary layers of CVs using the $Einstein$ IPC (0.2-4 keV) indicate that NL systems (5 detected systems) emit hard X-ray emission in this range with luminosities $\le$ a few$\times$10$^{32}$ (Patterson & Raymond 1985). NL systems have been detected in the early epochs of $ROSAT$ Observatory (about 11 NL systems; 0.1-2.4 keV) and optically thick soft X-ray components expected from these high accretion rate systems are not detected (van Teeseling et al. 1996). For these systems a hot optically thin X-ray source is found with plasma temperatures kT$<$ 6 keV. The X-ray luminosities are $<$ a few$\times$10$^{32}$ whereas the UV luminosities are in a range 10$^{31}$-10$^{35}$ . In addition, Greiner (1998) has done a comprehensive analysis of the $ROSAT$ detected NLs (see also Schlegel & Singh 1995) and have argued that $ROSAT$ data are also consistent with blackbody models yielding lower than thermal plasma models. These blackbody temperatures are calculated as (0.25-0.5) keV with very small emitting regions. Later, some NLs were studied with $ASCA$ and are found consistent with double MEKAL models at different temperatures (e.g., TT Ari and KT Aur, Mauche & Mukai 2002), and one with using multiple-temperature plasma and MEKAL models (Pratt et al. 2004). In all these observations the X-ray luminosities are $\le$ afew $\times$10$^{32}$ . Recent advances in UV spectroscopy have revealed that the high mass accretion disks in nova-likes are departing from the standard disk models especially in the inner disk (Puebla et al. 2007), where the boundary layer is, also, located. Puebla et al. (2007) have done a comprehensive UV modeling of accretion disks at high accretion rates in 33 CVs including many NLs and old novae. The UV findings indicate a necessity for improvement by incorporating a component from an extended optically thin region (e.g., wind, corona/chromosphere). This is evident from the strong emission lines and the P Cygni profiles observed in the UV spectra. Disk irradiation by the boundary layer or the central star, and nonstandard temperature profiles would help for better modeling in this band. An additional possibility to explain model discrepancies is disk truncation. Yet another possibility is to modify the viscosity law for example by using constraints on the viscosity following from numerical simulations of the magneto-rotational instability. A first step in improving the disk modeling has been to modify the inner disk to reflect the presence of a hot boundary layer (e.g. for QU Car: Linnell et al. 2008; for MV Lyr: Godon & Sion 2011) where an additional hot optically thick component was also found consistent with the data when modeling in the UV. The Nova-like Systems MV Lyrae, BZ Camelopardalis, and V592 Casiopeiae ====================================================================== In this paper, we present the XRT observations of three NLs, MV Lyr, BZ Cam, and V592 Cas. The important characteristics of the systems are presented in Table 1 and the relevant literature is summerized below. MV Lyr ------ MV Lyrae is a member of VY Scl subclass of NLs, mostly found in a a high state of accretion (similar to the outburst state of DNs but lasting much longer) with occasional drops (short-duration) in brightness. In these low states, the magnitude of MV Lyr drops from $V\approx 12-13$ to $V\approx 16-18$ [@hoa04]. The archival AAVSO data reveals that it stays in a high state for up to $\sim 5$yr, and following that for a period of a few years to $\sim10$yr it starts alternating between high state and low state on a time scale of a few months to a year. MV Lyrae has an orbital period of $P=3.19$hr and a mass ratio of $q = M_{2nd}/M_{wd} =0.4$ [@sch81; @ski95]. The inclination of the system is found in a range $i=10^{\circ}-13^{\circ}$, constrained with the small radial velocity measurements and lack of eclipses [@lin05]. The UV observations of MV Lyr obtained with the [International Ultraviolet Explorer ([IUE]{})]{} during its low state revealed the existence of a hot WD reaching 50,000K [@szk82] or higher [@chi82]. Later, the source was observed in a different low state with [FUSE]{} (in 2002) that lasted about 8 months. The analysis showed that the WD has a temperature of 47,000K, a gravity of $\log{g}=8.25$, a projected rotational velocity of $V_{rot} \sin{i} =200$ km s$^{-1}$, sub-solar abundances of $Z = 0.3 \times Z_{\odot}$, and a distance of $505 \pm 50$pc (see @hoa04). In this low state, the magnitude of MV Lyr was around $V\approx 18$ with a mass accretion rate about $\dot{M} \approx 3 \times 10^{-13}M_{\odot}$ yr$^{-1}$ [@hoa04]. Furthermore, @lin05 studied the different states of MV Lyr and its secondary with the aid of [IUE]{} and HST/[STIS]{} spectra, and found that the mass accretion rate was of the order of of $3 \times 10^{-9}M_{\odot}$ yr$^{-1}$ during the high state. [@god11] analyzed the FUSE spectrum of MV Lyr in its high state and found an accretion rate of about $2 \times 10^{-9}M_{\odot}$ yr$^{-1}$, assuming $i=10^{\circ} \pm 3^{\circ}$ [@sch81; @ski95; @lin05], a 50,000K WD and an extended 100,000K UV emitting region which can originate from an optically thick BL or it can be from an inner disk region irradiated by the boundary layer (i.e., optically thin BL). The Peculiar Nova-like BZ Cam ----------------------------- BZ Cam is a peculiar NL with rather special characteristics among cataclysmic variables. First, it is associated with a bow-shock nebula [@ell84; @kra87; @gre01]; second, as a wind-emitting system it manifests its outflow not only in the FUV resonance lines (as do wind-emitting CVs) but also in the Balmer and He[i]{} lines [@pat96]; third, the wind shows a bipolar nature as a highly unsteady and continuously variable $\sim$ 3000 km s$^{-1}$ supersonic outflow [@hon13]; last, the variability timescales seen in its optical wind-outflow features are much shorter (minutes to hours) than the shortest variability reported in the FUV of any other CV wind [@pat96]. For these reasons, BZ Cam has been the subject of many studies of the time-variability of its line profiles (e.g. [@kra87; @hol92; @woo92; @gri95; @pri00; @hon13]). It is classified as a VY Scl type NL system with a period of 221 min [@tho93; @pat96]. @gar88 [@gre01] present an analysis of its two optical low states. The source has an inclination of about $i=12^{\circ}-40^{\circ}$ [@lad91; @rin98] and a distance of $830\pm160$pc. Its WD mass is unknown but it is thought to contain a $0.3-0.4M_{\odot}$ main-sequence donor star [@luh85]. It is mostly seen at a magnitude of $V=12.0-12.5$ mag and during low state reaches $V=14.3$ mag [@gre01]. It was observed with ROSAT [@van96], IUE [@kra87; @woo90; @woo92; @gri95], HST/GHRS [@pri00] and FUSE [@fro12]. The UV and optical continuum is consistent with a $\sim$12,500 K Kurucz LTE model atmosphere [@pri00] with a small emitting area compared to the surface of a WD. In work presented elsewhere (Godon et al. 2014, in preperation), our analysis of the archival FUSE spectra of BZ Cam reveals a 50,000 K WD. V592 Cas -------- V592 Cas is an UX UMa subtype of Nova-like with an inclination of $i=28^{\circ}$$\pm$10$^{\circ}$ [@hub98], and a period of 2.76 hr [@tay98]. With a reddening of E(B-V)=0.22 [@car89], it has been placed at about 330-360pc [@hub98; @hoa09]. Its WD is about 0.75 M$_{\odot}$ with a temperature of 45,000K [@hoa09] and a mass accretion rate around 1$\times$10$^{-8}$ M$_{\odot}$ yr$^{-1}$ [@tay98; @hoa09]. V592 Cas has a bipolar wind outflow that has an episodic nature, with several events reaching velocities of 5000 km s$^{-1}$ in H$\alpha$ where the optical brightness variations and the strength of the outflow reveals no clear correspondance (Kafka et al. 2009). V592 Cas was observed with IUE [@tay98] and FUSE [@pri04] revealing Doppler shifts of the entire blueward absorption troughs in the ultraviolet resonance lines as the main source of variability characterized with an asymmetric and non-sinusoidal behaviour over the orbital phase. Furthermore, the outflowing wind does not show modulated variation on the superhump periods detected from V592 Cas, but show modulation only on the orbital phase (orbital period). This indicates that neither the precession of the disk nor the precession of the disk tilt (negative and positive superhumps) is affecting the outflow. In addition, V592 Cas reveals a short photometric oscillation of 22 mins [@kat02]. The Standard Star-Disk Boundary Layer ===================================== The standard disk theory [@sha73; @lyn74], predicts that viscous dissipation of energy is instantly radiated locally in the vertical z-direction. The disk total luminosity $L_{disk}$ is half the accretion luminosity $$L_{disk}=\frac{L_{acc}}{2} = \frac{G M_{} {\dot{M}}} {2R_{}},$$ where $G$ is the gravitational constant, $M_$ is the mass of the accreting star, $R_$ its radius and $\dot{M}$ is the mass accretion rate. Each face of the disk radiates $L_{disk}/2$. If the star is not rotating close to breakup (i.e. $\Omega_{} << \Omega_K(R_{})$r), then the theory predicts (e.g. @pri81) that the remaining rotational kinetic energy of the material at the inner edge of the disk (rotating at the local Keplerian speed) is dissipated in a small region as this material lands on the WD which is rotating more slowly at sub-Keplerian speeds. This region is refered as the boundary layer (BL) and up to half the accretion luminosity is liberated in this region. This remaining rotational kinetic energy given-off from the BL ($L_{BL}$) is [@klu87]: $$L_{BL}= L_{disk} \left( 1 - \frac{\Omega_{}}{\Omega_K(R_{})} \right)^2.$$ The BL luminosity is not reduced more than a factor of 1/4 due to rotational effects. At high mass accretion rates ($\dot{\rm M} \approx 10^{-9}-10^{-8}$ M$_{\odot}$ yr$^{-1}$) the BL is largely expected to be optically thick and emits in the soft X-ray band [@pri77; @pri79; @pop95; @god95; @hert13], and at lower mass accretion rates it expected to be optically thin and emit in the hard X-ray ($\sim$ 10 keV) band (Narayan & Popham 1993, Popham 1999). Furthermore, two-dimensional semi-analytical work [@pir04] shows that effects of the centrifugal force and pressure may result in, rings of enhanced brightness above and below the WD equator, called the “spreading layer” [@ino99]. At lower mass accretion rates than $ < 1.6\times10^{-8}$ M$_{\odot}$ yr$^{-1}$, however, @pir04 shows that the spreading is negligible which is the case for the three NL sources in this paper (see Table 1 for the accretion rates). The transition between an optically thin and an optically thick boundary layer does not only depend on the mass accretion rate, it also depends on the mass of the white dwarf, on its rotational velocity, and on the alpha viscosity parameter (disk viscosity $\nu$= $\alpha$c$_s$H where c$_s$ is the sound speed, H is the disk height and $\alpha$ is a free paramter in a range 0-1). Observations and Data ===================== was launched in November 2004 [@geh04] mainly designed to measure position, spectrum and brightness of gamma-ray bursts (GRBs). It has a Burst Alert Telescope (BAT; @bart04) with typically 100 sec reaction time and a field of view (FOV) of 100$\times$60 working in 15-150 keV range using a CdZnTe CCD. The narrow field instruments are the X-ray telescope (XRT; @bur05) and the Ultraviolet-Optical Telescope (UVOT; @rom05). The XRT operates in the 0.2-10 keV range with an 18 arcsec half-power diameter at 1.5 keV using CDDs that are similar to EPIC MOS on . It has an FOV of 23.6$\times$23.6 arcmin and an imaging resolution of 2.4 arcsec per pixel. Regular observations of MV Lyr, BZ Cam and V592 Cas were made with the spacecraft between 2012 June 8 and 2012 December 21 utilizing both the XRT and the UVOT using the UV grism. Table 2 gives the details and log of the observations. In this paper, we report on the analysis of the XRT data of the three NLs obtained in the photon counting (PC) mode for about 15 ksec each. Data were obtained, also, in the windowed timing mode (WT) using short exposures less than 5 min, this mode does not provide spatial resolution. Such short exposure times will not yield adequate S/N for any spectral or timing analysis, thus WT data have not been used for the analysis. The UV grism analysis will be discussed in a subsequent paper. PC mode is a frame transfer operation of an X-ray CCD retaining full imaging and spectroscopic resolution, but the time resolution is only 2.5 sec; the mode is used at low fluxes below 1 mCrab. We find XRT count rates of 0.069(3) c s$^{-1}$ for MV Lyr, 0.051(2) c s$^{-1}$ for V592 Cas, and 0.070(3) c s$^{-1}$ for BZ Cam. We cross checked the states of the sources using the existing AAVSO data and the UVW1 filter magnitudes and count rates, obtained simultaneously with the PC mode data. We find that all the three NLs were in a high state during the observations. The screened and pipeline-processed data (aspect-corrected, bias-subtracted, graded (0-12) and gain-calibrated event lists) are used for the analysis. Latest calibration file on the database (version, 20010101v013) is used, in accordance with the XRT data for the redistribution matrix. The ancillary response files are calculated for each source by merging exposure maps in [XIMAGE]{} created using $xrtexpomap$ task and finally running the task $xrtmkarf$. For light curve and spectrum generation, and further analysis, XSELECT V2.4b and HEASoft 6.13 (see http://heasarc.nasa.gov/lheasoft/) are utilized. In addition, we have used $ROSAT$ archival data of our target sources to derive blackbody temperature upper limits. The data for MV Lyr was obtained on 1992 November 4-8 using the PSPCB detector (20 ksec); OBSID rp300192n00. For BZ Cam, the observation was performed on 1992 September 28-29 using the PSPCB detector (6.0 ksec); OBSID rp300233n00. Finally for V592 Cas, the data was obtained during the RASS (all sky survey; 1990 December 29 to 1991 August 06) using the PSPCC detector (0.6 ksec); OBSID rs930701n00. The three $ROSAT$ data sets were aquired in high states, note that V592 Cas is a UX UMa type of NL which shows only high states. For details of $ROSAT$ data see van Teeseling et al. (1996), Greiner (1998), and Voges et al. (1999). Temporal Analysis ================= XRT light curves are extracted from the data sets of each NL sampling between 0.2-10 keV at 2.7 sec resolution and background subtracted. A circular region of radius 1.5 arc min is used for the photon extraction for both the source and the background (devoid of other contaminating sources). We searched for variations of the X-ray light curves over the orbit of the systems by folding each light curve on the orbital period using 10 phase bins. The results are displayed in Figure 1. For the orbital periods we used the spectroscopic periods derived for the three NLs: (1) For BZ Cam, we used the period and the ephemeris derived from the He I $\lambda$5876 line (T$_0$=2453654.008(2)$+$0.15353(4)$\times$E: Honeycutt et al. 2013); (2) For V592 Cas, we used the ephemeris derived from the He I $\lambda$5876 and $\lambda$6678 line (T$_0$=2450707.866(1)$+$ 0.115063(1)$\times$E: Taylor et al. 1998); (3) For MV Lyr, we used the ephemeris derived from the H$\alpha$ line (T$_0$=2449258.05$+$0.1329(4)$\times$E: Skillman et al. 1995). In Figure 1 each NL shows some variation of the X-ray count rates over the orbit of the system. V592 Cas shows complex behaviour, but looks in-phase with the H$\alpha$ variation (only the first narrow X-ray peak) and possibly out-of-phase with the He I $\lambda$5876 line. Both in BZ Cam and V592 Cas, the He I $\lambda$5876 line and H$\alpha$ line variations are out-of-phase. V592 Cas shows dips close to zero counts per sec and the X-ray variations seem more drastic compared to BZ Cam or MV Lyr. The second maximum of X-rays seem to be right before/at around when the wind strength maximizes (over the orbital phase) as the H$\alpha$ line equivalent widths maximize calculated from the blue shifted P Cygni absorption profiles. Note that for V592 Cas, both the He I $\lambda$5876 line and the H$\alpha$ line have a single peak variation over the orbit, whereas the X-rays show double peaks within the orbit relevant to the bipolar wind nature of the source. For BZ Cam, the maximum X-ray variation is about 67$\%$ in the X-ray count rate and the X-ray variation of MV Lyr is about 50$\%$ in the count rates. For these two sources, the relative phasing between the X-rays and the optical is not definitive because the error on the orbital periods fold close to one full phase since the given time zero. For V592 Cas, erorr on the relative phasing between the optical and the X-rays are less than 0.05. We cannot conclusively calculate the modulation amplitudes or significance of the X-ray variations over the orbit since the data is about 15 ksec for each NL and covers only less than two cycles of the orbit at best. For MV Lyr, a complete coverage does not exist. We notice that the X-ray variations for these sources do not follow a sinusoidal shape over the orbit. Analyses of energy dependence of X-ray variations are performed using two energy bands of 0.3-2.2 keV (or 0.3-1.0 keV) and 2.5-7.0 keV revealing no significant change in the shape of the mean light curves over these energy bands, given the statistical errors of the low count rates which indicates no sigificant energy dependence in variations over the XRT range. In addition, we looked for other periodicities or coherent QPOs (quasi-periodic oscillation) that could exist in the XRT data using fast fourier transform (FFT) analysis and/or averaging several power spectra by segregating data into several smaller time intervals. We found no significant periodicity or coherent QPO for BZ Cam, V592 Cas or MV Lyr given the S/N quality of the data. Spectral Analysis ================= A spectrum and a background spectrum was generated for each NL using all available data presented in Table 2 between 0.2-10 keV. A circular photon extraction region of radius 1.5 arc min is used for both the source and the background (devoid of other contaminating sources). Each spectrum was grouped to have a minimum of 30-60 counts in each bin to increase signal to noise and utilize good statistics. We emphasize that the data is of moderate spectral resolution and no significant line emission other than predicted by the fitted models was detected; particularly, no iron lines at 6.4 keV, 6.7 keV or 6.9 keV was found, and so no reflection component can be confirmed with this data. However, NLs have hot disks with winds, as opposed to cold disks of quiescent DN, which may yield complicated reflection effects that may not be revealed properly with data in the 0.2-10.0 keV band. Subsequently, these spectra are analyzed using single/double/triple thermal plasma emission models MEKAL/APEC (thermal plasma in collisional equilibrium) within XSPEC software or multi-temperature isobaric cooling flow (plasma) models CEVMKL or MKCFLOW in XSPEC (for references and model descriptions see Arnaud 1996, or https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/Models.html). To account for the absorption in the X-ray spectrum of interstellar or possible intrinsic origin, we utilized the $tbabs$ model in the fitting procedure (Wilms, Allen & McCray 2000). In general, the X-ray spectra of nonmagnetic CVs are well modelled with the multi-temperature isobaric cooling flow type plasma emission models as in MKCFLOW or CEVMKL (Mukai et al. 2003, Baskill et al. 2005, Pandel et al. 2005, Guver et al. 2006, Okada et al. 2008, Balman et al. 2011, Balman 2014). The X-ray spectra of nonmagnetic CVs show several different emission lines, revealing the existence of hot, optically thin plasma in these systems (see also Balman 2012). A wide range of temperatures is derived from these spectra revealed by the presence of detected H- and He-like emission lines from elements N to Fe, and some Fe L-shell lines. As the accreting material settles onto the WD through the BL region, it is expected to form a structure with continuous temperature distribution in the X-rays. This characteristic is consistent with an isobaric cooling flow plasma model which is a multi-temperature distribution of plasma with a differential emission measure assuming a power-law distribution of temperatures ($dEM=(T/T_{max})^{\alpha -1} dT/T_{max}$). In such a model, the emission measure at each temperature is proportional to the time the cooling gas remains at this temperature (Pandel et al. 2005). This type of plasma emission represent collisionaly ionized plasma in thermal equilibrium between ions/protons and electrons just like the standard MEKAL/APEC models assume. The XRT spectrum of BZ Cam was fitted using several plasma models within XSPEC. A single or double MEKAL model fits yield unacceptable values of 2.6 and 2.1, respectively. Increasing MEKAL components reduces the artificially, but approximates a multiple-temperature distribution plasma. Therefore, as discussed above, a more physical model CEVMKL which is a multi-temperature plasma emission model built from the $MEKAL$ code (Mewe et al. 1986) is used to fit the data. However, the data was found inconsistent with a of 2.7 . BZ Cam has strong wind outflows (in the UV and optical bands) as discussed in the introduction and it may be that there is an excessive component in the X-ray spectrum due to scattering off of the wind or an emission component due to an extended structure on the disk. We used a power law model to characterize this second component and fitted the spectrum with $tbabs$$\times$(CEVMKL+POWER) composite model. The resulting spectral parameters with an acceptable are given in Table 3 (middle column) and the fitted spectrum is displayed in Figure 2. We carefully checked that all spectral parameter ranges are consistent across spectral binning of 30-60 counts per bin for the fitted model with the minimum values. However, the values of the unacceptable fits in this paragraph are even higher when the spectrum with 60 counts per bin is used which is expected since the greater binning increases the discepancies between the model and data reducing statistical errors. The excess data are in the harder energy band. For fitting the XRT spectrum of MV Lyr, we assumed similar models. A single MEKAL model fit yields an unacceptable value of 2.64 . A double MEKAL model gives acceptable fits with a lower temperature of kT=0.2 keV and a higher temperature of kT=79.8 keV (a 2$\sigma$ lower limit on the high temperature MEKAL model is kT=29 keV). This signals a definite distribution of temperatures where the MEKAL components somehow reflect the lower and higher temperature limits. Thus, we fitted the spectrum with a CEVMKL model alone, which yielded a value of 1.7(dof.21). In order to look for any scattering effect of X-rays from a wind or a component relating to an extended structure, as in BZ Cam (note that this system is also known to have winds in the UV/optical), we added a power law component and calculated how much this composite model $tbabs$$\times$(CEVMKL+POWER) improves the fit performed using only the CEVMKL model. The spectral parameters and the value of this new/composite fit is given in Table 3 (left hand column), and the modeled spectrum is presented in Figure 3 . We tested the significance of adding the second power law component to the CEVMKL model with FTEST and found that it improves the fit at 98.6% confidence level (probab. 0.014), which is almost at 3$\sigma$ significance. Note that when the binning in the spectrum is increased to a minimum of 60 counts per bin, the single CEVMKL fit has a value of 2.0 which is then improved over 3$\sigma$ significance when the power law model is added to the fit. The power law model removes the excess data in the harder energy band. Increasing the binning improves the S/N of the spectral bins which in turn affects the statistics of spectral fits that do not model the spectrum adequately. This method reveals a correct spectral model that describes the spectrum, given the moderate spectral resolution and S/N of the data. Our final target, V592 Cas, was the lowest count rate source of the three NLs. We treated the modeling of the XRT spectrum in a similar manner as in the other two. However, in this case it is more difficult to chose between the models relying only on values due to the lower statistical quality of the data. A single MEKAL model fit yields a value of 1.5 with an X-ray temperature of kT=41 keV (a 2$\sigma$ lower limit is kT=15 keV). Adding a second MEKAL component yields two temperatures of 0.4 keV and 8.1 keV improving the of the fit at 92% confidence level (probab. 0.08 using FTEST). Inclusion of another MEKAL does improve the fit even more, but then it is redundant since it indicates existence of a distribution of temperatures. Thus, for the consistency of the modeling of V592 Cas with the other two NLs, we included a fit using the model $tbabs$$\times$CEVMKL yielding a value better than a single MEKAL fit at 94% confidence level (using FTEST). We present this fit in Table 3 (right hand column) and the fitted specrum is shown in Figure 4. In this case, inclusion of a secondary power law component does not improve the fit (yields value of 1.25). However, we include maximum limits on the power law flux and photon index for completeness. We caution that the non-detection of a significant power law component could be a result of the lower X-ray flux of the source. The unabsorbed X-ray flux and luminosities (calculated from the distances in Table 1) of the three NL systems are given in Table 3 along with the rest of the spectral parameters with errors stated at the 90% confidence level for a single parameter. We caution that due to the energy range of XRT and the hardness of the X-ray spectra, we may have not determined the power law photon indices as accurately and they may be in a steeper range as in 1.0-2.0 as opposed to 0.5-1.0 as seen from Table 3. Given the unabsorbed total X-ray fluxes of these sources (also, using the power law photon indices and fluxes) and the XRT count rates, BAT is not expected to detect these sources, with an expected BAT rate $\le$ 2$\times$10$^{-4}$ c s$^{-1}$. We have analyzed the BAT data and have not detected any of the sources, as expected. The range of neutral hydrogen column densities derived from the spectral fits (see Table 3), are checked using standard tools that calculate neutral hydrogen column in the line of sight: (1) COLDEN (using Dickey & Lockman 1990, http://cxc.harvard.edu/toolkit/colden.jsp); (2) $nhtot$ (using Willingale et al. 2013, http://www.swift.ac.uk/analysis/nhtot/index.php). Willingale et al. (2013) calculate the molecular hydrogen column density, N(H2), in the Milky Way Galaxy using a function that depends on the product of the atomic hydrogen column density, N(HI), and dust extinction, E(B-V), with the aid of the 21 cm radio emission maps and the GRB data. Using these software, we found in a range, 0.6-0.7$\times$10$^{21}$ cm$^{-2}$, 0.8-1.2$\times$10$^{21}$ cm$^{-2}$, and 3.1-4.3$\times$10$^{21}$ cm$^{-2}$ for MV Lyr, BZ Cam and V592 Cas, respectively. Our values are consistent with interstellar within 90% confidence level errors. The fits on Table 3 are performed over a range of binning between 30-60 counts per bin, checking the soft X-ray range and the tabulated values and errors of down to 0.4 keV and the hard X-ray band out to 7.5 keV. We note that there was a preprint of a conference proceeding put in the archive (astro-ph) by Zemko & Orio (2013) including some preliminary XRT analysis of two of the sources BZ Cam and MV Lyr in our study which was after the submission of this paper to the $Astrophysical\ Journal$. These authors use $Raymond$-$Smith$ and APEC models that are similar to the MEKAL models used in this study but do not use the expected multi-temperature distribution plasma models (e.g. CEVMKL, MKCFLOW) as discussed in this section. Finally, we note that though some acceptable fits may be achieved with higher (e.g. using partial covering absorbers ect.) than interstellar values using the data, such models with higher do not reproduce the detected $ROSAT$ count rates for these sources by factors of 3-100 times. Thus, they are not consistent with a global model for our targets in the 0.1-10 keV (or 0.1-50 keV) range. On the existence of blackbody emission components ------------------------------------------------- We do not find blackbody emission components consistent with our XRT spectra in our modeling. For completeness, we exploit the data to calculate 2$\sigma$ upper limits on temperatures and fluxes for blackbody emission consistent with BL emission in which we have reduced the binning in the data and included a softer energy range down to 0.4 keV. We calculate (1) in a range of kT$_{BB}$=(20-50) eV, an upper limit on the soft X-ray luminosity (blackbody) 6.6$\times$10$^{32}$ for MV Lyr in the 0.1-10.0 keV band; (2) in a range of kT$_{BB}$=(25-50) eV, a soft X-ray luminosity upper limit is 5.2$\times$10$^{32}$ in the 0.1-10.0 keV band consistent with V592 Cas; (3) For kT$_{BB}$=30 eV upper limit, we find a soft X-ray luminosity upper limit of 2.3$\times$10$^{33}$ in the 0.1-10.0 keV band for BZ Cam. We can not calculate upper limits on the soft X-ray flux/luminosity for effective temperatures lower than the given values/ranges iin the above calculation. For larger effective temperatures than the given values/ranges, the flux/luminosity upper limits are at lower values (i.e., $\le$ 1$\times$10$^{32}$ ) than these upper limits. We caution that these upper limits are not better than $ROSAT$ results in the literature due to the energy band width of and its soft sensitivity. We stress that the comprehensive $ROSAT$ analysis of the 11 nonmagnetic NL systems, with its better soft X-ray sensitivity in the 0.1-2.4 keV band, demonstrates that the hardness ratios of these sources are not compatible with blackbody emission (cf. Fig 2 from van Teeseling et al. 1996) and they are consistent with optically thin plasma emission. A joint $ROSAT$ and $EUVE$ data analysis of IX Vel (another NL) shows no blackbody emission (van Teeseling et al. 1995). Therefore, we aquired the archival $ROSAT$ data of MV Lyr (PSPCB detector), BZ Cam (PSPCB detector), and V592 Cas (PSPCC detector), and analyzed the data using the model of $tbabs$$\times$CEVMKL and $tbabs$$\times$(BBODY+CEVMKL) utilizing the response files [pspcc-gain1-256.rsp]{} and [pspcb-gain2-256.rsp]{}. We used observations obtained in the high states. The main goal of this analysis was to derive possible 2$\sigma$ temperature upper limits for any soft blackbody emission consistent with BLs. Our fits using the CEVMKL model yielded acceptable values (=1.0-1.2) with parameters close to the best-fit spectral parameters and in the error ranges given in Table 3. We do not present these since the derived results are similar with the ones in Table 3. The $\alpha$ parameter for the power law index of the temperature distribution in CEVMKL model was fixed at the best fit values in Table 3 for BZ Cam and V592 Cas. We note that we have not included the power law spectral component in the fits since the $ROSAT$ fits were conducted in 0.1-2.2 keV band, and this model will not be detected along with the thermal plasma emission component, in the narrow energy band. In order to calculate the 2$\sigma$ blackbody temperature upper limits, we added a second component of BBODY (in XSPEC) in the modeling. The fits yielded the following results: (1) MV Lyr, kT$_{BB}<$ 6.6 eV; (2) BZ Cam, kT$_{BB}<$ 5.4 eV; and (3) V592 Cas, kT$_{BB}<$ 7.1 eV. These 2$\sigma$ blackbody temperature upper limits are consistent with the detected or expected WD temperatures in these NLs and not with any optically thick boundary layer emission. We note that if single MEKAL models are used for the fitting, the derived temperature upper limits are very similar. Discussion ========== NLs have not been thoroughly investigated using the recent X-ray telescopes with higher sensitivity, wider energy band and better energy resolution in comparison with older mission studies of $ROSAT$ and $EINSTEIN$. We observed a selected group of non-magnetic NLs studied only with $ROSAT$ in a narrow energy band to further investigate their spectral and temporal characteristics using . X-ray observations of BLs in NL systems are important to study characteristics under the effect of high accretion rates, to investigate the geometry of the inner disk, to calculate the BL temperature, BL luminosity, and mass accretion rate. With simultaneous X-ray and UV observations, one can work on a self-consistent model of the BL and the accretion disk. We have presented the XRT spectral and temporal analysis of three NL systems, MV Lyr, BZ Cam and V592 Cas. The temporal analysis indicates that all systems show X-ray variations over the binary orbit as observed in the mean light curves. We suggest this may be related to scattering over the accretion disk. This may result from a modulated disk wind or an elevated disk rim in the accretion impact zone. However, the latter requires high inclination that is not consistent with these sources. Thus, scattering from disk winds that are episodic and modulated with the orbital period, is likely to create the X-ray variations over the orbit in accordance with the wind activity. However, it is important to note that the variations seem to follow the wind outflows as they maximize with the orbital modulation of the H$\alpha$ line which also shows P Cygni profiles (see V592 Cas). This indicates that the X-ray emitting regions may be at the base of these strong wind outflows. There is a need for better/longer X-ray time series data with simultaneous photometric light curves in the optical and UV for direct comparisons. The XRT spectral analysis of the three sources reveals hard X-ray emission with high X-ray temperatures in the 0.2-10.0 keV range. We found an X-ray temperature kT$_{max}$ $>$21 keV for MV Lyr, 33.3$^{+16.0}_{-14.0}$ keV for BZ Cam and 35.5$^{+19.7}_{-10.9}$ keV for V592 Cas. Amoung the three sources MV Lyr has the hardest X-ray spectrum, thus the maximum limit of the temperature was not constrained in the band (see Table 3 for a best fit value). Corresponding X-ray luminosities of the thermal plasma emission are 2$\times 10^{31}$ , 2.2$\times 10^{31}$ and 5.3$\times 10^{31}$ , respectively, calculated from the CEVMKL model (i.e., fitted model) parameters. The total X-ray luminosities (thermal+power law) are 1.7$\times 10^{32}$ for MV Lyr, 4.6$\times 10^{32}$ for BZ Cam, and 5.3$\times 10^{31}$ for V592 Cas in the 0.2-10.0 keV energy band. We calculated that in the 0.1-50 keV band (where bulk of the X-ray emission is) the X-ray luminosities of the thermal CEVMKL components are 1.9$\times 10^{32}$ , 5.0$\times 10^{32}$ , and 8.5$\times 10^{31}$ , for MV Lyr, BZ Cam, and V592 Cas, respectively. The fits for MV Lyr and BZ Cam involves a second spectral component. In order to double check the plausible under estimation of the thermal plasma emission luminosity, we assumed a single CEVMKL component fit for these two sources. We fitted the $tbabs$$\times$CEVMKL model fixing the maximum plasma emission temperatures from the fits in Table 3, which resulted acceptable . We re-calculated the thermal plasma emission luminosity in the 0.1-50 keV band for these sources and found 3.2$\times 10^{32}$ , and 7.0$\times 10^{32}$ for MV Lyr and BZ Cam, respectively. Thus, we are not underestimating the thermal plasma luminosities from the optically thin boundary layers. The X-ray flux and luminosity of the power law components, may change considerably in the 0.1-50 keV band due to the power law index and the cut-off energy from the thermal electron distribution. Thus, it needs to be determined from fits in a larger energy band width. We note that power law components are not part of the transfered accretion power of the systems but are related to additional scatterings. As summarized in the introduction, the X-ray emission from about 11 NLs have been studied with $ROSAT$ and found to show optically thin emission with a single MEKAL model at temperatures $<$6 keV (best fit values) and luminosities $<$10$^{32}$ . A few NLs were studied with $ASCA$ and modeled with double MEKAL models at different temperatures (e.g., TT Ari and KT Aur, Mauche & Mukai 2002), and one with using multiple-temperature plasma and MEKAL models (Pratt et al. 2004). Note that in all these observations the X-ray luminosities are $\le$ afew $\times$10$^{32}$ . Greiner (1998) found blackbody model of emission more consistent with $ROSAT$ spectra with (0.2-0.5) keV temperatures and hydrogen column densities of interstellar values however, with very small emitting regions. Other than these blackbody temperatures, $ROSAT$ (0.1-2.4 keV) does not reveal any blackbody component in NL spectra given the suitable soft X-ray energy band and its sensitivity. In addition, the $ASCA$ and data do not reveal these hot blackbodies or any blackbody emission. Most of these results are in accordance with our findings and we find hard X-ray spectra with multi-temperature optically thin plasma emission in the energy band without a blackbody emission, component consistent with $ROSAT$ 2$\sigma$ upper limits for blackbody emission with kT $<$ 7 eV. We have calculated the accretion rates in the hard X-ray emitting BLs of our systems (L$_{BL}$=GM$\dot{\rm M}$/2R$_{WD}$). The accretion rate in the boundary layer is 6.7$\times 10^{-11}$ yr$^{-1}$ for MV Lyr, 1.4$\times 10^{-10}$ yr$^{-1}$ for BZ Cam and 1.9$\times 10^{-11}$ yr$^{-1}$ for V592 Cas. For these calculations, values of M$_{WD}$ and R$_{WD}$ are assumed from Table 1 (for BZ Cam a similar WD mass and radius is taken as in MV Lyr) and thermal plasma emission luminosities in the 0.1-50 keV band have been used. It is of importance to compare our observational results with theoretical calculations of standard steady-state disk models, e.g. of Narayan & Popham (1993), and Popham & Narayan (1995). Taking the accretion rates calculated for these NLs in the optical and UV bands 1.3$\times 10^{-8}$ - 3.9$\times 10^{-9}$ (see Table 1), the standard disk models mostly predict optically thick BLs showing a blackbody emission spectrum with temperatures of 13-33 eV and L$_{soft}$$\ge$ 1$\times 10^{34}$ for an 0.8-1.0 WD, (WD rotation as fast as $\Omega_{}$=0.5$\Omega_K(R_{})$ is taken into account in L$_{soft}$, see Popham & Narayan (1995)). We underline that this blackbody temperature range is definitely larger than the 2$\sigma$ upper limits we derive from the $ROSAT$ data and the luminosity range seems larger than the luminosity range (0.1-10.0 keV) we calculated from the data (see section 6.1). The standard steady-state constant $\dot{\rm M}$ disk models predict (for optically thin emission) X-ray temperatures around 9-10 keV for a 1 WD at about 3.2$\times 10^{-10}$ (see Narayan & Popham 1993). Comparing the accretion disk luminosity from Table 1 and the X-ray luminosity of the thermal plasma emission from these three NLs, (1.8$\times 10^{32}$ , 5.0$\times 10^{32}$ and 8.5$\times 10^{31}$ , for MV Lyr, BZ Cam, and V592 Cas, respectively) we find that the ratio L$_{x}$/L$_{disk}$ $\sim$ 0.01-0.001 . The accretion rates and luminosities calculated using the X-ray data of these NLs resemble dwarf novae in quiescence. However, the optical and UV rates and luminosities for NL disks in the same brightness states resemble those of dwarf novae in outburst. For nonmagnetic systems, the earlier studies of the ratio of the X-ray flux to optical and/or UV flux, $F_x/F_{opt}$, decreased along the sequence SU UMa stars ($F_x/F_{opt}$$\sim$0.1)- U Gem stars- Z Cam stars ($F_x/F_{opt}$$\sim$0.01) - UX UMa stars and high state VY Scl stars ($F_x/F_{opt}$$\le$0.001) using the $EINSTEIN$ and $ROSAT$ results (see Kuulkers et al. 2006, van Teeseling et al. 1996, Patterson & Raymond 1985). This ratio is found to decrease with increasing P$_{orb}$ and increasing $\dot{\rm M}$ where a high $\dot{\rm M}$ causes the disk to emit more UV flux, but not as much X-ray flux (see review by Kuulkers et al. 2006). The virial temperature in the inner parts of the accretion disk (kT$_{virial}$=$\mu$m$_p$GM$_{WD}$/3R$_{WD}$ where $\mu$$\sim$ 0.6 and m$_p$ is the proton mass) limits the maximum plasma emission temperatures in nonmagnetic CVs (see also Pandel et al. 2005 for dwarf novae in quiescence). All the kinetic energy from the Keplerian motion of the accreting gas is converted into heat when virial temperatures are aquired by the flow. This gives the maximum energy per particle that can be dissipated and the temperature of standard 1-D boundary layers cannot be around the virial temperatures or material can not be confined to the disk. Given the WD properties on Table 2, if one assumes similar properties for the three NLs in question, we calculate T$_{virial}$=27 keV for these NLs. Energy budget of a radiative isobaric cooling flow indicates that T$_{max}$ is smaller than T$_{virial}$. The cooling flow releases an energy of (5/2)kT$_{max}$ per particle including kinetic and compressional components. The total thermal/kinetic energy at the inner edge of the disk available per particle is (3/2)kT$_{virial}$. This inturn yields another constraint that T$_{max}$/T$_{virial}$$<$ 3/5. Therefore, given T$_{virial}$=27 keV, then T$_{max}$$\le$ 16 keV. All of the best fit X-ray temperatures are above the calculated virial temperature and T$_{max}$ constraint. We also calculated the 2$\sigma$ lower limits of the X-ray temperatures to compare with our T$_{virial}$ and T$_{max}$ estimations and find that for BZ Cam the limit is $>$ 17 keV and for MV Lyr and BZ Cam, it is $>$ 21 keV. As a result, the boundary layers in these NLs are too hot, thus may not be confined to the disk and will expand/evaporate forming [*ADAF-like flows/X-ray coronal regions]{} where the accreting material will be advected onto the WD. We stress that all the multi-temperature plasma model fits show that the $\alpha$ parameter in these fits differ from 1.0 and that the cooling flows are non-standard as well revealing a different type of flow. ADAFs (Advection-dominated flows) correspond to a condition where the gas is radiatively inefficient and the accretion flow is underluminous (see Narayan & Mc Clintock 2008, Lasota 2008, Done et al. 2007). Advection-dominated accretion may be described in two different regimes. The first is when the accreting material has a very low density and a long cooling time (also referred as RIAF-radiatively inefficient accretion flow) with t$_{cool}$ $>$ t$_{acc}$ (t$_{cool}$ is the cooling time and t$_{acc}$ is the accretion time). This causes the accretion flow temperatures to be virialized in the ADAF region. Note that the standard ADAFs are two-temperature flows as studied in LMXBs where electrons and ions are at differing temperatures. The electron temperatures increase with decreasing accretion rate. For example, Atoll type relatively low accretion rate (compared to Z-sources) neutron star LMXBs (type-I bursters) show 1-200 keV spectra consistent with thermal Comptonization with a plasma electron temperature around 25-30 keV and some are found at even lower X-ray temperatures (see Barret et al. 2000 and Done et al. 2007). These systems show an optically thin BL as the main region of energy release merged with an ADAF. These sources have similar accretion rates to NLs if the ratio to $\dot{\rm M}_{\rm Edd}$ is considered in relative terms. Note that neutron stars in Atoll sources should be accreting substantial material as revealed in their X-ray bursts. For LMXBs hosting black holes the ADAFs have detected temperatures about 100 keV. For CVs virial temperatures in the disk are around 10-45 keV where the WDs are primaries as opposed to neutron stars or black holes (assuming 0.4 - 1.1 WDs). The second regime is such that the particles in the gas can cool effectively, but the scattering optical depth of the accreting material is large enough that the radiation can not escape from the system (see also “slim disk” model Abramowicz et al. 1988). The defining condition is then, t$_{diff}$ $>$ t$_{acc}$ (t$_{diff}$ is the diffusion time for photons). This regime requires high accretion rates of the order of 0.1$\dot{\rm M}_{\rm Edd}$. We stress that the mass accretion rates derived from the optical and UV observations for the three NLs in this paper are below this critical limit for a slim disk approach. In an RIAF-ADAF region (first regime) energy liberated by viscous dissipation remains in the gas and the energy is advected onto the compact star given some ratio of advection energy and viscous dissipation. Thus, the pressure and hence the sound speed are large. The accretion flow becomes geometrically thick, with high pressure support in the radial direction which causes the angular velocity to stay at sub-Keplerian values, and the radial velocity of the gas becomes relatively large with $\alpha$$\sim$0.1-0.3. This leads to a short accretion time t$_{acc}$$\sim$R/v$\sim$t$_{ff}$/$\alpha$ (t$_{ff}$: free fall timescale;see Narayan & Mc Clintock 2008). Finally, the gas with the large velocity and scale height will have low density, since the cooling time is long , and the medium will be optically thin. It is widely accepted that there is strong connection/association between ADAFs and outflows as in winds and jets/collimated outflows (see Narayan & Yi 1995, Blandford & Begelman 1999). This is largely because the ADAFs have positive Bernoulli parameter defined as the sum of the kinetic energy, potential energy and enthalpy, thus the gas is not well bound to the central star. We point out that alot of the NL systems have strong wind outflows and particularly, two of the NLs studied in this paper, BZ Cam and V592 Cas, have strong bipolar collimated wind outflows modulated with the orbital period (e.g., rotates obliquely to the observer‘s line of sight), as detected in the optical and UV bands with high velocities 3000-5000 km s$^{-1}$ (see Introduction). Given these characteristics, we suggest that the ADAF-like optically thin BLs may be the origin of these wind outflows from these systems. Mass loss rates of winds in NLs are about or less than 1% of the accretion rate, with acceleration length scale of about (20-100)R$_{WD}$ which are strongly affected by rotation (Kafka & Honeycutt 2004, Long & Knigge 2002). For V592 Cas, as noted earlier in the Introduction, optical brightness variations are not correlated with the strength of the wind outflows, and the outflow shows orbital variations and no variations relating to a disk tilt, disk precession or any superhump period of the system. Moreover, in the UV emitting inner disk, @fro12 finds that the continuum does not vary in a good fraction of the target NLs, revealing that changes in the wind structure are not tied to the accretion disk. These are a few hints that the origin of the wind outflow may be in the ADAF-like BL, not the disk itself. We note that BZ Cam is a CV embedded in a nebula not related to a nova explosion. It has been suggested that interactions of the BZ Cam wind with the interstellar medium produces the observed nebular bow shock (Hollis et al. 1992), but alternative origins for the nebula have also been proposed (e.g., Griffith et al. 1995; Greiner et al. 2001). Narayan & Popham (1993) show that the optically thin BLs of accreting WDs in CVs can be radially extended and that they advect part of the viscously dissipated energy as a result of their inability to cool, therefore indicating that optically thin BLs act as ADAF-like accretion flows. Partial radial pressure support by the hot gas also results in sub-Keplerian rotation profiles in these solutions. Advection can play an important role for the energy budget and the emission spectrum of the accretion flow in the BL. In addition, Popham & Narayan (1995) illustrate that the BL can stay optically thin even at high accretion rates for optical depth $\tau$ $<$ 1 together with $\alpha$ $>$ 0.1 . However, such models are not well investigated. For simplicity an ADAF around a WD can be described by truncating the ADAF solutions (as opposed to BHs) and the accretion energy is advected onto the WD heating it up. Accretion via a standard disk boundary layer is expected to spin-up a WD except when rotating near breakup (Popham & Narayan 1991), however a WD is spun down when accreting via ADAF-like hot flows (Medvedev & Menou 2002). Sion (1999) emphasizes that rapidly rotating WDs are rare in nonmagnetic CVs. For some preliminary modeling of ADAFs or hot settling flows (for CVs) see Medvedev & Menou (2002) and related references therein. The ADAF models in CVs should utilize reemission of the energy advected by the flow, modeled as a single-temperature blackbody of temperature: T$_{eff}=(L_{adv} / 4f\pi\sigma R_{wd}^2)^{1/4}$ where f is the fraction of the stellar surface that is emitting. In addition, we note that Godon & Sion (2003) have calculated that WDs in DN are not generally heated to more than 15% above their original temperature via standard steady-state disk accretion, even through multiple DN outbursts. Hence, other mechanisms are necessary (e.g. compressional heating, heating via advective accretion flows, irradiation via luminous optically thick boundary layer). Characteristics of ADAF-like flows may be different in CVs, since WDs are less compact than stellar-mass BHs, the hot flows will have less extend around a WD. Menou (2000) suggests that the gas in the hot flow is one-temperature (1-T implying plasma in collisional ionization equilibrium–CIE), because of the efficient Coulomb interactions at lower temperatures compared to BHs and that the energy advection is not dependent on the preferential heating of the ions by the viscous dissipation and lack of energy exchange between the ions and the electrons necessary for the two-temperature ADAFs in BH binaries (2-T implying a type of non-equilibrium ionization plasma–NEI). We note here that since there are no detailed theoretical calculations for ADAF flows in CVs, existence of 2-T ADAFs for CVs can not be completely ruled out which may yield a rather featureless X-ray spectra. The energy advected by the flow is lost through the event horizon in the BHs whereas in the CVs, it is expected to be re-radiated from the surface of the WD. The resulting EUV emission (and/or hard X-ray emission from the BL) could explain the strength of HeII $\lambda$4686 emission line as due to disk irradiation. One can compare the WD temperatures/heating in magnetic CVs (MCVs) and NL systems. This is only to calculate an approximate radiation efficiency for the BLs in NLs since MCVs particularly Polars, have different accretion geometry (they do not have accretion disk) and thus, does not accrete through advective hot flows. MCVs are known to have radiative shocks in accretion columns over the magnetic polar caps heating the WDs through basically radiative accetion flows. The Polar subtypes have been found to host cooler WDs compared with nonmagnetic CVs at the same orbital period (Sion 1999, Araujo-Betancor et al. 2005, Townsley & Gansicke 2009). The accreted material in these systems (possibly valid for IPs) are constained by the magnetic field to the polar regions down to a limited pressure depth. Strong lateral pressure gradients forces this material to spread over the surface. As a result, heat dissipated by compression up to a critical pressure is constrained to the polar regions, and compression at further higher pressures is spread over the whole surface heating the WD (Townsley & Gansicke 2009). The average WD temperature is about 50,000 K and about 16,000 K in NLs and MCVs (Polars), respectively above the period gap between 3-5 hrs (see Townsley & Gansicke 2009, Mizusawa et al. 2010). The relative luminosity difference of the WDs as a result of these temperatures is L$_{adv}$/L$_{rad}$ = (50000/16000)$^4$ (assuming similar R$_{WD}$). This ratio is about 96 which means that about 96 times more accretion power may be transferred to the WD in NLs. This will approximately result in about a factor of 96 times reduction in the radiation power from the BL since this power is advected onto the WD yielding a radiation inefficiency in the BL of about 0.01; we may denote this as $\epsilon_{adv}$. We note that if one assumes average temperature from the total samples of Polars and NLs, L$_{adv}$/L$_{rad}$ = (45000/13500)$^4$ =123, yielding an approximate $\epsilon_{adv}$ $\sim$ 0.008 (same average R$_{WD}$ assumed). As we have discussed in the previous paragraphs it has been proposed that ADAFs can have thermally-driven winds (Narayan & Yi 1995, Blandford & Begelman 1999) possibly modified/controlled by magnetic fields that would carry away some of the accretion power in the BL which is, then, not radiated by the BL. This does not necessarily imply that all NL winds are to originate from the BL. Therefore, the power/energy necessary to derive the wind outflows from BLs will also yield inefficiency in the radiation from the BL in the NLs ; we may denote this as $\epsilon_{wind}$. The wind luminosity, which represents the kinetic energy loss of the wind, is described as L$_{wind}$=$(\dot{M}_{wind}v_\infty^2)/2$. Not all the power used to derive the wind outflows appears as the kinetic power of the outflow and there is a certain efficiency factor associated with this conversion that changes from typically 10$^{-1}$-10$^{-4}$ for stellar winds (Lamers & Cassinelli 1999) or 10$^{-3}$-10$^{-4}$ for AGNs (Yuan & Narayan 2014). Assuming 1% of the power used to drive the wind outflows appears as the kinetic power of the outflow defined as the wind luminosity, L$_{outflow-BL}$=L$_{wind}$/0.01 . Using a wind speed of 3000 km s$^{-1}$ and a mass loss rate of 1$\times 10^{-10}$ yr$^{-1}$ (Kafka et al. 2009), L$_{wind}$ is 3$\times 10^{32}$ which will reduce the BL radiation by about 3$\times 10^{34}$ in V592 Cas. The same calculation using 5000 km s$^{-1}$ and 1$\times 10^{-11}$ yr$^{-1}$ (Honeycutt et al. 2013) yields a reduction in the BL radiation by about 8$\times 10^{33}$ in BZ Cam. The wind in MV Lyr is not as fast (see Linnell et al 2005) and does not yield strong reduction ($\le$2$\times 10^{31}$ ). Narayan & Yi (1995) predict that diminished winds/outflows may possibly occur when a hot (disk) corona forms (as viewed in the X-rays) with less effective advection-dominated flows. Overall, the efficiency of the BL radiation will be reduced by $\epsilon_{total}$=$\epsilon_{adv}$$\times$$\epsilon_{wind}$. It is possible that the soft X-ray to EUV emission from the region of the BL and/or the inner disk of a steady-state disk can be screened by the existing ADAF-like/X-ray coronal region and the soft radiation from the disk can be Compton-upscattered. Another possibility is that hard X-ray photons from the BL can Compton/scatter from the existing (strong) winds in these systems. Some of these winds are axisymmetric/bipolar in nature which may lead to non-isotropic scattering. Thus, the power law components derived from our fits are crucial and signals the existence of Comptonized or Compton up-scattered radiation in these systems as described above. It may be possible that both type of thermal Comptonization or scattering is occurring in these systems. We find power law luminosities of (0.2-2.4)$\times$10$^{32}$ (consistent with only some percent of the total X-ray luminosities of these NLs). We caution that the power law components also do not account for the disk luminosity of these systems (they are about 1-0.1% of the disk luminosity). There has been no definitive evidence that NLs are magnetic CVs; a few were suggested as intermediate polars (IP) (see [@fro12]), but have never been confirmed in X-rays. The UV and optical spectra of NLs are different from those of IPs. In particular, episodic bipolar strong wind production is not seen in the magnetic systems. The WDs in NL systems are very hot, 2-4 times hotter (see Townsley & Gansicke 2009, Mizusawa et al. 2010 and see also Table 1) in general compared with magnetic CVs revealing differences in the accretion physics, heating and geometry. The X-ray emission in IPs show virialized hot plasma from stand-off radiative shocks in small regions (accretion column) around the magnetic poles with temperatures 10-95 keV at 10$^{31-34}$ (Balman 2012 and references therein). On the other hand, the most important signature of IPs is the spin period of the WDs that manifests itself in the X-ray light curves regardless of the inclination of the systems. None of our systems (and most other NLs) have detected spin periods in the X-rays and we did not detect any (see Mukai 2011 for a catalog of IPs ). IP X-ray spectra show complex absorption phenomenon with partial covering absorption or warm absorbers showing energy dependent modulations where we only found interstellar absorption and no energy dependence of orbital variations atypical of IPs. Moreover, IPs with high accretion rates, as detected from the optical and UV bands of these NLs may be expected to show a component with a 40-120 eV blackbody temperature resulting from reprocessing of the X-rays, for which we find no evidence. About 30% of IPs indicate this soft component (Bernardini et al. 2012). In addition, IPs do not show major high and low states as in two of our NLs. Thus, there is no consistency with a magnetic CV picture for our targets. Conclusions =========== Observations of CV disk systems at low mass accretion rate (namely dwarf nova CVs in quiescence) have yielded the temperature and luminosity of the BLs from X-ray observations while the UV observations helped to determine the temperature and luminosity of the accreting white dwarfs, as they are the dominant component at low $\dot{M}$. The quiescent X-ray spectra are well characterised with a multi-temperature isobaric cooling flow model of plasma emission at kT$_{max}$=6-55 keV with accretion rates of 10$^{-12}$-10$^{-10}$ M$_{\odot}$ yr$^{-1}$ and the detected Doppler broadening in lines during quiescence is $<$750 km s$^{-1}$ at mostly sub-Keplerian velocities with electron densities $>$10$^{12}$ cm$^{-3}$. To further our understanding of the boundary layers in CVs and their spectral and temporal characterictics in the X-rays, we used a group of non-magnetic NLs (VY Scl and UX UMa sub-type systems) where NLs are mostly found in a state of high mass accretion rate with $\ge$1$\times$10$^{-9}$ yr$^{-1}$ and some showing occasional low states (VY Scls). X-ray observations of BLs in NL systems are important to derive parameters such as the BL temperature, BL luminosity, mass accretion rate and study accretion geometry. Using both X-ray and UV observations one can then build a self-consistent model of the BL and the accretion disk. In this work, we presented XRT observations of three non-magnetic NLs, MV Lyr (VY Scl type), BZ Cam (VY Scl type), and V592 Cas (UX UMa type), obtained in their high states. We find that these sources have X-ray spectra consistent with multi-temperature hot plasma emission that has a power law dependence in emission measure (i.e. temperature) distribution in a range kT$_{max}$=(21-50) keV. We calculate that the X-ray emitting plasma is virialized. The power-law index of the temperature distribution indicates that these plasmas depart from the predictions of isobaric cooling flow type models as opposed to low $\dot{M}$ systems like quiescent DN. We detect a second component in the X-ray spectra that is well modeled by a power law emission in the VY Scl type sources BZ Cam and MV Lyr. V592 Cas may be fitted with an additional power law model, but the fit is not significantly different than a fit without it. We do not find any periodicity in the XRT data for the three sources, but we find non-sinusoidal variation of X-ray emission over the orbital cycle of the systems without energy depence in the 0.2-10.0 keV band. Particularly, the mean light curve of V592 Cas indicates two seperated peaks that may be associated with a bipolar outflow structure. The ratio of the unabsorbed X-ray flux/luminosity of the three systems and the disk luminosities as calculated from UV and optical wavelengths is $\sim$0.01-0.001 where BZ Cam and MV Lyr are in the upper end and V592 Cas is in the lower end of this range indicating inefficiency of X-ray/BL radiation (note that we quote a lower limit for the disk luminosity of BZ Cam in Table 3). We do not find a blackbody emission component in the soft X-rays using our data and also the $ROSAT$ data with a 2$\sigma$ upper limit of kT$_{BB}$ $<$ 7 eV for the three systems, consistent with the WD temperatures but not with the optically thick boundary layer emission. As a result, we suggest that the BLs in NL systems may be optically thin hard X-ray emitting regions merged with ADAF-like flows and/or constitute X-ray corona regions on the inner disk close to the WD. We estimate that the high WD temperatures in the three NLs and others (NLs) may explain the efficiency reduction ${\epsilon}_{adv}$ in the optically thin BLs with a factor $\sim$ 0.01 . ADAF-like accretion flows in the BLs may help to explain the very fast collimated outflows from the NL systems because ADAFs have positive Bernoulli parameter (the sum of the kinetic energy, potential energy and enthalpy). Thus, the power lost to drive a wind from the ADAF-like BLs may result in even larger losses from the accretion power in these X-ray emitting BL regions. In addition, we note that analysis in the optical and the UV wavelengths (in the IR for some) indicate departure from the standard disk model for MV Lyr (Linnell et al. 2005), BZ Cam (Godon et al. 2014, in preparation), and V592 Cas (Hoard et al. 2009). We caution that our findings in the X-rays may not be optimally analogous to the case of LMXBs thus, ADAF-like flows (1-T in CIE or plausibly 2-T in NEI ) and formation of X-ray coronal regions in the BLs and/or the disk needs to be modeled for CVs in detail. We can not confirm at this stage that all NL systems have the type of BL/inner disk structure described for these three sources in this paper. acknowledgment {#acknowledgment .unnumbered} ============== The authors thank J. P. Lasota, J. Greiner and M. Revnivtsev for careful comments on the manuscript. PG is thankful to William Patrick Blair at the Henry Augustus Rowland Department of Physics and Astronomy at the Johns Hopkins University (Baltimore, MD), for his kind hospitality. This work was supported by the National Aeronautics and Space Administration (NASA) Under grant number NNX13AJ70G issued through the Astrophysics Division Office (SWIFT Cycle 8 Guest Investigator Program) to Villanova University. 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T. 2013 MNRAS, 431, 394 Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914 Woods, J.A., Drew, J.E., Verbunt, F. 1990, , 245, 323 Woods, J.A., Verbunt, F., Cameron, A.C., Drew, J.E., & Piters, A. 1992, , 255, 237 Wood, M.A. 1995, LNP Vol.443: White Dwarfs, 41 Yuan, F., & Narayan, R. 2014, pre-print, (2014arXiv1401.0586Y) Zemko, P., & Orio, M. 2013, pre-print, (2013arXiv1312.5122Z) [lllll]{} $M_{wd}$ & $M_{\odot}$ & 0.73-0.8$^{~(1,2)}$ & & 0.75$^{~(11)}$\ $R_{wd}$ & $km$ & 7,440$^{~(3)}$ & & 7,378$^{~(11)}$\ $M_{2nd}$ & $M_{\odot}$ & 0.3$^{~(2)}$ & & 0.21$^{~(11)}$\ $i$ & $deg$ & 10$\pm3^{~(4,5,6)}$ & 12-40$^{~(13,14)}$ & 28$\pm10^{~(10)}$\ $P$ & $hr$ & 3.19$^{~(5)}$ & 3.69$^{~(9,17,18)}$ & 2.76$^{~(8)}$\ $d$ & $pc$ & $505\pm 30^{~(1,2)}$ & 830$\pm$160$^{~(13)}$ & 330$^{~(10)}$-360$^{~(11)}$\ $T_{wd}$ & $K$ & $44,000\pm3000^{~(1,2)}$ & & $45,000^{~(11)}$\ $\dot{M}_{high}$ & $M_{\odot}$/yr$$ & $3 \times 10^{-9~(1,6)}$ & $\ge$$3 \times 10^{-9~(18)}$ & $\sim 1.3 \times 10^{-8~(11)}$\ V & min-max & 17.7-12.1 & 14.3-12.5 & 12.9-12.5\ L$_{disk}$ & & $\sim 2.7 \times 10^{34}$ & $\ge$$3 \times 10^{34}$ & $\sim 1.2 \times 10^{35}$\ $\Omega_$ & $\Omega_K$ & $\sim$0.28$^{~(1)}$ & &\ \ [cccccrcc]{} MV Lyr & 91443 & 4 & 2012-06-09 & 09:19:01 & 6600.00 & 0x122f & PC/WT\ MV Lyr & 91443 & 2 & 2012-06-08 & 09:04:01 & 7640.00 & 0x122f & PC/WT\ BZ Cam & 91441 & 2 & 2012-12-21 & 10:05:59 & 1240.00 & 0x122f & PC/WT\ BZ Cam & 91441 & 2 & 2012-12-21 & 10:09:59 & 13860.00 & 0x122f & PC/WT\ V592 Cas & 91442 & 4 & 2012-09-10 & 09:49:59 & 6865.00 & 0x122f & PC/WT\ V592 Cas & 91442 & 2 & 2012-09-09 & 04:44:45 & 6385.00 & 0x122f & PC/WT [lllll]{} CEVMKL & $N_H$(10$^{22}$atoms cm$^{-2}$) & $0.13^{+0.12} _{-0.06}$ & $0.30^{+0.07}_{-0.07} $ & $0.3^{+0.2}_{-0.2} $\ & $\alpha$ & $1.6^{+2.7}_{-0.4} $ & $0.13 ^{+0.16}_{-0.06} $ & $0.6^{+0.7}_{-0.3} $\ & $T_{max}$(keV) & $>$21$^{\dagger}$ & $ 33.0 ^{+16.0} _{-14.0} $ & $ 35.5^{+19.7}_{-10.9} $\ & $K_{CEVMKL}$ & $9.2^{+7.0}_{-4.8} \times 10^{-4}$ & $6.7 ^{+2.8}_{-1.0} \times 10^{-4}$ & $2.3^{+1.1}_{-1.0}\times 10^{-3}$\ Power law & PhoIndex$_{powerlaw}$ & $0.82 ^{+0.07}_{-0.07}$ & $0.40 ^{+0.1}_{-0.3}$ & $<$ 1.0\ & $K_{powerlaw}$ & $2.4 ^{+1.3}_{-0.20} \times 10^{-4}$ & $8.8^{+1.3}_{-4.4} \times 10^{-5}$ & $< 7.0\times 10^{-5}$\ & $\chi^2_{\nu} (\nu)$ & 1.17 (11) & 1.22 (10) & 1.17 (9)\ & Flux (10$^{-12}$erg cm$^{-2}$s$^{-1}$) & 5.4 & 5.8 & 3.4\ & Luminosity (10$^{32}$erg s$^{-1}$) & 1.7 & 4.6 & 0.5\ & Flux$_{cevmkl}$ (10$^{-12}$erg cm$^{-2}$s$^{-1}$) & 0.66 & 2.3 & 3.4\ & L$_{cevmkl}$ (10$^{32}$erg s$^{-1}$) & 0.2 & 1.9 & 0.5\ & Flux$_{power law}$ (10$^{-12}$erg cm$^{-2}$s$^{-1}$) & 4.8 & 2.9 & $<$ 1.3\ & L$_{power law}$ (10$^{32}$erg s$^{-1}$) & 1.5 & 2.4 & $<$ 0.2\ \ \ \
--- abstract: 'Let ${\boldsymbol}{\alpha}\in {\mathbb}{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{{\boldsymbol}{q}\in [-Q,Q]^N\cap{\mathbb}{Z}^N\backslash\{{\boldsymbol}{0}\}}\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\\|^{-1}$. Sharp upper bounds are known when $N=1$, using continued fractions or the three distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary $N$. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the sharpness of their lower bound $\gg Q^N\log Q$.' address: \| Department of Mathematics\ Royal Holloway, University of London\ TW20 0EX Egham\ UK author: - Reynold Fregoli bibliography: - 'Bibliography.bib' date: ', and in revised form ....' title: Sums of reciprocals of fractional parts --- Introduction ============ Sums of reciprocals of fractional parts have been studied by many authors in the light of their tight connections to, e.g., uniform distribution theory, metric Diophantine approximation, and lattice point counting (see [@Beck:ProbDioph], [@HarLi:SomeProblems1], [@HarLi:SomeProblems2], [@Kruse:Estimates], [@KuipersNiederr:UnifDistr], [@Schmidt:MetricalThms] and [@Shapira:ASolution]). In this note, we establish a new, sharp upper bound for such sums. We denote by $\\|x\\|$ the distance to the nearest integer of a real $x$. We write $A\ll B$ to mean that there exists a constant $c>0$ (absolute or depending only on the parameters indicated) such that $A\leq cB$. We use $\|\cdot\|_{2}$ to denote the Euclidean norm on ${\mathbb}{R}^{n}$ and $\|\cdot\|_{\infty}$ to denote the maximum norm. Let $N\in {\mathbb}{N}:=\{1,2,3,\ldots\}$, and let ${\boldsymbol}{x}\cdot{\boldsymbol}{y}:=\sum_{i=1}^N x_i y_i$ be the standard inner product on ${\mathbb}{R}^N$. Let ${\boldsymbol}{\alpha}$ be in ${\mathbb}{R}^{N}$ with $1,\alpha_1,\ldots,\alpha_N$ linearly independent over ${\mathbb}{Q}$, and suppose $Q_{1},\dotsc,Q_{N}\in(0,+\infty)$. Lê and Vaaler [@LeVaaler:Sumsof Corollary 1.2] proved that for $X:=[-Q_1,Q_1]\times\cdots\times[-Q_N,Q_N]$ and $Q:=(Q_1\cdots Q_N)^{1/N}\geq 1$ it holds $$\begin{aligned} 1\label{LV2} Q^N\log Q\ll \sum_{{\boldsymbol}{q}\in X\cap{\mathbb}{Z}^N\backslash\{{\boldsymbol}{0}\}}\\|{\boldsymbol}{\alpha}\cdot {\boldsymbol}{q}\\|^{-1}.\end{aligned}$$ They also showed that, whenever ${\boldsymbol}{\alpha}$ is multiplicatively badly approximable (see [@Bugeaud:Multiplicative] and [@LeVaaler:Sumsof (2.5)]), inequality (\[LV2\]) is sharp[^1], i.e., $$\begin{aligned} 1\label{LV3} \sum_{{\boldsymbol}{q}\in X\cap{\mathbb}{Z}^N\backslash\{{\boldsymbol}{0}\}}\\|{\boldsymbol}{\alpha}\cdot {\boldsymbol}{q}\\|^{-1}\ll_{{\boldsymbol}{\alpha}}Q^N\log Q\end{aligned}$$ for all $Q\geq 2$. However, if $N\geq 2$, every such ${\boldsymbol}{\alpha}$ yields a counterexample to the Littlewood conjecture. This follows from two facts: - a matrix is multiplicatively badly approximable if and only if its transpose is multiplicatively badly approximable (see [@German:Transference Corollary 1] and [@LeVaaler:Sumsof Theorem 2.2]); - every submatrix of a multiplicatively badly approximable matrix is itself multiplicatively badly approximable (see [@LeVaaler:Sumsof (2.6)]). To date, Littlewood’s conjecture is still open and the set of counterexamples to it, if not empty, has been shown to be extremely sparse. Indeed, Einsiedler, Katok, and Lindenstrauss [@EinseidlerKatokLindenstrauss:Invariant] have proved that its Hausdorff dimension is zero. Despite this, it can be shown that for some vectors in ${\mathbb}{R}^{N}$ we can reverse inequality (\[LV2\]). \[def:badlyapp\] Let $N\in{\mathbb}{N}$ and ${\boldsymbol}{\alpha}\in{\mathbb}{R}^{N}$. Let $\phi:(0,+\infty)\to(0,1]$ be a non-increasing[^2], real-valued function. We say that ${\boldsymbol}{\alpha}$ is a $\phi$-badly approximable vector if $$\|{\boldsymbol}{q}\|_{\infty}^{N}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|\geq\phi(\|{\boldsymbol}{q}\|_{\infty})$$ for all ${\boldsymbol}{q}\in{\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}$. If $\phi$ can be chosen constant, we simply say that ${\boldsymbol}{\alpha}$ is badly approximable. It was kindly pointed out to me by Victor Beresnevich that a simple gap-principle, already known in the literature (see [@KuipersNiederr:UnifDistr Proof of Lemma 3.3, p.123]), proves (\[LV3\]) for all badly approximable vectors ${\boldsymbol}{\alpha}$ in the special case $X=[-Q,Q]^N$. Note that the set of such vectors has full Hausdorff dimension in ${\mathbb}{R}^{N}$ (see [@Schmidt:BadlyApprox]). We recall this proof here. Let $Q\geq 2$ and suppose that ${\boldsymbol}{\alpha}\in{\mathbb}{R}^{N}$ is $\phi$-badly approximable. Then, for all distinct ${\boldsymbol}{q}_{1}, {\boldsymbol}{q}_{2}\in{\mathbb}{Z}^{N}\cap X$, we have $$\begin{aligned} 1\label{www} \\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{1}\pm{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{2}\\|=\\|{\boldsymbol}{\alpha}\cdot({\boldsymbol}{q}_{1}\pm{\boldsymbol}{q}_{2})\\|\geq \frac{\phi(2Q)}{(2Q)^{N}}.\end{aligned}$$ It follows that $$\left\|\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{1}\\|-\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{2}\\|\right\|\geq \frac{\phi(2Q)}{(2Q)^{N}}.$$ Therefore, none of the intervals $$\left[0,\frac{\phi(2Q)}{(2Q)^{N}}\right),\left[\frac{\phi(2Q)}{(2Q)^{N}},\frac{2\phi(2Q)}{(2Q)^{N}}\right),\left[\frac{2\phi(2Q)}{(2Q)^{N}},\frac{3\phi(2Q)}{(2Q)^{N}}\right),\ldots$$ contains more than one number of the form $\\|{\boldsymbol}{\alpha}\cdot {\boldsymbol}{q}\\|$ (${\boldsymbol}{q}\in {\mathbb}{Z}^{N}\cap X$), and no such number lies in the first interval. Hence, $$\begin{aligned} 1\label{????} \sum_{{\boldsymbol}{q}\in X\cap{\mathbb}{Z}^N\backslash\{{\boldsymbol}{0}\}}\\|{\boldsymbol}{\alpha}\cdot {\boldsymbol}{q}\\|^{-1}\leq \sum_{j=1}^{\|X\cap{\mathbb}{Z}^N\backslash\{{\boldsymbol}{0}\}\|}\frac{(2Q)^{N}}{j\phi(2Q)}\ll_N \frac{Q^N\log Q}{\phi(2Q)}.\end{aligned}$$ For $N=1$, the above upper bound can be improved on to $$\begin{aligned} 1\label{??} \ll Q\log Q+\frac{Q}{\phi(Q)}\end{aligned}$$ using the theory of continued fractions (see [@Lang:IntrotoDiophApprox Theorem 2, p.37-40]). The bound given in (\[??\]) is best possible. However, if one allows the upper bound to be expressed in terms of the (least) denominator $q_{K}$ of the $K$-th convergent of $\alpha$, an even more precise result can be obtained via the three distance theorem, as shown by Beresnevich and Leong [@BeresnevichLeong:Somsof Corollary 1]. Nevertheless, neither the techniques based on continued fractions nor those of Beresnevich and Leong using the three distance theorem seem to generalise in an obvious way to higher dimension. In this note we introduce yet another method, based on a recent counting result for weakly admissible lattices, which allows us to extend (\[??\]) to arbitrary dimension $N$. \[thm:theorem3\] Let $X:=[-Q,Q]^N$ and let ${\boldsymbol}{\alpha}\in{\mathbb}{R}^{N}$ be a $\phi$-badly approximable vector. Then, we have $$\label{eq:theorem3-1} Q^{N}\log\left(Q\phi(Q)\right)\ll_{N}\sum_{\substack{\mathbf{q}\in X \cap{\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\ll_{N}Q^{N}\log Q+\frac{Q^{N}}{\phi(Q)}$$ for all $Q\geq 1$. Clearly, the lower bound is superseded by (\[LV2\]) (and non trivial only if $\log(Q\phi(Q))\geq 1$), but we have decided to include it in Theorem (\[thm:theorem3\]) nonetheless. This is because both bounds in (\[eq:theorem3-1\]) can be proved almost at once and the method we use is substantially different from Lê and Vaaler’s. Note that the upper bound, in conjunction with the Khintchine-Groshev Theorem, implies that for every $\varepsilon>0$ the set of ${\boldsymbol}{\alpha}\in {\mathbb}{R}^N$ for which $$\sum_{\substack{\mathbf{q}\in X\cap {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\ll_{{\boldsymbol}{\alpha}}Q^{N}(\log Q)^{1+\varepsilon},$$ has full Lebesgue measure. Moreover, the upper bound in (\[eq:theorem3-1\]) is sharp, in the sense that for all $N\geq 1$ there exists a sequence of positive integers $Q_{i}\to +\infty$ such that $$\label{eq:sharpness} \sum_{\substack{\mathbf{q}\in X_{i}\cap {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\gg Q_{i}^{N}\log Q_{i}+\frac{Q_{i}^{N}}{\phi(Q_{i})},$$ where $X_{i}:=[-Q_{i},Q_{i}]^{N}$. To show this, let $\phi$ be chosen maximal, i.e., $$\phi(x)=\min\{\|{\boldsymbol}{q}\|_{\infty}^{N}\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\\|\ \|\ 0<\|{\boldsymbol}{q}\|_{\infty}\leq x,\ {\boldsymbol}{q}\in{\mathbb}{Z}^{N}\}$$ for $x\geq 1$ and $\phi(x)=1$ for $x<1$. Let ${\boldsymbol}{q}_{i}\in{\mathbb}{Z}^{N}$ be a sequence of pairwise distinct vectors such that $\phi(\|{\boldsymbol}{q}_{i}\|_{\infty})=\|{\boldsymbol}{q}_{i}\|_{\infty}^{N}\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{i}\\|$, and set $Q_{i}:=\|{\boldsymbol}{q}_{i}\|_{\infty}$. Then, $\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}_{i}\\|^{-1}=Q^{N}_{i}/\phi(Q_{i})$. This implies that $$\label{eq:sharpness2} \sum_{\substack{\mathbf{q}\in X_{i}\cap {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\geq\frac{Q_{i}^{N}}{\phi(Q_{i})}.$$ Hence, (\[eq:sharpness\]) follows from (\[LV2\]) and (\[eq:sharpness2\]). The main tool to prove Theorem \[thm:theorem3\] is Proposition \[prop:mainestimate1\], which gives a precise estimate for the size of the set $$\begin{aligned} 1 \label{eq:M1} M({\boldsymbol}{\alpha},\varepsilon, Q):=\left\{(p,{\boldsymbol}{q})\in{\mathbb}{Z}^{1+N}\setminus\{{\boldsymbol}{0}\}\ \right\|\ \left.\left\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}+p\right\|\leq\varepsilon,\ \|{\boldsymbol}{q}\|_{\infty}\leq Q\right\},\end{aligned}$$ where $0<\varepsilon\leq 1/2$ and $Q\geq 1$. \[prop:mainestimate1\] Let ${\boldsymbol}{\alpha}\in{\mathbb}{R}^{N}$ be a $\phi$-badly approximable vector. Let $Q\geq 1$ and $0<\varepsilon\leq 1/2$. Then, we have $$\left\|\left\|M({\boldsymbol}{\alpha},\varepsilon, Q)\right\|-2^{N+1}\varepsilon Q^{N}\right\|\ll_{N}\left(\frac{\varepsilon Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}.$$ With Proposition \[prop:mainestimate1\] at hand, Theorem \[thm:theorem3\] can be derived from a simple dyadic summation. Moreover, Proposition \[prop:mainestimate1\] is a straightforward consequence of a recent, general lattice point counting result, due to Widmer [@Widmer:WeakAdmiss Theorem 2.1], which we recall in the next section. Weakly admissible lattices and counting lattice points ====================================================== To state [@Widmer:WeakAdmiss Theorem 2.1], we need to introduce some notation. We follow the notation used in [@Widmer:WeakAdmiss], except that we write ${\mathcal}{N}$ (instead of $N$) for $\sum_{i=1}^{n}m_{i}$. We assume throughout that ${\mathcal}{N}\geq 2$. Let $n$ be a positive integer and let ${\mathcal}{S}:=({\boldsymbol}{m},{\boldsymbol}{\beta})$, where ${\boldsymbol}{m}:=(m_{1},\dotsc,m_{n})\in{\mathbb}{N}^{n}$ and ${\boldsymbol}{\beta}:=(\beta_{1},\dotsc,\beta_{n})\in(0,+\infty)^{n}$. Let $${\textup{Nm}}_{{\boldsymbol}{\beta}}\left(\underline{{\boldsymbol}{x}}\right):=\prod_{i=1}^{n}\left\|{\boldsymbol}{x}_{i}\right\|_{2}^{\beta_{i}}$$ be the multiplicative norm induced by ${\boldsymbol}{\beta}$ on ${\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}\ni\underline{{\boldsymbol}{x}}:=({\boldsymbol}{x}_{1},\dotsc,{\boldsymbol}{x}_{n})$. Let $I$ be a non-empty subset of $\{1,\dotsc,n\}$ and let also $$C=C_{I}:=\left\{\left.\underline{{\boldsymbol}{x}}\in{\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}\ \right\|\ {\boldsymbol}{x}_{i}={\boldsymbol}{0}\ \ \forall i\in I\right\}.$$ We fix the couple $({\mathcal}{S},C)$ and for any $\Gamma\subset{\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}$ we consider the quantity $$\nu\left(\Gamma,\varrho\right):=\inf\left\{\left. {\textup{Nm}}_{{\boldsymbol}{\beta}}(\underline{{\boldsymbol}{x}})^{\frac{1}{t}}\ \right\|\ \underline{{\boldsymbol}{x}}\in\Gamma\setminus C,\ \left\|\underline{{\boldsymbol}{x}}\right\|_{2}<\varrho\right\},$$ where $t:=\beta_{1}+\dotsb+\beta_{n}$. We observe that $\nu(\Gamma,\cdot)$ is a decreasing function of $\varrho$, bounded by below from $0$. Hence, we have the following definition from [@Widmer:WeakAdmiss]. \[def:weakadmiss\] A full rank lattice $\Lambda$ in ${\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}$ is said to be weakly admissible for the couple $({\mathcal}{S},C)$, if $\nu\left(\Lambda,\varrho\right)>0$ for all $\varrho\in(0,+\infty)$. Before stating the counting theorem, we require some more notation. For any $\Gamma\subset{\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}$ we define $$\lambda_{1}\left(\Gamma\right):=\inf\left\{\ \|{\boldsymbol}{x}\|_{2}\ \|\ {\boldsymbol}{x}\in\Gamma\setminus\{{\boldsymbol}{0}\}\right\},$$ and we set $$\mu\left(\Gamma,\varrho\right):=\min\left\{\lambda_{1}\left(\Gamma\cap C\right),\nu\left(\Gamma,\varrho\right)\right\}.$$ We can now state a simplified version of [@Widmer:WeakAdmiss Theorem 2.1], streamlined for our application. \[thm:2.1\] Let $n\in{\mathbb}{N}$ and let $({\mathcal}{S},C)$ be a couple as in Definition (\[def:weakadmiss\]). For\ ${\boldsymbol}{Q}:=\left(Q_{1},\dotsc,Q_{n}\right)\in(0,+\infty)^{n}$ we set $$\overline{Q}:=\left(Q_{1}^{\beta_{1}}\dotsm Q_{n}^{\beta_{n}}\right)^{\frac{1}{t}}$$ and $Q_{\max}:=\max\{Q_{1},\dotsc,Q_{n}\}$. Let $$Z_{{\boldsymbol}{Q}}:=\prod_{i=1}^{n}[-Q_{i},Q_{i}]^{m_{i}}\subset{\mathbb}{R}^{m_{1}}\times\dotsb\times{\mathbb}{R}^{m_{n}}$$ and let $\Lambda$ be a weakly admissible lattice for the couple $({\mathcal}{S},C)$. Then, there exists a real constant $c=c({\mathcal}{N})>0$, only depending on the quantity ${\mathcal}{N}:=\sum_{i=1}^{n}m_{i}$, such that $$\left\|\left\|Z_{{\boldsymbol}{Q}}\cap\Lambda\right\|-\frac{{\textup{Vol}}\left(Z_{{\boldsymbol}{Q}}\right)}{\det\Lambda}\right\|\leq c\!\!\!\!\!\!\displaystyle\inf_{\ \ \ \ 0<B\leq Q_{\max}}\left(\frac{\overline{Q}}{\mu(\Lambda,B)}+\frac{Q_{\max}}{B}\right)^{{\mathcal}{N}-1},$$ where ${\textup{Vol}}(Z_{{\boldsymbol}{Q}})$ denotes the volume of the set $Z_{{\boldsymbol}{Q}}$ and $\det\Lambda$ denotes the determinant of the lattice $\Lambda$. Proofs ====== Proof of Proposition \[prop:mainestimate1\] ------------------------------------------- First we note that if $(p,{\boldsymbol}{q})\in M({\boldsymbol}{\alpha},\varepsilon,Q)$, then $$\label{eq:emptycase} \varepsilon Q^{N}\geq\left\\|{\boldsymbol}{q}\cdot{\boldsymbol}{\alpha}\right\\|\|{\boldsymbol}{q}\|_{\infty}^{N}\geq\phi(Q).$$ Suppose that $\varepsilon Q^{N}/\phi(Q)< 1$. Then, $M({\boldsymbol}{\alpha},\varepsilon,Q)=\emptyset$, by (\[eq:emptycase\]). Moreover, $$2^{N+1}\varepsilon Q^{N}\ll_{N}\varepsilon Q^{N}\leq\frac{\varepsilon Q^{N}}{\phi(Q)}\leq\left(\frac{\varepsilon Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}.$$ Hence, Proposition \[prop:mainestimate1\] holds true whenever $\varepsilon Q^{N}/\phi(Q)< 1$. From now on, we can assume that $$\label{eq:notemptycase} \frac{\varepsilon Q^{N}}{\phi(Q)}\geq 1.$$ Let $$A_{{\boldsymbol}{\alpha}}:=\left(\begin{array}{@{}c\|ccc@{}} 1 & \alpha_{1} & \dots & \alpha_{N} \\\hline 0 & & & \\ \vdots & & \phantom{\scriptstyle{N}}\scalebox{1.5}{$\text{I}$}_{N} & \\ 0 & & & \end{array}\right).$$ Define $\Lambda_{{\boldsymbol}{\alpha}}:=A_{{\boldsymbol}{\alpha}}{\mathbb}{Z}^{N+1}\subset{\mathbb}{R}^{N+1}$ and $Z_{\varepsilon,Q}:=[-\varepsilon,\varepsilon]\times[-Q,Q]^{N}\subset{\mathbb}{R}^{N+1}$. Then, $$\label{eq:cardinalities} \left\|M({\boldsymbol}{\alpha},\varepsilon, Q)\right\|=\left\|\Lambda_{{\boldsymbol}{\alpha}}\cap Z_{\varepsilon,Q}\right\|-1,$$ since ${\boldsymbol}{0}\in\Lambda_{{\boldsymbol}{\alpha}}\cap Z_{\varepsilon,Q}$. Therefore, to prove Proposition \[prop:mainestimate1\], it suffices to estimate the quantity $\left\|\Lambda_{{\boldsymbol}{\alpha}}\cap Z_{\varepsilon,Q}\right\|$. To this end, we use Theorem \[thm:2.1\]. Let $n=2$ and let ${\boldsymbol}{m}={\boldsymbol}{\beta}:=(1,N)$. Let $C:=C_{I}$, with $I:=\{2\}$. Then, all vectors ${\boldsymbol}{v}\in\Lambda\setminus C$ have the form $${\boldsymbol}{v}=A_{{\boldsymbol}{\alpha}}\begin{pmatrix} p \\ {\boldsymbol}{q} \end{pmatrix}=\begin{pmatrix} {\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}+p \\ {\boldsymbol}{q} \end{pmatrix},$$ where ${\boldsymbol}{q}\in{\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}$ and $p\in{\mathbb}{Z}$. Recall now that ${\boldsymbol}{\alpha}$ is a $\phi$-badly approximable vector. Hence, for all ${\boldsymbol}{v}\in\Lambda_{{\boldsymbol}{\alpha}}\setminus C$ it holds $${\textup{Nm}}_{{\boldsymbol}{\beta}}({\boldsymbol}{v})=\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}+p\|\|{\boldsymbol}{q}\|_{2}^{N}\geq\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\\|\|{\boldsymbol}{q}\|_{\infty}^{N}\geq\phi\left(\left\|{\boldsymbol}{q}\right\|_{\infty}\right)\geq\phi(\|{\boldsymbol}{v}\|_{2}).$$ Since $\phi$ is non-increasing, we can conclude that $$\label{eq:weakadmiss} \nu\left(\Lambda_{{\boldsymbol}{\alpha}},\varrho\right)\geq\phi(\varrho)^{\frac{1}{N+1}}>0$$ for all real $\varrho\geq\lambda_{1}(\Lambda_{{\boldsymbol}{\alpha}}\setminus C)$. However, if $\varrho<\lambda_{1}\left(\Lambda_{{\boldsymbol}{\alpha}}\setminus C\right)$, then $\nu(\Lambda_{{\boldsymbol}{\alpha}},\varrho)= +\infty$ and (\[eq:weakadmiss\]) trivially holds true. This shows that $\Lambda_{{\boldsymbol}{\alpha}}\subset{\mathbb}{R}\times{\mathbb}{R}^{N}$ is weakly admissible for the couple $(({\boldsymbol}{m},{\boldsymbol}{\beta}),C)$. We can thus apply Theorem \[thm:2.1\], with $\Lambda=\Lambda_{{\boldsymbol}{\alpha}}$ and $Z_{{\boldsymbol}{Q}}=Z_{\varepsilon,Q}$. By choosing $B:=Q_{\max}=Q$, we get $$\label{eq:estimate} \left\|\left\|Z_{\varepsilon,Q}\cap\Lambda_{{\boldsymbol}{\alpha}}\right\|-\frac{{\textup{Vol}}\left(Z_{\varepsilon,Q}\right)}{\det\Lambda_{{\boldsymbol}{\alpha}}}\right\|\leq c\left(\frac{\left(\varepsilon Q^{N}\right)^{\frac{1}{N+1}}}{\mu(\Lambda,Q)}+1\right)^{N}.$$ Since $\det\Lambda_{{\boldsymbol}{\alpha}}=1$ and ${\textup{Vol}}\left(Z_{\varepsilon,Q}\right)=2^{N+1}\varepsilon Q^{N}$, to conclude the proof, we just need to estimate the right-hand side of (\[eq:estimate\]). We observe that $\lambda_{1}(\Lambda_{{\boldsymbol}{\alpha}}\cap C)=\lambda_{1}\left({\mathbb}{Z}\times\{{\boldsymbol}{0}\}\right)=1$. Hence, $$\mu(\Lambda,Q)\geq\min\left\{1,\phi(Q)^{\frac{1}{N+1}}\right\}=\phi(Q)^{\frac{1}{N+1}},$$ by (\[eq:weakadmiss\]). Combining (\[eq:cardinalities\]) and (\[eq:estimate\]), and using (\[eq:notemptycase\]), we find $$\left\|\left\|M({\boldsymbol}{\alpha},\varepsilon, Q)\right\|-2^{N+1}\varepsilon Q^{N}\right\|\ll_{N}\left(\left(\frac{\varepsilon Q^{N}}{\phi(Q)}\right)^{\frac{1}{N+1}}+1\right)^{N}\ll_{N}\left(\frac{\varepsilon Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}.$$ This completes the proof. Proof of Theorem \[thm:theorem3\] --------------------------------- We start by observing that $$\begin{aligned} \label{eq:coreq1} \sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1} & \leq\sum_{k=1}^{\infty}2^{k+1}\left\|\left\{\left.(p,{\boldsymbol}{q})\in{\mathbb}{Z}^{N+1}\setminus\{{\boldsymbol}{0}\}\ \right\| \right.\right.\ 2^{-k-1}<\left\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}+p\right\| \left.\left.\leq 2^{-k},\ \left\|{\boldsymbol}{q}\right\|_{\infty}\leq Q\right\}\right\|\nonumber \\ & \leq\sum_{k=1}^{\infty}2^{k+1}\left\|M({\boldsymbol}{\alpha},2^{-k}, Q)\right\|=\sum_{k=1}^{\left\lfloor\log_{2}(Q^{N}/\phi(Q))\right\rfloor}2^{k+1}\left\|M({\boldsymbol}{\alpha},2^{-k}, Q)\right\|,\end{aligned}$$ where the last equation is due to (\[eq:emptycase\]). Now, by Proposition \[prop:mainestimate1\], we know that $$\label{eq:asymptotics} \left\|\left\|M({\boldsymbol}{\alpha},2^{-k}, Q)\right\|-2^{N+1-k}Q^{N}\right\|\ll_{N}\left(\frac{2^{-k}Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}.$$ Hence, (\[eq:coreq1\]) yields $$\begin{aligned} \sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1} & \ll_{N}\sum_{k=1}^{\left\lfloor\log_{2}(Q^{N}/\phi(Q))\right\rfloor}2^{k+1}\left(2^{N+1-k}Q^{N}+\left(\frac{2^{-k}Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}\right)\nonumber \\ & \ll_{N}\sum_{k=1}^{\left\lfloor\log_{2}\left(Q^{N}/\phi(Q)\right)\right\rfloor}\left(Q^{N}+2^{\frac{k}{N+1}}\left(\frac{Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}\right)\nonumber \\[10pt] & \ll_{N}Q^{N}\log_{2}\left(\frac{Q^{N}}{\phi(Q)}\right)+\left(\frac{Q^{N}}{\phi(Q)}\right)^{\frac{1}{N+1}}\left(\frac{Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}}\label{eq:coreq1.5} \\[15pt] & \ll_{N} Q^{N}\log Q+\frac{Q^{N}}{\phi(Q)}\label{eq:coreq1.75},\end{aligned}$$ where (\[eq:coreq1.5\]) follows from the trivial estimate $\sum_{k=1}^{K}2^{k/(N+1)}\leq 2^{K/(N+1)+1}$ and (\[eq:coreq1.75\]) is due to the fact that $1/\phi(Q)\geq\log\left(1/\phi(Q)\right)$. This proves the upper bound. To prove the lower bound, we notice that $$\begin{aligned} \label{eq:coreq2} \sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1} & \geq\sum_{k=1}^{\infty}2^{k}\left\|\left\{\left.(p,{\boldsymbol}{q})\in{\mathbb}{Z}^{N+1}\setminus\{{\boldsymbol}{0}\}\ \right\| \right.\right.\ 2^{-k-1}<\left\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}+p\right\| \left.\left.\leq 2^{-k},\ \left\|{\boldsymbol}{q}\right\|_{\infty}\leq Q\right\}\right\|\nonumber \\ & \geq\sum_{k=1}^{\infty}2^{k}\left(\left\|M({\boldsymbol}{\alpha},2^{-k}, Q)\right\|-\left\|M({\boldsymbol}{\alpha},2^{-k-1}, Q)\right\|\right).\end{aligned}$$ From Proposition \[prop:mainestimate1\], we also know that for all $k\geq 1$ and $Q\geq 1$ $$\label{eq:coreq3} \left\|\left\|M({\boldsymbol}{\alpha},2^{-k}, Q)\right\|-2^{N+1-k}Q^{N}\right\|\leq c_{N}\left(\frac{2^{-k}Q^{N}}{\phi(Q)}\right)^{\frac{N}{N+1}},$$ where $c_{N}$ is a positive constant. Hence, whenever $1\leq k\leq\log_{2}\left(Q^{N}\phi(Q)^{N}/c_{N}^{N+1}\right)=:K$, we have $$\left(2^{N+1}-1\right)2^{-k}Q^{N}\leq \left\|M({\boldsymbol}{\alpha},2^{-k},Q)\right\|\leq\left(2^{N+1}+1\right)2^{-k}Q^{N}.$$ This, in turn, shows that $$\label{eq:coreq5} \left\|M({\boldsymbol}{\alpha},2^{-k},Q)\right\|-\left\|M({\boldsymbol}{\alpha},2^{-k-1},Q)\right\|\geq 2^{-k}Q^{N},$$ when $1\leq k\leq K-1$. Therefore, provided $K-1\geq 1$, we can plug (\[eq:coreq5\]) into (\[eq:coreq2\]) and restrict the sum to $k\leq\left\lfloor K-1\right\rfloor$. This yields the lower bound $$\label{eq:coreq6.5} \sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\geq\lfloor K-1\rfloor Q^{N}\geq (K-2)Q^{N},$$ which, of course, remains true for $K-1< 1$. Now, a trivial lower bound for the sum of the reciprocals is $$\sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\gg_{N}Q^{N},$$ since $\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\geq 2$ for all ${\boldsymbol}{q}\neq 0$. Hence, recalling that $K=\log_{2}\left(Q^{N}\phi(Q)^{N}/c_{N}^{N+1}\right)$, we conclude that $$\sum_{\substack{\mathbf{q}\in[-Q,Q]^{N}\\ \cap\ {\mathbb}{Z}^{N}\setminus\{{\boldsymbol}{0}\}}}\left\\|{\boldsymbol}{\alpha}\cdot{\boldsymbol}{q}\right\\|^{-1}\gg_{N}\left(K-2\right)Q^{N}+\log_{2}\left(4c_{N}^{N+1}\right)Q^{N}\gg_N Q^{N}\log\left(Q\phi(Q)\right).$$ Acknowledgements {#acknowledgements .unnumbered} ================ I’m most grateful to my supervisor, Martin Widmer, for his encouragement and precious advice. It has been a pleasure to discuss some aspects of this note with Victor Beresnevich, who provided some very useful feedback on it. I would like to thank Royal Holloway, University of London, for funding my position here, and my office mates for their support and cheerful presence. [^1]: Note that Lê and Vaaler assumed $X:=[0,Q_1]\times\cdots\times[0,Q_N]$, but this makes no difference. Suppose indeed that the sum taken over, e.g., the set $[-Q_1,0]\times[0,Q_2]\times\cdots\times[0,Q_N]$ is big, then, we can multiply the first coordinate of ${\boldsymbol}{\alpha}$ by $-1$, obtaining a multiplicatively badly approximable vector such that the sum over $X$ is now big. [^2]: We say that $\phi$ is non-increasing if $\phi(x)\geq\phi(y)$, whenever $x<y$.
--- abstract: 'The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the ’bypass’ path to turbulence. Initially the so-called ’linear optimal perturbation problem’ is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous, sub-optimal disturbances are focussed upon; nonlinear quadratic interactions are immediately evoked by such initial perturbations and an unstable streamwise-homogeneous large amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the edge between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.' bibliography: - 'SqDuct.bib' date: 11 July 2007 and in revised form 7 November 2007 title: Transition to turbulence in duct flow --- \[sec:intro\] Introduction ========================== Transition to turbulence in ducts is still an unsolved issue despite the 120-plus years since the observations by Osborne Reynolds that led to the definition of a similarity parameter, the ratio of the viscous to the convective time scale, capable of broadly separating the cases where the flow state was laminar from those where turbulence prevailed. Recent years have seen a resurgence of interest in the topic, spurred by new developments in linear and nonlinear stability theories. As is now well known, classical small perturbation theory is uncapable to provide an explanation for the onset of transition in ducts and pipes ([@Gill1965; @Salwenetal1980; @TatsumiYoshimura1990]). Current understanding ascribes the failure of classical theory to its focus on the asymptotic behavior of individual modes; when a small disturbance composed by a weighted combination of linear eigenfunctions is considered, there is the potential for very large short-time amplification of perturbation energy, even in nominally stable flow conditions. This behavior has been reported by [@Landahl1980] and [@BobergBrosa1988] and has been given the name of algebraic instability (and later ’transient growth theory’), since the initial rapid growth in time of small disturbances goes like $t$. The property is related to the non-normality of the linearized stability operator (which does not commute with its adjoint). Despite the appeal and elegance of transient growth theory, it was realized that the fully nonlinear Navier-Stokes equations need to be used to understand transition phenomena. In this context, we mention the work of [@Nagata1990; @Nagata1997], [@Waleffe1997; @Waleffe1998; @Waleffe2003], [@FaisstEckhardt2003] and [@WedinKerswell2004]. These works present traveling wave and equilibrium solutions of the Navier-Stokes equation for channel and pipe flows that are possibly related to transition. Experimental investigations along these lines are due to [@Hofetal2004; @Hofetal2005]. In dynamical systems’ terminology it is argued that unstable travelling waves appear through saddle node bifurcations in phase space; the travelling waves found so far are all saddle points with low dimensional unstable manifolds. The saddles act by attracting the flow from the vicinity of the laminar state, and then repelling it away. For transitional or turbulent, yet moderate, values of $Re$ the flow wanders in phase space between few repelling states, spending much time in their vicinities, before being abruptly ejected away, so that experimental observations yield recurrent sequences of familiar patterns ([@Artuso; @Kerswell2005]). A yet unanswered issue concerns the initial conditions that are most suited to yield such unstable states. Traditional emphasis on so-called optimal perturbations may be misplaced. In fact, there is but a weak connection between the flow structures that grow the most during the linear transient phase and the chaotic flows found at large times. Such a connection for the case of the pipe flow concerns the so-called edge state which sits on a separatrix between laminar and turbulent flows ([@Eckhardt2007; @PringleKerswell]). This state, made up by two asymmetric vortices in the cross-section, resembles the optimal disturbance of transient growth theory ([@Bergstrom1993]). For the motion in a square duct there seems to be no connection at all: the low-$Re$ turbulent flow, when averaged in time and space, is characterized by eight secondary vortices symmetric about diagonals and bisection lines. It seems reasonable to argue that such secondary structures represent the skeleton of the unstable periodic orbits, but the disturbances that grow the most in the linear transient phase are formed by two vortices, symmetric about a diagonal ([@GallettiBottaro2004]). In both configurations, pipe and square-duct, the optimal perturbation is a stationary pseudo-mode, elongated in the streamwise direction, and not a travelling wave. This is a generic occurrence in wall-bounded shear flows, and it does not bode well for the establishment of a simple, direct relation between small amplitude disturbances (excited in an initial receptivity phase) and finite amplitude wavelike states. Although nonlinear effects can be pinpointed right away as the culprit for the missing link between early stage of transition and late stages, there is scope for a receptivity analysis focussed on transiently growing initial conditions, followed by nonlinear simulations. A motivation for the search of wavelike structures during the early stages of transition is also provided by recent careful experiments ([@PeixinhoMullin2006]) on the reverse transition in pipe flow, where modulated wave trains are found to emerge from long-term transients. The present paper starts by comparing the efficiency of optimal and sub-optimal perturbations in triggering transition to turbulence at a value of the Reynolds number $Re$ close to the threshold between laminar and turbulent flow; it further shows that the turbulent motion oscillates around edge states which display an intriguing resemblance to optimal disturbances, before relaminarization occurs. Finally, an interpretation of the results is provided after projecting them onto a suitably defined phase space. Model configuration {#modconf} =================== The incompressible flow in a duct of square cross-section is an appealing configuration for the presence of geometrical symmetries capable to strongly constrain the patterns of motion. Countless studies have been devoted to the formation of secondary vortices in the turbulent regime (see [@Gavrilakis1992] for a direct numerical simulation approach) and, more recently, an attempt has been made to link the appearance of such large-scale coherent states to the vortices appearing during the initial, optimal transient phase of disturbance growth ([@GallettiBottaro2004; @Bottaroetal2006]). The longitudinal laminar flow velocity component has an analytic form $U(y,z)$ available, for example, in [@TatsumiYoshimura1990], with $y$ and $z$ cross-stream axes. After normalizing distances with the channel height $h$, velocities with the friction velocity $u_\tau$, with $u_\tau^2=-\frac{h}{4\rho} \frac{dP}{dx}$, time with $h/u_\tau$ and pressure with $\rho u_\tau^2$, the following equations are found to govern the behaviour of the developed flow in an infinite duct: $$\begin{aligned} \begin{array}{ll} u_x + v_y + w_z = 0,\\ u_t +uu_x + vu_y + wu_z = -p_x + \frac{1}{Re_\tau}\Delta u + 4,\\ v_t +uv_x + vv_y + wv_z = -p_y + \frac{1}{Re_\tau}\Delta v,\\ w_t +uw_x + vw_y + ww_z = -p_z + \frac{1}{Re_\tau}\Delta w, \end{array} \label{ndimomeg}\end{aligned}$$ with $\Delta=\partial_{xx}+\partial_{yy}+\partial_{zz}$ and $Re_\tau=u_\tau~h/\nu$. By using $u_\tau$ as velocity scale we fix the pressure gradient, rather than the flow rate. An incompressible pseudo-spectral solver, based on Chebyshev collocation in $y$ and $z$ and Fourier transform along $x$, has been employed to solve these equations. For time-integration a third-order semi-implicit backward differentiation/Adams-Bashforth scheme is used. In order to compute a pressure unpolluted by spurious modes, the pressure is approximated by polynomials ($P_{N-2}$) of two units lower-order than for the velocity ($P_N$). Only one collocation grid is used, and no pressure boundary condition is needed. The accuracy and stability properties of the method are discussed by [@Botella]. An adequate grid at $Re_\tau=150$ has been found to be composed by $51 \times 51$ Chebyshev points, with $N_x=128$ streamwise grid points or $84$ Fourier modes after de-aliasing. The $x$-length of the domain has been chosen equal to $4 \pi$ to accommodate a sufficiently large range of wavenumbers $\alpha$, with periodic boundary conditions. A finer grid resolution has also been used for fully developed turbulent flow with $71 \times 71 \times 256$ physical grid points, or $71\times 71 \times 170$ spectral modes and a streamwise length $L_x=6\pi$[^1]. The finer resolution run provides a slightly larger value of the threshold energy for transition, but integral quantities such as disturbance energy or skin friction factor are only marginally affected. Since the threshold value is, in any case, a function also of the shape of the initial condition, we do not deem indispensable to pursue expensive calculations to determine it exactly. For all cases studied, an adequate time step has been found to be $\Delta t=5\times 10^{-4}$. The mean value of the generic function $g$ is defined as $\overline{g}(y,z) = \frac{1}{L_x~T}\int_{xt} g(x,y,z,t)~dxdt$. The addition of time averaging is necessary because of the finite (relatively low) streamwise length. The bulk velocity is $U_b=\int_{yz}\overline{u}~dydz$ and the centreline velocity is $U_c=\overline{u}(0.5,0.5)$. The friction factor for the square duct can be written as $f = 8~u_\tau^2 / U_b^2$. Some representative results are given in Table \[tab\], while the secondary flow field at $Re_\tau = 150$, averaged over forty units of time, is shown in figure \[turb\]. It displays a very regular pattern with eight vortices, despite the fact that no averaging over quadrants has been performed. ----------------- -------- ----------- -------- -------- $f$ $U_c/U_b$ $Re_b$ $Re_c$ \[3pt\] laminar 0.018 2.0963 3163 6630.4 turbulent 0.0415 1.53 2084 3188 ----------------- -------- ----------- -------- -------- : \[tab\] Comparison of some numerical values for laminar and fully-developed turbulent flow at $Re_\tau=150$. The subscript $b$ refers to bulk and $c$ to centreline. The skin friction $f$ given by the empirical correlation by [@Jones] is $f=0.0481$. ![\[turb\] Turbulent mean cross flow vortices and streamwise flow contours; isolines are spaced by $4~ u_\tau$.](turbUVW.eps){width="7cm"} Optimal perturbations ===================== The Navier-Stokes equations, linearized around the ideal laminar flow, are: $$\begin{aligned} \label{LST2dLNS} i \alpha u +v_y+w_z & =&0, \\ u_t +i\alpha U u + vU_y+wU_z &=& -i\alpha p + (- \alpha^2 u + u_{yy}+u_{zz} )/Re_\tau,\\ v_t +i\alpha U v &=& - p_y + (- \alpha^2 v + v_{yy}+v_{zz} )/Re_\tau,\\ w_t +i\alpha U w &=& - p_z + (- \alpha^2 w + w_{yy}+w_{zz} )/Re_\tau,\end{aligned}$$ associated to boundary conditions $u=v=w=0$ on the walls; $\alpha$ is the streamwise wavenumber. The equations are integrated from a given initial condition at $t=0$ up to a final target time $t=T$. To identify the flow state at $t=0$ producing the largest disturbance growth at any given $T$, a variational technique, based on the repeated numerical integration of direct and adjoint stability equations, is used, coupled with transfer and optimality conditions ([@CorbettBottaro2000]). The functional for which optimization is sought is based on an energy-like norm and reads: $$\begin{aligned} G(T)=\frac{E(T)}{E(0)},\quad \mbox{with} \quad E=\frac{1}{2} \int_y \int_z (u^* u + v^* v+ w^* w)\quad dy~dz, $$ with $$ denoting complex conjugate. As a preliminary result the energy stability limit for this flow is determined; the minimal Reynolds number below which disturbances decrease monotonically is found to be $Re_\tau=23.22$ for $\alpha=0$. In terms of Reynolds number based on centreline velocity and half channel height, this limiting value is equal to $Re_c=79.44$. For comparison, in the case of Poiseuille flow, the Reynolds number is $Re_c\simeq49$, with $\alpha=0$ and spanwise wavenumber $\beta\simeq2$ (cf.* [@SchmidHenningson2000]). Then, we compute optimal perturbations at $Re_\tau=150$, for 3 different Fourier modes: $\alpha=0,~ 1,~ 2$. The gain and the corresponding cross-flow optimal disturbances at $t=0$ are presented in figure \[gain\]. ![\[gain\] Cross-flow velocity vectors for the optimal disturbances for $\alpha = 0, 1$ and $2$. Note that for $\alpha \ne 0$ the streamwise velocity fluctuations do not vanish at $t=0$.](OptPert.eps){width="9cm"} A global optimal solution ($G = 873.11$) is found for $\alpha=0$ at a time $T = 1.31$ and a comparable gain ($G=869.03$) is found at a later time ($T = 2.09$) for a solution which is topologically different. These two solutions, which we call ’global optimals’ represent streamwise vortices of vanishing streamwise disturbance velocity; they evolve downstream producing streaks of high and low longitudinal velocity. A global optimal state with two cells arranged along a duct diagonal, was obtained by [@GallettiBottaro2004] in the context of a spatial, rather than temporal, optimization strategy. The existence of a four-cell optimal was not reported. It will be shown below that the two- and the four-cell states are very robust: when they are used as initial conditions for nonlinear simulations the flow trajectory is uncapable of evolving away from them towards a different topology which eventually exploits other symmetries, except when exceedingly large disturbance energies are given as input. For non-zero streamwise wavenumber, sub-optimal perturbations are found which take the form of modulated travelling waves. For $\alpha=1$, $G=381.41$ at $T=0.384$ and for $\alpha=2$, $G=237.05$ at $T=0.246$. To represent a wave train the temporal dependence of the generic disturbance can be written as the product of an exponential wave-part and an envelope function slowly modulated in time: $q(x,y,z,t)=\tilde{q}(y,z,t)exp^{i\alpha(x-ct)}$. The phase velocity is found to be quasi-constant with time and close to the bulk velocity: $c(\alpha=1)=1.1117$ and $c(\alpha=2)=1.1536$, both scaled with $U_b$. The study of the temporal (rather than the spatio-temporal) evolution of disturbances is acceptable when flow structures travel at a well-defined speed within the duct. Such an approximation appears to be reasonably well satisfied in experiments on equilibrium puffs in pipe flow ([@Hofetal2005]), which are found to be advected downstream at speeds slightly larger than $U_b$, for values of the Reynolds number $Re_b = U_b D / \nu$ ($D$ pipe diameter) exceeding $Re_b \approx 1800$. Below such a threshold, turbulence can no longer be maintained autonomously. Nonlinear evolution {#Nle} =================== Temporally evolving simulations have been conducted in a periodic duct of length $4\pi$ for a variety of initial conditions at $Re_\tau=150$; such a value is close to the threshold of self-sustained turbulence, as confirmed very recently by [@Uhlmann]. For each direct numerical simulation the initial state consists of the quasi-parabolic base flow profile plus a optimal or sub-optimal perturbation, normalized with prescribed energy $E_0$, plus random noise. The complex amplitude of the noise in Fourier space varies between $\pm10^{-10}$; the noise is necessary to fill the spectrum. ![\[skin-friction\] Skin friction for the four different optimal perturbations. For $\alpha=0$, the initial energy $E_0$ is equal to $10^{-1}$, for $\alpha=1$, $E_0=7.8\times 10^{-3}$ and for $\alpha=2$, $E_0=4.4\times 10^{-3}$.](F.eps){width="6cm"} In figure \[skin-friction\] the time evolution of the skin friction factor $f$ is plotted for four initial conditions of different initial energies. When the simulations are initiated with one of the two global optimal solutions of figure \[gain\] the ensuing behavior is uneventful ([cf.]{} figure \[skin-friction\]), and even for a rather large initial disturbance amplitude, $E_0 = 0.1$, no instability appears to modify the flow which returns slowly to the laminar condition with $f=0.018$. The results are more suggestive when linear travelling waves are used to initiate the nonlinear computations. For the case in which the condition at $t=0$ is the sub-optimal state with $\alpha = 1$ or $2$, the initial growth of $f$ is slower than for the global optimal case but, by time $t=2$, a strong mean flow deviation is created by nonlinear interactions, leading to a value of the friction factor which oscillates around the turbulent mean value $f = 0.0415$. It is notable that the energies $E_0$ of the sub-optimal initial conditions sufficient to trigger transition are very much lower than $10^{-1}$. We have explored in more details the long time behaviour of the flow when the sub-optimal initial perturbation with $\alpha=1$ is used to trigger transition; the results are condensed in figure \[seuils\]. ![\[seuils\] Evolution in time of the fluctuation energy, initial condition is given by the sub-optimal perturbation with $\alpha=1$. Different curves correspond to different initial energy values: continuous line corresponds to $E_0=7.8\times 10^{-3}$, dash-dotted line to $E_0=7.7\times 10^{-3}$; dashed line to $E_0=5\times 10^{-3}$ and dotted line represents the linear computation. The energy is normalized with its initial value $E_0$. The streamwise averaged vortices are drawn at times $t=0.8,~20,~50,~70$ and $80$.](Seuils.eps){width="13cm"} As a function of the value of $E_0$ the Navier-Stokes calculations permit to recover the linear behaviour (when $E_0 \leq 5\times 10^{-3}$) or to depart from it. The threshold for transition is found for $E_0=7.8\times10^{-3}$; below such an initial energy value the flow returns rapidly (within a few units of time) to the laminar state, above it the flow becomes turbulent. The energy gain of figure \[seuils\] shows that when $E_0=7.8\times 10^{-3}$ there is, at $t=0.8$, a sudden increase of the fluctuation energy, likely linked to an instability of the distorted mean flow. There is an interesting connection here with the recent theory of ’minimal defects’[@BLC; @Biau1; @Gavarini; @BenDov1]. The secondary flow at $t=0.8$ shown in the same figure displays symmetries about bisectors and diagonals, but this is not a generic occurrence and different initial conditions generate mean flow defects with other symmetries. For $E_0=7.7\times 10^{-3}$ the growth is followed by decay and rapid relaminarization (cf. also figure \[spectrum\] left); when $E_0=7.8\times 10^{-3}$ the growth which starts at $t=0.8$ is followed by a rapid filling of the spectrum with a peak in the intensity of fluctuations at $t=1.8$. Such a filling is made clear by figure \[spectrum\] (right): first the modes with $\alpha=n$, $n\in \mathbb{N}$, grow because of quadratic interactions, then the modes with $\alpha=(2n-1)/2$, $n\in \mathbb{N}$, emerge out of the random noise (visible in the figure after $t=5$). Figure \[spectrum\] (left) focusses on the modes $\alpha=0$ and $\alpha=2$ which are first to be produced by nonlinearities. The streamwise-independent mode remains amplified for a long time because of lift-up effect, its appearance out of the $\alpha=\pm 1$ fundamental mode is the analogue of the so-called oblique transition process in channel flow ([@oblique]). By time $t=3$ the turbulent flow can be considered as fully developed. Such an edge state persists until $t\approx78$ (cf. \[seuils\]); it remains dynamically connected to the laminar base flow solution since relaminarization is abruptly reached at $t \approx 80$, after coalescence of the smaller vortices into a pair and then into a single large vortex. While turbulence is maintained, the secondary patterns displayed at $t=20,~50$ and $70$ in figure \[seuils\] resemble the four-cell global optimal disturbance. At $t=20$, two pairs of vortices are clearly visible in the cross-section; they are close to the two vertical walls which can thus be defined as ’active’ since it is there that the turbulent wall cycle operates ([@Uhlmann]). This 4-cells state oscillates while maintaining remarkable coherence for some twenty units of time. Averaging over $t$ yields the same flow pattern with four regular vortices discovered very recently by [@Uhlmann]. Past $t=50$, the active walls shift and the large scale vortices lean on the horizontal surfaces, despite the presence of smaller intermittent features that can be found near the left vertical wall at $t=50$ and the right wall at $t=70$. At $t \approx 80$ the flow relaminarizes; tests performed with different lengths of the computational domain show that the threshold $E_0$ for transition remains unaffected, but relaminarization occurs later both for a shorter computational box ($t \approx 90$ when $L_x = 2 \pi$) and a longer box ($t \approx 150$ when $L_x = 6 \pi$), for the same numerical grid density. Interestingly, for the case of the short box the oscillations of the variables around the mean display enhanced amplitudes, highlighting the fact that a distorted dynamics could be caused by constraining too much the flow structures. ![\[spectrum\] Left: temporal behavior of three different Fourier modes ($\alpha=0,~1,~2$) for the nonlinear simulation initiated by the optimal $\alpha = 1$ initial condition with $E_0=7.8\times 10^{-3}$ (continuous lines) and $E_0=7.7\times 10^{-3}$ (dashed lines), plus low amplitude noise. On the right the long time evolution of the fluctuations ($\alpha=0.5,~1,~1.5,...$) for the case $E_0=7.8\times 10^{-3}$ is represented. The power density spectrum $\kappa_n$ is defined by $\kappa_n = \frac{1}{N^2}\int_{yz} (\mathbf{\tilde{u}}^\mathbf{\tilde{u}})_n + (\mathbf{\tilde{u}}^\mathbf{\tilde{u}})_{N-n}~dydz$, with $\mathbf{\tilde{u}}_n\left(\alpha_n,y,z,t \right)$ the $x$-Fourier transform of $\mathbf{u}$, from which the laminar profile has been substracted.](spectrum.eps){width="13cm"} For a geometrical description of the transition process, we choose the phase subspace spanned by two observables: the Reynolds number (based on bulk velocity) and the energy of the streamwise-averaged flow, noted $E_U=0.5\int_{yz} U^2+V^2+W^2 dydz$, with $(U,~V,~W)$ the velocity vector after streamwise averaging. ![\[PhaseDiag\] ’Phase diagram’ representation of the transition process: time evolution in Reynolds number, based on bulk velocity, and mean flow energy $E_U$. The simulations start from the laminar flow solution (open circle in the left figure) plus a (sub-)optimal perturbation. The cross with arrows provides the approximate position of an unstable saddle node. The time averaged values of $E_U$ and $Re_b$ in the fully developed turbulent regime are indicated by dashed lines in the zoom at right which refers only to the initial disturbance perturbations with $\alpha=1$.](PhaseDiag.eps){width="14cm"} Figure \[PhaseDiag\] shows some trajectories between the laminar ($Re_b=3163,~E_U=306.45$) and the turbulent ($Re_b=2084,~E_U=116$) fixed points (the latter is simply defined by the temporal average of $E_U$ and $Re_b$ in the fully developed turbulent regime). The paths for the initial conditions with $\alpha=0$ correspond to homoclinic orbits in phase space. For the two cases which follow a non-trivial branch ($\alpha=1$ and $2$) the flow approaches the unstable saddle node (qualitatively sketched in the figure) before departing away from it along its unstable manifold. The trajectory in figure \[PhaseDiag\] (right) then starts circling around the point which characterizes the fully developed turbulent state, with orbits of increasing size. Before the end of the fifth orbit, it escapes through the unstable manifold of the saddle node towards the laminar fixed point. Each orbit has an ellipsoidal shape and is made up by two portions with very small local radii of curvature, around which the flow spends most of its time, and two long portions of large radii of curvature which are spanned very rapidly by the flow. Relaminarization is consistent with the emerging picture of shear flow turbulence which considers it as a transient event with a characteristic lifetime increasing exponentially with Reynolds number ([@nature]). Discussion and conclusions {#sec:concl} ========================== Although non-normality and transient growth are important issues, the traditional emphasis on optimal disturbances may be misleading when trying to predict the onset of transition. The results found here suggest that optimal initial perturbations in the form of steady streamwise vortices play mostly a passive role, while rapidly growing travelling wave packets have the potential to induce transition past a reasonably low threshold value of the initial disturbance energy. This suggests that an energy-like functional is possibly not the most pertinent objective of the optimization procedure. In the simulations performed here we have found that key to transition is the establishment of a disturbed mean flow profile (mode $\alpha=0$), susceptible to an exponential or algebraic instability which causes enhanced growth of the travelling wave mode with $\alpha = \pm 1$. Such an early stage can be described by a weakly nonlinear triadic interaction model. Once turbulence is established, its sustainment depends on non-normality, capable to produce rapid transient growth of the disturbance energy, and non-linearity, which enables directional redistribution of the amplified disturbances. The coupling between transient growth and directional redistribution of energy causes the flow to orbit around the turbulent fixed point in the phase-space of figure \[PhaseDiag\]. Near the edge of chaos the lifetime of turbulence is finite and beyond a given value of $t$ (function of the Reynolds number and of the streamwise dimension of the periodic domain) relaminarization occurs. Interestingly, the edge state resembles the optimal disturbance; this fact has recently been observed in pipe flow ([@Eckhardt2007; @PringleKerswell]) and still awaits for an explanation. The financial support of the EU (program Marie Curie EST <span style="font-variant:small-caps;">FLUBIO</span> 20228-2006) and of the Italian Ministry of University and Research (PRIN 2005-092015-002) are gratefully acknowledged. [^1]: With this resolution we obtain an excellent match with the results by [@Gavrilakis1992] for $Re_\tau=300$.
--- abstract: 'The radius of convexity of two normalized Bessel functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic functions, the Hadamard factorization of some Dini functions, properties of the zeros of Dini functions via Lommel polynomials and some inequalities for complex and real numbers.' address: - 'Department of Economics, Babeş-Bolyai University, 400591 Cluj-Napoca, Romania' - 'Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary' - 'Department of Mathematics and Informatics, Sapientia Hungarian University of Transylvania, 540485 Târgu-Mureş, Romania' author: - Árpád Baricz - Róbert Szász title: The radius of convexity of normalized Bessel functions --- Introduction and the Main Results ===================================== Let $\mathbb{D}(z_0,r)=\{z\in\mathbb{C}:\|z-z_0\|<r\}$ denote the open disk centered at $z_0$ and of radius $r>0.$ Let $\mathcal{A}$ be the class of analytic and univalent functions which are defined on the disk $\mathbb{D}(0,r),$ and are normalized by the conditions $f(0)=f'(0)-1=0$. Note that a function $f\in\mathcal{A}$ is convex in $\mathbb{D}(0,r)$ if and only if $${\operatorname{Re}}\left(1+\frac{zf''(z)}{f'(z)}\right)>0 \ \mbox{for all}\ z\in \mathbb{D}(0,r).$$ Moreover, we say that $f$ is a convex function of order $\alpha\in[0,1)$ in $\mathbb{D}(0,r)$ if $${\operatorname{Re}}\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha \ \mbox{for all}\ z\in \mathbb{D}(0,r),$$ and the real number $$r^c_{f}(\alpha)=\sup\left\{r\in(0,\infty):\ {\operatorname{Re}}\left(1+\frac{zf''(z)}{f'(z)}\right) >\alpha,\ \mbox{for all}\ z\in{\mathbb{D}}(0,r)\right\}$$ is called the radius of convexity of order $\alpha$ of the function $f$. The real number $r^{c}_f(0)=r^{c}_f$ is in fact the largest radius for which the image domain $f(\mathbb{D}(0,r^c_f))$ is convex. The Bessel function of the first kind is defined by the following power series $$J_{\nu}(z)=\sum_{n\geq0}\frac{(-1)^{n}\left(z/2\right)^{2n+\nu}}{n!\Gamma(n+\nu+1)}.$$ The radii of convexity (and of starlikeness) of the next three kind of normalized Bessel functions $$f_{\nu}(z)=(2^{\nu}\Gamma(1+\nu)J_{\nu}(z))^{{1}/{\nu}},\nu\neq{0},$$ $$g_{\nu}(z)=2^{\nu}\Gamma(1+\nu)z^{1-\nu}J_{\nu}(z),$$ $$h_{\nu}(z)=2^{\nu}\Gamma(1+\nu)z^{1-\nu/2}J_{\nu}(z^{\frac{1}{2}})$$ were investigated in the papers [@bks; @bs; @br; @sz], see also the references therein. The paper [@bs] contains the radius of convexity of the functions $f_{\nu},$ $g_{\nu}$ and $h_{\nu}$ in the case when $\nu>-1.$ In the proof of the main results of [@bs] it was essential that the Bessel functions of the first kind have only real zeros and also the fact that some Dini functions of the form $z\mapsto aJ_{\nu}(z)+bzJ'_{\nu}(z)$ have only real zeros too. This time we deal with the convexity of $g_{\nu}$ and $h_{\nu}$ in the case when $\nu\in(-2,-1).$ In this case the function $z\mapsto aJ_{\nu}(z)+bzJ'_{\nu}(z)$ has complex zeros too and this complicates the problem. If $\nu\in(-2,-1),$ then the method which has been used in [@bs] is not applicable directly. To face the difficulty we first prove some results on the zeros of Dini functions when $\nu\in(-2,-1).$ We follow the method of Hurwitz and we use the Lommel polynomials, see Lemma \[lemroots\] and its proof. Since our method in this paper cannot be used for the function $f_{\nu},$ we note that it would of interest to see the radius of convexity of $f_{\nu}$ when $\nu\in(-1,0)$ or $(-2,-1).$ The paper is organized as follows: at the end of this section we present the main results; section 2 contains the preliminary results and their proofs; while section 3 contains the proofs of the main results. It is worth to mention that Lemma \[lemroots\], Lemma \[lemg\] and Lemma \[lemh\] are the key tools of the main results, but actually they are of independent interest and may be useful in other problems related to Bessel functions and zeros of Bessel functions. Here and in the sequel $I_{\nu}$ denotes the modified Bessel function of the first kind and order $\nu.$ Our main results of the paper are the following theorems. \[th1\] If $\nu\in(-2,-1)$ and $\alpha\in[0,1),$ then the radius of convexity of order $\alpha$ of the function $g_{\nu}$ is the smallest positive root of the equation $$\label{k2ac7zdl0m} 1+r\frac{rI_{\nu+2}(r)+3I_{\nu+1}(r)}{I_{\nu}(r)+rI_{\nu+1}(r)}=\alpha.$$ \[th2\] If $\nu\in(-2,-1)$ and $\alpha\in[0,1),$ then the radius of convexity of order $\alpha$ of the function $h_{\nu}$ is the smallest positive root of the equation $$\label{neweq}1+\frac{rI_{\nu+2}(r^{\frac{1}{2}})+4r^{\frac{1}{2}}I_{\nu+1}(r^{\frac{1}{2}})}{4I_{\nu}(r^{\frac{1}{2}})+ 2r^{\frac{1}{2}}I_{\nu+1}(r^{\frac{1}{2}})}=\alpha.$$ Preliminaries ================= In order to prove the main results we need the following preliminary results. The first lemma contains some know results of Hurwitz on Lommel polynomials (see [@hu; @wa1] and [@wa p. 305]) $$g_{2m,\nu}(z)=\sum_{n=0}^m(-1)^n\frac{(2m-n)!}{n!(2m-2n)!}\frac{\Gamma(\nu+2m-n+1)}{\Gamma(\nu+n+1)}z^n.$$ \[lemHur\] Let $m,s\in\mathbb{N}$ and $\nu\in\mathbb{R}.$ The following statements are valid: 1. If $\nu>-1,$ then the Lommel polynomial $g_{2m,\nu}(z)$ has only positive real zeros. 2. If $\nu\in(-2s-2,-2s-1),$ then the polynomial $g_{2m,\nu}(z)$ has $2s$ complex zeros and $m-2s$ real zeros. Between the real zeros a root is negative and $m-2s-1$ are positive. 3. If $\nu\in(-2s-1,-2s),$ then $g_{2m,\nu}(z)$ has $2s$ complex zeros and $m-2s$ positive real zeros. The next result is very important in the proof of the main results and may be of independent interest. We note that Lemma \[lemroots\] complement the result of Spigler [@spigler], who proved that if $\nu>-1$ and $-a/b>\nu,$ then the equation $aJ_{\nu}(z)+bzJ_{\nu}'(z)=0$ has an infinite sequence of positive roots and two purely imaginary roots. \[lemroots\] If $\alpha\geq0,$ then the following assertions are true: 1. If $\nu>-1,$ then all the zeros of the equation $J_\nu(z)+\alpha{z}J'_\nu(z)=0$ are real. 2. If $\nu\in(-2,-1),$ then the function $z\mapsto J_\nu(z)+\alpha{z}J'_\nu(z)$ has two purely imaginary zeros and all the other zeros are real. We have to discuss the case $\alpha>0,$ as the case $\alpha=0$ is well-known due to Hurwitz’s result on zeros of Bessel functions of the first kind, see [@hu]. Let $g_{m,\nu}$ be the $m$-th Lommel polynomial. According to Lemma \[lemHur\] if $\nu>-1,$ then $g_{2m,\nu}(z)$ has only positive real zeros; and if $\nu\in(-2,-1),$ then $g_{2m,\nu}(z)$ has only real zeros, of which exactly one is negative and all the others are positive. We consider the function $\omega:\mathbb{C}\rightarrow\mathbb{R},$ defined by $\omega(z)=z^\frac{1}{\alpha}g_{2m,\nu}(z).$ Since $\omega(0)=0,$ the Rolle theorem implies that the equation $\omega'(z)=0$ has only positive real zeros if $\nu\in(-1,\infty),$ and for $\nu\in(-2,-1)$ has a negative zero and $m-1$ positive different zeros. The equation $\omega'(z)=0$ is equivalent to $g_{2m,\nu}(z)+\alpha{z}g_{2m,\nu}'(z)=0.$ Thus, the polynomial $h_{m,\nu}(z)=g_{2m,\nu}(z)+\alpha{z}g_{2m,\nu}'(z)$ of degree $m$ has only positive zeros if $\nu\in(-1,\infty),$ and has exactly one negative zero and $m-1$ positive zeros for $\nu\in(-2,-1).$ If $x_1<x_2<{\dots}<x_m$ denote the zeros of $g_{2m,\nu}(z)$ and $y_1<y_2<{\dots}<y_m$ the zeros of $h_{m,\nu}(z),$ then the following inequalities hold $$\label{f1s0a8d662wehl14} x_1<y_1<0<y_2<x_2<y_3<x_3<{\dots}<x_{m-1}<y_m<x_m.$$ On the other hand, we have [@wa p. 484] $$\lim_{m\rightarrow\infty}\frac{h_{m,\nu}(z)}{\Gamma(\nu+2m+1)}=f_\nu(z)+\alpha{z}f_\nu'(z),$$ where $$f_\nu(z)=2^{\nu}z^{-\nu}J_{\nu}(2\sqrt{z})=\sum_{n\geq0}\frac{(-1)^nz^n}{n!\Gamma(\nu+n+1)},$$ and the convergence is uniform on compact subsets of $\mathbb{C}.$ Thus, it follows that $z\mapsto f_\nu(z)+\alpha{z}f_\nu'(z)$ has exactly one negative real zero and all the other zeros are real and positive. This means that if we denote by $\{\pm\alpha_{\nu,n}\|n\in\mathbb{N^}\}$ the set of the zeros of the equation $J_\nu(z)+\alpha{z}J'_{\nu}(z)=0,$ where $\alpha_{\nu,1}=ia,$ $a>0$ and $0={\operatorname{Re}}\alpha_{\nu,1}<\alpha_{\nu,2}<\alpha_{\nu,3}<\ldots,$ then (\[f1s0a8d662wehl14\]) implies that $$0<a<b, \ 0<j_{\nu,2}<\alpha_{\nu,2}<j_{\nu,3}<\alpha_{\nu,3}<j_{\nu,4}<\alpha_{\nu,4}<\ldots,$$ where $j_{\nu,1}=ib, \ 0<j_{\nu,2}<j_{\nu,3}<\ldots$ are the zeros of $J_{\nu}(z)=0.$ The next two lemmas have been proved in [@bs] provided that $\nu>-1.$ The main tool in the proofs of these lemmas were the following estimations: if $H_\nu^{(1)}$ and $H_\nu^{(2)}$ denote the Bessel functions of the third kind then the following asymptotic expansions hold $$H_\nu^{(1)}(w)=\left(\frac{2}{\pi{w}}\right)^{\frac{1}{2}}e^{i(w-\frac{1}{2}\nu\pi-\frac{1}{4}\pi)}(1+\eta_{1,\nu}(w)),\ H_\nu^{(2)}(w)=\left(\frac{2}{\pi{w}}\right)^{\frac{1}{2}}e^{-i(w-\frac{1}{2}\nu\pi-\frac{1}{4}\pi)}(1+\eta_{2,\nu}(w)),$$ where $\eta_{1,\nu}(w)$ and $\eta_{2,\nu}(w)$ are $\mathcal{O}(1/w),$ when $\|w\|$ is large. Since these estimations hold for every $\nu\in\mathbb{C}$ (see [@wa p. 198]) and the condition $\nu>-1$ has not been used, it follows that the following two lemmas hold for every $\nu\in\mathbb{R}.$ \[lemg\] Let $z\in\mathbb{C}$ and let $\alpha_{\nu,n},$ $n\in\mathbb{N},$ denote the $n$-th zero of the equation $J_\nu(z)-zJ_{\nu+1}(z)=0,$ where $0\leq{\operatorname{Re}}\alpha_{\nu,1}\leq{\operatorname{Re}}\alpha_{\nu,2}\leq{\dots}\leq{\operatorname{Re}}\alpha_{\nu,n}\leq{\dots}.$ The following development holds for every $z\neq{\alpha_{\nu,n}}$ $$\label{bbb4nhs4569033}\frac{g''_{\nu}(z)}{g'_{\nu}(z)}=\frac{zJ_{\nu+2}(z)-3J_{\nu+1}(z)}{J_\nu(z)-zJ_{\nu+1}(z)} =-\sum_{n\geq1}\frac{2z}{\alpha^2_{\nu,n}-z^2}.$$ \[lemh\] Let $z\in\mathbb{C}$ and let $\beta_{\nu,n},$ $n\in\mathbb{N},$ denote the $n$-th zero of the equation $(2-\nu)J_\nu(z)+zJ_\nu'(z)=0,$ where $0\leq{\operatorname{Re}}\beta_{\nu,1}\leq{\operatorname{Re}}\beta_{\nu,2}\leq{\dots}\leq{\operatorname{Re}}\beta_{\nu,n}\leq{\dots}.$ The following development holds for each $z\neq\beta_{\nu,n}$ $$\label{bbb4nhs456x1s2d5f69033}z\frac{h_{\nu}''(z)}{h_{\nu}'(z)}= \frac{\nu(\nu-2)J_\nu(z^{\frac{1}{2}})+(3-2\nu)z^{\frac{1}{2}}J_\nu'(z^{\frac{1}{2}})+ zJ_\nu''(z^{\frac{1}{2}})}{2(2-\nu)J_\nu(z^{\frac{1}{2}})+2z^{\frac{1}{2}}J_\nu'(z^{\frac{1}{2}})}=-\sum_{n\geq1}\frac{z}{\beta^2_{\nu,n}-z}.$$ To prove the main results we will need also the next results. If $v\in{\mathbb{C}},$ $\delta\in{\mathbb{R}}$ and $\delta>\|v\|,$ then $$\label{a} \frac{\|v\|}{\delta-\|v\|}\geq {{\operatorname{Re}}}\left(\frac{v}{\delta-v}\right)\geq\frac{-\|v\|}{\delta+\|v\|},$$ $$\ \label{abc2s4d} \frac{\|v\|}{\delta+\|v\|}\geq{\operatorname{Re}}\left(\frac{v}{\delta+v}\right)\geq\frac{-\|v\|}{\delta-\|v\|}.$$ Moreover, if $v\in{\mathbb{C}},$ $\gamma,\delta\in{\mathbb{R}}$ and $\gamma\geq\delta>r\geq\|v\|,$ then $$\label{4l5h6ddg89u}\frac{r^2}{(\delta-r)(\gamma+r)}\geq {\operatorname{Re}}\left(\frac{v^2}{(\delta+v)(\gamma-v)}\right).$$ In order to prove (\[a\]), we denote $v=re^{i\theta}$ and we consider the function $u:[0,2\pi]\rightarrow\mathbb{R},$ defined by $$u(\theta)={\operatorname{Re}}\frac{v}{\delta-v}=\frac{r\delta\cos\theta-r^2}{\delta^2+r^2-2r\delta\cos\theta}.$$ Since $$u'(\theta)=\frac{\delta{r}(r^2-\delta^2)\sin\theta}{(\delta^2+r^2-2r\delta\cos\theta)^2},$$ the function $u$ is decreasing on $[0,\pi]$ and increasing on $[\pi,2\pi],$ which implies that $$\frac{\|v\|}{\delta-\|v\|}=u(0)=u(2\pi)\geq {\operatorname{Re}}\left(\frac{v}{\delta-v}\right)\geq{u(\pi)}=\frac{-\|v\|}{\delta+\|v\|}.$$ The inequality (\[abc2s4d\]) can be proved in a similar way. Now, in order to prove (\[4l5h6ddg89u\]) we note that it is enough to prove this inequality in the case $v=re^{i\theta},$ $\theta\in[0,2\pi].$ Thus, we have to show that $$w(\pi)\geq{w(\theta)}, \ \textrm{where} \ w(\theta)=\frac{\gamma\delta\cos2\theta+r(\gamma-\delta)\cos\theta-r^2}{(\gamma^2+r^2-2\gamma{r}\cos\theta)(\delta^2+r^2+2\delta{r}\cos\theta)},\ \theta\in[0,2\pi].$$ Denoting $t=\cos\theta$ we obtain that this inequality is equivalent to $w_1(-1)\geq{w_1(t)},$ where $t\in[-1,1]$ and $$w(\theta)=w_1(t)=\frac{2\gamma\delta{t^2}+r(\gamma-\delta)t-r^2-\gamma\delta}{(\gamma^2+r^2-2\gamma{r}t)(\delta^2+r^2+2\delta{r}t)},$$ and this can be rewritten as $w_2(t)\geq0,$ where $t\in[-1,1]$ and $$w_2(t)=(\gamma^2+r^2-2\gamma{r}t)(\delta^2+r^2+2\delta{r}t)- (2\gamma\delta{t^2}+r(\gamma-\delta)t-r^2-\gamma\delta)(\gamma+r)(\delta-r).$$ We note that $w_2(t)$ is a polynomial of degree two, and its roots satisfy $t_1=-1$ and $$t_1t_2=-\frac{(\gamma^2+r^2)(\delta^2+r^2)+(r^2+\gamma\delta)(\gamma+r)(\delta-r)}{4\gamma\delta{r^2}+2\gamma\delta(\gamma+r)(\delta-r)}.$$ Thus, we get $$t_2=\frac{(\gamma^2+r^2)(\delta^2+r^2)+(r^2+\gamma\delta)(\gamma+r)(\delta-r)}{4\gamma\delta{r^2}+2\gamma\delta(\gamma+r)(\delta-r)}.$$ Consequently, the inequality $w_2(t)\geq0$ is equivalent to $(1+t)(t_2-t)\geq0,$ $t\in[-1,1].$ In order to finish the proof we have to show that $t_2\geq1.$ A short calculation shows that this inequality is equivalent to $$r(\gamma\delta-r^2)(\gamma-\delta)+r^2(\gamma-\delta)^2\geq0,$$ and with this the proof is done. Proofs of the Main Results* ============================== By using $$z\frac{g_{\nu}''(z)}{g_{\nu}'(z)}=\frac{\nu(\nu-1)J_\nu(z)+2(1-\nu)zJ_\nu'(z)+z^2J_\nu''(z)}{(1-\nu)J_\nu(z)+zJ_\nu'(z)},$$ the fact that $J_{\nu}$ is a particular solution of the Bessel differential equation, and the recurrence formula $zJ'_\nu(z)=\nu{J_\nu(z)}-zJ_{\nu+1}(z),$ it follows that $$z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}=z\frac{zJ_{\nu+2}(z)-3J_{\nu+1}(z)}{J_{\nu}(z)-zJ_{\nu+1}(z)}.$$ In view of we obtain $$1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}=1-2\sum_{n\geq1}\frac{z^2}{\alpha_{\nu,n}^2-z^2}.$$ By using Lemma \[lemroots\] for $\alpha=(1-\nu)^{-1},$ the condition $\nu\in(-2,-1)$ implies $\alpha_{\nu,1}=ia,$ $a>0$ and $\alpha_{\nu,n}>0$ for $n\in\{2,3,\dots\}.$ Thus, we get $$1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}=1+\frac{2z^2}{a^2+z^2}-2\sum_{n\geq2}\frac{z^2}{\alpha_{\nu,n}^2-z^2} =1+\frac{1}{a^2}\frac{2z^2}{1+\frac{z^2}{a^2}}-2\sum_{n\geq2}\frac{z^2}{\alpha_{\nu,n}^2-z^2}.$$ On the other hand, the convergence of the function series in (\[bbb4nhs4569033\]) is uniform on every compact subset of $\mathbb{C}\setminus\{\alpha_{\nu,n}\|n\in\mathbb{N}\}.$ Integrating both sides of the equality (\[bbb4nhs4569033\]), it follows that $$\begin{aligned} \label{bbb4ngkmhs4569mnd033}{g'_{\nu}(z)}=\prod_{n\geq1}\left(1-\frac{z^2}{\alpha^2_{\nu,n}}\right).\end{aligned}$$ The convergence of the infinite product is uniform on every compact subset of $\mathbb{C}.$ Comparing the coefficients of $z^2$ on both sides of (\[bbb4ngkmhs4569mnd033\]) we get the following equality $$\label{ed15s4z7ax4x4x} \sum_{n\geq1}\frac{1}{\alpha^2_{\nu,n}}=\frac{3}{4(\nu+1)}.$$ The equality (\[ed15s4z7ax4x4x\]) implies $$\frac{1}{a^2}=-\frac{3}{4(\nu+1)}+\sum_{n\geq2}\frac{1}{\alpha_{\nu,n}^2}$$ and using this we obtain that $$1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}=1-\frac{3a^2}{2(\nu+1)}\frac{z^2}{a^2+z^2}- 2\sum_{n\geq2}\frac{\alpha_{\nu,n}^2+a^2}{\alpha_{\nu,n}^2}\frac{z^4}{(\alpha_{\nu,n}^2-z^2)(a^2+z^2)}.$$ On the other hand, we have $-\frac{3a^2}{2(\nu+1)}>0,$ and taking $v=z^2$ in the inequality (\[abc2s4d\]) we get $${\operatorname{Re}}\frac{z^2}{a^2+z^2}\geq\frac{-\|z\|^2}{a^2-\|z\|^2}\geq\frac{-r^2}{a^2-r^2},$$ for all $\|z\|\leq{r}<a.$ Moreover, taking $v=z^2$ in inequality (\[4l5h6ddg89u\]) it follows that $${\operatorname{Re}}\frac{z^4}{(\alpha_{\nu,n}^2-z^2)(a^2+z^2)}\leq\frac{r^4}{(\alpha_{\nu,n}^2+r^2)(a^2-r^2)},$$ for all $\|z\|\leq{r}<a<\alpha_{\nu,n}$ and $n\in\{2,3,\dots\}.$ Summarizing, provided that $\|z\|\leq{r}<a,$ the following inequality holds $${\operatorname{Re}}\left(1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}\right)\geq1+\frac{3a^2}{2(\nu+1)}\frac{r^2}{a^2-r^2}- 2\sum_{n\geq2}\frac{\alpha_{\nu,n}^2+a^2}{\alpha_{\nu,n}^2}\frac{r^4}{(\alpha_{\nu,n}^2+r^2)(a^2-r^2)}=\left. \left[{\operatorname{Re}}\left(1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}\right)\right]\right\|_{z=ir}.$$ This means that $${\inf_{z\in\mathbb{D}(0,r)}}{{\operatorname{Re}}}\left(1+z\frac{g''_{\nu}(z)}{g'_{\nu}(z)}\right)= 1+ir\frac{g''_{\nu}(ir)}{g'_{\nu}(ir)}=1-\frac{2r^2}{a^2-r^2}+2\sum_{n\geq2}\frac{r^2}{\alpha_{\nu,n}^2+r^2},$$ for all $r\in(0,a).$ Now, let us consider the function $\varphi:(0,a)\rightarrow\mathbb{R},$ defined by $$\varphi(r)=1-\frac{2r^2}{a^2-r^2}+2\sum_{n\geq 2}\frac{r^2}{\alpha_{\nu,n}^2+r^2}=1+ir\frac{g''_{\nu}(ir)}{g'_{\nu}(ir)}.$$ This function satisfy $\lim_{r\searrow0}\varphi(r)=1>\alpha, \ \lim_{r\nearrow{a}}\varphi(r)=-\infty,$ and $$\varphi'(r)=-\frac{4ra^2}{(a^2-r^2)^2}+\sum_{n\geq 2}\frac{4r\alpha_{\nu,n}^2}{(\alpha_{\nu,n}^2+r^2)^2}< -\frac{4ra^2}{(a^2-r^2)^2}+\sum_{n\geq 2}\frac{4r}{\alpha_{\nu,n}^2}=-\frac{4ra^2}{(a^2-r^2)^2}+\frac{4r}{a^2}+\frac{3r}{\nu+1}<\frac{3r}{\nu+1}.$$ In other words, the function $\varphi$ maps $(0,a)$ into $(-\infty,1)$ and is strictly decreasing. Thus the equation $1+ir\frac{g''_{\nu}(ir)}{g'_{\nu}(ir)}=\alpha$ has exactly one root in the interval $(0,a),$ and this equation is equivalent to (\[k2ac7zdl0m\]). If we denote by $r_2\in(0,a)$ the unique root of the equation $1+ir\frac{g''_{\nu}(ir)}{g'_{\nu}(ir)}=\alpha,$ then by using the minimum principle of harmonic functions we have that $${\operatorname{Re}}\left(1+\frac{zg_{\nu}''(z)}{g_{\nu}'(z)}\right)>\alpha\ \ \ \mbox{for all}\ \ \ z\in{\mathbb{D}(0,r_2)},$$ $$\inf_{{z\in{\mathbb{D}(0,r_2)}}}{\operatorname{Re}}\left(1+\frac{zg_{\nu}''(z)}{g_{\nu}'(z)}\right)=\alpha,\ \ \inf_{{z\in{\mathbb{D}(0,r)}}}{\operatorname{Re}}\left(1+\frac{zg_{\nu}''(z)}{g_{\nu}'(z)}\right)<\alpha, \ \ \textrm{for} \ \ r>r_2,$$ and the proof is done. Lemma \[lemh\] implies that $$z\frac{h_{\nu}''(z)}{h_{\nu}'(z)}=\frac{\nu(\nu-2)J_\nu(z^{\frac{1}{2}})+(3-2\nu)z^{\frac{1}{2}}J_\nu'(z^{\frac{1}{2}})+ zJ_\nu''(z^{\frac{1}{2}})}{2(2-\nu)J_\nu(z^{\frac{1}{2}})+2z^{\frac{1}{2}}J_\nu'(z^{\frac{1}{2}})}=-\sum_{n\geq1}\frac{z}{\beta^2_{\nu,n}-z}.$$ By using Lemma \[lemroots\] for $\alpha=(2-\nu)^{-1},$ if $\nu\in(-2,-1),$ then $\beta_{1,\nu}=ic$ and $0<\beta_{\nu,2}<\beta_{\nu,3}<{\dots}<\beta_{\nu,n}<{\dots},$ and we infer $$1+z\frac{h_{\nu}''(z)}{h_{\nu}'(z)}=1+\frac{z}{c^2+z}-\sum_{n\geq2}\frac{z}{\beta^2_{\nu,n}-z}.$$ On the other hand, by integrating both sides of the equality (\[bbb4nhs456x1s2d5f69033\]), it follows that $$\label{bbb4nhg45d1s456x1s2d5f69033}h_\nu(z)=\prod_{n\geq1}\left(1-\frac{z}{\beta^2_{\nu,n}}\right).$$ Comparing the coefficients of $z$ on both sides of (\[bbb4nhg45d1s456x1s2d5f69033\]) we get the following equality $$\sum_{n\geq1}\frac{1}{\beta^2_{\nu,n}}=\frac{1}{2(\nu+1)},$$ which in turn implies that $$-\frac{1}{c^2}+\sum_{n\geq 2}\frac{1}{\beta^2_{\nu,n}}=\frac{1}{2(\nu+1)}.$$ Thus, we get $$\label{x1c28f4sj146669}1+z\frac{h_{\nu}''(z)}{h_{\nu}'(z)}= 1-\frac{c^2}{2(\nu+1)}\frac{z}{c^2+z}+\sum_{n\geq2}\left[-\frac{c^2+\beta^2_{\nu,n}}{\beta^2_{\nu,n}} \frac{z^2}{(c^2+z)(\beta^2_{\nu,n}-z)}\right].$$ If $c^2>r\geq\|z\|,$ then the second inequality of (\[abc2s4d\]) implies that $${\operatorname{Re}}\left[-\frac{c^2}{2(\nu+1)}\frac{z}{c^2+z}\right]\geq\frac{c^2}{2(\nu+1)}\frac{r}{c^2-r}.$$ Since for $n\in\{2,3,\dots\}$ we have $\beta_{\nu,n}^2>c^2>r\geq\|z\|,$ by using the inequality (\[4l5h6ddg89u\]) it follows that $${\operatorname{Re}}\left[-\frac{c^2+\beta^2_{\nu,n}}{\beta^2_{\nu,n}}\frac{z^2}{(c^2+z)(\beta^2_{\nu,n}-z)}\right] \geq-\frac{c^2+\beta^2_{\nu,n}}{\beta^2_{\nu,n}}\frac{r^2}{(c^2-r)(\beta^2_{\nu,n}+r)}.$$ If $c^2>r\geq\|z\|,$ then these two inequalities and (\[x1c28f4sj146669\]) together imply that $${\operatorname{Re}}\left[1+z\frac{h_{\nu}''(z)}{h_{\nu}'(z)}\right] \geq1+\frac{c^2}{2(\nu+1)}\frac{r}{c^2-r}-\sum_{n\geq2}\frac{c^2+\beta^2_{\nu,n}}{\beta^2_{\nu,n}} \frac{r^2}{(c^2-r)(\beta^2_{\nu,n}+r)}=1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)}.$$ Thus, we obtain $${\inf_{z\in\mathbb{D}(0,r)}}{\operatorname{Re}}\left(1+z\frac{h''_{\nu}(z)}{h'_{\nu}(z)}\right)=1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)} \ \textrm{for \ all} \ \ r\in(0,c^2).$$ Now, consider the function $\phi:(0,c^2)\rightarrow\mathbb{R},$ defined by $$\phi(r)=1+\frac{c^2}{2(\nu+1)}\frac{r}{c^2-r}-\sum_{n\geq2}\frac{c^2+\beta^2_{\nu,n}}{\beta^2_{\nu,n}}\frac{r^2}{(c^2-r)(\beta^2_{\nu,n}+r)} =1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)}.$$ Since $$\lim_{r\searrow0}\phi(r)=1, \ \lim_{r\nearrow{c^2}}\phi(r)=-\infty,$$ it follows that the equation $1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)}=\alpha$ has at last one real root in the interval $(0,c^2).$ Let $r_3$ denote the smallest positive real root of the equation $1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)}=\alpha.$ We have $${\operatorname{Re}}\left(1+\frac{zh_{\nu}''(z)}{h_{\nu}'(z)}\right)>\alpha, \ z\in{\mathbb{D}(0,r_3)},$$ and if $r>r_3$ then $$\inf_{z\in{\mathbb{D}(0,r)}}{\operatorname{Re}}\left(1+\frac{zh_{\nu}''(z_0)}{h_{\nu}'(z)}\right)<\alpha.$$ In order to finish the proof we remark that $1-r\frac{h_{\nu}''(-r)}{h_{\nu}'(-r)}=\alpha$ is equivalent to . 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--- abstract: 'A $(3+1)$-dimensional Einstein-Gauss-Bonnet theory of gravity has been recently formulated as the $D \to 4$ limit of the higher dimensional field equations after the rescaling of the coupling constant. This theory is a nontrivial generalization of General Relativity, it bypasses the Lovelock’s theorem, and avoids the Ostrogradsky instability. This approach has been extended to the four-dimensional Einstein-Lovelock gravity. Here we study the eikonal gravitational instability of asymptotically flat, de Sitter and anti-de Sitter black holes in the four dimensional Einstein-Gauss-Bonnet and Einstein-Lovelock theories. We find parametric regions of the eikonal instability for various orders of the Lovelock gravity, values of coupling and cosmological constants, and share the code which allows one to construct the instability region for an arbitrary set of parameters.' author: - 'R. A. Konoplya' - 'A. Zhidenko' title: '(In)stability of black holes in the $4D$ Einstein-Gauss-Bonnet and Einstein-Lovelock gravities' --- Introduction ============ Stability of a black-hole metric against small (linear) perturbations of spacetime is the necessary condition for viability of the black-hole model under consideration. Therefore, a number of black-hole solutions in various alternative theories of gravity were tested for stability [@Konoplya:2011qq]. One of the most promising approaches to construction of alternative theories of gravity is related to the modification of the gravitational sector via adding higher curvature corrections to the Einstein action. This is well motivated by the low energy limit of string theory. Among higher curvature corrections, the Gauss-Bonnet term (quadratic in curvature) and its natural generalization to higher orders of curvature in the Lovelock form [@Lovelock:1971yv; @Lovelock:1972vz] play an important role. The Lovelock theorem states that only metric tensor and the Einstein tensor are divergence free, symmetric, and concomitant of the metric tensor and its derivatives in four dimensions [@Lovelock:1971yv; @Lovelock:1972vz]. Therefore, it was concluded that the appropriate vacuum equations in $D=4$ are the Einstein equations (with the cosmological term). In $D>4$ the theory of gravity is generalized by adding higher curvature Lovelock terms [@Lovelock:1971yv; @Lovelock:1972vz] to the Einstein action. Black hole in the $D>4$ Einstein-Gauss-Bonnet gravity and its Lovelock generalization were extensively studied and various peculiar properties were observed. For example, the life-time of the black hole whose geometry is only slightly corrected by the Gauss-Bonnet term is characterized by a much longer lifetime and a few orders smaller evaporation rate [@Konoplya:2010vz]. The eikonal quasinormal modes in the gravitational channel break down the correspondence between the eikonal quasinormal modes and null geodesics [@Cardoso:2008bp; @Konoplya:2017wot]. However, apparently the most interesting feature of higher curvature corrected black hole is the gravitational instability: When the coupling constants are not small enough, the black holes are unstable and the instability develops at high multipoles numbers [@Dotti:2005sq; @Gleiser:2005ra; @Konoplya:2017lhs; @Konoplya:2017zwo; @Yoshida:2015vua; @Takahashi:2011qda; @Konoplya:2008ix; @Cuyubamba:2016cug; @Takahashi:2012np]. Therefore, it was called the eikonal instability [@Konoplya:2017lhs; @Konoplya:2017zwo]. Recently, there was found the way to bypass the Lovelock’s theorem [@Glavan:2019inb] by performing a kind of dimensional regularization of the Gauss-Bonnet equations and obtain a four-dimensional metric theory of gravity with diffeomorphism invariance and second order equations of motion. The approach was first formulated in $D > 4$ dimensions and then, the four-dimensional theory is defined as the limit $D \to 4$ of the higher-dimensional theory after the rescaling of the coupling constant $\alpha \to \alpha/(D-4)$. The properties of black holes in this theory, such as (in)stability, quasinormal modes and shadows, were considered in [@Konoplya:2020bxa], while the innermost circular orbits were analyzed in [@Guo:2020zmf]. The generalization to the charged black holes and an asymptotically anti-de Sitter and de Sitter cases in the $4D$ Einstein-Gauss-Bonnet theory was considered in [@Fernandes:2020rpa] and to the higher curvature corrections, that is, the $4D$ Einstein-Lovelock theory, in [@Konoplya:2020qqh; @Casalino:2020kbt]. Some further properties of black holes for this novel theory, such as axial symmetry and thermodynamics, were considered in [@Wei:2020ght; @Rahul-Kumar; @Hegde:2020xlv]. Here we will study the linear stability of asymptotically flat, de Sitter and anti-de Sitter black holes in the $4D$ Einstein-Lovelock gravity. We will plot the parametric regions of the eikonal instability for the $4D$ Einstein-Gauss-Bonnet-(aniti-)de Sitter black holes and various examples for its Lovelock extension. We will show that the negative values of the coupling constants allows for a larger parametric region of stability than the positive coupling constants. Some inequalities determining the instability region are derived. Our paper is organized as follows. In Sec. \[sec:blackhole\] we briefly describe the static black hole solution in the $4D$ Einstein-Lovelock theory. Sec. \[sec:perturbations\] discusses the gravitational perturbations and the eikonal instability. Finally, in Conclusions, we summarize the obtained results. Static black holes in the four-dimensional Lovelock theory {#sec:blackhole} ========================================================== The Lagrangian density of the Einstein-Lovelock theory has the form [@Lovelock:1971yv]: $$\begin{aligned} \label{Lagrangian} \mathcal{L} &=& -2\Lambda+\sum_{m=1}^{\m}\frac{1}{2^m}\frac{\alpha_m}{m} \delta^{\mu_1\nu_1\mu_2\nu_2 \ldots\mu_m\nu_m}_{\lambda_1\sigma_1\lambda_2\sigma_2\ldots\lambda_m\sigma_m}\,\\\nonumber &&\times R_{\mu_1\nu_1}^{\phantom{\mu_1\nu_1}\lambda_1\sigma_1} R_{\mu_2\nu_2}^{\phantom{\mu_2\nu_2}\lambda_2\sigma_2} \ldots R_{\mu_m\nu_m}^{\phantom{\mu_m\nu_m}\lambda_m\sigma_m},\end{aligned}$$ where $\delta^{\mu_1\mu_2\ldots\mu_p}_{\nu_1\nu_2\ldots\nu_p}$ is the generalized totally antisymmetric Kronecker delta, $R_{\mu\nu}^{\phantom{{\mu\nu}}\lambda\sigma}$ is the Riemann tensor, $\alpha_1=1/8\pi G=1$ and $\alpha_2,\alpha_3,\alpha_4,\ldots$ are arbitrary constants of the theory. The Euler-Lagrange equations, corresponding to the Lagrangian density (\[Lagrangian\]) read [@Kofinas:2007ns]: $$\begin{aligned} \nonumber \Lambda\delta^{\mu}_{\nu} &=& R^{\mu}_{\nu}-\frac{R}{2}\delta^{\mu}_{\nu}+\sum_{m=2}^{\m}\frac{1}{2^{m+1}}\frac{\alpha_m}{m} \delta^{\mu\mu_1\nu_1\mu_2\nu_2 \ldots\mu_m\nu_m}_{\nu\lambda_1\sigma_1\lambda_2\sigma_2\ldots\lambda_m\sigma_m} \\ && \times R_{\mu_1\nu_1}^{\phantom{\mu_1\nu_1}\lambda_1\sigma_1} R_{\mu_2\nu_2}^{\phantom{\mu_2\nu_2}\lambda_2\sigma_2} \ldots R_{\mu_m\nu_m}^{\phantom{\mu_m\nu_m}\lambda_m\sigma_m}\,. \label{Lovelock}\end{aligned}$$ The antisymmetric tensor is nonzero only when the indices $\mu,\mu_1,\nu_1,\mu_2,\nu_2,\ldots\mu_m,\nu_m$ are all distinct. Thus, the general Lovelock theory is such that $2\m <D$. In particular, for $D=4$, we have $\m=1$ corresponding to the Einstein theory [@Lovelock:1972vz]. When $D=5$ or $6$, $\m=2$ and one has the (quadratic in curvature) Einstein-Gauss-Bonnet theory with the coupling constant $\alpha_2$. Following [@Konoplya:2017lhs], we introduce $$\label{amdef} \a_m=\frac{\alpha_m}{m}\frac{(D-3)!}{(D-2m-1)!}=\frac{\alpha_m}{m}\prod_{p=1}^{2m-2}(D-2-p)$$ and consider the limit $D\to 4$ while $\a_m$ remain constant. In this way, we obtain the regularized $4D$ Einstein-Lovelock theory formulated in [@Konoplya:2020qqh], which generalizes the approach of [@Glavan:2019inb] used for the Einstein-Gauss-Bonnet theory. In the Einstein-Gauss-Bonnet case ($\m=2$) the above equation reads $$\alpha_2 = \frac{2 \a_2}{(D-3)(D-4)}.$$ Then, taking the limit $D\to 4$ we see that $$\alpha_2 \to \frac{2 \a_2}{D-4}.$$ Notice that our units differ by a factor of $2$ from those used in [@Glavan:2019inb] and coincide with the units of [@Fernandes:2020rpa]. Prior to [@Glavan:2019inb] the dimensional regularization of the Einstein-Gauss-Bonnet theory was suggested by Y. Tomozawa [@Tomozawa:2011gp]. Although the Lagrangian (\[Lagrangian\]) diverges in the limit $D \to 4$, no singular terms appear in the Einstein-Lovelock equations for any $D\geq3$. In particular, following [@Konoplya:2020qqh] one can find the four-dimensional static and spherically symmetric metric, described by the metric $$\label{Lmetric} ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2 + r^2 (d\theta^2+d \sin^2\theta d\phi^2).$$ The metric function $f(r)$ is defined through a new variable $\psi(r)$, $$\label{Lfdef} f(r)=1-r^2\,\psi(r),$$ which satisfies the algebraic equation $$\label{MEq} P[\psi(r)]\equiv\psi(r)+\sum_{m=2}^{\m}\a_m\psi(r)^m=\frac{2M}{r^{3}}+\frac{\Lambda}{3}\,,$$ where $M$ is the asymptotic mass [@Myers:1988ze]. The Gauss-Bonnet theory ($\m=2$) leads to the two branches [@Fernandes:2020rpa]: $$f(r)=1-\frac{r^2}{2\a_2}\left(-1\pm\sqrt{1+4\a_2\left(\frac{2M}{r^3}+\frac{\Lambda}{3}\right)}\right),$$ one of which, corresponding to the “+” sign, is perturbative in $\a_2$, while for the “-” the metric function $f(r)$ goes to infinity when $\a_2 \to 0$. Notice that the above solution was also obtained in [@Cai:2009ua; @Cognola] in a different context, when discussing quantum correction to entropy. The higher-order Lovelock corrections result in more branches, only one of which is perturbative in $\a_m$. Following [@Konoplya:2017lhs], we consider here only the perturbative branch, so that we recover the Einstein theory in the limit $\a_m\to0$. In particular, for $\m=3$ and $\a_3\geq\a_2^2/3$, $$f(r)=1-\frac{\a_2r^2}{3\a_3}\left(A_+(r)-A_-(r)-1\right),$$ where $$\begin{aligned} \nonumber A_{\pm}(r)&=&\sqrt[3]{\sqrt{F(r)^2+\left(\frac{3\a_3}{\a_2^2}-1\right)^3}\pm F(r)},\\\nonumber F(r)&=&\frac{27\a_3^2}{2\a_2^3}\left(\frac{2M}{r^3}+\frac{\Lambda}{3}\right)+\frac{9\a_3}{2\a_2^2}-1\,.\end{aligned}$$ It is convenient to measure all dimensional quantities in units of the horizon radius $r_H$. For the asymptotic mass we obtain $$\label{Mdef} 2M=r_H\left(1+\sum_{m=2}^{\m}\frac{\a_m}{r_H^{2m-2}}\right)-\frac{\Lambda r_H^2}{3}.$$ For $\Lambda>0$ the perturbative branch is asymptotically de Sitter, so that $\Lambda$ can be expressed in terms of the de Sitter horizon $r_C$ as follows $$\begin{aligned} \label{LdSdef} \frac{\Lambda}{3}&=&\frac{1}{r_C^2+r_Cr_H+r_H^2}+\sum_{m=2}^{\m}\a_m \frac{r_C^{3-2m}-r_H^{3-2m}}{r_C^3-r_H^3}.\end{aligned}$$ When $\Lambda<0$, we introduce the AdS radius $R$, by assuming that the metric function has the following asymptotic $f(r)\to r^2/R^2$ as $r\to\infty$, and the cosmological constant is given by $$\label{LAdS} \frac{\Lambda}{3}=-\frac{1}{R^2}+\sum_{m=2}^{\m}\frac{(-1)^m\a_m}{R^{2m}}.$$ The metric function $f(r)$ for the perturbative branch of the general Einstein-Lovelock black hole can be obtained numerically [@Konoplya:2020qqh].[^1] Gravitational perturbations and the eikonal (in)stability {#sec:perturbations} ========================================================= In [@Takahashi:2010] it was shown that after the decoupling of angular variables and some algebra, the gravitational perturbation equations of the higher dimensional Einstein-Gauss-Bonnet theory can be reduced to the second-order master differential equations $$\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial r_^2}+V_i(r_)\right)\Psi_{i}(t,r_)=0,$$ where $\Psi_i$ are the wave functions, $r_$ is the tortoise coordinate, $$dr_\equiv \frac{dr}{f(r)}=\frac{dr}{1-r^2\psi(r)},$$ and $i$ stands for $t$ (tensor), $v$ (vector), and $s$ (scalar) types of gravitational perturbations, according to their transformations respectively the rotation group on a $(D-2)$-sphere. Using the definition of constants $\a_m$ (\[amdef\]) and taking the limit $D\to4$ in the perturbation equations, one can see that the tensor-type perturbations are pure gauge, so that only the vector-type (axial) and scalar-type (polar) perturbations possess physical degrees of freedom. The effective potentials $V_s(r)$, $V_v(r)$ are given by the following expressions: $$\begin{aligned} \nonumber V_v(r)&=&\frac{(\ell-1)(\ell+2)}{rT(r)}\frac{d}{dr_}T(r)+R(r)\frac{d^2}{dr_^2}\Biggr(\frac{1}{R(r)}\Biggr),\\ V_s(r)&=&\frac{\ell(\ell+1)}{rP(r)}\frac{d}{dr_}P(r)+\frac{P(r)}{r}\frac{d^2}{dr_^2}\left(\frac{r}{P(r)}\right), \label{pot}\end{aligned}$$ where $\ell=2,3,4,\ldots$ is the multipole number, $$T(r)\equiv r P'[\psi(r)]=r\left(1+\sum_{m=2}^{\m}m\a_m\psi(r)^{m-1}\right),$$ and $$\begin{aligned} \nonumber R(r)&=&r\sqrt{\|T'(r)\|},\\\nonumber P(r)&=&\frac{2(\ell-1)(\ell+2)-2r^3\psi'(r)}{\sqrt{\|T'(r)\|}}T(r).\end{aligned}$$ Notice that, since we consider the perturbative branch of solutions, then we have $T(r)>0$ for $r>r_H$ [@Konoplya:2017lhs]. Although we can formally obtain expressions for the effective potentials when $T'(r)\leq0$, the kinetic term of perturbations in such points has a wrong (negative) sign. The perturbations are linearly unstable, and this phenomenon was called the ghost* instability [@Takahashi:2010]. Usually, we used to believe that if a gravitational instability takes place, it happens at the lowest $\ell=2$ multipole, while higher multipoles increase the centrifugal part of the effective potential and make the potential barrier higher, so that, usually,, higher $\ell$ are more stable. The eikonal instability we observe here is qualitatively different: higher $\ell$ leads not only to the higher height of the barrier, but also increases the depth of the negative gap near the event horizon. Then, at some sufficiently large $\ell$ the negative gap becomes so deep, that the bound state with negative energy becomes possible, which signifies the onset of instability. For large $\ell$ the effective potential for the vector-type perturbations $$V_{v} = \ell^2 \left(\frac{f(r)rT'(r)}{T(r)} + {\cal O}\left(\frac{1}{\ell}\right)\right)$$ becomes positive-definite in the parametric regime, which is free from the ghost instability. Therefore, the eikonal instability exists only in the scalar channel. For large $\ell$ the effective potential reads $$V_{s}=\ell^2 \left(\frac{rf(r)(2T'(r)^2-T(r)T''(r))}{2T'(r)T(r)} + {\cal O}\left(\frac{1}{\ell}\right)\right),$$ giving the following instability condition [@Takahashi:2010], $$\label{instability-cond} 2T'(r)^2-T(r)T''(r))<0.$$ This condition can be test for each black-hole configuration. In the Einstein-Gauss-Bonnet theory ($\m=2$) it sufficient to test if (\[instability-cond\]) at $r=r_H$ [@Konoplya:2017lhs]. Therefore, the problem is reduced to the polynomial inequality of fourth order in $\a_2$, which can be solved analytically. We find that the eikonal instability occurs if $$\label{GBinstability} \frac{\a_2}{r_H^2}>\frac{\sqrt{6\sqrt{3}-10+\lambda^2}-\lambda}{2},$$ where $\lambda=(2\sqrt{3}-3)\Lambda r_H^2-1$. In the asymptotically flat case ($\Lambda=0$, $\lambda=1$), the black hole has the eikonal instability in the scalar sector for $$\frac{\a_2}{r_H^2}>\frac{\sqrt{6\sqrt{3}-9}-1}{2}\approx0.09.$$ It is possible to show that the ghost instability takes place for larger values of $\a_2$. By substituting (\[LdSdef\]) and (\[LAdS\]) into (\[GBinstability\]) one can find the instability condition in the geometrized units. We show the parametric region of the eikonal instability on Fig. \[fig:Gauss-Bonnet\]. Indeed, we see that the asymptotically flat and (anti-)de Sitter black holes in the Einstein-Gauss-Bonnet theory are stable for the whole range of valid parameters when $\alpha$ is negative and for sufficiently small values of positive $\alpha$. For the higher-order Einstein-Lovelock theory it is not sufficient to test (\[instability-cond\]) in the point $r=r_H$. In order to obtain the region of instability we have used the Wolfram Mathematica code developed in [@Konoplya:2017lhs], which is attached to the arXiv preprint as an ancillary material. From Figs. \[Lovelock-dS\] we see that in the Lovelock theory of the third order ($\m=3$) the positive cosmological constant modifies the (in)stability region relatively softly. As for the four-dimensional Einstein-Gauss-Bonnet black holes, we see the $\Lambda$-term slightly increases the region of stability. For the asymptotically AdS black holes (Figs. \[Lovelock-AdS\]) we see that the region of stability shrinks as we decrease their size in units of the AdS radius $R$. It is interesting to note that for the four-dimensional black holes the ghost instability occurs for $\a_3<\a_2^2/3$ and scalar-type eikonal instability exists for $\a_3<0$. 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--- abstract: 'We illustrate the formation of a thin-long structure of heavy nuclei by three-nucleus simultaneous collisions within time-dependent density functional theory. The impact parameter dependence for such formation is systematically demonstrated through clarifications of the difference between binary and ternary collision events. A new method for producing thin-long heavy nuclei in the laboratory is suggested, as well as the possible formation of the thin-long structure in hot dense matter such as that encountered in core collapse supernovae.' author: - 'Yoritaka Iwata$^{1}$' - 'Kei Iida$^{2}$' - 'Naoyuki Itagaki$^{3}$' title: 'Synthesis of thin-long heavy nuclei in ternary collisions' --- Generally, synthesis of superheavy elements and many other exotic nuclei comes from collisions between two stable nuclei, whereas simultaneous collisions among more than two stable nuclei, which can occur in principle, have not been taken seriously. In experiments using accelerators, it seems difficult to construct a setup to make three nuclei collide simultaneously. Nevertheless it is reasonable to expect that a fixed target can be bombarded by two beams moving in the opposite directions. In astrophysical circumstances, on the other hand, there seem many chances to see three-body reactions such as triple-alpha reactions [@clayton]. The triple-alpha reactions, if occurring through two-body resonances as in the Hoyle picture, are similar to radioactive ion beam experiments [@rnb7] in the sense that two reactions occur non-simultaneously in terms of a time scale of the strong interaction. Even in stars and supernova cores, however, simultaneous three-nucleus collisions within the strong interaction time scale are usually presumed to be rather hard to encounter, let alone more than three nucleus collisions. How hard they are remains to be examined thoroughly, although the possible influence of simultaneous triple-alpha fusion reactions on carbon production in stars has attracted much attention [@ogata]. Two-nucleus fusion reactions in stars and supernova cores are often investigated in two steps [@ichimaru; @sawyer]. One starts with fundamental two-nucleus fusion reactions in vacuum and then incorporates medium (plasma) effects. For example, in the famous Gamow approach, one solves the elementary tunneling problem for given relative kinetic energy and then obtain the temperature-dependent reaction rate by averaging the resulting kinetic-energy-dependent reaction rate over the Boltzmann distribution of the kinetic energy. In this work, we concentrate on the fundamental binary and ternary collisions within the time-dependent density functional theory (TDDFT). Among multi-nucleus collisions, ternary collisions are particularly worth investigating in the sense that there always exists one plane containing all the three center-of-mass coordinates of colliding pairs. Such a geometric restriction facilitates the formation of a low-dimensional quantum system. As we shall see, a thin-long structure of the fusion products can be stabilized by rotation in the case of non-central binary and ternary collisions. The TDDFT approach, which was originally proposed by P. A. M. Dirac [@dirac], is useful for describing nuclear collisions at low energies except the reactions below the Coulomb barrier. In the present case, we perform three-dimensional TDDFT calculations with Skyrme-type effective nucleon-nucleon interactions (SLy6 [@Chabanat-Bonche] and SKI3 [@reinhard]). One can then derive various quantities from the many-nucleon wave function self-consistently obtained in the form of the Slater determinant. We remark that the colliding system would break into a few fragments with various neutron/proton ratios [@iwata-prl] for the collision energy per nucleon of order the nucleon Fermi energy, and multi-fragmentation takes place for rather higher energies. Multi-fragmentation, in which two-nucleon collisions are usually expected to play an important role [@PTEP], is out of our scope. ![\[fig2\] (Color online) The initial positions of three identical $^{56}$Fe nuclei, Nucleus I, Nucleus II, and Nucleus III, which are set to $(\|{\bm b}\|,0,15~{\rm fm})$, $(0,0,0)$, and $(-\|{\bm b}\|,0,-15~{\rm fm})$, respectively. The velocity vector of each nucleus is given as $(0,0,\|{\bm v}\|)$, $(0,0,0)$, and $(0,0,-\|{\bm v}\|)$, respectively. The volume of a box inside which the simulation is performed is $48 \times 48 \times 48$ fm$^3$. ](fig2-1.eps){width="6cm"} \(a) $\|{\bm b}\| = 1$ fm\ ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b01-1.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b01-2.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b01-3.eps "fig:"){width="2.cm"}\ (b) $\|{\bm b}\| = 3$ fm\ ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b03-1.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b03-2.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b03-3.eps "fig:"){width="2.cm"}\ (c) $\|{\bm b}\| = 5$ fm\ ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b05-1.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b05-2.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b05-3.eps "fig:"){width="2.cm"}\ (d) $\|{\bm b}\| = 7$ fm\ ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b07-1.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b07-2.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b07-3.eps "fig:"){width="2.cm"}\ (e) $\|{\bm b}\| = 9$ fm\ ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b09-1.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b09-2.eps "fig:"){width="2.cm"} ![\[fig3\](Color online) Ternary collisions with different impact parameters that start with the initial condition shown in Fig. \[fig2\] (the SLy6 parameter set is taken). The velocity is set to be $1.95 \times 10^{22}$ fm/s. Snapshots at 1.33$\times10^{-22}$ s (left), 13.3$\times10^{-22}$ s (middle), and 26.6$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1, 3, 5, 7 fm, while those at 1.33$\times10^{-22}$ s, 13.3$\times10^{-22}$ s, and 20.0$\times10^{-22}$ s are shown for $\|{\bm b}\|$ = 9 fm.](fig2-b09-3.eps "fig:"){width="2.cm"}\ Since $^{56}$Fe can be abundant in dense stellar matter, we consider heavy element synthesis due to multiple $^{56}$Fe collisions within $\sim 33.3 \times 10^{-22}$ s, which corresponds to the typical duration time of low-energy heavy-ion reactions. We first show an example of ternary collisions with the initial condition shown in Fig. \[fig2\] and the incident energy of $1.47$ MeV per nucleon in the center-of-mass frame, which is illustrated in Fig. \[fig3\]. In this case, the fusion product is $^{168}$Pt. In contrast with reactions producing light elements, the corresponding reactions are endothermic and hence require additional energy even after the contact. For the impact parameters that allow Nuclei I and II as well as Nuclei II and III to touch with each other, corresponding to the cases of $\|{\bm b}\|=1,3,5,7$ fm, the final products have a thin-long structure of length 25-30 fm, which is stabilized by rotation. For example, in the case of $\|{\bm b}\| = 7$ fm, rotation up to $\pi$ rad is achieved in a few 10$^{-21}$ s, which corresponds to the total rotational energy of order $2.10$ MeV. Note that thin-long nuclei can occur at optimal values of the incident energy, below and above which fusion is drastically suppressed. Note also that the stabilization of such a thin-long structure by rotation can be seen in exotic clustering of light nuclei [@ichikawa11]. It has been shown here that thin-long structures are produced by non-central ternary collisions for a broad range of the impact parameter and that the rotation speed becomes faster for larger $\|{\bm b}\|$ as it should. Note that thin-long structures turn into more spherical shapes within $33.3 \times 10^{-22}$ s if there are no rotations, which suggests that the rotation plays an indispensable role in keeping the fusion products thin-long. For comparison, we consider two-nucleus collisions by starting with the same initial configurations as in Fig. \[fig2\] except for the absence of Nucleus II. The corresponding incident energy is $2.20$ MeV per nucleon in the center-of-mass frame, and the fusion product is $^{112}$Te. We can observe from Fig. \[fig4\] that a similar but shorter structure occurs in the case of two-nucleus collisions. The degree of deformation from a spherical case is significantly larger in the case of ternary collisions (see Table \[table1\]). We have thus quantitatively confirmed that ternary collisions are more efficient than binary collisions in producing thin-long structures. Note that fusion is not only one of the most efficient ways of producing heavy elements in stars, but also the main method for producing superheavy nuclei in the laboratory. The present TDDFT calculations suggest that fusion reactions of two or three identical $^{56}$Fe, which lead to the synthesis of chemical elements heavier than iron, can be easier to occur with the help of rotations. \(a) $\|{\bm b}\| = 1$ fm\ ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b01-1.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b01-2.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b01-3.eps "fig:"){width="2.cm"}\ (b) $\|{\bm b}\| = 3$ fm\ ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b03-1.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b03-2.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b03-3.eps "fig:"){width="2.cm"}\ (c) $\|{\bm b}\| = 5$ fm\ ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b05-1.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b05-2.eps "fig:"){width="2.cm"} ![\[fig4\] (Color online) Two-nucleus collisions with different impact parameters. The difference from Fig. \[fig3\] lies solely in the absence of Nucleus II in the initial configurations illustrated by Fig. \[fig2\]. Snapshots at 3.33$\times10^{-22}$ s (left), 6.66$\times10^{-22}$ s (middle), 13.3$\times10^{-22}$ s (right) are shown for $\|{\bm b}\|$ = 1,3,5 fm.](fig4-b05-3.eps "fig:"){width="2.cm"}\ ------------------------ ------------- ------------- --------------- ${\mathcal R}/A^{1/3}$ 4.98 fm 3.63 fm $\sim$2.40 fm ------------------------ ------------- ------------- --------------- : \[table1\] Length of the thin-long products by ternary collision at 26.6$\times10^{-22}$ s and by binary collision at 13.3$\times10^{-22}$ s that are shown in Figs. \[fig3\] and \[fig4\]. The diameter for a minimum sphere that contains each product is calculated and then averaged over all the cases in which the product is fused. The resultant length, denoted by ${\mathcal R}$, is divided by $A^{1/3}$. For comparison, a typical diameter for a spherical nucleus is also given. The efficiency of multi-nucleus collisions in the production of thin-long heavy nuclei can be essentially understood by the competition between the surface tension and the Coulomb repulsion. Let us describe this competition within the framework of an incompressible liquid-drop model, which can characterize a specific geometry of the early stage of multi-nucleus collisions (see Fig. \[fig2-1\] (a)). In this model we restrict ourselves to the situation in which the total volume of the system is fixed to $4 \pi r_0^3 A/3$ with $r_0 = 1.20$ fm and the number of nucleons $A$. Let us now consider a string of $n$ identical nuclei of radius $R = r_0 (A/n)^{1/3}$, which we regard as a precursor of a thin-long heavy nucleus as long as $n>1$ (Fig. \[fig2-1\] (a)). Then, we obtain the length $R_z$ of this string as $$R_z = 2nR = 2r_0 n^{2/3} A^{1/3}.$$ The liquid-drop model allows one to quantify the competition between the surface tension and the Coulomb repulsion, which is essential to the formation of thin-long heavy nuclei. The surface and Coulomb energies of the string can be written as $$E_{\rm Surf} = a_{s} \frac{4n \pi R^2}{4 \pi r_0^2} = a_{s} ~ n^{1/3} A^{2/3}$$ and $$\begin{aligned} E_{\rm Coul} && =\sum_{i>j} a_{c} \frac{(A/ \kappa n)^2}{r_{ij}} + \frac{3}{5} n a_c \frac{(A/\kappa n)^2}{R} \nonumber \\ && = \frac{2 a_{c}}{\kappa^2 r_0} \left[\frac{4}{5} n^{-2/3} + (2^{-n}-1)n^{-5/3} \right] A^{5/3},\end{aligned}$$ where $\kappa=A/Z$ with the number of protons $Z$, $r_{ij}$ is the distance between the centers of the $i$-th and $j$-th nuclei that constitute the string, and $a_{s}$ and $a_{c}$ are set to $17.0$ MeV and $1.38$ MeV fm, respectively, in a manner that is consistent with empirical masses of stable nuclei. Accordingly, the competition is characterized by the sum $$\begin{aligned} \label{liquid-energy} E(\kappa,n,A) &\equiv& E_{\rm Surf} + E_{\rm Coul} \nonumber \\ &=& 17.0~ n^{1/3} A^{2/3} +\frac{2.30}{\kappa^{2}} \left[\frac{4}{5} n^{-2/3} \right. \nonumber \\ && + \left. (2^{-n}-1)n^{-5/3} \right] A^{5/3}~({\rm MeV}). \end{aligned}$$ Although $n$ and $A$ are integers, for optimization we take $n$ and $A$ as real numbers. The extremal condition for given $\kappa$ and $A$ can then be obtained from $$\begin{aligned} &&\partial_n E(\kappa,n,A) \nonumber \\ &=&\frac{850\kappa^2n^22^{n} -[(184n-575)2^n+239n+575]A}{150\kappa^2n^{8/3} 2^n A^{-2/3}} \nonumber \\ && ({\rm MeV}).\end{aligned}$$ \(a) Model\ ![\[fig2-1\] (Color online) An incompressible liquid-drop model for multi-nucleus collisions. (a) Linearly connected $n$ spherical nuclei with the total and proton numbers $(A,Z)$. (b) $\partial_n E(\kappa=2,n,A)$ is depicted as a function of $n$ ($A =$ 162, 199, 212, 225, 262). A state with $n = 1$ loses its global stability at $A = 225$, which is a little bit larger than $A_{0}(\kappa=2)=212$, and, for $A>225$, a state with $n > 2$ is stabilized. ](fig2-1-2.eps "fig:"){width="8cm"}\ (b) Optimal solution\ ![\[fig2-1\] (Color online) An incompressible liquid-drop model for multi-nucleus collisions. (a) Linearly connected $n$ spherical nuclei with the total and proton numbers $(A,Z)$. (b) $\partial_n E(\kappa=2,n,A)$ is depicted as a function of $n$ ($A =$ 162, 199, 212, 225, 262). A state with $n = 1$ loses its global stability at $A = 225$, which is a little bit larger than $A_{0}(\kappa=2)=212$, and, for $A>225$, a state with $n > 2$ is stabilized. ](fig6-2.eps "fig:"){width="8cm"}\ ---------------- -------------- -------------- ------------ $E(2.5,n,400)$ 3.28 GeV 3.30 GeV 3.32 GeV $E(2.5,n,500)$ 4.38 GeV 4.45 GeV 4.55 GeV ---------------- -------------- -------------- ------------ : \[table2\] The energy $E(\kappa,n,A)$ calculated from Eq. as function of $n$. We take $A=400,500$ and $\kappa = 2.5$, which are typical values expected for ternary collisions of heavy nuclei in the laboratory. The mass number satisfying $ \partial_n E(\kappa,n,A_{\rm crit}) = 0$ corresponds to $$A_{\rm crit}(\kappa,n) = \frac{850\kappa^2n^22^{n}}{(184n-575)2^n+239n+575},$$ where the denominator is positive for any $n$. At least one set of the numbers $(\kappa,n,A)$ satisfying $\partial_n E(\kappa,n,A) = 0$ exists only when $A > A_{\rm crit}(\kappa,n)$. Otherwise, $\partial_n E(\kappa,n,A) > 0$ holds. Since, for given $\kappa$, $A_{\rm crit}(\kappa,n)$ has the global minimum value $A_0(\kappa)=212 (\kappa/2)^2$, we can conclude that the value of $A_0(\kappa)$ gives a rough criterion for the occurrence of the stability transition from the state with $n=1$ to the state with $n \ge 2$. In fact, for $A<A_0(\kappa)$ where $\partial_n E(\kappa,n,A) > 0$ is satisfied for any $n$, a single sphere ($n=1$) is energetically preferred, while for $A>A_0(\kappa)$, as shown in Fig. \[fig2-1\](b), there can be more than one value of $n$ that fulfills $\partial_n E(\kappa,n,A) = 0$. As a result of detailed analyses of the energy landscape, we find that a state with $n=1$ remains energetically optimal for mass number up to a value that is a little bit larger than $A_0(\kappa)$. Above this value, the optimal state is not preferably realized in two-nucleus collisions ($n = 2$) but in multi-nucleus collisions ($n \ge 3$). This implies that for superheavy synthesis, production of thin-long nuclei by ternary collisions ($n=3$) can be more efficient than the production by usual binary collisions leading to the states with $n=1$ or $n=2$ (Table \[table2\]). In summary we have found a thin-long structure of heavy nuclei as a result of simultaneous three-nucleus collisions within the TDDFT approach. The validity of the calculations using the SLy6 interaction has been confirmed by using the other interaction (SKI3 [@reinhard]) in terms of the realization and rotational stabilization of thin-long heavy nuclei. This study is expected to provide a motivation for designing a new accelerator and detector system for superheavy synthesis in which three-nucleus simultaneous collisions can take place. In fact, in addition to the existing methods for superheavy synthesis that are based on “binary” fusion and multi-nucleon transfer reactions, “ternary” fusion reactions and subsequent rotational stabilization of compound nuclei could provide a novel method for the superheavy science. It is also interesting to consider possible astrophysical implications. For example, the three-nucleus fusion rate in stars and supernova cores can be estimated by allowing for the average over the initial configurations through the Boltzmann factor and the plasma effects, i.e., electron screening effects and many-body Coulomb correlation effects between ionic nuclei [@ichimaru]. The latter effects act to reduce the Coulomb barrier between the colliding nuclei and thus to enhance the fusion rate. If the plasma is relatively dilute and hot as in the Sun, the plasma effects can be safely ignored. In this case, an extension of the usual Gamow rate to three-nucleus fusion applies, leading to the fusion rate as a function of the plasma temperature $T$. Once the plasma becomes dense and thus strongly coupled as in supernova cores [@bethe], the plasma effects should manifest themselves in the fusion rate through the three-particle static correlation function for electron-screened ions. However, it is a challenging problem to accurately obtain the three-particle correlation function as a function of $T$ and the plasma density $\rho$ [@SPPII]. Recall that there are optimal values of the incident energy for three-nucleus fusion in vacuum. In matter, the optimal values can be lowered by the plasma effects, leading to enhancement of the three-nucleus fusion rate through the Boltzmann factor. If the inverse of this rate at the values of $\rho$ and $T$ relevant for stars and supernova cores is shorter than the corresponding evolutionary time scale, one can expect that thin-long heavy nuclei occur in such celestial objects. This work was supported in part by the Helmholtz Alliance HA216/EMMI and in part by Grants-in-Aid for Scientific Research on Innovative Areas through No. 24105008 provided by MEXT. Y.I. and K.I. acknowledge the hospitality of the Yukawa Institute for Theoretical Physics, where this work was initiated during the international molecule “Physics of structure and reaction of neutron-rich nuclei and surface of neutron stars studied with time-dependent Hartree-Fock approach”. Y. I. thanks Profs. T. Otsuka and J. A. Maruhn for encouragement. [1]{} D. D. Clayton, [Principles of Stellar Evolution and Nucleosynthesis]{} (The University of Chicago Press, Chicago, 1983). 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--- abstract: 'We consider the equation $$-\Delta u = \|x\|^{\alpha} \|u\|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B,$$ where $B \subset {\mathbb R}^2$ is the unit ball centered at the origin, $\alpha \geq0$, $p>1$, and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets $n\geq1$ of the solution $u_{\alpha,n}$, we prove that the Morse index $m(u_{\alpha,n})$ is monotone non-decreasing with respect to $\alpha$. Secondly, we provide a lower bound for the Morse indices $m(u_{\alpha, n})$, which shows that $m(u_{\alpha, n}) \to +\infty$ as $\alpha \to + \infty$.' address: \| Instituto de Ciências Matemáticas e de Computação\ Universidade de São Paulo, CEP 13560-970 - São Carlos - SP - Brazil author: - Wendel Leite da Silva and Ederson Moreira dos Santos title: Monotonicity of the Morse index of radial solutions of the Hénon equation in dimension two --- [^1] Introduction ============ The Hénon equation [@henon] was proposed as a model to study stellar distribution in a cluster of stars with the presence of a black hole located at the center of the cluster. Besides its application to astrophysics, Hénon-type equations also model steady-state distributions in other diffusion processes; see the introduction in [@pacella-indiana] and the references therein for a more precise description on applications. Apart from its applications, the Hénon equation is an excellent prototype for the study of some important problems on the qualitative analysis of solutions of elliptic partial differential equations. For example, the symmetry of least energy solutions and least energy nodal solutions [@smets-su-willem; @BWW; @SW] and some concentration phenomena [@cao-peng-yan; @BW; @pacella-indiana]. In this paper we present some results on the Morse index of radially symmetric solutions. Consider the equation $$\label{1} -\Delta u = g(\|x\|,u) \textrm{ in } \Omega, \quad u =0 \textrm{ on } \partial \Omega.$$ where $\Omega \subset \R^N$, $N\geq 2$, is either a ball or an annulus centered at the origin, $g:[0,+\infty)\times \R \rightarrow \R$ is such that $r\mapsto g(r,u)$ is $C^{0,\beta}$ on bounded sets of $[0,+\infty)\times \R$, $u\mapsto g_u(r,u)$ is $C^{0,\gamma}$ on bounded sets of $[0,+\infty)\times \R$, where $g_u$ denotes the derivative of $g$ with respect to the variable $u$. Given any continuous function $u : \Omega \rightarrow \R$ we will denote by $n(u)$ the number of nodal sets of $u$, i.e. of connected components of $\{x \in\Omega; u(x)\neq 0\}$. The Morse index $m(u)$ of a solution $u$ of is the maximal dimension of a subspace of $H^1_0(\Omega)$ in which the quadratic form $$\label{Q} H^1_0(\Omega) \ni w\mapsto Q_u(w):=\int_{\Omega}\|\nabla w(x)\|^2 dx - \int_{\Omega}g_u(\|x\|,u(x))w^2(x) dx$$ is negative definite. Since we are considering the case of bounded domains, $m(u)$ coincides with the number of negative eigenvalues, counted with their multiplicity, of the linearized operator $L_u := -\Delta - g_u(\|x\|, u)$ in the space $H^1_0(\Omega)$. When the solution $u$ is radial, we will denote by $m_{rad}(u)$ the radial Morse index of $u$, i.e. the maximal dimension of a subspace of $H^1_{0,rad}(\Omega)$ in which the quadratic form $Q_u$ is negative definite or, alternatively, $m_{rad}(u)$ is the number of negative eigenvalues, counted with their multiplicity, of $L_u$ in the space $H^1_{0,rad}(\Omega)$. In case the nonlinear term $g$ does not depend on the space variable, Aftalion and Pacella [@aftalion] obtained some lower bounds on the Morse index of sign changing radial solutions of , which recently were improved by De Marchis, Ianni and Pacella [@pacella Theorem 2.1]. [Theorem A]{}\[Autonomous problems\]\[ThA\] Let $u$ be a radial nodal solution of with $g(\|x\|, u) = f(u)$, $f \in C^1$. Then $$m_{rad}(u) \geq n(u)-1\quad \text{and}\quad m(u) \geq m_{rad}(u) + N(n(u)-1).$$ Moreover, if $f$ is superlinear, i.e. satisfies the condition $$\label{superlinear} f'(s)>\frac{f(s)}{s} \quad \forall\, s\in \R \backslash \{0\},$$ then $$m_{rad}(u)\geq n(u) \ \ \text{and hence} \ \ m(u) \geq n(u) + N(n(u)-1).$$ In this paper we consider the non-autonomous equation $$\label{f} \left\{ \begin{array}{l} \begin{aligned} -\Delta u &= \|x\|^\alpha f(u) &\textrm{in}&\ \ \Omega, \vspace{0.3 cm}\\ u &= 0 &\textrm{on}&\ \ \partial \Omega, \end{aligned} \end{array} \right.$$ where $\Omega \subset \R^2$ is either a ball or an annulus centered at the origin, $\alpha>0$ and $f:\R \rightarrow \R$ is $C^{1,\beta}$ on bounded sets of $\R$. We recall the following estimates obtained in [@ederson Theorems 1.1 and 1.2], which were used to prove that least energy nodal solutions of are not radially symmetric. [Theorem B]{}\[ThB\] Let $u$ be a radial sign changing solution of . Then $u$ has Morse index greater than or equal to $3$. Moreover, if holds, then the Morse index of $u$ is at least $n(u) + 2$. In case $\alpha$ is even, then these lower bounds can be improved, namely they become $\alpha+ 3$ and $n(u)+ \alpha+2$, respectively. Very recently these lower bounds were improved in [@amadori Theorem 1.1] by characterizing the Morse index in terms of a singular one dimensional eigenvalue problem. We also mention the paper [@weth] where its proved that the Morse index of radial solutions goes to infinity as $\alpha \to \infty$. Given any $\beta \in \R$, we set $[\beta]:=\max \{n\in \Z; n\leq \beta\}$. [Theorem C]{}\[ThC\] Let $\alpha \geq0$ and $u$ be a radial nodal solution of . Then $$m_{rad}(u) \geq n(u)-1\quad \text{and}\quad m(u) \geq m_{rad}(u) + \left( n (u) -1\right) \left(2\left[\frac{\alpha}{2}\right] + 2\right).$$ Moreover, if holds, then $$m_{rad}(u)\geq n(u) \ \ \text{and hence} \ \ m(u) \geq n(u) + \left( n (u) -1\right) \left(2\left[\frac{\alpha}{2}\right] + 2\right).$$ We obtain an improvement for these lower bounds. \[newtheo\] Let $\alpha \geq0$ and $u$ be a radial nodal solution of . Then $$m_{rad}(u) \geq n(u)-1\quad \text{and}\quad m(u) \geq m_{rad}(u) + \left( m(u_0) - m_{rad}(u_0)\right) \left(\left[\frac{\alpha}{2}\right] + 1\right),$$ where $u_0$ is a radial solution with $n(u_0)=n(u)$ of the autonomous problem $$\label{u_0} -\Delta u_0 = \left(\frac{2}{\alpha+2}\right)^2 f(u_0) \ \ \text{in} \ \ \Omega_{\frac{2}{\alpha+2}}:=\{\|x\|^{\frac{\alpha}{2}}x; x\in \Omega\},\quad u_0=0 \ \ \text{on} \ \ \partial \Omega_{\frac{2}{\alpha+2}}.$$ Moreover, if holds, then $$m_{rad}(u)\geq n(u) \ \ \text{and hence} \ \ m(u) \geq n(u) + \left( m(u_0) - m_{rad}(u_0)\right) \left(\left[\frac{\alpha}{2}\right] + 1\right).$$ Observe that $$\frac{m(u_0)- m_{rad}(u_0)}{2} \geq n(u)-1.$$ Indeed, the above inequality is equivalent to $$m(u_0) \geq m_{rad}(u_0) + 2 (n(u)-1) = m_{rad}(u_0) + 2 (n(u_0)-1),$$ and this is guaranteed by Theorem A. Also observe that this inequality can be strict in case $f(s) = \|s\|^{p-1}s$, $p \gg1$, $n(u)=2$, since $\frac{m(u_0)- m_{rad}(u_0)}{2} =5$ by [@pacella12 Theorem 1.1]. Next, we consider the particular case of the Hénon equation $$\label{alpha} \left\{ \begin{array}{l} \begin{aligned} -\Delta u &= \|x\|^{\alpha}\|u\|^{p-1}u &\textrm{in}&\ \ B, \vspace{0.3 cm}\\ u &= 0 &\textrm{on}&\ \ \partial B, \end{aligned} \end{array} \right.\tag{$P_{\alpha}$}$$ where $B \subset \R^2$ is the unit open ball centered at the origin, $\alpha \geq 0$ is a parameter and $p > 1$. Fixed the number of nodal sets $n$, we prove that the Morse index of a radial nodal solution of with $n$ nodal sets is monotone non-decreasing with respect to $\alpha$. \[theo1\] Let $u_\alpha$ and $u_\beta$ be radial solutions of and $(P_\beta)$, respectively, with the same number $n\geq1$ of nodal sets. If $0\leq\alpha\leq\beta$, then $m(u_\alpha)\leq m(u_\beta)$ and $m_{rad}(u_\alpha)=m_{rad}(u_\beta)=n$. The case of $N=2$ is special. We may use the change of variables with $\kappa = \frac{\beta+2}{\alpha+2}$ to establish a correspondence between the radial solutions of with the radial solutions of $(P_{\beta})$ with the same number of nodal sets. Although such transformation is not available for dimensions higher than two, we conjecture that Theorem \[theo1\] should also hold for $N\geq 3$. An auxiliary eigenvalue problem =============================== In this section we recall an important decomposition for some singular eigenvalue problems. Let $N\geq 2$ and consider the sphere $S^{N-1} \subset \mathbb{R}^N$. We recall that the spherical harmonics on $S^{N-1}$ are the eigenfunctions of the Laplace-Beltrami operator $-\Delta_{S^{N-1}}$. Indeed, the operator $-\Delta_{S^{N-1}}$ admits a sequence of eigenvalues $0=\lambda_0<\lambda_1\leq\ldots$ and corresponding eigenfunctions $(Y_k)$ which form a complete orthonormal system for $L^2(S^{N-1})$. More precisely, each $Y_k$ satisfies $$-\Delta_{S^{N-1}}Y_k(\theta)=\lambda_k Y_k(\theta),\quad \text{for}\ \theta \in S^{N-1},$$ and each eigenvalue $\lambda_k$ is given by the formula $$\label{eq:SH} \lambda_k = k(k+N-2),\quad k=0,1,\ldots$$ whose multiplicity is $$N_0=1\quad \text{and}\quad N_k=\frac{(2k+N-2)(k+N-3)!}{(N-2)!k!}\ \ \text{for}\ \ k\geq1.$$ Let $\Omega \subset \mathbb{R}^N$ be either a ball or an annulus centered at the origin and consider the problem $$\label{eigenfunction} -\Delta \psi + a(x)\psi = \lambda \frac{\psi}{\|x\|^2}\ \textrm{ in }\ \Omega\backslash\{0\}, \quad \psi = 0 \textrm{ on } \partial \Omega,$$ where $a(x)$ is a radial function in $L^{\infty}(\Omega)$. Set $$\mathcal{H}_0 := \left\{ u \in H^1_0(\Omega); \int_{\Omega} \frac{u^2}{\|x\|^2} < \infty\right\}.$$ Then, endowed with inner product $$(u,v)_{\mathcal{H}_0 }:= \int_\Omega \nabla u \nabla v + \frac{uv}{\|x\|^2} dx, \quad u, v \in \mathcal{H}_0,$$ $\mathcal{H}_0$ is a Hilbert space. We say that $\psi \in \mathcal{H}_0\backslash\{0\}$ is an eigenfunction of , if $$\int_{\Omega} \nabla \psi \nabla \varphi + a(x) \psi \varphi dx = \lambda \int_{\Omega} \frac{\psi \varphi}{\|x\|^2} dx \ \ \forall\, \varphi \in \mathcal{H}_0.$$ We recall the following result on the decomposition of eigenvalues of ; see [@amadori Proposition 4.1] or [@bartsch Lemma 3.1]. \[decomposition\] Let $\lambda < \left(\frac{N-2}{2}\right)^2$ be an eigenvalue of . Then, there exists $k\geq0$ such that $$\label{lambda} \lambda=\lambda_{rad}+\lambda_k\, ,$$ where $\lambda_{rad}$ is a radial eigenvalue of and $\lambda_k$ as in . Conversely, if holds and $\psi_{rad}$ is an eigenfunction associated to $\lambda_{rad}$, then $\psi=\psi_{rad}(r)Y_k(\theta)$ is an eigenfunction of associated to $\lambda$. Proofs of the main results ========================== Given $\kappa>0$, set $$\label{Tk} T_\kappa:\R^2\rightarrow \R^2, \quad T_\kappa(0)= 0 \ \ \text{and} \ \ T_\kappa(y):=\|y\|^{\kappa-1}y \ \ \text{for} \ \ y \neq0.$$ We perform the change of variable $x\leftrightarrow y$ putting $x=T_\kappa(y)$ and we observe that, see [@ederson Lemma 2.1], $T_{\kappa}$ has the following properties: - $T_\kappa$ is a diffeomorphism between $\R^2\backslash \{0\}$ and $\R^2\backslash \{0\}$ whose inverse is $$\label{Tk-1} T_\kappa^{-1}(x)=\|x\|^{\frac{1}{\kappa}-1}x,\ \ i.e.\ \ T_\kappa^{-1}=T_{\frac{1}{\kappa}}.$$ - In cartesian coordinates, $$\label{JTk} \|\det J_{T_\kappa}(y)\|=\kappa\|y\|^{2\kappa-2},\ \ \forall \, y\neq 0.$$ Let $\Omega \subset \R^2$ be either a annulus or a ball centered at the origin and set $\Omega_\kappa:= T_\kappa^{-1}(\Omega)$. \[lemma 1\] The map $$S_\kappa:H^{1}_{0}(\Omega_\kappa)\rightarrow H^1_0(\Omega),\ \ S_\kappa \psi:=\psi\circ T_\kappa^{-1},$$ is a continuous linear isomorphism. Moreover, with $\varphi=\psi\circ T_\kappa^{-1}$, $$\min\left\{\kappa,\frac{1}{\kappa}\right\}\int_{\Omega}\|\nabla\varphi(x)\|^2dx \leq \int_{\Omega_\kappa}\|\nabla\psi(y)\|^2dy \leq \max\left\{\kappa,\frac{1}{\kappa}\right\}\int_{\Omega}\|\nabla\varphi(x)\|^2dx, \ \ \forall \, \psi \in H^1_0(\Omega_\kappa),$$ $$\kappa \int_{\Omega} \|\nabla \varphi (x)\|^2 dx = \int_{\Omega_{\kappa}} \|\nabla \psi(y)\|^2 dy, \ \ \forall \ \psi \in H^1_{0,{\rm{rad}}}(\Omega_{\kappa}).$$ Given a radial function $u : \Omega \to \R$, set $v : \Omega_\kappa \to \R$ by $v(y) = u(T_\kappa y)$. Then $v$ is radially symmetric and, by [@ederson eq. (2.13)], $$\label{radiallaplacian} \Delta v(y) = \kappa^2 \|x\|^{\frac{2\kappa - 2}{\kappa}}\Delta u(x).$$ Thus, if $u$ is a radial solution of , then $v : \Omega_\kappa \to \R$ satisfies $$-\Delta v(y) = \kappa^2 \|y\|^{2\kappa-2+\kappa \alpha} f(v(y)), \quad y \in \Omega_\kappa, \quad v = 0 \quad \rm{on}\ \partial \Omega_\kappa.$$ Now, given any $\beta \geq 0$, we choose $\kappa$ so that $$\label{kab} 2\kappa-2+\kappa \alpha = \beta, \quad \rm{i.e.}\quad \kappa := \frac{\beta+2}{\alpha+2}\,.$$ and so $$\label{beta} -\Delta v(y) = \left(\frac{\beta+2}{\alpha+2}\right)^2 \|y\|^{\beta} f(v(y)), \quad y \in \Omega_\kappa, \quad v = 0 \quad \rm{on}\ \partial \Omega_\kappa.$$ In the particular case with $f(s)=\|s\|^{p-1}s$, setting $w(y) = \left(\frac{\beta+2}{\alpha+2}\right)^{\frac{2}{p-1}}v(y)$, we get $$-\Delta w(y) = \|y\|^{\beta} \|w(y)\|^{p-1}w(y), \quad y \in \Omega_\kappa, \quad v = 0 \quad \rm{on}\ \partial \Omega_\kappa.$$ Therefore, we have proved the following result. \[characterization\] $u_\alpha$ is a radial solution of in $\Omega$ with $n$ nodal sets if, and only if, $$\label{eq:related} u_{\beta}(y) = \left(\frac{\beta+2}{\alpha+2}\right)^{\frac{2}{p-1}}u_\alpha\left(\|y\|^{\frac{\beta-\alpha}{\alpha+2}}y\right),\quad y \in \Omega_\kappa,\quad \kappa = \frac{\beta+2}{\alpha+2},$$ is a radial solution of $(P_\beta)$ in $\Omega_\kappa$ with $n$ nodal sets. Given $\alpha \geq0$, $n \in \N$, we know that there exists a unique solution of (up to multiplication by $-1$) with $n$ nodal sets; see [@ederson Theorem 1.3 (i)]. Let $u_\alpha$ and $u_\beta$ be radial solutions of and $(P_\beta)$, respectively, with $n(u_\alpha)=n(u_\beta)$. Then $u_\alpha$ and $u_\beta$ are related by and $m(u_\alpha)$ is the maximal dimension of a subspace of $H^1_0(B)$ in which the quadratic form $$H^1_0(B) \ni w\mapsto Q_\alpha(w):=\int_{B}\|\nabla w(x)\|^2 dx - p\int_{B}\|x\|^\alpha \|u_\alpha(x)\|^{p-1}w^2(x) dx$$ is negative definite. Similarly we can compute $m(u_\beta)$. The crucial point for the proof of Theorem \[theo1\] is the following result. \[prop1\] If $0 \leq \alpha \leq \beta$, then $$Q_{\beta}(w_\kappa)\leq \kappa \, Q_{\alpha}(w),\quad \forall \, w \, \in H^1_0(B), \ \ Q_{\beta}(w_\kappa)= \kappa \, Q_{\alpha}(w),\quad \forall \, w \, \in H^1_{0, rad}(B),$$ where $w_\kappa(y) = w \circ T_\kappa(y)$, $T_\kappa$ is defined as in with $\kappa = \frac{\beta+2}{\alpha+2}$. By , up to multiplication by $-1$, $$u_{\beta}(y) = \kappa^{\frac{2}{p-1}}u_\alpha (T_\kappa y).$$ Hence $$\begin{aligned} Q_{\beta}(w_\kappa) &= \int_{B}\|\nabla w_\kappa(y)\|^2dy - \kappa ^2 p\int_{B}\|y\|^{\beta} \|u_\alpha(T_\kappa y)\|^{p-1}w_\kappa^2(y)dy. \end{aligned}$$ Since $\kappa \geq 1$, it follows from Lemma \[lemma 1\] that $$\int_{B}\|\nabla w_\kappa(y)\|^2dy \leq \max\left\{\kappa,\frac{1}{\kappa}\right\}\int_{B}\|\nabla w(x)\|^2dx = \kappa \int_{B}\|\nabla w(x)\|^2dx \ \ \forall \, w \, \in H^1_0(B),$$ $$\int_{B}\|\nabla w_\kappa(y)\|^2dy = \kappa \int_{B}\|\nabla w(x)\|^2dx \quad \forall \, w \, \in H^1_{0, rad}(B).$$ Now, putting $x=T_\kappa(y)$, it follows from and that $dy = \kappa^{-1}\|x\|^{\frac{2-2\kappa}{\kappa}}dx$. Thus $$\kappa ^2 p \! \int_{B}\|y\|^{\beta} \|u_\alpha(T_\kappa y)\|^{p-1}w_\kappa^2(y)dy \!= \!\kappa p \!\int_{B}\|x\|^{\frac{\beta-2\kappa+2}{\kappa}} \|u_\alpha(x)\|^{p-1}w^2(x)dx \!=\!\kappa p \! \int_{B}\|x\|^\alpha \|u_\alpha(x)\|^{p-1}w^2(x)dx,$$ since $\frac{\beta-2\kappa+2}{\kappa}=\alpha $. Therefore, $$Q_{\beta}(w_\kappa) \leq \kappa \int_{B}\|\nabla w(x)\|^2dx - \kappa p \int_{B}\|x\|^\alpha \|u_\alpha(x)\|^{p-1}w^2(x)dx =\kappa \, Q_{\alpha}(w) \ \ \forall \, w \, \in H^1_0(B)$$ and $$Q_{\beta}(w_\kappa) = \kappa \, Q_{\alpha}(w) \ \ \forall \, w \, \in H^1_{0,rad}(B). \qedhere$$ Let $u_\alpha$ and $u_{\beta}$ be radial solutions of and $(P_{\beta})$, respectively, with $n(u_\alpha)=n(u_\beta)=n$, $\alpha \leq \beta$. From Proposition \[prop1\], we have $$Q_{\beta}(w_\kappa)\leq \kappa \, Q_{\alpha}(w),\quad \forall \, w \, \in H^1_0(B), \ \ Q_{\beta}(w_\kappa)= \kappa \, Q_{\alpha}(w),\quad \forall \, w \, \in H^1_{0, rad}(B).$$ Therefore, if $V$ is a subspace of $H^1_0(B)$ in which the quadratic form $Q_{\alpha}$ is negative definite, then $Q_{\beta}$ is also negative definite in the subspace $V_{\kappa}:=\{w\circ T_\kappa; w\in V\}$. Moreover, $Q_{\alpha}$ is negative definite in a subspace $V$ of $H^1_{0,rad}(B)$ if, and only if, $Q_{\beta}$ is negative definite in subspace $V_{\kappa}$. Since $V$ and $V_{\kappa}$ have the same dimension, we infer that $m(u_\alpha)\leq m(u_\beta)$ and $m_{rad}(u_\alpha)=m_{rad}(u_\beta)$. Finally, we know from [@Harrabi Proposition 2.9] that $m_{rad}(u_0)=n$ and we conclude the proof of Theorem \[theo1\]. To prove Theorem \[newtheo\], we start with the particular case with $\alpha$ even. \[prop2\] Let $\alpha > 0$ be even and let $u$ be a radial nodal solution of . Then $$m_{rad}(u)\geq n(u)-1\quad \text{and}\quad m(u) \geq m_{rad}(u) + (m(u_0)-m_{rad}(u_0))\left(\frac{\alpha + 2}{2}\right),$$ where $u_0$ is a radial solution of with $n(u_0)=n(u)$. If in addition holds, then $$\label{estimate} m_{rad}(u)\geq n(u)\quad \text{and hence}\quad m(u) \geq n(u)+(m(u_0)-m_{rad}(u_0))\left(\frac{\alpha + 2}{2}\right).$$ Let $u$ be a radial nodal solution of , with $\alpha = 2(m-1)$ even and $\kappa=\frac{1}{m}$. Then $\kappa = \frac{2}{\alpha+2}$ and, by ($\beta=0$ in this case), the function $u_0=u\circ T_{\kappa}$ is a radial nodal solution of the autonomous problem $$-\Delta u_0 = \frac{1}{m^2} f(u_0) \ \textrm{ in } \ \Omega_\kappa, \quad u_0 = 0 \ \textrm{ on }\ \partial \Omega_\kappa.$$ Therefore, by [@gladiali Lemma 2.6], the singular eigenvalue problem $$\label{eig-no-weight} -\Delta \psi - \frac{1}{m^2} f'(u_0)\psi = \lambda \frac{\psi}{\|y\|^2} \ \textrm{ in } \ \Omega_\kappa \backslash\{0\}, \quad \psi = 0 \ \textrm{ on } \ \partial \Omega_\kappa,$$ has $m(u_0)-m_{rad}(u_0)$ negative eigenvalues associated to nonradial eigenfunctions, counted with their multiplicity. We count the negative radial eigenvalues of as $\lambda_{1}=\lambda_{1}^{rad} < \lambda_{2}^{rad}<\ldots<\lambda_{m_{rad}(u_0)}^{rad}$ whose corresponding eigenfunctions we denote by $\psi_n^{rad}$. By Proposition \[decomposition\], every negative nonradial eigenvalue of has the decomposition $\lambda_n^{rad}+k^2$, for some $n=1,\ldots,m_{rad}(u_0)$ and $k\in\N$. For each $n=1,\ldots,m_{rad}(u_0)$, consider $\mathcal{N}_n:=\{k\in\N; \lambda_n^{rad}+k^2<0\}$. Thus $$\label{eq:DES1} m(u_0)-m_{rad}(u_0) = 2\sum_{n=1}^{m_{rad}(u_0)}\mathcal \# \mathcal{N}_n.$$ Moreover, using eq. with $\kappa=\frac{2}{\alpha+2}$, we have that the functions $\varphi_n^{rad}=\psi_n^{rad}\circ T_{\kappa}^{-1}$, $n=1,\ldots,m_{rad}(u_0)$, are the radial eigenfunctions of $$\label{eigenvalue} -\Delta \varphi - \|x\|^\alpha f'(u)\varphi = \lambda \frac{\varphi}{\|x\|^2} \ \textrm{ in } \ \Omega\backslash\{0\}, \quad \varphi = 0 \ \textrm{ on } \ \partial \Omega,$$ with $\lambda = m^2\lambda_n^{rad}<0$. Then $m_{rad}(u)=m_{rad}(u_0)$ and with $\mathcal{N}_n^{\alpha}:=\{k\in\N; m^2\lambda_n^{rad}+k^2<0\}$, $$\label{eq:DES2} m(u)-m_{rad}(u) = 2\sum_{n=1}^{m_{rad}(u_0)}\mathcal \# \mathcal{N}_n^{\alpha}\, .$$ We claim that $$\label{eq:DES3} \# \mathcal{N}_n^{\alpha} \geq m (\# \mathcal{N}_n) \quad \forall\, n= 1, \ldots, m_{rad}(u_0).$$ Indeed, if $k \in \mathcal{N}_n$, then $\lambda_n^{rad} + k^2 < 0$, whence $m^ 2\lambda_n^{rad} + (mk)^2 < 0$. The latter shows that $mk \in \mathcal{N}_n^{\alpha}$ and this proves . Therefore, from , , , we infer that $$m(u) - m_{rad}(u) \geq (m(u_0)-m_{rad}(u_0)) m = (m(u_0)-m_{rad}(u_0))\left(\frac{\alpha + 2}{2}\right).$$ Now, with respect to the radially symmetric eigenfunctions, it is proved [@pacella Theorem 2.1] that has at least $n(u)-1$ negative eigenvalues associated to radial eigenfunctions, and this number becomes $n(u)$ if holds. Again using eq. with $\kappa=\frac{2}{\alpha+2}$, we have that $\lambda\mapsto m^2 \lambda$ is a bijection between radial eigenvalues of and we obtain the lower bounds for $m_{rad}(u)$. Next we use Proposition \[prop2\] to prove Theorem \[newtheo\]. Let $\alpha>0$ and $u$ be a radial nodal solution of . Then, by , for all $\gamma\geq 0$, the function $v=u\circ T_\kappa$ is a radial nodal solution of $$-\Delta v = \kappa^2 \|y\|^{\gamma} f(v) \ \textrm{ in } \ \Omega_\kappa, \quad v = 0 \ \textrm{ on } \ \partial \Omega_\kappa, \ \ \textrm{ with } \ \kappa = \frac{\gamma+2}{\alpha+2}.$$ Hence, if $\gamma \leq \alpha$, i.e. $\kappa \leq 1$, setting $w_\kappa = w\circ T_\kappa$, it follows from Lemma \[lemma 1\] and that $$\int_{\Omega}\|\nabla w(x)\|^2 dx - \int_{\Omega}\|x\|^{\alpha}f'(u(x))w^2(x) dx \leq \frac{1}{\kappa} \left[\int_{\Omega_\kappa}\|\nabla w_\kappa(y)\|^2 dy - \kappa^2 \int_{\Omega_\kappa}\|y\|^{\gamma}f'(v(y))w^2(y) dy \right],$$ for all $w \in H^1_0(\Omega)$ and the equality holds for all $w \in H^1_{0, rad}(\Omega)$. Consequently, $m_{rad}(u) = m_{rad}(v)$ and $m(u) \geq m(v)$. In particular, taking $\gamma = 2[\frac{\alpha}{2}]$ we can use Proposition \[prop2\] for $v$ to obtain $$m_{rad}(v) \geq n(v)-1=n(u)-1,\ \ m_{rad}(v) \geq n(u)\ \text{if \eqref{superlinear} holds} \ \ \text{and}$$ $$m(v) \geq m_{rad}(v) + (m(u_0)-m_{rad}(u_0))\left(\frac{\gamma+2}{2}\right) = m_{rad}(u) + (m(u_0)-m_{rad}(u_0))\left(\left[\frac{\alpha}{2}\right] + 1\right),$$ where $u_0$ is radial solution of with $n(u_0)=n(u)$. Observe that the key argument in the proof of Theorem \[newtheo\] is the monotonicity of the Morse indices $m(u) \geq m(v)$ proved above, thanks to $\gamma \leq \alpha$. [10]{} A. Aftalion and F. Pacella. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. , 339(5):339–344, 2004. A. Amadori and F. Gladiali. On a singular eigenvalue problem and its applications in computing the morse index of solutions to semilinear pde’s. , 2018. T. Bartsch, M. Clapp, M. Grossi, and F. Pacella. Asymptotically radial solutions in expanding annular domains. , 352(2):485–515, 2012. T. Bartsch, T. Weth, and M. Willem. Partial symmetry of least energy nodal solutions to some variational problems. , 96:1–18, 2005. J. Byeon and Z.-Q. Wang. On the [H]{}[é]{}non equation: asymptotic profile of ground states. [I]{}. , 23(6):803–828, 2006. D. Cao, S. Peng, and S. Yan. Asymptotic behaviour of ground state solutions for the [H]{}[é]{}non equation. , 74(3):468–480, 2009. F. De Marchis, I. Ianni, and F. Pacella. Exact [M]{}orse index computation for nodal radial solutions of [L]{}ane-[E]{}mden problems. , 367(1-2):185–227, 2017. F. De Marchis, I. Ianni, and F. Pacella. A [M]{}orse index formula for radial solutions of [L]{}ane-[E]{}mden problems. , 322:682–737, 2017. E. M. dos Santos and F. Pacella. Hénon-type equations and concentration on spheres. , 65(1):273–306, 2016. F. Gladiali, M. Grossi, and S. L. N. Neves. Symmetry breaking and [M]{}orse index of solutions of nonlinear elliptic problems in the plane. , 18(5):1550087, 31, 2016. A. Harrabi, S. Rebhi, and A. Selmi. Existence of radial solutions with prescribed number of zeros for elliptic equations and their [M]{}orse index. , 251(9):2409–2430, 2011. M. H[é]{}non. Numerical experiments on the stability of spherical stellar systems. , 24:229–238, 1973. Z. Lou, T. Weth, and Z. Zhang. Symmetry breaking via morse index for equations and systems of h[é]{}non-schr[ö]{}dinger type. , 2018. E. Moreira dos Santos and F. Pacella. Morse index of radial nodal solutions of [H]{}énon type equations in dimension two. , 19(3):1650042, 16, 2017. D. Smets and M. Willem. Partial symmetry and asymptotic behavior for some elliptic variational problems. , 18(1):57–75, 2003. D. Smets, M. Willem, and J. Su. Non-radial ground states for the [H]{}[é]{}non equation. , 4(3):467–480, 2002. [^1]: Wendel Leite da Silva was partially supported by CNPq and CAPES. Ederson Moreira dos Santos was partially supported by CNPq grant 307358/2015-1 and FAPESP grant 2015/17096-6.
--- abstract: 'According to a celebrated result of L. Caffarelli, every optimal mass transportation mapping pushing forward the standard Gaussian measure onto a log-concave measure $e^{-W} dx$ with $D^2 W \ge \mbox{Id}$ is $1$-Lipschitz. We present a short survey of related results and various applications.' author: - 'Alexander V. Kolesnikov' title: Mass transportation and contractions --- Keywords: [optimal transportation, Monge–Amp[è]{}re equation, log-concave measures, Gaussian measures, isoperimetric inequalities, Sobolev inequalities]{} Introduction ============ Given a positive number $\alpha$ we say that a mapping $T : {{\ensuremath{\mathbb{R}}}}^d \to {{\ensuremath{\mathbb{R}}}}^d$ is $\alpha$-Lipschitz if $$\|T(x)-T(y)\| \le \alpha \|x-y\|.$$ For a smooth $T$ this is equivalent to the following: $$\sup_{x \in {{\ensuremath{\mathbb{R}}}}^d}\\| DT (x)\\| \le \alpha,$$ where $\\| \cdot \\|$ is the operator norm. For the case $\alpha=1$ we say that $T$ is a contraction. Similarly, a mapping $T : X \to Y$ between metric spaces is called contraction, if $\rho_Y(T(x_1), T(x_2)) \le \rho_X(x_1, x_2)$. Let $\mu$ be a Borel measure on a metric space $(M, \rho)$. Given a Borel set $A \subset M$ we define the corresponding boundary measure $\mu^+$ of $\partial A$ $$\mu^{+}(\partial A) = \underline{\lim}_{h \to 0} \frac{\mu(A_h) - \mu(A)}{h},$$ where $A_h = \{x: \rho(x,A) \le h\}$. A set $A$ is called isoperimetric if it has the minimal surface measure among of all the sets with the same measure $\mu(A)$. The isoperimetric profile $\mathcal{I}_{\mu}$ of $\mu$ is defined as the following function $$\mathcal{I}_{\mu}(t) = \inf \{ \mu^+(\partial A): \ \mu(A)=t\}.$$ Generally, isoperimetric sets are not possible to find. Nevertheless, bounds for isoperimetric functions (the so-called isoperimetric inequalities) have many applications in analysis, geometry and probability theory. It is well-known, for instance, that isoperimetric inequalities imply Sobolev-type inequalities. See more in [@Gromov], [@MilSch], [@Ledoux], [@Ros], [@Vill]. Numerous applications of contractions in analysis, probability and geometry rely on the following fact: [ Let $X$, $Y$ be two metric spaces and $X$ is equipped with a measure $\mu$. Assume that there exists a contraction $T : X \to Y$ between metric spaces $X$ and $Y$. Then the image measure $\nu = \mu \circ T^{-1}$ has a better isoperimetric profile ]{} $$\mathcal{I}_{\nu} \ge \mathcal{I}_{\mu}.$$ In this paper we study mainly a special case of optimal transportations of measures. Given two Borel probability measures $\mu$ and $\nu$ we consider the optimal transportation map $T$ minimizing the cost $$W^2_2(\mu,\nu) = \int \|x - T(x)\|^2 \ d \mu$$ among of all the maps pushing forward $\mu$ to $\nu$. The latter means that $\mu \circ T^{-1}(A) = \nu(A)$ for every Borel $A$. If $\mu = \rho_0 \ dx$ and $\nu = \rho_1 \ dx$ are absolutely continuous, then $T$ does exist and can be obtained from the solution to the corresponding Monge-Kantorovich transportation problem. Moreover, this map is $\mu$-unique and has the form $T = \nabla \Phi$, where $\Phi$ is convex (see [@Vill]). Assuming smoothness of $\Phi$, one can easily verify that $\Phi$ solves the following nonlinear PDE (the Monge–Amp[è]{}re equation): $${\rho_1(\nabla \Phi)} \det D^2 \Phi = {\rho_0}.$$ This paper contains an overview of the results related to the contractivity of optimal transportation mappings. The first result in this direction has been established by L. Caffarelli (see [@Caf]). Let $\mu$ be the standard Gaussian measure $\mu = \frac{1}{(2\pi)^{d/2}} e^{-\frac{x^2}{2}} \ dx$ and $\nu = e^{-W} \ dx$ with $D^2 W \ge \mbox{Id}$, then the corresponding $T$ is a contraction. This observation implies immediately the Bakry-Ledoux comparison theorem and various functional inequalities, including the log-Sobolev inequality for uniformly log-concave measures. Among of other applications let us mention the Gaussian correlation conjecture and the Brascamp-Lieb inequality. We discuss several extensions of this result and some open problems. Caffarelli’s contraction theorem ================================ The Theorem \[contr1\] and Theorem \[contr2\] below will be both referred to as “Caffarelli’s contraction theorem”. Note, however, that the original formulation is given in Theorem \[contr2\]. \[contr1\] [(L. Caffarelli) ]{} Let $T=\nabla \Phi$ be the optimal transportation mapping pushing forward a probability measure $\mu = e^{-V} dx$ onto a probability measure $\nu = e^{-W} dx$. Assume that $V$ and $W$ are twice continuously differentiable and $D^2 W \ge K$. Then for every unit vector $e$ $$\sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} \Phi^2_{ee} \le \frac{1}{K} \sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} V_{ee}.$$ In particular, if $\mu$ is the standard Gaussian measure and $K \ge 1$, then $T$ is a contraction. [Sketch of the proof:]{} [1) Maximum principle proof.]{} The proof based on the maximum principle is formal but elegant. Functons $V, W$ and $\Phi$ are assumed to be sufficiently regular. Note that smoothness of $\Phi$ can be justified in some favorable situations ($V,W$ are smooth and satisfy certain growth assumptions, see Theorem 4.14 of [@Vill]). By the change of variables formula $$e^{-V} = e^{-W(\nabla \Phi)} \det D^2 \Phi.$$ Taking the logarithm of both sides we get $$V = W(\nabla \Phi) - \log \det D^2 \Phi.$$ We fix some unit vector $e$ and differentiate this formula twice along $e$. To this end we apply the following fundamental relation $$\partial_{e} \ln \det D^2 \Phi = \frac{\partial_{e} \det D^2 \Phi}{\det D^2 \Phi} = \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{e}.$$ Differentiating this formula along another direction $v$ and using that $$D^2 \Phi_{v} (D^2\Phi)^{-1} + D^2 \Phi \bigl[ (D^2\Phi)^{-1} \bigr]_{v}=0$$ we obtain $$\partial_{e v} \ln \det D^2 \Phi = \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{e v} - \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{e} (D^2 \Phi)^{-1} D^2 \Phi_{v} \Bigr].$$ Coming back to the change of variables formula we get $$V_e = \langle \nabla W(\nabla \Phi), D^2 \Phi \cdot e \rangle - \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{e}$$ and $$\begin{aligned} V_{ee} & = \langle D^2 W(\nabla \Phi) D^2 \Phi \cdot e, D^2 \Phi \cdot e \rangle + \langle \nabla W(\nabla \Phi), \nabla \Phi_{ee} \rangle \\& - \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{ee} + \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{e}\Bigr]^2.\end{aligned}$$ Now assume that $\Phi_{ee}$ attains its maximum at $x_0$. Then $$\nabla \Phi_{ee}(x_0)=0, \ D^2 \Phi_{ee} \le 0.$$ Note that $\mbox{Tr} \bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{e}\bigr]^2 >0 $ because it equals to $\mbox{Tr} C^2 $, where $$C =(D^2 \Phi)^{-1/2} D^2 \Phi_{e} (D^2 \Phi)^{-1/2}$$ is a symmetric matrix. Clearly, $\mbox{\rm Tr} (D^2 \Phi(x_0))^{-1} D^2 \Phi_{ee}(x_0) \le 0$ and one gets $$V_{ee}(x_0) \ge K \\| D^2 \Phi(x_0) \cdot e\\| \ge K \Phi^2_{ee}(x_0).$$ Hence $$\sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} \Phi^2_{ee} \le \frac{1}{K} \sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} V_{ee}(x_0).$$ [2) Incremental quotients proof]{} Instead of differentiating the Monge-Amp[è]{}re equation we consider the incremental quotient $$\delta_2 \Phi(x) = \Phi(x+th) + \Phi(x-th) -2\Phi(x) \ge 0$$ for some fixed vector $h \in {{\ensuremath{\mathbb{R}}}}^d$ with $\|h\|=1$. By approximation, one can assume that $\mbox{\rm supp}(\nu)$ is a bounded convex domain and $V$, $W$ are locally H[ö]{}lder. Caffarelli’s regularity theory assures that $\Phi \in C^{2, \alpha}_{loc}({{\ensuremath{\mathbb{R}}}}^d)$. In addition, again by approximation, one can assume that $\mu$ has at most Gaussian decay, meaning that $V(x) \le C_1 + C_2 \|x\|^2$ for some $C_1, C_2 \ge 0$. Then the following lemma holds (see Lemma 4 in [@Caf]) $\lim_{x\to \infty} \delta_2 \Phi(x) = 0$. Thus there exists a maximum point $x_0$ of $\delta_2 \Phi(x)$. Differentiating at $x_0$ yields $$\label{maxim1} \nabla \Phi(x_0+th) + \nabla \Phi(x_0-th) =2 \nabla \Phi(x_0),$$ $$D^2 \Phi(x_0+th) + D^2 \Phi(x_0-th) \le 2 D^2 \Phi(x_0).$$ It follows from the concavity of the determinant that $$\begin{aligned} \det D^2 \Phi(x_0) & \ge \det \Bigl( \frac{D^2 \Phi(x_0+th) + D^2 \Phi(x_0-th)}{2} \Bigr) \\& \ge \Bigl( \det D^2 \Phi(x_0+th) \ \det D^2 \Phi(x_0-th)\Bigr)^{\frac{1}{2}}.\end{aligned}$$ Applying the change of variables formula $ \det D^2 \Phi = e^{W(\nabla \Phi) -V} $ one finally gets $$\begin{aligned} \label{V-W} V(x_0 + th) + V(x_0 - th) & - 2 V(x_0) \ge \\& \nonumber W(\nabla \Phi(x_0 + th)) + W(\nabla \Phi(x_0 -th)) - 2 W(\nabla \Phi(x_0)).\end{aligned}$$ It follows from (\[maxim1\]) that $ v:= \nabla \Phi(x_0+th) - \nabla \Phi(x_0) = \nabla \Phi(x_0) - \nabla \Phi(x_0-th). $ Hence we get by (\[V-W\]) that $$\sup V_{hh} \cdot t^{2} \ge K \|\nabla \Phi(x_0+th) - \nabla \Phi(x_0)\|^{2} = K \| \nabla \Phi(x_0-th) - \nabla \Phi(x_0)\|^{2} = K \|v\|^{2}.$$ By convexity of $\Phi$ $$\begin{aligned} \Phi(x_0+th) + \Phi(x_0-th) - 2\Phi(x_0) & \le t \langle \nabla \Phi(x_0+th) - \nabla \Phi(x_0-th), h \rangle \\& = 2t \langle v,h \rangle \le 2t \|v\|.\end{aligned}$$ Finally $$\frac{\sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} V_{hh}}{K} \ge \Bigl( \frac{\delta_2 \Phi}{2t^2}\Bigr)^{2}.$$ This clearly implies $$\Phi_{hh} \le 2 C$$ with $C = \sqrt{\frac{\sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} V_{hh}}{K}}.$ But this estimate is worse that the desired one. To get the sharp estimate we repeat the arguments and use the additional information that $\Phi_{hh} \le a_0 C$, where $a_0 = 2$. Apply the identity $$\Phi(x_0+th) + \Phi(x_0-th) - 2\Phi(x_0) = \int_{0}^{t} \langle \nabla \Phi(x_0 + sh) - \nabla \Phi(x_0 -s h), h \rangle \ ds.$$ By convexity of $\Phi$ $\langle \nabla \Phi(x_0 + sh) - \nabla \Phi(x_0 -s h), h \rangle \le \langle \nabla \Phi(x_0 + th) - \nabla \Phi(x_0 - t h), h \rangle$. One has $$\Phi(x_0+th) + \Phi(x_0-th) - 2\Phi(x_0) \le \int_0^t \min \bigl( 2 a_0 C s, 2 \|v\| \bigr) \ ds.$$ Computing the right-hand side and taking into account that $\|v\| \le Ct$, we get that $$\Phi(x_0+th) + \Phi(x_0-th) - 2\Phi(x_0) \le a_1 Ct^2.$$ where $a_1 = \frac{3}{2}$. Hence $\Phi_{hh} \le a_1 C$. Repeating this arguments infinitely many times we get that $\Phi_{hh} \le a_n C$ and $\lim_n a_n =1$. The proof is complete. [3) Proof via $L^p$-estimates]{} See Section 6. We note that the original result from [@Caf] was slightly different from the result stated above. Here is the exact statement proved by Caffarelli. \[contr2\] [(L. Caffarelli)]{} Let $\mu = e^{-Q} \ dx$ be any Gaussian measure. Then for any measure $\nu = e^{-Q - P} \ dx$, where $P$ is convex, the corresponding optimal transportation $T$ is a contraction. [Sketch of the proof:]{} Let us apply the maximum principle arguments. We are looking for a maximum of $\Phi_{ee}(x)$ among of all unit $e$ and $x \in {{\ensuremath{\mathbb{R}}}}^d$. Apply the relation obtained above $$\begin{aligned} Q_{ee} & = \langle D^2 (Q+P)(\nabla \Phi) D^2 \Phi \cdot e, D^2 \Phi \cdot e \rangle + \langle \nabla (Q+P)(\nabla \Phi), \nabla \Phi_{ee} \rangle \\& - \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{ee} + \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{e}\Bigr]^2.\end{aligned}$$ By the same reasons as above $$\begin{aligned} Q_{ee} \ge \langle D^2 (Q+P)(\nabla \Phi) D^2 \Phi \cdot e, D^2 \Phi \cdot e \rangle.\end{aligned}$$ Now take into account that $P$ is convex and, in addition, $e$ must be an eigenvector of $D^2 \Phi$. Hence we obtain $$Q_{ee} \ge \Phi^2_{ee} \cdot Q_{ee}(\nabla \Phi).$$ Taking into account that $Q_{ee}$ is constant, we obtain the claim. General (uniformly) log-concave measures ======================================== The incremental quotients proof can be easily extended to the case of measures which are uniformly log-concave in a generalized case. The latter means that the potential $W$ satisfies $$W(x+y)+W(x-y) - W(x) \ge \delta(\|y\|)$$ for some increasing function $\delta$. The following result has been proved in [@Kol]. \[hoelder\] Assume that $V$ and $W$ satisfy $$V(x+ y) + V(x-y) -2V(x) \le A_p \|y\|^{p+1},$$ $$W(x+ y) + W(x-y) -2W(x) \ge A_q \|y\|^{q+1},$$ for some $0 \le p \le 1$, $1 \le q$, $A_p>0$, $A_q>0$. Then $\Phi$ satisfies $$\label{var-Hoelder} \Phi(x+th) + \Phi(x-th) -2\Phi(x) \le 2\Bigl( \frac{A_p}{A_q}\Bigr)^{\frac{1}{q+1}} t^{1+\alpha}$$ for every unit vector $h \in {{\ensuremath{\mathbb{R}}}}^d$ with $\alpha = \frac{p+1}{q+1}$. The constant in (\[var-Hoelder\]) is not optimal in general. It follows from (\[var-Hoelder\]) that $\nabla \Phi$ is globally H[ö]{}lder. This fact is actualy true without any convexity assumption on $\Phi$, but the convex case is more simple and the result follows from the following lemma communicated to the authors by Sasha Sodin. \[Sodin-lem\] For every convex $f$ and unit vector $h$ one has $$\|\nabla f(x+th) -f(x)\| \le \frac{2}{t} \sup_{v: \|v\|=1} \Bigl( f(x+2tv) + f(x-2tv) - 2 f(x) \Bigr).$$ Using this lemma one can extend the H[ö]{}lder regularity result. \[MS-conc\] Assume that $$V(x+ y) + V(x-y) -2V(x) \le \|y\|^{2},$$ and $$W(x+y)+W(x-y) - W(x) \ge \delta(\|y\|)$$ with some non-negative increasing function $\delta$. Then $$\|\nabla \Phi(x) - \nabla \Phi(y) \| \le 8 \delta^{-1}(4\|x-y\|^2).$$ Applying this estimate one can transfer the famous Gaussian Sudakov-Tsirelson isoperimetric inequality to any (generalized) uniform log-concave measure. Recall (see [@Bo]), that the standard Gaussian measure $\gamma$ satisfies the Gaussian isoperimetric inequality $$\gamma(A^r) \ge \Phi(\Phi^{-1}(\gamma(A)) +r),$$ where $A^r = \{x\in {{\ensuremath{\mathbb{R}}}}^d: \exists a \in A: \|a-x\|<r\}$, $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-\frac{t^2}{2}} dt$. Consequently, applying Theorem \[MS-conc\] to $\mu=\gamma$ and $\nu = e^{-W} dx$ with $W$ satisfying $$W(x+y)+W(x-y) - W(x) \ge \delta(\|y\|),$$ we get $$\nu\bigl( A_{r} \bigr) \ge \Phi\Bigl(\Phi^{-1}(\nu(A)) + \frac{1}{2} \sqrt{\delta(r/8)} \Bigr).$$ In particular, $\nu$ admits the following dimension-free concentration property: $$\nu\bigl( A_{r} \bigr) \ge 1- \frac{1}{2} \exp \Bigl( -\frac{1}{8} \ \delta(r/8) \Bigr)$$ with $\nu(A) \ge 1/2$. A similar result has been established by S. Sodin and E. Milman in [@MilSod] by localization arguments. Note that according to results of E. Milman [@Milman08] concentration and isoperimetric inequalities are in a sence equivalent for log-concave measures. Lebesgue measure on a convex set ================================ In this section we discuss the following problem. Given a nice (product) probability measure $\mu$ (e.g. Gaussian or exponential) estimate effectively the Lipschitz constant of the optimal mapping pushing forward $\mu$ onto the normalized Lebesgue measure on a convex set $K$. This problem was motivated in particular by the famous Kannan-Lov[á]{}sz-Simonovits conjecture (KLS-conjecture). Recall that the Cheeger $C_{ch}(K)$ constant of a convex body $K$ is the smallest constant such that the inequality $$\int_K \Bigl\|f - \frac{1}{\lambda(K)} \int_K f dx \Bigr \| \ dx \le C_{ch}(K) \int_{K} \|\nabla f\| \ dx$$ holds for every smooth $f$. [KLS conjecture]{}. There exists an universal constant $c$ such that $$C_{ch}(K) \le c$$ for every convex $K \subset {{\ensuremath{\mathbb{R}}}}^d$ satisfying $$\int_K x_i \ dx = 0, \ \ \frac{1}{\lambda(K)} \int_K x_i x_j \ dx = \delta_{i}^{j}.$$ More on the KLS conjecture see in [@KLS], [@Bob07], [@Milman08]. Some estimates of the Lipschitz constant for optimal transportation of convex bodies have been obtained in [@Kol]. The arguments below generalize the maximum principle proof of Caffarelli. Let $\nabla \Phi$ be the optimal transportation mapping pushing forward $e^{-V} dx$ to $\frac{1}{\lambda(K)} \lambda\|_{K}$. Let us fix a unit vector $h$ . We are looking for a function $\psi$ such that $$\psi(\Phi_h) + \log \Phi_{hh}$$ is bounded from above. Assume that $x_0$ is the maximum point. One has at this point $$\label{max-grad} \psi'(\Phi_h) \nabla \Phi_h + \frac{1}{\Phi_{hh}} \nabla \Phi_{hh}=0$$ $$\label{max-hess} \psi''(\Phi_h) \nabla \Phi_h \oplus \nabla \Phi_h + \psi'(\Phi_h) D^2 \Phi_h + \frac{1}{\Phi_{hh}} D^2 \Phi_{hh} - \frac{1}{\Phi^2_{hh}} \nabla \Phi_{hh} \oplus \nabla \Phi_{hh} \le 0.$$ Differentiation the change of variables formula gives (see Section 1) $$V_h = - \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{h},$$ $$V_{hh} = - \mbox{\rm Tr} (D^2 \Phi)^{-1} D^2 \Phi_{hh} + \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr]^2.$$ Multiply (\[max-hess\]) by $(D^2 \Phi)^{-1}$, take the trace and plug in the expression for $V_{hh}$ into the formula. One obtains $$\begin{aligned} V_{hh} & \ge - \frac{1}{\Phi_{hh}} \mbox{\rm Tr} \bigl[ (D^2 \Phi)^{-1} \cdot \nabla \Phi_{hh} \oplus \nabla \Phi_{hh} \bigr] + {\Phi_{hh}} \cdot \psi^{''}(\Phi_h) \mbox{\rm Tr} \bigl[ (D^2 \Phi)^{-1} \cdot \nabla \Phi_{h} \oplus \nabla \Phi_{h} \bigr] \\& + \Phi_{hh} \cdot \psi'(\Phi_h) \mbox{\rm Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr] + \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr]^2.\end{aligned}$$ Remark that $ \mbox{\rm Tr} \bigl[ (D^2 \Phi)^{-1} \cdot \nabla \Phi_{h} \oplus \nabla \Phi_{h} \bigr] =\Phi_{hh}$. One obtains from (\[max-grad\]) that $ \nabla \Phi_{hh} =- \Phi_{hh} \cdot \psi'(\Phi_h) \nabla \Phi_h$. Plugging this into the inequality for $V_{hh}$ one gets $$V_{hh} \ge \Phi^2_{hh} \bigl[ \psi^{''} - (\psi')^2\bigr] \circ \Phi_h + \Phi_{hh} \cdot \psi'(\Phi_h) \mbox{\rm Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr] + \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr]^2.$$ Note that $$\mbox{\rm Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr] = \mbox{\rm Tr} C, \ \ \mbox{Tr} \Bigl[ (D^2 \Phi)^{-1} D^2 \Phi_{h}\Bigr]^2 = \mbox{Tr} C^2,$$ where $$C = (D^2 \Phi)^{-1/2} (D^2 \Phi_{h}) (D^2 \Phi)^{-1/2}$$ is a symmetric matrix. Hence, by the Cauchy inequality $$V_{hh} \ge \Phi^2_{hh} \Bigl[ \psi^{''} - \Bigl(1 + \frac{d}{4}\Bigr) (\psi')^2\Bigr] \circ \Phi_h.$$ Now assume that $V_{hh}$ is bounded from above by a constant $C$. Let $\psi$ be a function satisfying $$\psi^{''} - \Bigl(1 + \frac{d}{4}\Bigr) (\psi')^2 \ge e^{2 \psi}.$$ Then we get $$C \ge \Phi^2_{hh}(x_0) e^{2 \psi(\Phi_h(x_0))} = \sup_{x \in {{\ensuremath{\mathbb{R}}}}^d} \Phi^2_{hh} e^{2 \psi(\Phi_h)}.$$ In particular, choosing carefully $\psi$ one can obtain the following statement (see [@Kol] for details). \[set-image\] 1) Optimal transportation $T$ of the standard Gaussian measure $\gamma$ onto $\frac{1}{\lambda(K)} \lambda\|_{K}$, where $K$ is convex, satisfies $$\\|DT\\| \le c \sqrt{d} \ \mbox{\rm diam}(K),$$ where $c$ is an universal constant and $\mbox{\rm diam}(K)$ is the diameter of $K$. 2\) Optimal transportation $T$ between $\mu = e^{-V} \ dx$ and $\frac{1}{\lambda(K)} \lambda\|_{K}$, with $ V_{hh} \le C $, $\|V_h\| \le C$ for some $C$, satisfies $$\\|DT\\| \le c \ \mbox{\rm diam}(K),$$ where $c$ depends only on $C$. Unfortunately, estimates of Theorem \[set-image\] are not strong enough to recover even known results on the Cheeger constant for convex bodies. This gives raise to the following problem. Does there exist any dimension-free estimate for $\\|DT\\|$, when $\mu = \gamma$ and $\nu = \frac{1}{\lambda(K)} \lambda\|_{K}$? The same for the case when $\mu$ is the product of exponentional distributions. Note, that it would be enough for our purpose to have a integral norm estimate $\int \\|DT\\|^p \ d\gamma$, $p \ge 1$. This follows form the result of E. Milman [@Milman08] about equivalence of norms for log-concave measures. Contraction for the mass transport generated by semigroups ========================================================== A contraction result for another type of mass transport has been obtained recently in [@KimMilman] by Y.-H Kim and E. Milman. The idea of the construction of this transportation mapping goes back to J. Moser. Consider the diffusion semigroup $P_t = e^{tL}$ denerated by $$L = \Delta- \langle \nabla V, \nabla \rangle = e^{V} \mbox{div} (e^{-V} \cdot \nabla)$$ and the flow of probability measures $$\nu_t =P_t (e^{-W+V}) \cdot \mu.$$ Clearly, $\mu$ is the invariant measure for $P_t$, $\nu_0 = \nu$, and $\nu_{\infty} = \mu$. Let us write the transport equation for $\nu_t$: $$\frac{d}{dt} \nu_t = L P_t (e^{-W+V}) \cdot \mu = \mbox{div} \bigl[ \nabla P_t (e^{-W+V}) \cdot e^{-V} \bigr] = \mbox{div} \bigl[ \nabla \log P_t (e^{-W+V}) \cdot \nu_t \bigr].$$ The corresponding flow of diffeomorphisms is governed by the equation $$\label{Lagr} \frac{d}{dt} S_t = -\nabla \log P_t (e^{-W+V}) \circ S_t, \ \ S_0 = \mbox{Id},$$ where $\nu_t$ and $S_t$ are related by $${\nu_t} = \nu \circ S^{-1}_t.$$ In particular, the limiting map $S_{\infty} = \lim_{t \to \infty} S_t$ pushes forward $\nu$ to $\mu$. We denote the inverse mappings by $T_t$: $$T_t \circ S_t = \mbox{Id}, \ \ T = \lim_{t \to \infty} T_t.$$ The contraction property for $T=S^{-1}$ is equivalent to the expansion property of $S$. It is sufficient to show that $(DS_t)^* DS_t \ge \mbox{Id}$. Using (\[Lagr\]) one gets $$\frac{d}{dt} DS_t(x) = -DW_t(S_t) \cdot DS_t, \ \ W_t = \nabla \log P_t (e^{-W+V}).$$ Hence $$\frac{d}{dt} (DS_t)^* DS_t = 2 (DS_t)^* \cdot DW_t(S_t) \cdot DS_t.$$ Clearly, if $$DW_t(S_t) = -D^2 \log P_t (e^{-W+V}) \ge 0,$$ then $S_t$ has the desired expansion property. Assume now that the function $U$ defined by $$\nu = e^{-U} \cdot \mu, \ U = W-V,$$ is convex. Then the property $- D^2 \log P_t (e^{-W+V}) = - D^2 \log P_t e^{-U} \ge 0$ means that $P_t$ [preserves log-concave functions]{}. Thus we obtain Assume that $U$ is convex. If $U_t = -\log P_t e^{-U}$ is a convex function for every $t \ge 0$, then every $T_t$ is a $1$-contraction. It should be noted that by a resulf from [@Kol2001] the property to preserve [all]{} log-concave fuinctions do admit only diffusion semigroups with Gaussian kernels. Nevertheless, Kim and Milman were able to show under certain symmetry assumptions log-concavity is preserved. The proof is based on the application of the maximum principle. They get, in particularly, the following result (see [@KimMilman] for a more general statement). Assume that $\mu$ is a product mesure, $V$ and $U$ are convex functions, $U$ is unconditional $U(x_1, \cdots, x_n) = U(\pm x_1, \cdots, \pm x_n)$, and $V(x) = \sum_{i=1}^d \rho_i(\|x_i\|)$ with $\rho_i''' \le 0$. Then $T$ is a contraction. In addition, the optimal transportation mapping $T_{opt}$ pushing forward $\mu$ onto $\nu$ is a contraction too. Let us very briefly explain the idea of the proof. Let $t_0$ be the first moment when the convexity of $U_t$ fails. Assume that the minimum of $\partial_{ee} U_{t_0}$ is attained at some point $x_0$ for some direction $e$. Then $(d/dt - \Delta) \partial_{ee} U_{t} \|_{t_0, x_0} \le 0$. In addition, $\nabla \partial_e U_t =0$ and $\nabla \partial_{ee} U_t =0$. Using this one can show that $$(d/dt - \Delta) \partial_{ee} U_{t}\|_{t_0, x_0} = - \langle \nabla U_t, \nabla V_{ee}\rangle\|_{t_0, x_0}.$$ At the time $t_0$ the function $U_t$ is still convex and it is easy to see that the right-hand side schould be non-negative. This leads to a contradiction. $L^p$-contractions ================== In this section we discuss an $L^p$-generalization of the Caffarelli’s theorem (see [@Kol2010]). The proof below is obtained with the help of the so-called above-tangent formalism (see [@Kol2010]). The huge advantage of this approach is that no a priori regularity of the function $\Phi$ is required. See [@Kol2010] for details and relations to the transportation inequalities. The estimates obtined in this section can be considered as global dimenion-free Sobolev a priori estimates for the optimal transportation problem. In particular, they can be generalized fo infinite-dimensional measures. \[lp-est\] Assume that $D^2 W \ge K \cdot \mbox{\rm Id}$. Then for every unit $e$, $p \ge 1$, one has $$K \\| \Phi^2_{ee} \\|_{L^{p}(\mu)} \le \\| (V_{ee})_{+} \\|_{L^{p}(\mu)},$$ $$K \\| \Phi^2_{ee} \\|_{L^{p}(\mu)} \le \frac{p+1}{2} \\| V^2_{e} \\|_{L^{p}(\mu)}.$$ Fix unit vector $e$. According to a result of McCann [@McCann2] the change of variables formula $$V(x) = W(\nabla \Phi(x)) - \log \det D^2_a \Phi$$ holds $\mu$-almost everywhere Here $ D^2_a \Phi$ is the absolutely continuous part of the second distributional derivative $D^2 \Phi$ (Alexandrov derivative). One has $$V(x+ te) - V(x) = W(\nabla \Phi(x+te))- W(\nabla \Phi(x)) - \log \Bigl[ ({\det}_a D^2 \Phi(x))^{-1} \cdot {\det}_a D^2\Phi(x+te)\Bigr].$$ By the uniform convexity of $W$ $$\begin{aligned} V(x+ te) - V(x) & \ge \langle \nabla \Phi(x+te)- \nabla \Phi(x), \nabla W(\nabla \Phi(x)) \rangle \\& + \frac{K}{2} \|\nabla \Phi(x+te)- \nabla \Phi(x)\|^2 - \log \Bigl[ ({\det}_a D^2 \Phi(x))^{-1} \cdot {\det}_a D^2\Phi(x+te)\Bigr].\end{aligned}$$ Multiply this identity by $(\delta_{te} \Phi)^p $, where $p \ge 0$ and $$\delta_{te} \Phi = \Phi(x+te) + \Phi(x-te) -2 \Phi(x)$$ and integrate over $\mu$. We apply the following simple lemma. Let $\varphi: A \to {{\ensuremath{\mathbb{R}}}}$, $\psi: B \to {{\ensuremath{\mathbb{R}}}}$ be convex functions on convex sets $A$, $B$. Assume that $\nabla \psi(B) \subset A$. Then $$\mbox{\rm div} (\nabla \varphi \circ \nabla \psi) \ge \mbox{\rm Tr}\bigl[ D^2_a \varphi (\nabla \psi) \cdot D^2_a \psi \bigr] \ dx \ge 0,$$ where $\mbox{\rm div}$ is the distributional derivative. Integrating by parts and applying this lemma we get $$\begin{aligned} \int \langle \nabla \Phi(x+te) & - \nabla \Phi(x), \nabla W(\nabla \Phi(x)) \rangle (\delta_{te} \Phi)^p \ d\mu \\& = \int \langle \nabla \Phi(x+te) \circ (\nabla \Psi)- x, \nabla W(x) \rangle (\delta_{te} \Phi)^p \circ (\nabla \Psi) \ d\nu \\& \ge \int \Bigl( \mbox{Tr} \bigl[ D^2 _a\Phi(x+te) \cdot (D^2_a \Phi)^{-1} \bigr] \circ (\nabla \Psi) - d \Bigr) (\delta_{te} \Phi)^p \circ (\nabla \Psi) \ d\nu \\& + p \int \Big\langle \nabla \Phi(x+te) \circ (\nabla \Psi)- x, (D^2 \Psi) \nabla \delta_{te} \Phi \circ (\nabla \Psi) \Big\rangle (\delta_{te} \Phi)^{p-1}\circ (\nabla \Psi) \ d\nu.\end{aligned}$$ We note that $$\mbox{Tr} A - d - \log \det A \ge 0$$ for any $A$ of the type $A=BC$, where $B$ and $C$ are symmetric and positive. Indeed, $$\mbox{Tr} A - d - \log \det A =\mbox{Tr} C^{1/2} B C^{1/2} - d - \log \det C^{1/2} B C^{1/2} = \sum_i \lambda_i - 1 - \log \lambda_i,$$ where $\lambda_i$ are eigenvalues of $C^{1/2} B C^{1/2}$. Consequently $$\begin{aligned} \int \bigl( V(x+ te) - & V(x) \bigr) (\delta_{te} \Phi)^p d\mu \ge \frac{K}{2} \int \|\nabla \Phi(x+te)- \nabla \Phi(x)\|^2 (\delta_{te} \Phi)^p \ d\mu \\& + p \int \Big\langle \nabla \Phi(x+te) - \nabla \Phi(x), (D^2 \Psi) \circ \nabla \Phi(x) \nabla \delta_{te} \Phi \Big\rangle (\delta_{te} \Phi)^{p-1} \ d\mu.\end{aligned}$$ Applying the same inequality to $-te$ and taking the sum we get $$\begin{aligned} \int & \bigl( V(x+ te) + V(x-te) - 2V(x) \bigr) (\delta_{te} \Phi)^p d\mu \\& \ge \frac{K}{2} \int \|\nabla \Phi(x+te)- \nabla \Phi(x)\|^2 (\delta_{te} \Phi)^p \ d\mu + \frac{K}{2} \int \|\nabla \Phi(x-te)- \nabla \Phi(x)\|^2 (\delta_{te} \Phi)^p \ d\mu \\& + p \int \Big\langle \nabla \delta_{te} \Phi , (D^2_a \Phi)^{-1} \nabla \delta_{te} \Phi \Big\rangle (\delta_{te} \Phi)^{p-1} \ d\mu.\end{aligned}$$ Note that the last term is non-negative. Dividing by $t^{2p}$ and passing to the limit we obtain $$\label{lp-sd+} \int V_{ee} \Phi_{ee}^p \ d \mu \ge K \int \\| D^2 \Phi \cdot e\\|^2 \Phi_{ee}^p \ d \mu + p \int \langle (D^2 \Phi)^{-1} \nabla \Phi_{ee}, \nabla \Phi_{ee} \rangle \Phi_{ee}^{p-1} \ d \mu.$$ For the proof of the first part we note that $$\int V_{ee} \Phi^p_{ee} \ d\mu \ge K \int \Phi_{ee}^{p+2} \ d \mu.$$ Applying the H[ö]{}lder inequality one gets $$\\| (V_{ee})_{+}\\|_{L^{(p+2)/2}(\mu)} \\| \Phi_{ee}^p\\|_{L^{(p+2)/p}(\mu)} \ge \int V_{ee} \Phi^p_{ee} \ d\mu .$$ This readily implies the result. To prove the second part we integrate by parts the left-hand side $$\begin{aligned} \int V_{ee} \Phi^{p}_{ee} \ d \mu & = -p \int V_e \Phi_{eee} \Phi^{p-1}_{ee} \ d\mu + \int V^2_e \Phi^{p}_{ee} \ d \mu \\& = - p \int \langle \nabla \Phi_{ee}, V_e \cdot e \rangle \Phi^{p-1}_{ee} d\mu + \int V^2_e \Phi^{p}_{ee} \ d \mu.\end{aligned}$$ By the Cauchy inequality the latter does not exceed $$p \int \langle (D^2 \Phi)^{-1} \nabla \Phi_{ee}, \nabla \Phi_{ee} \rangle \Phi_{ee}^{p-1} \ d \mu + \frac{p}{4} \int V^2_e \langle D^2 \Phi e, e \rangle \Phi_{ee}^{p-1} \ d \mu + \int V^2_e \Phi^{p}_{ee} \ d \mu.$$ Inequality (\[lp-sd+\]) implies $$\frac{p+4}{4} \int V^2_e \Phi_{ee}^{p} \ d \mu \ge K \int \|\nabla \Phi_e\|^2 \Phi_{ee}^p \ d \mu \ge K \int \Phi_{ee}^{p+2} \ d \mu.$$ The rest of the proof is the same as in the first part. In the limit $p\to\infty$ we obtain the contraction theorem of Caffarelli $$K \\| \Phi_{ee} \\|^2_{L^{\infty}(\mu)} \le \\| (V_{ee})_{+} \\|_{L^{\infty}(\mu)}.$$ A more difficult estimate for the operator norm $\\| \cdot \\|$ has been also obtained in [@Kol2010]. Assume that $D^2 W \ge K \cdot \mbox{\rm Id}$. Then for every $r \ge 1$ one has $$K \Bigl( \int \\| D^2 \Phi\\|^{2r} \ d \mu \Bigr)^{\frac{1}{r}} \le \Bigl( \int \\| (D^2 V)_{+} \\|^r \ d \mu \Bigr)^{\frac{1}{r}}.$$ Contractions for infinite measures ================================== In this section we investigate contractions of infinite measures. Let us stress that unlike the probability case we don’t have a natural probabilistic normalization of the total volume. We start with the following $1$-dimensional example Let $d=1$ and $\mu= \lambda\|_{{{\ensuremath{\mathbb{R}}}}^+}$, $\nu = I_{[0,+\infty)} \rho \ dx$ and $\rho \ge 1$. Then the standard monotone transportation $T$ is a contraction. Indeed, this follows immediately from the explicit representation of $T$ $$\int_{0}^{T} \rho \ dx = x.$$ Let us investigate what happens for $d=2$ if the image measure is rotationally symmetric. [(F. Morgan)]{} Let $d=2$ and $\mu= \lambda$, $\nu = \Psi(r) \ dx$. A natural transport mapping has the form $$T(x) = \varphi(r) \cdot n, \ \ n = \frac{x}{r}$$ Clearly $$\nu(T(B_r)) = 2 \pi \int_{0}^{\varphi(r)} \ s \Psi(s) \ dr= \pi r^2 = \mu(B_r) .$$ Let us compute $DT$ in the frame $(n,v)$, where $v = \frac{(-x_2,x_1)}{r}$. One has $$\partial_n T = \varphi' \cdot n \ \ \ \partial_v T = \frac{\varphi}{r} \cdot v.$$ Clearly, a necessary and sufficient condition for $T$ to be a contraction is the following: $$\varphi' \le 1$$ or $\psi' \ge 1$ for $\psi = \varphi^{-1}$. From the change of variables formula we obtain $$\psi(r) = \sqrt{2 \int_0^r s \Psi(s) \ ds}.$$ Condition $\psi' \ge 1$ is equivalent to $ \int_0^r s \Psi (s) \ ds \le \frac{(r \Psi(r))^2}{2}. $ The latter holds, for instance, if $$(s \Psi (s))' \ge 1.$$ Indeed, in this case $$\int_0^r s \Psi (s) \ ds \le \int_0^r s \Psi (s) (s \Psi (s))' \ ds =\frac{(r \Psi(r))^2}{2}.$$ Similarly in dimension $d$, a sufficient condition for the transportation mapping $T = \varphi(r) \frac{x}{r}$ between $\lambda$ and $\Psi(r) \ dx$ to be a contraction n is that $$(r \Psi^{\frac{1}{d-1}}(r))' \ge 1.$$ In $d$-dimensional Euclidean space with density $\Psi(r)$ satisfying $ (r \Psi^{\frac{1}{d-1}}(r))' \ge 1$, the Euclidean isoperimetric inequality holds. Some example of contraction mappings arise naturally in differential geometry (see [@MaurMorg], Propositions 1.1 and 2.1). Let $M$ be the plane equipped with the metric $$dr^2 + g^2(r) r^2 d\theta^2$$ (surface of revolution), $g \ge 1$. Then the identity mapping form $M$ to the Euqlidean plane with measure $g \ dx$ is a volume preserving contraction. In particular, $\cosh^2(r) \ dx$ is a Lipschitz image of $H^2$ (with metric $dr^2 + \cosh^2(r) d \theta^2$). The following comparison result has been proved in [@KoZh]. It turns out that a natural model measure for the one-dimensional log-convex distributions has the following form: $$\nu_{A} = \frac{dx}{\cos Ax}, \ -\frac{\pi}{2A} < x < \frac{\pi}{2A}.$$ Its potential $V$ satisfies $ V'' e^{-2V} =A^2. $ Using a result [@RCBM] on symmetricity of the isoperimetric sets one can compute the isoperimetric profile of $\nu_A$: $$\mathcal{I}_{\nu_A}(t) = e^{At/2}+e^{-At/2}.$$ Let $\mu = e^{W} dx$ be a measure on ${{\ensuremath{\mathbb{R}}}}^1$ with even convex potential $W$. Assume that $$W''e^{-2W} \ge A^2,$$ and $W(0)=0$. Then $\mu$ is the image of $\nu_A$ under a $1$-Lipschitz increasing mapping. Without loss of generality one can assume that $W$ is smooth and $W''e^{-2W} > A^2$. Let $\varphi$ be a convex potential such that $T=\varphi'$ sends $\mu$ to $\nu_A$. In addition, we require that $T$ is antisymmetric. Clearly, $\varphi'$ satisfies $$e^{W} = \frac{\varphi''}{\cos A \varphi'}.$$ Assume that $x_0$ is a local maximum point for $\varphi''$. Then at this point $$\varphi^{(3)}(x_0)=0, \ \ \varphi^{(4)}(x_0) \ge 0.$$ Differentiating the change of variables formula at $x_0$ twice we get $$W'' = \frac{\varphi^{(4)}}{\varphi''} - \Bigl( \frac{\varphi^{(3)}}{\varphi''}\Bigr)^2 + \frac{A^2}{\cos^2 A \varphi'} (\varphi'')^2 + A \frac{\sin A \varphi'}{\cos A \varphi'} \varphi^{'''}.$$ Consequently one has at $x_0$ $$W'' \le \frac{A^2}{\cos^2 A \varphi'} (\varphi'')^2 = A^2 e^{2W}.$$ But this contradicts to the main assumption. Hence $\varphi''$ has no local maximum. Note that $\varphi$ is even. This implies that that $0$ is the global minimum of $\varphi''$. Hence $ \varphi'' \ge \varphi''(0)=1. $ Clearly, $T^{-1}$ is the desired mapping. Other results and applications ============================== An immediate consequence of the contraction theorem is the Bakry-Ledoux comparison theorem, which is a probabilistic analog of the L[é]{}vy-Gromov comparison theorem for Ricci positive manifolds. Assume that $\mu =e^{-V} dx$, where $D^2 V \ge \mbox{\rm{Id}}$, is a probability measure on ${{\ensuremath{\mathbb{R}}}}^d$. Then $$\mathcal{I}_{\mu} \ge \mathcal{I}_{\gamma},$$ where $\gamma$ is the standard Gaussian measure. In the same way the contraction theorem implies different functional and concentration inequalities for uniformly log-concave measures (log-Sobolev, Poincar[é]{} etc.). The following unsolved problem is known as the ”Gaussian correlation conjecture”. [Gaussian correlation conjecture.]{} Let $A$ and $B$ be symmetric convex sets and $\gamma$ be the standard Gaussian measure. Then $$\label{gcc} \gamma(A \cap B) \ge \gamma(A) \gamma(B).$$ The Gaussian correlation conjecture has quite a long history. This problem arose in 70th. The positive solution is known for two-dimensional sets and for the case when one of the sets is an ellipsoid. The ellipsoid case was proved by G. Harg[é]{} [@Harge] by semigroup arguments. \[ellipsoid\] Let $B$ be an ellipsoid. Then (\[gcc\]) holds. Applying a linear transfromation of measures one can reduce the proof to the case when $B$ is a ball and $\gamma$ is a (non-standard) Gaussian measure. Consider the optimal transportation $T$ between $\gamma$ and $\gamma_A = \frac{1}{\gamma(A)} \gamma_{\|A}$. By Theorem \[contr2\] $T$ is a contraction and by the symmetry reasons $T(0)=0$. Hence $T(B) \subset B$ and $$\frac{\gamma(A \cap B)}{\gamma(A)} = \gamma_A(B) = \gamma(T^{-1}(B)) \ge \gamma(B).$$ This completes the proof. The following beautiful observation [@Harge2] follows from the contraction theorem and properties of the Ornstein-Uhlenbeck semigroup $$P_t f(x) = \int f(x e^{-t} + \sqrt{1-e^{-2t}} y) \ d \gamma(y).$$ If $\gamma$ is a standard Gaussian measure, $g$ is symmetric convex and $f $ is symmetric log-concave, then $$\int f g \ d \gamma \le \int f d \gamma \cdot \int g \ d \gamma.$$ Let $T(x) =x + \nabla \varphi(x)$ be the optimal transportation of $\gamma$ onto $\frac{f \cdot \gamma}{\int f d\gamma}$. Thus we need to prove that $$\int g(x + \nabla \varphi(x)) \ d \gamma \le \int g \ d \gamma.$$ Set: $$\psi(t) = \int g(x + P_t (\nabla \varphi(x)) ) \ d \gamma,$$ where $P_t = e^{tL}$ is the Ornstein-Uhlenbeck semigroup generated by $L = \Delta - \langle x, \nabla \rangle$. Note that $$\frac{\partial}{\partial t} \psi(t) = \frac{\partial}{\partial t} \int g(x + P_t (\nabla \varphi(x))) \ d \gamma = \int \langle \nabla g(x + P_t (\nabla \varphi(x))), L P_t (\nabla \varphi(x)) \rangle \ d \gamma.$$ Integrating by parts we get $$\frac{\partial}{\partial t} \psi(t) = - \int \mbox{Tr} \Bigl[ D^2 g (x + P_t (\nabla \varphi(x))) \cdot (I + M) M\Bigr] \ d \gamma,$$ where $$M = D P_t (\nabla \varphi(x)) = e^{-t/2} P_t ( D^2 \varphi).$$ Clearly, by the contraction theorem $I + M \ge 0$ and $M \le 0$. Hence $\mbox{Tr} \Bigl[ D^2 g \cdot (I + M) M \Bigr] \le 0$ and $\psi(t)$ is increasing. Note that $P_{+\infty}(\nabla \varphi) = \int \nabla \varphi \ d\gamma = \frac{\int x f \ d \gamma}{\int f \ d \gamma} = 0$. Hence $ \int g(x + \nabla \varphi(x)) \ d \gamma \le \psi(+ \infty) = \int g \ d \gamma $. The proof is complete. Some other applications to correlation inequalities have been obtained in [@CorEr], [@KimMilman]. A generalization of Theorem \[ellipsoid\] to non-Gaussian measures have been obtained in [@KimMilman] (see Corollary 4.1). Other applications obtained in [@Caf], [@CorEr], [@Harge], [@KimMilman] concern inequalities of the type $$\int \Gamma(x) \ d\mu \le \int \Gamma(x) \ d \nu,$$ where $\Gamma(x)$ is convex (moment inequalities etc.). The following theorem was obtained in [@CEFM] with the help of the contraction theorem. In particular, it solves the so-called (B)-conjecture from the theory of Gaussian measures. Let $K$ be a symmetric convex set and $\gamma$ is a standard Gaussian measure. Then the function $$t \to \gamma(e^t K)$$ is log-concave. In particular, $\gamma(\sqrt{ab} K)^2 \ge \gamma(aK) \gamma(bK)$ for every $a>0, b>0$. [Sketch of the proof.]{} Since $\gamma(e^{t_1 + t_2} K) = \gamma(e^{t_1} (e^{t_2} K))$, it is sufficient to show that $g(t) = \gamma(e^t K)$ is log-concave at zero. This is equivalent to the inequality $g''(0) g(0) \le (g'(0))^2$. Computing the derivatives of $g$ we get that this is equivalent to $$\int \|x\|^4 \ d \gamma_K - \bigl( \int \|x\|^2 \ d \gamma_K\bigr)^2 \le 2 \int x^2 \ d \gamma_K,$$ where $\gamma_K = \frac{1}{\gamma(K)} I_{K} \cdot \gamma$. Let us prove a more general relation: if $\mu = e^{-W} \ dx$ is a log-concave measure with $D^2 W \ge \mbox{Id}$ and $f$ is a function, satisfying $\int f \ d\mu = 0, \int \nabla f \ d \mu=0$, then the following Poincar[é]{}-type inequality holds: $$\label{strongPoin} \int f^2 \ d \mu \le \frac{1}{2} \int \\| \nabla f \\|^2 \ d \mu.$$ Applying (\[strongPoin\]) to $f = \|x\|^2 - \int \|x\|^2 \ d \mu$, we get the desired inequality for $\mu$. Then it remains to approximate $\gamma_K$ by measures of this type. Note that by the Caffarelli’s theorem it is sufficient to prove inequality (\[strongPoin\]) only for the standard Gaussian measure. But in this case (\[strongPoin\]) is well-known and can be obtained from the expansion of $f$ on the basis formed by the Hermite polynomials. The proof is complete. Note that apart from the observations of the previous section nothing is known about contractions of manifolds. The following result was obtained by S. I. Valdimarsson (see [@Vald]). For every nonnegative symmetric $M$ let us denote by $\gamma_M$ the Gaussian measure with density $$\sqrt{\det M} e^{-\pi \langle M x, x \rangle}.$$ Let $A$, $G$ and $B$ are positive definite symmetric linear transformations, $A<G$, $GB=BG$, $H$ is a convex function, and $\mu_0$ is a probability measure. The optimal transportation $T = \nabla \Phi$ between probability measures $$\mu = \gamma_{B^{-1/2} G B^{-1/2}} \ast \mu_0 \ \mbox{and} \ \nu = C e^{-H} \cdot\gamma_{B^{-1/2} A^{-1} B^{-1/2}}$$ satisfies $$D^2 \Phi \le G.$$ A particular form of the measure $\mu$ allows Valdimarsson (after F. Barthe [@Barthe]) to obtain by the transportation arguments a new form of the well-known Brascamp-Lieb inequality. See [@Vald] for details. We finish with the following observation from [@BarKol]. Let $\mu = I_{[0,+\infty)} e^{-x} \ dx $ be the one-sided exponential measure and $\nu = e^{g} \cdot \mu$ with $\|g'\| \le c$ for some $c<1$. Then the monotone map $T$ which transports $\nu$ to $\mu$ satisfies $$T'(x) \in [1-c, 1+c]$$ for all $x \in [0, \infty)$. The inverse map $S = T^{-1}$ is a $\frac{1}{1-c}$-contraction. The result follows from the explicit representation of $T$ but can heuristically proved by the maximum principle arguments applied to $S$. Indeed $$g(S) -S + \log S' = - x.$$ If $x_0$ is the maximum point for $S'$, one has $S''(x_0) = 0$. In addition, $$g'(S(x_0)) S'(x_0)-S'(x_0) + \frac{S''(x_0)}{S'(x_0)} = -1.$$ Clearly $S'(x_0) = \frac{1}{1-g'(S(x_0)} \le \frac{1}{1-c}$. 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--- abstract: 'We study the projectivity of the free Banach lattice generated by a lattice $\mathbb{L}$ in two cases: when the lattice is finite, and when the lattice is an infinite linearly ordered set. We prove that in the first case it is projective while in the second case it is not.' address: - 'Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo 30100 Murcia, Spain.' - 'Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo 30100 Murcia, Spain.' author: - Antonio Avilés - José David Rodríguez Abellán title: Projectivity of the free Banach lattice generated by a lattice --- [^1] Introduction ============ Free and projective Banach lattices were introduced in [@dPW15]. The free Banach lattice $FBL(A)$ generated by a set $A$ is a Banach lattice characterized by the property that every bounded map $T:A{\longrightarrow}X$ into a Banach lattice $X$ extends to a unique Banach lattice homomorphism $\hat{T}:FBL(A){\longrightarrow}X$ with the same norm. This idea was generalized in [@ART18] and [@ARA18], where the free Banach lattice generated by a Banach space $E$ and by a lattice $\mathbb{L}$ are respectively studied. By a lattice we mean here a set $\mathbb{L}$ together with two operations $\wedge$ and $\vee$ that are the infimum and supremum of some partial order relation on $\mathbb{L}$, and a lattice homomorphism is a function between lattices that commutes with those two operations. \[FBLGBL\] Given a lattice $\mathbb{L}$, the free Banach lattice generated by $\mathbb{L}$ is a Banach lattice $F$ together with a lattice homomorphism $\phi: \mathbb{L} \longrightarrow F$ such that for every Banach lattice $X$ and every bounded lattice homomorphism $T: \mathbb{L} \longrightarrow X$, there exists a unique Banach lattice homomorphism $\hat{T}: F \longrightarrow X$ such that $\|\| \hat{T} \|\| = \|\| T \|\|$ and $T = \hat{T} \circ \phi$. Here, the norm of $T$ is ${\left\VertT\right\Vert} := \sup {\left\{{\left\VertT(x)\right\Vert}_X : x \in \mathbb{L}\right\}}$, while the norm of $\hat{T}$ is the usual norm of an operator acting between Banach spaces. This definition determines a Banach lattice that we denote by $FBL\langle \mathbb{L}\rangle$ in an essentially unique way. When $\mathbb{L}$ is a distributive lattice the function $\phi$ is injective and we can view $FBL\langle \mathbb{L} \rangle$ as a Banach lattice which contains a subset lattice-isomorphic to $\mathbb{L}$ in a way that its elements work as free generators modulo the lattice relations on $\mathbb{L}$, cf. [@ARA18]. To see that, it is well known that a lattice $\mathbb{L}$ is distributive if, and only if, $\mathbb{L}$ is lattice-isomorphic to a bounded subset of a Banach lattice. Thus, it is clear that if $\phi$ is inyective then $\mathbb{L}$ is distributive. On the other hand, if $\mathbb{L}$ is distributive, we have a bounded injective lattice homomorphism $T : \mathbb{L} \longrightarrow X$ for some Banach lattice $X$. Using the definition of being the free Banach lattice generated by the lattice $\mathbb{L}$, there is $\hat{T}$ such that $\hat{T} \circ \phi = T$. Since $T$ is inyective, $\phi$ is also inyective. The notions of free and projective objects are closely related in the general theory of categories. In the context of Banach lattices, de Pagter and Wickstead [@dPW15] introduced projectivity in the following form: \[projdef\] A Banach lattice $P$ is projective if whenever $X$ is a Banach lattice, $J$ a closed ideal in $X$ and $Q : X \longrightarrow X/J$ the quotient map, then for every Banach lattice homomorphism $T : P \longrightarrow X/J$ and $\varepsilon > 0$, there is a Banach lattice homomorphism $\hat{T} : P \longrightarrow X$ such that $T = Q \circ \hat{T}$ and $\\|\hat{T}\\| \leq (1 + \varepsilon){\left\VertT\right\Vert}$. Some examples of projective Banach lattices given in [@dPW15] include $FBL(A)$, $\ell_1$, all finite dimensional Banach lattices and Banach lattices of the form $C(K)$, where $K$ is a compact neighborhood retract of $\mathbb{R}^n$. But we are still far from understanding what the projective Banach lattices are. Such basic questions as whether $c_0$, $\ell_2$ or $C([0,1]^\mathbb{N})$ are projective were left open in [@dPW15]. Since the canonical projective Banach lattice is the free Banach lattice $FBL(A)$, it is natural to think that its variants $FBL[E]$ (the free Banach lattice generated by the Banach space $E$) and $FBL\langle \mathbb{L}\rangle$ may also be projective at least in some cases. In this paper we focus on the case of $FBL\langle \mathbb{L}\rangle$. We prove that $FBL\langle \mathbb{L}\rangle$ is projective whenever $\mathbb{L}$ is a finite lattice, while it is not projective when $\mathbb{L}$ is an infinite linearly ordered space. If $\mathbb{L}$ is a finite lattice, $FBL\langle \mathbb{L}\rangle$ is a renorming of a Banach lattice of continuous functions $C(K)$ on a compact neighborhood retract $K$ of $\mathbb{R}^n$, which is projective [@dPW15]. Projectivity, however, is not preserved under renorming, because of the $(1+\varepsilon)$ bound required in Definition \[projdef\]. Getting this bound will be the key point in the proof. In the infinite case, we considered only linearly ordered sets, as they are easier to handle than general lattices. We do not know if there is some infinite lattice $\mathbb{L}$ such that $FBL\langle \mathbb{L}\rangle$ is projective. Preliminaries ============= Absolute neighborhood retracts. ------------------------------- An absolute neighborhood retract (ANR) is a topological space $X$ with the property that whenever $X$ is a subspace of $Y$, then there is an open subset $V$ of $Y$ such that $X\subset V \subset Y$ and $X$ is a retract of $V$, meaning that there is a continuous function $r:V{\longrightarrow}X$ such that $r(x)=x$ for all $x\in X$. The following are two basic facts of the theory that can be found in [@vMill] as Theorems 1.5.1 and 1.5.9: - Every closed convex subset of $\mathbb{R}^n$ is ANR. - If $X_1$, $X_2$ are closed subsets of $X$, and $X_1$, $X_2$ and $X_1\cap X_2$ are ANR, then $X_1\cup X_2$ is also ANR. From the two facts above, one can easily prove that every finite union of closed convex subsets of $\mathbb{R}^n$ is ANR, by induction on the number of convex sets in that union. Free Banach lattices. --------------------- We collect the necessary facts and definitions about free Banach lattices from [@dPW15; @ART18; @ARA18]. An explicit construction of the free Banach lattice $FBL(A)$ generated by a set $A$ is as follows. For $x \in A$, let $\delta_x:[-1,1]^A \longrightarrow [-1,1]$ be the evaluation function given by $\delta_x(x^) = x^(x)$ for every $x^* \in [-1,1]^A$, and for $f:[-1,1]^A \longrightarrow \mathbb{R}$ we define $$\\|f\\| = \sup {\left\{\sum_{i = 1}^n {\left\vert r_if(x_{i}^{\ast})\right\vert} : \text{}r_i \in \mathbb{R}, \text{ } x_i^{\ast} \in [-1,1]^A, \text{ }\sup_{x \in A} \sum_{i=1}^n {\left\vertr_ix_i^{\ast}(x)\right\vert} \leq 1 \right\}},$$ which we will denote by $\\| f \\|$ or $\\| f \\|_{FBL(A)}$. The Banach lattice $FBL(A)$ is the Banach lattice generated by the evaluation functions $\delta_x$ inside the Banach lattice of all functions $f:[-1,1]^A{\longrightarrow}\mathbb{R}$ with finite norm. The natural identification of $A$ inside $FBL(A)$ is given by the map $u: A \longrightarrow FBL(A)$ where $u(x) = \delta_x$. Since every function in $FBL(A)$ is an uniform limit of such functions, they are all continuous and positively homogeneous (they commute with multiplication by positive scalars). When $A$ is finite, then $FBL(A)$ consists of all continuous and positively homogeneous functions on $[-1,1]^A$, or equivalently in this case, all positively homogeneous functions on $[-1,1]^A$ that are continuous on the boundary $\partial [-1,1]^A$. Thus, when $A$ is finite, $FBL(A)$ is a renorming of the Banach lattice of continuous functions on $\partial [-1,1]^A$. We can describe of $FBL \langle \mathbb{L} \rangle$ as the quotient of $FBL(\mathbb{L})$ (the free Banach lattice generated by the underlying set of the lattice $\mathbb{L}$) by the closed ideal $\mathcal{I}$ of $FBL(\mathbb{L})$ generated by the set $${\left\{\delta_x \vee \delta_y - \delta_{x \vee y},\ \ \delta_x \wedge \delta_y - \delta_{x \wedge y}\ : \ x,y \in \mathbb{L}\right\}}.$$ In [@ARA18] we prove that, $FBL(\mathbb{L})/\mathcal{I}$, together with the map $\phi: \mathbb{L} \longrightarrow FBL(\mathbb{L})/\mathcal{I}$ given by $\phi(x) = \delta_x + \mathcal{I}$ is the free Banach lattice generated by the lattice $\mathbb{L}$. Also in [@ARA18] there is a different description of $FBL\langle \mathbb{L} \rangle$ as a space of functions. The construction is analogous to that of $FBL(A)$ but taking into account the lattice structure. Namely, if we see $[-1,1]$ as a lattice, define $$\mathbb{L}^{\ast} = {\left\{x^{\ast}: \mathbb{L} \longrightarrow [-1,1] : x^{\ast} \text{ is a lattice-homomorphism}\right\}}.$$ For every $x \in \mathbb{L}$ consider the evaluation function $\dot{\delta}_x : \mathbb{L}^{\ast} \longrightarrow [-1,1]$ given by $\dot{\delta}_x(x^{\ast}) = x^{\ast}(x)$, and for $f \in \mathbb{R}^{\mathbb{L}^{\ast}}$, define $${\left\Vertf\right\Vert}_\ast = \sup {\left\{\sum_{i = 1}^n {\left\vert r_if(x_{i}^{\ast})\right\vert} : \text{ }r_i \in \mathbb{R}, \text{ } x_i^{\ast} \in \mathbb{L}^{\ast}, \text{ }\sup_{x \in \mathbb{L}} \sum_{i=1}^n {\left\vertr_ix_i^{\ast}(x)\right\vert} \leq 1 \right\}}.$$ Let $FBL_\langle \mathbb{L} \rangle$ be the Banach lattice generated by the evaluations ${\left\{\dot{\delta}_x : x \in \mathbb{L}\right\}}$ inside the Banach lattice of all functions $f \in \mathbb{R}^{\mathbb{L}^{\ast}}$ with $\\|f\\|_\ast<\infty$, endowed with the norm $\\|\cdot\\|_\ast$ and the pointwise operations. This, together with the assignment $\phi(x)=\dot{\delta}_x$ is the free Banach lattice generated by $\mathbb{L}$. Thus, we have two alternative constructions of the free Banach lattice generated by $\mathbb{L}$ that we are denoting as $FBL\langle \mathbb{L} \rangle = FBL(\mathbb{L})/\mathcal{I}$ and $FBL_\langle \mathbb{L} \rangle$, respectively. There is a natural Banach lattice homomorphism $R:FBL(\mathbb{L}){\longrightarrow}FBL_\langle \mathbb{L} \rangle$ given by restriction $R(f) = f\|_{\mathbb{L}^}$. This is surjective and its kernel is the ideal $\mathcal{I}$, and thus $R$ induces the canonical isometric Banach lattice isomorphism between $FBL\langle \mathbb{L} \rangle$ and $FBL_\langle \mathbb{L} \rangle$. Projective Banach lattices. --------------------------- We state here a variation of [@dPW15 Theorem 10.3]: \[quotientofprojective\] Let $P$ be a projective Banach lattice, $\mathcal{I}$ an ideal of $P$ and $\pi:P{\longrightarrow}P/\mathcal{I}$ the quotient map. The quotient $P/\mathcal{I}$ is projective if and only if for every $\varepsilon>0$ there exists a Banach lattice homomorphism $u_\varepsilon:P/\mathcal{I}{\longrightarrow}P$ such that $\pi\circ u_\varepsilon = id_{P/\mathcal{I}}$ and $\\|u_\varepsilon\\|\leq 1+\varepsilon$. If $P/\mathcal{I}$ is projective, then we can just apply Definition \[projdef\]. On the other hand, if we have the above property and we want to check Definition \[projdef\], take $\varepsilon_0>0$, a quotient map $Q:X\mapsto X/J$ and a Banach lattice homomorphism $T:P/\mathcal{I}{\longrightarrow}X/J$. Take $\varepsilon$ with $(1+\varepsilon)^2\leq 1+\varepsilon_0$. Since $P$ is projective we can find $S:P{\longrightarrow}X$ with $Q\circ S = T\circ\pi$ and $\\|S\\|\leq (1+\varepsilon)\\|T\circ\pi\\| = (1+ \varepsilon)\\|T\\|$. If we take $\hat{T}= S\circ u_\varepsilon$, then $Q\circ \hat{T} = Q\circ S \circ u_\varepsilon = T\circ \pi\circ u_\varepsilon = T$ and $\\|\hat{T}\\| \leq (1+\varepsilon)^2\\|T\\| \leq (1+\varepsilon_0)\\|T\\|$ as desired. Since $FBL(\mathbb{L})$ is projective [@dPW15 Proposition 10.2], and the restriction map described above $R:FBL(\mathbb{L}){\longrightarrow}FBL_\ast\langle \mathbb{L}\rangle$ is a quotient map [@ARA18], we get, as a particular instance of Proposition \[quotientofprojective\], \[caractProyectividad\] Let $\mathbb{L}$ be a lattice and let $R: FBL(\mathbb{L}) \longrightarrow FBL_\langle \mathbb{L} \rangle$ be the restriction map $R(f) = f\vert_{\mathbb{L}^}$. The Banach lattice $FBL_\langle \mathbb{L} \rangle$ is projective if, and only if, for every $\varepsilon > 0$ there exists a Banach lattice homomorphism $u_{\varepsilon}: FBL_\langle \mathbb{L} \rangle \longrightarrow FBL(\mathbb{L})$ such that $\\| u_{\varepsilon} \\| \leq 1+\varepsilon$ and $R \circ u_{\varepsilon} = id_{FBL_\langle \mathbb{L} \rangle}$. Projectivity of the free Banach lattice generated by a finite lattice ===================================================================== We are going to prove that if $\mathbb{L}$ is a finite lattice, then $FBL_\langle \mathbb{L} \rangle$ is a projective Banach lattice. \[LEstrellaANR\] If $\mathbb{L} = {\left\{0, \ldots, n-1\right\}}$ is a finite lattice, then $\mathbb{L}^ \cap \partial [-1,1]^n$ is $ANR$. Clearly, $\partial [-1,1]^n$ is a finite union of closed convex subsets of $\mathbb{R}^n$. On the other hand, let $$A_{ijk} = {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_i \vee x_j = x_k\right\}}$$ and $$B_{ijk} = {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_i \wedge x_j = x_k\right\}}.$$ It is clear that $$A_{ijk} = {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_i = x_k,\, x_j \leq x_i\right\}} \cup {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_j = x_k,\, x_i \leq x_j\right\}},$$ and $$B_{ijk} = {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_i = x_k,\, x_j \geq x_i\right\}} \cup {\left\{(x_1, \ldots, x_n) \in \mathbb{R}^n : x_j = x_k,\, x_i \geq x_j\right\}}$$ are union of two closed convex sets. Since $$\mathbb{L}^* = \left(\bigcap_{i \vee j = k} A_{ijk}\right) \bigcap \left(\bigcap_{i \wedge j = k} B_{ijk}\right),$$ we have that $\mathbb{L}^\ast $ is also a finite union of closed convex subsets of $\mathbb{R}^n$. We conclude that $\mathbb{L}^* \cap \partial [-1,1]^n$ is a finite union of closed convex subsets of $\mathbb{R}^n$ and thus ANR. In the context of compact metric spaces, the retractions in the definition of ANR can be taken arbitrarily close to the identity. We state this fact as a lemma in the particular case that we need: \[closeretraction\] Let $\mathbb{L} = {\left\{0, \ldots, n-1\right\}}$ be a finite lattice. Then, given $\varepsilon > 0$ , there exist an open set $V_{\varepsilon} = V_{\varepsilon}(\mathbb{L}^)$ with $\mathbb{L}^ \cap \partial [-1,1]^n \subset V_{\varepsilon} \subset \mathbb{R}^n$ and a continuous map $\varphi: V_{\varepsilon} \longrightarrow \mathbb{L}^* \cap \partial [-1,1]^n$ such that $\varphi \vert_{\mathbb{L}^* \cap \partial [-1,1]^n} = id_{\mathbb{L}^* \cap \partial [-1,1]^n}$ and $d(x^, \varphi(x^)) < \varepsilon$ for every $x^* \in V_{\varepsilon}$, where $d$ is the Euclidean distance in $\mathbb{R}^n$. As $\mathbb{L}^* \cap \partial [-1,1]^n$ is an ANR by Proposition \[LEstrellaANR\], we cand find a bounded neighborhood $V$ of $\mathbb{L}^* \cap \partial [-1,1]^n$ in $\mathbb{R}^n$ and a retraction $\psi : V \longrightarrow \mathbb{L}^* \cap \partial [-1,1]^n$. Let us take an open set $W$ such that $\mathbb{L}^* \cap \partial [-1,1]^n \subset W \subset \overline{W} \subset V \subset \mathbb{R}^n$. Now, $\psi \vert_{\overline{W}} : \overline{W} \longrightarrow \mathbb{L}^* \cap \partial [-1,1]^n$ is a continuous map between compact metric spaces, so it is uniformly continuous. Given $\varepsilon > 0$, there exists $\delta > 0$ such that $d(\psi(x^), \psi (y^)) < \varepsilon/2$ if $x^, y^ \in \overline{W}$ and $d(x^,y^) < \delta$. Put $\eta = \min (\varepsilon/2, \delta)$ and take $$V_{\varepsilon} = {\left\{x^* \in W : \text{there exists }y^* \in \mathbb{L}^* \cap \partial [-1,1]^n \text{ with }d(x^,y^) < \eta\right\}},$$ and $\varphi = \psi\|_{ V_{\varepsilon}}: V_{\varepsilon} \longrightarrow \mathbb{L}^* \cap \partial [-1,1]^n$. Clearly, $\varphi$ is continuous and $\varphi \vert_{\mathbb{L}^* \cap \partial [-1,1]^n} = id_{\mathbb{L}^* \cap \partial [-1,1]^n}$. Let $x^* \in V_{\varepsilon}$, and let $y^* \in \mathbb{L}^* \cap \partial [-1,1]^n$ such that $d(x^,y^) < \eta$. Then, $$d(x^, \varphi(x^)) \leq d(x^,y^) + d(y^, \varphi (x^)) = d(x^,y^) + d(\varphi (y^), \varphi (x^)) < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.$$ If $\mathbb{L}$ is a finite lattice, then $FBL_* \langle \mathbb{L} \rangle$ is a projective Banach lattice. Let $n$ be the cardinality of $\mathbb{L}$. We may suppose that $\mathbb{L} = \{0,\ldots,n-1\}$ with some lattice operations, and in this way we identity $[-1,1]^\mathbb{L}$ with $[-1,1]^n$. We fix $\varepsilon > 0$, and we will construct the map $u_{\varepsilon}: FBL_\langle \mathbb{L} \rangle \longrightarrow FBL(\mathbb{L})$ of Lemma \[caractProyectividad\]. Let $V_\varepsilon$ and $\varphi$ be given by Lemma \[closeretraction\]. By Urysohn’s lemma, we can find a continuous function $1_{\varepsilon}: \partial[-1,1]^n \longrightarrow [0,1]$ such that $1_{\varepsilon}(x^) = 1$ if $x^* \in \mathbb{L}^* \cap \partial[-1,1]^n$, and $1_{\varepsilon}(x^) = 0$ if $x^ \not\in V_{\varepsilon}$. We define $u_{\varepsilon}(f)(x^) = {1_{\varepsilon}}(x^)\cdot f(\varphi(x^))$ if $x^ \in V_{\varepsilon}$, and $u_{\varepsilon}(f)(x^) = 0$ if $x^ \notin V_{\varepsilon}$, for every $f \in FBL_\langle \mathbb{L} \rangle$ and $x^ \in \partial[-1,1]^n$. We extend the definition for elements $x^\ast\in [-1,1]^n \setminus \partial [-1,1]^n$ in such a way that $u_\varepsilon(f)$ is positively homogeneous. Since $\mathbb{L}$ is finite, the fact that $u_\varepsilon(f)$ is continuous on $\partial [-1,1]^n$ and positively homogenous guarantees that $u_\varepsilon(f)\in FBL(\mathbb{L})$. It is easy to check that $u_{\varepsilon}$ is a Banach lattice homomorphism and that $R \circ u_{\varepsilon} = id_{FBL_\langle \mathbb{L} \rangle}$. It would remain to check that $\\| u_{\varepsilon} \\| \leq 1+\varepsilon$. We will prove instead that for this $u_\varepsilon$ we have $\\|u_\varepsilon\\|\leq 1 + n\varepsilon$, which is still good enough. We know that $$\\| u_{\varepsilon} \\| = \sup {\left\{\\| u_{\varepsilon}(f) \\| : f \in FBL_ \langle \mathbb{L} \rangle, \\| f \\|_* \leq 1\right\}},$$ where $$\\| u_{\varepsilon}(f) \\| = \sup {\left\{\sum_{i = 1}^m {\left\vertr_i u_{\varepsilon}(f)(x_{i}^{\ast})\right\vert} : x_i^{\ast} \in \partial[-1,1]^n, \text{ } r_i \in \mathbb{R},\text{ }\sup_{x \in \mathbb{L}} \sum_{i=1}^m {\left\vertr_i x_i^{\ast}(x)\right\vert} \leq 1 \right\}}.$$ So we fix $f \in FBL_\langle \mathbb{L} \rangle$ with ${\left\Vertf\right\Vert}_ \leq 1$, where $${\left\Vertf\right\Vert}_\ast = \sup {\left\{\sum_{i = 1}^m {\left\verts_i f(y_{i}^{\ast})\right\vert} : y_i^{\ast} \in \mathbb{L}^, \text{ } s_i \in \mathbb{R},\text{ }\sup_{x \in \mathbb{L}} \sum_{i=1}^m {\left\verts_i y_i^{\ast}(x)\right\vert} \leq 1 \right\}},$$ and we want to prove that $\\|u_\varepsilon(f)\\|\leq 1+n\varepsilon$. Using the expression of $\\|u_\varepsilon(f)\\|$ as a supremum, we pick $x_1^, \ldots, x_m^* \in \partial[-1,1]^n$, $r_1, \ldots, r_m \in \mathbb{R}$ such that $\sup_{x \in \mathbb{L}} \sum_{i=1}^m {\left\vertr_i x_i^{\ast}(x)\right\vert} \leq 1$, and we want to prove that $$\sum_{i = 1}^m {\left\vertr_i u_{\varepsilon}(f)(x_{i}^{\ast})\right\vert}\leq 1 + n\varepsilon.$$ The first estimation is that $$\sum_{i = 1}^m {\left\vertr_i u_{\varepsilon}(f)(x_{i}^{\ast})\right\vert} = \sum_{x_i^* \in V_{\varepsilon}} {\left\vertr_i {1_{\varepsilon}}(x_i^)f(\varphi(x_i^))\right\vert} \leq \sum_{x_i^* \in V_{\varepsilon}} {\left\vertr_i f(\varphi(x_i^))\right\vert}.$$ If we write ${y_i}^\ast := \varphi(x_i^)$, the inequality above becomes $$(\star)\ \ \sum_{i = 1}^m {\left\vertr_i u_{\varepsilon}(f)(x_{i}^{\ast})\right\vert} \leq \sum_{x_i^* \in V_{\varepsilon}} {\left\vertr_i f({y_i}^\ast)\right\vert}.$$ On the other hand, if $x \in \mathbb{L}$ then $$\begin{split} \sum_{x_i^* \in V_{\varepsilon}}{\left\vertr_i {y_i^}(x)\right\vert} &= \sum_{x_i^ \in V_{\varepsilon}}{\left\vertr_i \varphi(x_i^)(x)\right\vert} \\ &\leq \sum_{x_i^ \in V_{\varepsilon}}{\left\vertr_i x_i^(x)\right\vert} + \sum_{x_i^ \in V_{\varepsilon} }{\left\vertr_i\right\vert} {\left\vert\varphi(x_i^)(x) - x_i^(x)\right\vert}\\ &\leq 1 + \varepsilon \sum_{x_i^* \in V_{\varepsilon} }{\left\vertr_i\right\vert} \leq 1 + \varepsilon n.\\ \end{split}$$ The last inequality is because $x_i ^\ast\in \partial [-1,1]^n$, and therefore $$\sum_{i=1}^m\|r_i\| = \sum_{i=1}^m\|r_i\|\sup_{x\in\mathbb{L}}\|x_i^\ast(x)\| \leq \sum_{x\in\mathbb{L}}\sum_{i=1}^m \|r_i\| \|x_i^\ast(x)\| \leq \|\mathbb{L}\|\cdot 1 = n.$$ Taking $s_i = \frac{r_i}{1 + n\varepsilon}$, we have that, for all $x\in\mathbb{L}$, $$\sum_{x_i^* \in V_{\varepsilon}}{\left\verts_i {y_i^}(x)\right\vert} = \sum_{x_i^ \in V_{\varepsilon}}{\left\vert\frac{r_i}{1 + n\varepsilon} {y_i^}(x)\right\vert} \leq 1.$$ Thus, the $s_i$ and the $y_i$ are as in the supremum that defines $\\|f\\|_\ast \leq 1$. Therefore $$\sum_{x_i^ \in V_{\varepsilon}}{\left\verts_i f({y_i^})\right\vert} \leq 1,$$ and getting back to our initial estimation ($\star$), we get $$\sum_{i = 1}^m {\left\vertr_i u_{\varepsilon}(f)(x_{i}^{\ast})\right\vert} \leq \sum_{x_i^ \in V_{\varepsilon}}{\left\vertr_i f({y_i^})\right\vert} \leq 1 + n\varepsilon.$$ Projectivity of the free Banach lattice generated by an infinite linear order ============================================================================= Now, we are going to prove that if $\mathbb{L}$ is an infinite linear order, then $FBL_\langle \mathbb{L} \rangle$ is not projective. This will be a direct consequence of the fact that the free Banach lattices generated by the set of the natural numbers and the set of the natural numbers together with $+ \infty$ as linearly ordered sets are not projective. In the proof, we will use the following: \[funcionesenFBLN\] Suppose that $\varphi_i: [-1,1]^{\mathbb{N}} \longrightarrow \mathbb{R}$, $i = 1,2, \ldots$, are continuous functions such that, for every $i$, 1. $\varphi_i((x_n)_{n \in \mathbb{N}}) = x_i$ whenever $x_1 \leq x_2 \leq \ldots$, 2. $\varphi_i(x) \leq \varphi_{i+1}(x)$ for all $x \in [-1,1]^{\mathbb{N}}$. Then, when we view the $\varphi_i$’s as elements of the free Banach lattice $FBL(\mathbb{N})$, the sequence of norms $\\| \varphi_{i} \\|_{FBL(\mathbb{N})}$ is unbounded. Let $\pi_i : [-1,1]^{\mathbb{N}} \longrightarrow [-1,1]$ be the projection on the $i$-th coordinate. Consider the set $M := {\left\{(x_n)_{n \in \mathbb{N}} \in [-1,1]^{\mathbb{N}} : x_1 \leq x_2 \leq \ldots\right\}} \subset [-1,1]^{\mathbb{N}}$. Since $M$ is closed and $[-1,1]^{\mathbb{N}}$ with the product topology is compact, we have that $M$ is compact. Condition [(1)]{} in the lemma means that $\varphi_i\vert_M = \pi_i\vert_M$ for all $i$. Using the compactness of $M$ and the continuity of $\varphi_i$ and $\pi_i$, this implies that there exists a neighborhood $U_{i}$ of $M$ such that $$d(\varphi_i\vert_{U_{i}}, \pi_i\vert_{U_{i}}) = \sup_{x \in U_{i}}\| \varphi_i(x) - \pi_i(x) \| < \frac{1}{2}.$$ For an integer $k \geq 3$, let $$W_{k} := {\left\{(x_n)_{n \in \mathbb{N}} \in [-1,1]^{\mathbb{N}} : x_i < x_j + k^{-1} \text{ whenever }i<j<k\right\}}.$$ The family ${\left\{W_{k} : k \geq 3\right\}}$ is a neighborhood basis of $M$. We define inductively an increasing sequence of natural numbers $k_0<k_1<k_2<k_3<\cdots$, and a sequence of points $y^1,y^2,\ldots\in [-1,1]^\mathbb{N}$ as follows. We take $k_0=1$ as a starting point of the induction. Suppose that we have defined $k_j$ for $j < n$. We choose $k_{j+1}>k_j$ such that $W_{k_{j+1}}\subset U_{k_j}$, and we define $y^{j+1} : \mathbb{N} \longrightarrow [-1,1]$ to be the map given by $$y^{j+1}(n)= \left\{ \begin{array}{lcc} 0 & if & n < k_j, \\ \\ 1 & if & k_j \leq n < k_{j+1}, \\ \\ 0 & if & n \geq k_{j+1}. \end{array} \right.$$ We have $y^{j+1} \in W_{k_{j+1}}$, so $\| \varphi_{k_j}(y^{j+1}) - \pi_{k_j}(y^{j+1}) \| = \|\varphi_{k_j}(y^{j+1}) - 1 \| < \frac{1}{2}$ and $\varphi_{k_j}(y^{j+1}) > \frac{1}{2}$. When $j+1\leq m$, using condition (1) of the Lemma, we get that $$\varphi_{k_m}(y^{j+1}) \geq \varphi_{k_{j}}(y^{j+1}) > \frac{1}{2}.$$ Remember how the norm is defined: $$\\| \varphi \\|_{FBL(\mathbb{N})} = \sup {\left\{ \sum_{j = 1}^m \|r_j \varphi (x_j)\| : r_j \in \mathbb{R},\, x_j \in [-1,1]^{\mathbb{N}},\, \sup_{n \in \mathbb{N}} \sum_{j=1}^m \|r_j x_j(n)\| \leq 1 \right\}}.$$ We have that $\sup_{n \in \mathbb{N}} \| y^1(n) + \cdots + y^m(n) \| = 1$, therefore $$\\| \varphi_{k_m} \\|_{FBL(\mathbb{N})} \geq \|\varphi_{k_m}(y^1)\| + \cdots + \|\varphi_{k_m}(y^m)\| > \frac{m}{2}.$$ Now, let $\mathbb{N}^+ = \mathbb{N} \cup \lbrace +\infty \rbrace$. \[Nnotproj\] $FBL_* \langle \mathbb{N} \rangle$ and $FBL_* \langle \mathbb{N}^+ \rangle$ are not projective. First, if $FBL_* \langle \mathbb{N} \rangle$ was projective, then for $\varepsilon>0$ we would have a map $u_{\varepsilon}: FBL_\langle \mathbb{N} \rangle \longrightarrow FBL(\mathbb{N})$ like in Proposition \[caractProyectividad\]. Remember that if $i \in \mathbb{N}$, $\dot{\delta_i}: \mathbb{N}^ \longrightarrow \mathbb{R}$ is the map given by $\dot{\delta_i}(x^) = x^(i)$ for every $x^* \in \mathbb{N}^$, that is an element of $FBL_\ast\langle\mathbb{N}\rangle$. We consider $\varphi_i = u_\varepsilon(\dot{\delta}_i)\in FBL(\mathbb{N})$, that we view as continuous functions $\varphi_i:[-1,1]^\mathbb{N}{\longrightarrow}\mathbb{R}$. The fact that $u_\varepsilon$ is a lattice homomorphism gives condition (2) of Lemma \[funcionesenFBLN\], while the fact that $R\circ u_\varepsilon = id_{FBL_\ast\langle\mathbb{L}\rangle}$ gives condition (1) of Lemma \[funcionesenFBLN\]. The fact that $\\|u_\varepsilon\\|\leq 1+\varepsilon$ contradicts the conclusion of Lemma \[funcionesenFBLN\]. On the other hand, if $FBL_ \langle \mathbb{N}^+ \rangle$ was projective, then for $\varepsilon>0$ there would exists a map $w_{\varepsilon}: FBL_* \langle \mathbb{N}^+ \rangle \longrightarrow FBL ( \mathbb{N}^+ )$ like in Proposition \[caractProyectividad\]. For every $i \in \mathbb{N}^+$, let $\psi_{i} = w_{\varepsilon}(\dot{\delta_i})$. Then again $\psi_i((x_n)_{n \in \mathbb{N}^+}) = x_i$ if $x_1 \leq x_2 \leq \ldots \leq x_{+\infty}$, and $\psi_1 \leq \psi_2 \leq \ldots \leq \psi_{+\infty}$. Fix $\mathcal{U}$ a nonprincipal ultrafilter on $\mathbb{N}$ and define $\varphi_i((x_n)_{n \in \mathbb{N}}) = \psi_i((x_1, x_2, \ldots, \lim_{\mathcal{U}}x_n))$ for every $i \in \mathbb{N}$. Then the functions $\varphi_i$ are as in Lemma \[funcionesenFBLN\], so $\\| \varphi_i \\|_{FBL(\mathbb{N})}$ is unbounded. We check now that $\\| \varphi_i \\|_{FBL(\mathbb{N})} \leq \\| \psi_i \\|_{FBL(\mathbb{N}^+)} \leq 1 + \varepsilon$, a contradiction. Take $\sum_{j=1}^m \|r_j\varphi_i(x_j)\|$ one of the sums that appear in the definition of $\\| \varphi_i \\|_{FBL(\mathbb{N})}$ as a supremum. Consider $y_j = (x_j(1), x_j(2), \ldots, \lim_{\mathcal{U}}x_j(n))$. Then, $\sum_{j=1}^m \|r_j\varphi_i(x_j)\| = \sum_{j=1}^m \|r_j\psi_i(y_j)\|$ and this is one of the sums that appears in the supremum defining $\\| \psi_i \\|_{FBL(\mathbb{N}^+)}$ because $$\sum_{j=1}^m \|r_jy_j(n)\| = \sum_{j=1}^m \|r_jx_j(n)\| \leq 1$$ if $n \in \mathbb{N}$ and $$\sum_{j=1}^m \|r_jy_j(+\infty)\| = \sum_{j=1}^m\|r_j \lim_{\mathcal{U}}x_j(n)\| = \lim_{\mathcal{U}}\sum_{j=1}^m \|r_j x_j(n)\| \leq 1.$$ The following fact can be viewed as a corollary of Proposition \[quotientofprojective\], but we state if for convenience: \[projcomplemented\] Let $P$ and $P'$ be Banach lattices, and let $\tilde{\pi}:P{\longrightarrow}P'$ and $\tilde{\imath}:P'{\longrightarrow}P$ be Banach lattice homomorphisms such that $\\|\tilde{\imath}\\| = \\|\tilde{\pi}\\|=1$ and $\tilde{\pi}\circ\tilde{\imath} = id_{P'}$. If $P$ is projective, then $P'$ is projective. In order to check the projectivity of $P'$, let $Q:X{\longrightarrow}X/J$, $T':P'{\longrightarrow}X/J$ and $\varepsilon > 0$ be as in Definition \[projdef\]. Then we can apply the projectivity of $P$ considering $T=T'\circ\tilde{\pi}$, so we get $\hat{T}:P{\longrightarrow}X$ such that $Q\circ \hat{T} = T'\circ\tilde{\pi}$ and $\\|\hat{T}\\|\leq (1 + \varepsilon) \\|T\\|\leq (1 + \varepsilon) \\|T'\\|$. The desired lift is $\hat{T}' = \hat{T} \circ \tilde{\imath}$. On the one hand $\\|\hat{T}'\\| \leq \\|\hat{T}\\| \leq (1 + \varepsilon) \\|T'\\|$, and on the other hand $Q\circ \hat{T} \circ \tilde{\imath} = T'\circ \tilde{\pi} \circ \tilde{\imath} = T'$. Let $\mathbb{L}$ be an infinite linearly ordered set. Then, $FBL_* \langle \mathbb{L} \rangle$ is not projective. $\mathbb{L}$ contains either an increasing or a decreasing sequence. Let us suppose without loss of generality that it contains an increasing sequence $x_1<x_2<x_3<\cdots$. First, suppose that it has no upper bound. The map $\imath : (\mathbb{N}, \leq) \longrightarrow (\mathbb{L}, \preceq)$ given by $\imath(n) = x_n$ for every $n \in \mathbb{N}$ is a lattice homomorphism. Let $\pi : \mathbb{L} \longrightarrow \mathbb{N}$ be the map given by $$\pi(x)= \left\{ \begin{array}{lcc} 1 & \text{if} & x < x_2, \\ n & \text{if} & x \in [x_n, x_{n+1}) \text{ for any } n \geq 2 . \end{array} \right.$$ Notice that $\pi$ is also a lattice homomorphism and $\pi\circ \imath = id_{\mathbb{N}}$. We are going to use the universal property of the free Banach lattice over a lattice as stated in Definition \[FBLGBL\]. Let $\phi_{\mathbb{L}}$ and $\phi_{\mathbb{N}}$ be the canonical inclusion of $\mathbb{L}$ and $\mathbb{N}$ into $FBL_\langle \mathbb{L} \rangle$ and $FBL_\langle \mathbb{N} \rangle$, respectively, and let $\tilde{\imath} : FBL_* \langle \mathbb{N} \rangle \longrightarrow FBL_* \langle \mathbb{L} \rangle$ and $\tilde{\pi} : FBL_\langle \mathbb{L} \rangle \longrightarrow FBL_ \langle \mathbb{N} \rangle$ be the corresponding extensions of $\phi_{\mathbb{L}} \circ \imath$ and $\phi_{\mathbb{N}} \circ \pi$ according to Definition \[FBLGBL\]. The composition $\tilde{\pi}\circ\tilde{\imath}$ and the identity mapping $FBL_\langle \mathbb{N} \rangle \longrightarrow FBL_ \langle \mathbb{N} \rangle$ are both extensions of $\phi_{\mathbb{N}}$ so by the uniqueness in Definition \[FBLGBL\], $\tilde{\pi}\circ\tilde{\imath} = id_{FBL_\langle\mathbb{N}\rangle}$. We can apply Lemma \[projcomplemented\], so if $ FBL_\langle \mathbb{L} \rangle$ was projective, then $FBL_\langle \mathbb{N} \rangle$ would also be projective, in contradiction with Lemma \[Nnotproj\]. On the other hand, if the sequence $x_1<x_2<x_3<\cdots$ has an upper bound, $x_{+\infty}$, we can take $\imath : (\mathbb{N}^+, \leq) \longrightarrow (\mathbb{L}, \preceq)$ given by $\imath(n) = x_n$ for every $n \in \mathbb{N}^+$ and $\pi : \mathbb{L} \longrightarrow \mathbb{N}^+$ given by $$\pi(x)= \left\{ \begin{array}{lcc} 1 & \text{if} & x < x_2, \\ n & \text{if} & x \in [x_n, x_{n+1}) \text{ for any } n \geq 2 ,\\ +\infty &\text{if} & x>x_n \text{ for all } n. \end{array} \right.$$ and apply the same reasoning substituting $\mathbb{N}$ by $\mathbb{N}^+$. [999]{} A. Avilés, J. D. Rodríguez Abellán, The free Banach lattice generated by a lattice, Positivity, 23 (2019), 581–597. A. Avilés, J. Rodríguez, P. Tradacete, The free Banach lattice generated by a Banach space, J. Funct. Anal. 274 (2018), 2955–2977. J. van Mill, Inifinite-Dimensional Topology. Prerequisites and Introduction, North-Holland mathematical library, v. 43, 1989. B. de Pagter, A. W. Wickstead, Free and projective Banach lattices*, Proc. Royal Soc. Edinburgh Sect. A, 145 (2015), 105–143. [^1]: Authors supported by project MTM2017-86182-P (Government of Spain, AEI/FEDER, EU) and project 20797/PI/18 by Fundación Séneca, ACyT Región de Murcia. Second author supported by FPI contract of Fundación Séneca, ACyT Región de Murcia.
--- abstract: 'The multi dimensional string objects are introduced as a new alternative for an application of string models for time series forecasting in trading on financial markets. The objects are represented by open string with 2-endpoints and D2-brane, which are continuous enhancement of 1-endpoint open string model. We show how new object properties can change the statistics of the predictors, which makes them the candidates for modeling a wide range of time series systems. String angular momentum is proposed as another tool to analyze the stability of currency rates except the historical volatility. To show the reliability of our approach with application of string models for time series forecasting we present the results of real demo simulations for four currency exchange pairs.' author: - Erik Bartoš - Richard Pinčák bibliography: - 'timeSeries.bib' title: 'Identification of market trends with string and D2-brane maps' --- Introduction {#sec:intro} ============ Trading and predicting foreign exchange in the forex market [@Fang2003369] has become one of the intriguing topic and is extensively studied by researchers from different fields due to its commercial applications and attractive benefits that it has to offer. Algorithmic trading with large amount of processed data, the ticks in millisecond scale, requires new physical methods to describe the statistics of the return intervals on the short and large scales [@Lux:2000] and new geometric representation of data , e. g., new view on data statistics in higher dimensions [@Kabin:2016]. Moreover, the global markets consist of a large number of interacting units and their time-averaged dynamics resemble the systems with many-body effects. The classical statistical instruments which treats the market as a whole, like the returns and volatility distributions [@Wang:2009; @Wang:2009b], must be enhanced by new phenomena from informational and social sciences. Theoretical interest is also oriented to the distribution of the occurrence of rare extreme events in historical time series data [@Bunde:2005; @Hartmann:2004]. Their clustering in data records indicates the existence of a long-term memory dependencies in financial time series, which is intensively studied [@Bogachev:2008], i. e., by multiplicative random cascade models. New approaches covering the findings of the long memory effects of forex data [@Nacher:2012] and its stochastic features in the presence of nonstationarity [@Anvari:2013], the renormalization group approach [@Zamparo:2013], exploitation of genetic algorithms [@Venkatesan2002625], open novel perspectives. We work on the concept which approach the string theory [@Zwiebach:2009] to the field of time series forecast and data analysis through a transformation of currency rate data to the topology of physical strings and branes [@Horvath:2012zz; @Pincak:2013aa; @Pincak:2014ba]. The ideas have been practically demonstrated by a novel prediction method based on string invariants [@Bundzel:2015bc] with genetic algorithm for optimization of method’s parameters. The method has been tested on competition and real world data, its performance compared to artificial neural networks and support vector machines algorithms. Another interesting application has been the construction of trading algorithm based on 1-endpoint strings and the demonstration of model properties on real online trade system [@Pincak:2015hha]. Stability of the algorithm on transaction costs for long trade periods has been confirmed and compared to benchmark prediction models and trading strategies. The aim of this paper is to outline new possibilities how the prediction models in trading on financial market can be enhanced in the framework of a string theory. We propose to proceed from simple 1-endpoint and 2-endpoints strings to more complex objects, D2-branes. The D2-branes have the ability to smooth the movement of prices on the market and to process the preserved market memory with better efficiency than in the case of the strings, the study of a statistics of momenta of string objects reveal the perspectives of D2-branes. However, the simulations with prediction models based on string approach show that one can profit only on the regions with high stability. In real data, the fluctuations of forex market prices brake the statistics of the predictions and one must build into the models various trading brakes, to deal with the rapid changes. The evaluation of a volatility [@Wang:2006; @Gallo2015620] serves as one of the sources of analyzing tools in pricing strategies. We introduce new methodics based on the analogy with the angular momentum in the string theory. Its application into trading models can serve as complementary financial instrument in addition to a volatility. The changes of Regge slope parameter or the string tension can identify trends on the market, their understanding allows us to dynamically change an intra-string characterization (reduction a string length for a short period) and better predict the movement of prices. Especially in large market fluctuations, their exploitation needs further experimental verification. The rest of the paper is organized as follows. Section \[sec:strings\] formulates the general models of multi dimensional string models. Their properties and comparison with previous models are discussed. In Section \[sec:regge\], we introduce the Regge alpha slope for the investigation of the stability of currency rates. The obtained results are summarized in the last section. In Appendix \[app:B\] we demonstrate the application of our model for the real demo sessions for currency pairs EUR/USD, CHF/JPY, AUD/CAD, AUD/JPY. From simple to complex strings {#sec:strings} ============================== The concept of string maps is based on the connection of the currency quotes and the string objects. For the defined time series of currency exchange rates for the ask $p_{{\mathrm {ask}}}(\tau)$ and bid $p_{{\mathrm {bid}}}(\tau)$ values in time $\tau$ one can construct the string maps with the typical length $l_s$. These non-local objects serve as the basic objects for further operations. In contrast to classical time series forecasting methods, e. g., autoregressive and moving average models, which forecast the variable of interest using a linear combination of past values or errors of the variable, the string maps carry the larger price history, thereafter the trends of irregular or untypical price changes can be caught with better accuracy. In the works [@Horvath:2012zz; @Pincak:2013aa] the $q$-deformed prediction model based on the deviations from benchmark string sequence of 1-endpoint string map $P_{q}^{(1)}(\tau,h)$ was thoroughly studied. The momentum $M$ of the string (the predictor) were proposed for the study of deviations of string maps from benchmark string sequence in the form $$\begin{gathered} \label{eq:mp} M(l_s, m, q, \varphi) =\\ \left(\frac{1}{l_s+1} \sum_{h=0}^{l_s}\Big\| P_{N}(\tau,h) - F_{{\mathrm {CS}}}(h,\varphi)\Big\|^q\right)^{1/q},\end{gathered}$$ for $m,\;q>0$ , $l_s$ is the string length. $P_{N}(\tau,h)$ represents the generalized $N$-points string map, in previous case $P_{N}(\tau,h) \to P_{q}^{(1)}(\tau,h)$. The regular function $F_{{\mathrm {CS}}}(h,\varphi)$ could be substituted by various periodic functions. In [@Pincak:2015hha] we have used the form $$\label{eq:rf} F_{{\mathrm {CS}}}(h,\varphi) = \frac{1}{2}\big(1+\cos(\tilde{\varphi})\big),\quad \tilde{\varphi} = \frac{2\pi m h}{l_s + 1} + \varphi.$$ We have shown that the final form of Eq. (\[eq:rf\]) depends on the trading strategy, when one looks the regions in the time series market with almost invariant values of $M$ and tries to predict the increase or decrease of prices with the best efficiency. Such model yields to the momenta values depicted in Fig. \[fig:pOS1ep\]. Despite the simple approach, the results of the simulations with the prediction model in the OANDA market have demonstrated the stability of the proposed trading algorithm on the transaction costs for the long trade periods. However, the values of $M$ have not allowed to find the wide regions of invariants even for the higher $q$ or other periodic functions as in Eq. (\[eq:rf\]), which lead us to find more complex solutions (more details in Appendix \[app:B\]). \ Open string with two endpoints ------------------------------ In the next we study the influence of more complex string objects on the momentum behavior. At first we propose to incorporate a long-term trend by the nonlinear map corresponding to an open string with 2-endpoints $$\begin{gathered} \label{eq:POS2ep} P_{q}^{(2)}(\tau,h) = f_{q}\bigg(\bigg(\frac{p(\tau+h) - p(\tau)}{p(\tau+h)}\bigg)\\ \times\bigg(\frac{p(\tau+l_s) - p(\tau+h)}{p(\tau+l_s)}\bigg)\bigg),\end{gathered}$$ with $h\in {\mathinner\langle}0,l_s{\mathinner\rangle}$, $q$ deformation $f_{q}={\mathrm {sign}}(x)\|x\|^{q}$ and $p(\tau)$ value represents the mean value $p(\tau) = (p_{{\mathrm {ask}}}(\tau) + p_{{\mathrm {bid}}}(\tau))/2$. The $P_{q}^{(2)}(\tau,h)$ fulfills boundary conditions of Dirichlet type $$P_{q}^{(2)}(\tau,0)= P_{q}^{(2)}(\tau,l_s)=0\,,\qquad \mbox{at all ticks}\,\,\, \tau\,. \label{eq:Dir}$$ Practically, one can replace $P_{N}(\tau,h)\to P_{q}^{(2)}(\tau,h)$ in Eq. (\[eq:mp\]) and look at the values of $M$. Fig. \[fig:pOS2ep\] shows that the effect of the regularization is notable in comparison with previous case of 1-endpoint open string, even for low values of $q$ parameter. It allow us to focus on the predictor values which determine the stability of the algorithm or in other words they reflect the price changes on the scale of string length. Open polarized string with two endpoints ---------------------------------------- Further modification of the string map to include spread-adjusted currency return $(p_{{\mathrm {bid}}}(\tau) - p_{{\mathrm {ask}}}(\tau))/(p(\tau,h))$ is rather straightforward, it is an analogy with a charged string polarized by an external field. The formula has the form $$\begin{aligned} \label{eq:POPS2ep} P_{q}^{{\mathrm {ab}}}(\tau,h) =& f_{q}\bigg(\bigg(\frac{p_{{\mathrm {bid}}}(\tau+h) - p_{{\mathrm {ask}}}(\tau)}{p(\tau+h)}\bigg){\nonumber}\\ &\times \bigg(\frac{p_{{\mathrm {bid}}}(\tau+l_s) - p_{{\mathrm {ask}}}(\tau+h)}{p(\tau+l_s)}\bigg)\bigg)\,.\end{aligned}$$ The violation of the Dirichlet boundary condition is restored, for instance, by the subtraction $\tilde{P}_{q}^{{\mathrm {ab}}}(\tau,h) \equiv P_{q}^{{\mathrm {ab}}}(\tau,h) - P_{q}^{{\mathrm {ab}}}(\tau,0)$. For the polarized string mapping, i. e., the replacement $P_{N}(\tau,h)\to P_{q}^{{\mathrm {ab}}}(\tau,h)$, the regularized and nonregularized values of the momenta $M$ looks identically to the previous case of open string (see Fig. \[fig:pOS2ep\]) and the simulations yield to the similar results. To quantify the received predictor statistics one can construct the histograms for a spectrum of $M$ momenta as shown in Fig. \[fig:histOS1ep\]. Broader peaks of the distributions for regularized values of $M$ for 1-endpoint and 2-endpoints strings suggests that the values are more smoothed than in the unregularized case and the aims to forecast the market trends are based on the sharper values of $M$, i. e., only the highest changes of a price on the market are taken into account and in this way they facilitate the evaluation of buy/sell orders. \ \ \ D2-brane model -------------- More interesting way how to go beyond a string model is to extent the string lines towards the more complex maps, the membranes called D2-branes. Practically it can be realized with the mapping in the form $$\begin{aligned} \label{eq:PD2q} P_{{\mathrm {D2}},q}(\tau, h_1, h_2) =& f_q\Bigg(\bigg(\frac{p^{{\mathrm {ask}}}(\tau + h_1) - p^{{\mathrm {ask}}}(\tau)}{p^{{\mathrm {ask}}}(\tau + h_1)} \bigg){\nonumber}\\ &\times\bigg(\frac{ p^{{\mathrm {ask}}}(\tau+l_s) - p^{{\mathrm {ask}}}(\tau+h_1)}{p^{{\mathrm {ask}}}(\tau +l_s)} \bigg) {\nonumber}\\ &\times \bigg(\frac{p^{{\mathrm {bid}}}(\tau) - p^{{\mathrm {bid}}}(\tau+h_2) }{p^{{\mathrm {bid}}}(\tau)}\bigg) \\ &\times \bigg(\frac{p^{{\mathrm {bid}}}(\tau+h_2) - p^{{\mathrm {bid}}}(\tau+l_s)}{p^{{\mathrm {bid}}}(\tau+h_2)} \bigg)\Bigg) \,. {\nonumber}\end{aligned}$$ with the coordinates $(h_1, h_2) \in \langle 0,l_{\rm s}\rangle \times \langle 0,l_{\rm s}\rangle$ which vary along two extra dimensions. The mapping satisfies the Dirichlet boundary conditions $$\begin{aligned} P_{{\mathrm {D2}},q}(\tau, h_1, 0) &= P_{{\mathrm {D2}},q}(\tau, h_1, l_s) ={\nonumber}\\ P_{{\mathrm {D2}},q}(\tau, 0, h_2) &= P_{{\mathrm {D2}},q}(\tau, l_s, h_2).\end{aligned}$$ The momentum of D2-brane model can be modified to $$\begin{gathered} \label{eq:mpD2} M(l_s, m, q, \varphi, \varepsilon) = \Bigg(\frac{1}{(l_s+1)^2}\\ \times \sum_{h_1=0}^{l_s}\sum_{h_2=0}^{l_s}\Big\| P_{{\mathrm {D2}},q}(\tau,h_1,h_2) - F_{{\mathrm {D2}}}(h,\varphi,\varepsilon)\Big\|^q\Bigg)^{1/q},\end{gathered}$$ the regular function depends also on more variables, e. g., it can has the form $$\begin{aligned} F_{{\mathrm {D2}}}(h,\varphi,\varepsilon) = \frac{1}{2}\big(\sin(\tilde{\varphi}^2)\cos(\tilde{\varepsilon}^2)\big),\, \tilde{\varepsilon} = \frac{2\pi m h}{l_s + 1} + \varepsilon.\end{aligned}$$ ![Not regularized (blue) and regularized (red) values of the momenta for D2-brane. The sample of 10 thousand time series ticks, $q=1$ and $q=8$. \[fig:pD2\]](pD2Als100Q1 "fig:"){width="\columnwidth"}\ ![Not regularized (blue) and regularized (red) values of the momenta for D2-brane. The sample of 10 thousand time series ticks, $q=1$ and $q=8$. \[fig:pD2\]](pD2Als100Q8 "fig:"){width="\columnwidth"} \ \ \ The effect of higher dimension D2-branes onto the $M$ values in Eq. (\[eq:mpD2\]) is visible in Fig. \[fig:pD2\]. In comparison with 1-endpoint and 2-endpoints open strings the unregularized values are more smoothed. The regularization does not improve the spectrum so significantly as in the previous case of string models as it is visible from the histograms shown in Fig. \[fig:histMix\]. One can conclude that even the D2-branes model with basic configuration is suitable to capture the dynamic changes of prices on the financial market. As another tool for evaluating of the different approaches represented by the string and D2-branes models can server the return volatility $\sigma_{ls/2}$. In contrast to a historical volatility (the standard deviation of currency returns), the return volatility acts at the time scale $l_s/2$ as string statistical characteristic. It is defined as $$\begin{aligned} \sigma_r(l_s/2) &= \sqrt{r_2(l_s/2) - r_1^2(l_s/2)},\\ {\nonumber}r_m(l_s/2) &= \sum_{h=1}^{l_s/2}\big[(p(\tau+h) - p(\tau+h-1))/(p(\tau + h))\big]^{m},\end{aligned}$$ for $m=1,2$. The scatterplot in Fig. \[fig:return\] shows the relationship of return volatility at the scale of $l_s/2$ to the changes in the price trends represented by the string amplitudes for 2-endpoints string $P_{i}^{(2)}(\tau,l_s/2)$ and D2-brane $P_{D2,i}(\tau,l_s/2,l_s/2)$, $i=1,8$. The impact of high $q$ to identify the rare events of volatility is visible in both cases, nevertheless, if one decides or does not decide to use the $q$-deformed model in favor of D2-branes it depends also on the technical conditions of real time calculations, because to receive the statistics and to make predictions with D2-branes requires more computing power. ![image](returnVolatility2OSQ1ls1000.png){width="\columnwidth"} ![image](returnVolatility2OSQ8ls1000.png){width="\columnwidth"}\ ![image](returnVolatilityD2Q1ls1000.png){width="\columnwidth"} ![image](returnVolatilityD2Q8ls1000.png){width="\columnwidth"} Comparision {#sec:regression} ----------- For the purpose to demonstrate the impact of different types of string maps on the net asset value (NAV) we performed numerical simulations with open strings with one and two endpoints, D2-branes and ARMA(p,q) type forecasting models on trade online system (more in Appendix \[app:B\]) with build-in derived algorithms. The plot in Fig. \[fig:comparo\] presents the results of the simulations for EUR/USD currency pair. In the simulations we have tried to keep all parameters the same as possible, the impact of string length $l_s$ was tested on final result, OS1ep and OS2ep models have the same regularization function with $q=8$, D2-brane model is not regularized. The study revealed the incapability of ARMA models to keep even zero profit. On the contrary, the results of the string models revealed improvement of NAV with the transition from 1-endpoint to to 2-endpoints open string and D2-branes. Moreover, the higher efficiency for the string models may be achieved by longer string $l_s$ lengths. ![image](simulation_01){width="\linewidth"}\ ![image](simulation_02){width="\linewidth"} Regge slope parameter {#sec:regge} ===================== In this section we closely look at another quantity which has origin in the string theory, so called Regge slope parameter $\alpha'$. The connection of the slope parameter and the angular momentum makes it suitable for the investigation of the stability of currency rates as shown below. For rotating open string, the parameter $\alpha'$ or inverse of the string tension, is the constant that relates the angular momentum of the string $J$ to the square of its energy $E$ $$\alpha'=\frac{J}{\hbar E^2}.$$ In our analogy we introduce the slope parameter in terms of the angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$. For the time series of open-high-low-close (OHLC) values of currency rates $p(\tau)$ one can construct separated ask and bid strings, in our case we use the open string with 2-endpoints and string length $l_s$, introduced via the nonlinear map in Eq. (\[eq:POS2ep\]). Then the momentum distance function $d_{q}^{{\mathrm {ab}}}(\tau)$ between the ask string $P_{q,{\mathrm {ask}}}^{(2)}(\tau,h) \equiv P_{q}^{(2)}(\tau,h)\big\|_{p\to p_{{\mathrm {ask}}}}$ and bid string $P_{q,{\mathrm {bid}}}^{(2)}(\tau,h) \equiv P_{q}^{(2)}(\tau,h)\big\|_{p\to p_{{\mathrm {bid}}}}$ has the form $$\begin{gathered} d_{q}^{{\mathrm {ab}}}(t) = \frac{1}{l_s+1} \sum\limits_{h=0}^{l_s} \Big\| P_{q,{\mathrm {ask}}}^{(2)}(\tau,h) - P_{q,{\mathrm {bid}}}^{(2)}(\tau,h) \Big\|.\end{gathered}$$ In case of rotating open string, the nonvanishing component of angular momentum is $M_{12}$, and its magnitude is denoted by $J=\|M_{12}\|$ (more in [@Zwiebach:2009]) $$M_{12}=\int_{0}^{\sigma_1}(X_1P_{2}^{\tau} - X_2P_{1}^{\tau}) {{\mathrm {d}}}\sigma$$ for space and conjugate components $P_{i}$, $X_{i}$, $i=1,2$ and $\sigma_1=E/T_{0}$. It leads to the relation connecting the slope parameter $\alpha'$ and $T_{0}$ as the string tension $T_{0}$ $$\label{eq:aT} T_{0} = \frac{1}{2\pi\,\alpha'\,\hbar c}.$$ In our notation, the angular momentum can be written as $$\begin{gathered} M_{q}^{{\mathrm {ab}}}(\tau) = \sum\limits_{h=0}^{l_s} \Big[P_{q,{\mathrm {ask}}}^{(2)}(\tau,h)X_{q,{\mathrm {bid}}}^{(2)}(\tau,h) \\ -P_{q,{\mathrm {bid}}}^{(2)}(\tau,h)X_{q,{\mathrm {ask}}}^{(2)}(\tau,h)\Big]\end{gathered}$$ with the conjugate variable $X_{q}^{(2)}(\tau,h)$ received by the recurrent summation from $P_{q}^{(2)}(\tau,h)$ in Eq. (\[eq:POS2ep\]), following the relation $\dot X_{q}^{(2)}(\tau,h) = P_{q}^{(2)}(\tau,h)$ $$\begin{gathered} X_{q}^{(2)}(\tau,h+1) = X_{q}^{(2)}(\tau,h) \\ + P_{q}^{(2)}(\tau,h-1)[t(\tau+h)-t(\tau + h -1)],\end{gathered}$$ $t(\cdot)$ denotes the timestamp for the time $\tau$ (see [@Horvath:2012zz]). The slope parameter has final form $$\begin{aligned} \alpha_{q}' &= \frac{{\mathinner\langle}{\mathinner\vert}M_{q}^{{\mathrm {ab}}}(\tau){\mathinner\vert}{\mathinner\rangle}}{2\pi l_s^2}.\end{aligned}$$ Table \[tab:tension\] presents the typical values of slope parameter $\alpha_{q}'$ together with the mean of string amplitude $P_{1}^{(2)}(l_s/2)$ for 2-endpoints string mapping, the string tension $T_0$ is estimated with the help of Eq. (\[eq:aT\]) ($\hbar c = 1$). It is obvious that each currency pair operates with the own characteristic inter-string values. From the theory of D-branes is known generalized formula for the tension of $D_{p}$-brane [@Johnson2003d] $$T_{D_{p}} = \frac{1}{g_s(2\pi)^p\, l_s^{p+1}},$$ $l_s$ is the familiar string length and $g_s$ is the string coupling, which can be used for higher dimensions not considered in this work. ------------------- ----------------------------------------------------------------- -------------------------------- ------------------- Currency pair $\bm{{\mathinner\langle}P_{1}^{(2)}(l_s/2){\mathinner\rangle}}$ $\bm{\alpha_{1}'}$ $\bm{T_0}$ $[\times10^{-7}]$ $[\times10^{-13} (2\pi)^{-1}]$ $[\times10^{12}]$ AUD/CAD $3.6841$ $8.9764$ $1.1140$ EUR/USD $0.3539$ $2.1890$ $4.5684$ GBP/USD $-5.0099$ $5.4474$ $1.8357$ USD/CAD $8.6794$ $12.0247$ $0.8316$ USD/CHF $10.6082$ $10.6185$ $0.9418$ USD/JPY $28.2180$ $6.9397$ $1.4410$ ------------------- ----------------------------------------------------------------- -------------------------------- ------------------- -------------- -------- -------- -------- -------- String length 5 10 40 60 10 0.5175 0.6072 0.3759 0.3531 20 0.3949 0.4574 0.4307 0.3865 30 0.4726 0.5022 0.5398 0.4935 40 0.4460 0.5098 0.6579 0.5843 50 0.4384 0.4184 0.5230 0.5240 -------------- -------- -------- -------- -------- : The Pearson product-moment correlation coefficients between the angular momentum (dependent on a string length $l_s$) and the historical volatility (dependent on a time window) calculated for close ask ticks of EUR/USD exchange rate on December 4th, 2015.[]{data-label="tab:corr"} Discussion and conclusions {#sec:con} ========================== ![Plot shows the close ask value of EUR/USD exchange rate (red) for ticks on December 4th, 2015. The historical volatility in 5, 10, 40, 60 min. windows (green) is compared with the angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$ (blue) for $q=1$ and $l_s=10$ min. \[fig:regge1\]](regge_slope_05 "fig:"){width="\columnwidth"}\ ![Plot shows the close ask value of EUR/USD exchange rate (red) for ticks on December 4th, 2015. The historical volatility in 5, 10, 40, 60 min. windows (green) is compared with the angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$ (blue) for $q=1$ and $l_s=10$ min. \[fig:regge1\]](regge_slope_10 "fig:"){width="\columnwidth"}\ ![Plot shows the close ask value of EUR/USD exchange rate (red) for ticks on December 4th, 2015. The historical volatility in 5, 10, 40, 60 min. windows (green) is compared with the angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$ (blue) for $q=1$ and $l_s=10$ min. \[fig:regge1\]](regge_slope_40 "fig:"){width="\columnwidth"}\ ![Plot shows the close ask value of EUR/USD exchange rate (red) for ticks on December 4th, 2015. The historical volatility in 5, 10, 40, 60 min. windows (green) is compared with the angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$ (blue) for $q=1$ and $l_s=10$ min. \[fig:regge1\]](regge_slope_60 "fig:"){width="\columnwidth"} In the study we have introduced new string mappings to transform the currency quotes to multidimensional string objects, represented by open strings with 2-endpoints and D2-branes. The proposed objects enhance string model algorithm [@Pincak:2015hha] used in the real market conditions on the online trade system. We have investigated the influence of not regularized and regularized mappings on the final spectrum of momenta values for the objects (Figs. \[fig:pOS\], \[fig:pD2\]). The regularization have been obtained by the addition of the regular function to the original mapping function and $q$ parameter for the deformation. The effect of flattening of momenta values for 2-endpoints strings is notable even for low $q$ values (Fig. \[fig:return\]), in contrast to the 1-endpoint string, where the effect is visible only for high $q$ values. For D2-branes is the situation more favorable, the flattening is achieved already for not regularized momenta values, due to the properties of brane mapping itself. For completeness, we have mentioned also open polarized strings with 2-endpoints, but the obtained statistics is very similar to previous mappings and we have not investigated it in detail. According the obtained corresponding numerical simulations (Fig. \[fig:comparo\]), the improvement of NAV with the proposed models is significant according to 1-endpoint open strings, moreover the dependence of results on string length suggests the possibility to optimize the parameters through parallel computations or evolutionary algorithms. We also propose to apply the values of angular momentum $M_{q}^{{\mathrm {ab}}}(\tau)$ as complementary tool to analyze the stability of currency rates except the historical volatility, as they are compared in Fig. \[fig:regge1\]. For short string lengths the angular momentum indicates the same sharp changes in exchange rate. The correlation between measures is highest for equal values of a string length and time window parameters, see Table \[tab:corr\]. Although there exists a certain relation between those measures, for instance, the similar sensitivity in time, the memory effect of angular momentum seems to be lower. Therefore, it may provide a helpful indication of market changes or to serve as a trade brake in algorithms. In connection with a slope parameter $\alpha'$ and a string tension $T_0$ we have compared their values for a set of currency pairs in Table \[tab:tension\] (we have chosen the main six trading pairs). One can deduce that an increase of slope parameter values (or decrease of tension) indicates the changes on a market and a volatility is increasing. Although the fall in prices can last for a short time, the trading algorithms must immediately respond on the situation to avoid large losses. In Appendix \[app:B\] we have outlined the possible way how to deal with identified trends in real conditions. Moreover, each currency pair needs special treatment, which raises the requirement of parallel computing with genetic algorithms. We leave this as an open question for future work. A combination of theoretical models based on geometrical description with the current financial data-driven disclosures may lead to serious revelation in traditional econometric methods. The models based on D2-branes mapping can serve as the starting point for the next generation of optimization of trade parameters in the evolutionary processes. Sharpe ratio {#app:A} ============ The Sharpe ratio for calculating risk-adjusted return $$\label{eq:sr} S=\frac{\mathrm{E}[r_a-r_f]}{\sigma}=\frac{\mu-r_f}{\sigma},$$ where $r_a$ is asset return, $r_f$ is risk free rate of return, $\mathrm{E}[r_a]$ is mean asset return, $\mathrm{E}[r_a-r_f] = \mu - r_f$ is the expected value of the excess of the asset return over the benchmark return with standard notation $$\Big\{x_i\Big\}_{i=1}^{N},\; \mu=\frac{1}{N}\sum\limits_{i=1}^{N}x_i,\; \sigma = \sqrt{\frac{1}{N}\sum\limits_{i=1}^{n}(x_i - \mu)^2}$$ $\mu$ is the mean, $\sigma$ is the standard deviation. The Sharpe ratio formula for the modified value at risk $$\label{eq:mvar} S_{\mathrm{MVaR}}=\frac{\mu-r_f}{\mathrm{MVaR}},$$ with $$\begin{aligned} \mathrm{MVaR} =& -(\mu + \sigma z_{cf}), {\nonumber}\\ {\nonumber}z_{cf} =& z_c+\frac{1}{6}\Big[(z_c^2-1)S\Big]+\frac{1}{24}\Big[(z_c^3-3z_c)K\Big]\\ &-\frac{1}{36}\Big[(2z_c^3-5z_c)S^2\Big],\end{aligned}$$ $z_{c}$ is the $c$-quantile of the standard normal distribution, $S$ is the skewness of asset return and $K$ is the excess kurtosis of asset return. Utilization of the model for real trading {#app:B} ========================================= To demonstrate the behavior of physical ideas under the real trading conditions, we have constructed the trading algorithm, which has been intensively developed and tested. The algorithm version StringAlgo v.15 has demonstrated the financial forecasting on the OANDA real data for the PMBCS model [@Pincak:2015hha]. To see the differences and perspectives of the proposed model from the Section \[sec:strings\], we present the real, i. e., not theoretical, results from first demo sessions on the Interactive Brokers (IB) and LMAX Exchange (LMAX) market accounts, which were done through the Librade online trade system [@IB; @LMAX; @Librade]. The chosen currency pairs EUR/USD, CHF/JPY, AUD/CAD, AUD/JPY were simulated with new algorithm version and thereafter compared with the results from demo sessions (see Tab. \[tab:demo1\]). The algorithm StringAlgo v.16 has builtin new proposed string maps Eq. (\[eq:POS2ep\]), (\[eq:POPS2ep\]), (\[eq:PD2q\]), as well the modified Sharpe ratio (Appendix \[app:A\]) which serves as new statistical quantity to evaluate the value at risk. Fig. \[fig:demo1\] shows net asset value (NAV) plots for this real demo trading results and the simulation results (the NAV scales differ as they were initialized with different trade volumes). One can observe that all demo results for currency pairs follow the main trend of simulations for chosen time period, i. e., nearly two months. The best coincidence is clearly visible for currency pair EUR/USD (IB-test-12 and LMAX-test-16 accounts), which was in the center of our interest. Also we found nice candidate for currency pair AUD-CAD as one can see in the case of LMAX-test-14 account. The encouraging results with evolutionary algorithm for the parametric optimization [@Bundzel:2015bc] lead us to enhance the algorithm with a module for parallel evaluation of string moment values on short time scales. The trends in price change, identified either with volatility or angular momentum (Fig. \[fig:regge1\]), yield to dynamic change of the parameters as a string length and a trade altitude. They are not keep constant, e. g., the trade altitude is lowered, so the algorithm can profit even under new conditions. The evolutionary algorithm has predefined limits within which it selects the most suitable combination of parameters leading ultimately to buy/sell orders. Intensive tests of new module are ongoing and we expect the first results in the near future. Demo session Currency pair Start of session End of session Simulation -------------------- --------------------- ------------------------ ---------------------- ----------------------- IB-test-12 EUR/USD 2015-09-28 2015-11-23 SIM16-P009 LMAX-test-16 EUR/USD 2015-10-13 2015-12-03 SIM16-P010 SIM16-P019 SIM16-P026 LMAX-test-13 CHF/JPY 2015-10-09 2015-12-01 SIM16-P046-CHF-JPY SIM16-P047-CHF-JPY SIM16-P052-CHF-JPY LMAX-test-14 AUD/CAD 2015-10-12 2015-12-05 SIM16-P058-AUD-CAD-R1 LMAX-test-15 AUD/JPY 2015-10-12 2015-12-03 SIM16-P066-AUD-JPY SIM16-P069-AUD-JPY ![image](demo_IB-test-12-01){width="1.5\columnwidth"}\ ![image](demo_IB-test-12-02){width="1.5\columnwidth"}\ ![image](demo_LMAX-test-13-01){width="1.5\columnwidth"}\ ![image](demo_LMAX-test-14-01){width="1.5\columnwidth"}\ ![image](demo_LMAX-test-15-01){width="1.5\columnwidth"}\ ![image](demo_LMAX-test-16-02){width="1.5\columnwidth"} The work was partially supported by Slovak Research and Development Agency SRDA and Slovak Grant Agency for Science VEGA under the grants No. APVV-0463-12, VEGA 1/0158/13 and VEGA 2/0009/16. The authors thank the TH division for warm hospitality during their visits at CERN.
--- abstract: 'Edge machine learning involves the deployment of learning algorithms at the network edge to leverage massive distributed data and computation resources to train artificial intelligence (AI) models. Among others, the framework of federated edge learning (FEEL) is popular for its data-privacy preservation. FEEL coordinates global model training at an edge server and local model training at edge devices that are connected by wireless links. This work contributes to the energy-efficient implementation of FEEL in wireless networks by designing joint computation-and-communication resource management ($\text{C}^2$RM). The design targets the state-of-the-art heterogeneous mobile architecture where parallel computing using both a CPU and a GPU, called heterogeneous computing, can significantly improve both the performance and energy efficiency. To minimize the sum energy consumption of devices, we propose a novel $\text{C}^2$RM framework featuring multi-dimensional control including bandwidth allocation, CPU-GPU workload partitioning and speed scaling at each device, and $\text{C}^2$ time division for each link. The key component of the framework is a set of equilibriums in energy rates with respect to different control variables that are proved to exist among devices or between processing units at each device. The results are applied to designing efficient algorithms for computing the optimal $\text{C}^2$RM policies faster than the standard optimization tools. Based on the equilibriums, we further design energy-efficient schemes for device scheduling and greedy spectrum sharing that scavenges “spectrum holes" resulting from heterogeneous $\text{C}^2$ time divisions among devices. Using a real dataset, experiments are conducted to demonstrate the effectiveness of $\text{C}^2$RM on improving the energy efficiency of a FEEL system.' author: - 'Qunsong Zeng, Yuqing Du, Kaibin Huang, and Kin K. Leung [^1]' bibliography: - 'Energy\_Efficient.bib' title: 'Energy-Efficient Resource Management for Federated Edge Learning with CPU-GPU Heterogeneous Computing' --- Introduction ============ Realizing the vision of exploiting the enormous data distributed at edge devices (e.g., smartphones and sensors) to train artificial intelligence (AI) models has been pushing machine learning from the cloud to the network edge, called edge learning [@gxzhu_2018_edge_learning]. Currently, arguably the most popular edge learning framework is federated learning that preserves users’ privacy by distributed learning over devices [@gxzhu2018FEEL; @deniz2019federated_edge_learning; @sqwang_2018_edge_learning]. Each round of the iterative learning process involves the broadcasting of a global model to devices, their uploading of local model updates computed using local datasets, and a server’s aggregation of the received local updates for updating the global model. Executing complex learning tasks on energy and resource constrained devices is a main challenge faced in the implementation of edge learning. To tackle the challenge, the next-generation mobile systems-on-chip (SoC) will feature a heterogeneous architecture comprising a central processing unit (CPU) and a graphics processing unit (GPU) \[and even a digital processing unit (DSP) in some designs\] [@R9]. Experiments have demonstrated its advantages in terms of performance and energy efficiency. In this work, we address the issue of energy-efficient implementation of federated edge learning (FEEL) in a wireless system comprising next-generation devices capable of heterogeneous computing. To this end, a novel framework is proposed for energy-efficient joint management of computation-and-communication ($\text{C}^2$) resources at devices. Edge Implementation of Federated Learning ----------------------------------------- The implementation of FEEL in wireless networks faces two key challenges among others. As mentioned, the distributed learning process requires potentially many edge devices to periodically upload high-dimensional local model updates to an edge server. This includes the first challenge that the excessive communication overhead generated in the learning process can overwhelm the air interface that has finite radio resources but needs to support FEEL as well as other services. Designing techniques for suppressing the overhead forms a vein of active research, called communication-efficient FEEL. Diversified approaches have been proposed such as device scheduling [@nishio2018background; @yang2018background], designing customized multi-access technologies [@gxzhu2018FEEL; @deniz2019federated_edge_learning], and optimizing uploading frequencies [@sqwang_2018_edge_learning]. The second challenge faced by FEEL is how to execute power hungry learning tasks (e.g., training AI models each typically comprising millions of parameters) on energy constrained devices. Tackling the challenge by designing energy-efficient techniques leads to the emergence of an active research theme, called energy-efficient FEEL, which is the topic of current investigation. While there exists a rich literature of energy-efficient techniques for resource management (RM) in radio access networks (see e.g., [@liye]), the designs of their counterparts for FEEL, which are the main research focus in energy-efficient FEEL, are different due to the changes on the system objective and operations. In particular, FEEL systems aim at improving the learning performance instead of providing a radio access service and performing update aggregation instead of decoupling multiuser data. In early works [@QS; @chen2019joint; @sun2019energyaware], researchers proposed radio-RM techniques (e.g., bandwidth allocation and device scheduling) to improve the tradeoff between devices’ transmission energy consumption and learning performance. More recent research accounts for computation energy consumption which usually constitutes a substantial part of a device’s total consumption in the learning process, motivating the design of $\text{C}^2$RM techniques in [@tran2019energy; @yang2019energy; @mo2020energyefficient]. In [@tran2019energy], the tradeoffs between learning performance (in terms of accuracy and latency) and devices’ energy consumption are optimized by balancing the communication and computation latencies under a total latency constraint, referred hereafter as $\text{C}^2$ time division. In a standard FEEL design such as that in [@tran2019energy], the updates by devices are usually assumed synchronized. It arises from the operation of gradient aggregation and refer to the requirement that all local updates need to be received by the server before the global model can be updated. Consequently, all devices are allowed the same duration (per-round latency) for uploading their local gradients and thus synchronized in update transmission. The assumption on synchronized updates is relaxed in [@yang2019energy] where the learning latency is measured by a weighted sum of individual devices’ heterogeneous latencies and then the learning-energy tradeoffs similar to those in [@tran2019energy] are optimized. On the other hand, the idea of clock frequency control, or called dynamic voltage and frequency scaling (DVFS) [@R15], was explored in [@mo2020energyefficient] for energy-efficient FEEL based on the assumption that each device has a CPU featuring DVFS. Then the $\text{C}^2$RM was optimized where the communication RM is based on either time-division multiple access (TDMA) or non-orthogonal multiple access (NOMA) and the computation RM is based on dynamic frequency scaling. While prior work assumes single-processor devices, CPU-GPU heterogeneous computing discussed in the sequel is a new paradigm of mobile computing supporting applications with intensive data crunching. The area of energy-efficient FEEL based on heterogeneous computing is uncharted and explored in this work. CPU-GPU Heterogeneous Computing ------------------------------- The mentioned next-generation heterogeneous SoC are capable of supporting diversified workloads such as communication, signal processing, inference, and learning, which arise from a wide range of new mobile applications. Examples of such chips include Snapdragon by Qualcomm, R-series by AMD, and Kirin 970 by Huawei. Via the cooperation of the CPU and GPU on the same chip for executing a single task, heterogeneous computing on such SoC fully utilizes the computation resources of both processors to optimize the computation performance and energy efficiency [@R16]. In particular, the new paradigm has been demonstrated by experiments to accelerate the running of deep neural networks on smartphones [@R8; @R18]. Existing research on heterogeneous computing focuses on several main design issues including workload partitioning [@R9; @R11; @R14; @R15], dynamic frequency scaling [@R5; @R20], and memory access scheduling [@R10]. The first two issues are addressed in this work. Specifically, workload partitioning refers to dividing and allocating workload of a task over the integrated CPU and GPU according to the task requirements and the processors’ states (e.g., temperatures) and characteristics (e.g., speeds and energy efficiencies) so as to optimize the overall performance and efficiency [@R9; @R11; @R14; @R15]. On the other hand, frequency scaling (or DVFS) previously considered for a single CPU [@tran2019energy; @mo2020energyefficient] becomes the more sophisticated management in heterogeneous computing due to the joint CPU-GPU control [@R20]. While prior work on heterogeneous computing focuses on a single device, we study the joint control of workload partitioning and DVFS in the context of a large system comprising multiple devices and furthermore explore their integration with radio-RM. Contributions ------------- The objective of this work is not to contribute any new learning technique but to focus on $\text{C}^2$RM to facilitate the implementation of a standard FEEL technique in a wireless network. To this end, we consider a FEEL system consisting of one edge server and multiple edge devices. Its main difference from those in existing work (see e.g., [@yang2019energy]) is that each device is equipped with a CPU-GPU platform enabling heterogeneous computing. The design objective is to minimize the sum energy consumption at devices under a guarantee on learning performance (latency and accuracy) by jointly controlling $\text{C}^2$RM in the following four dimensions: - Bandwidth allocation: Allocating bandwidths to devices for transmission of local model updates under a constraint on the total uplink bandwidth; - $\text{C}^2$ time division: Dividing the allowed per-round latency for the computation and communication of each device; - CPU-GPU workload partitioning: Partitioning and allocating the computation workload for each device to its CPU and GPU; - CPU-GPU frequency scaling: Controlling the CPU-GPU frequencies/speeds at each device. While the RM control in the first two dimensions are considered in prior work as discussed, the other two are unique for heterogeneous computing. Their joint control over multiple devices is a challenging and open problem. To the best of the authors’ knowledge, this work represents the first attempt on studying heterogeneous computing in the context of energy-efficient FEEL. The main contributions are described as follows. - Equilibrium based $\text{C}^2$RM framework: It is mathematically proved that the control policies are optimal if and only if they achieve the following equilibriums: - The CPU-GPU pair of every device has equal energy-workload rates, defined as the increase in energy consumption per additional workload for each processing unit; - A similar equilibrium also exists with respect to the processing units’ computation speeds as their optimal values are proved to be proportional to the corresponding workloads. - All devices have equal energy-time rates, defined as the energy rates with respect to the communication/computation latency; - All devices have equal energy-bandwidth rates, defined as the energy rates with respect to the allocated bandwidth. The equilibriums are applied to obtaining the optimal polices for 1) computation RM, 2) communication RM, and 3) joint $\text{C}^2$RM. They are either derived in closed-form or computed using low-complexity algorithms. In particular, for low-complexity joint $\text{C}^2$RM, the problem is decomposed into one master problem for achieving the equilibrium in energy-time rates, and two sub-problems for separately achieving the two equilibriums in energy-workload rates and energy-bandwidth rates. The resultant algorithm is shown to have much lower complexity than a standard solution method such as block coordinate descent (BCD). - $\text{C}^2$ aware scheduling: Building on the equilibrium framework, an energy-efficient scheduling scheme is designed to select a fixed number of devices for participating in FEEL. The novelty lies in the design of $\text{C}^2$ aware scheduling metric that balances both the channel state and computation capacity of each device. Specifically, given the objective of sum energy minimization, the metric is designed to be the energy consumption of a device given the derived optimal $\text{C}^2$ time division and equal bandwidth allocation. In addition, we analyze the effect of the number of scheduled devices on model convergence. - Greedy spectrum sharing: For the preceding designs, we assume fixed bandwidth allocation in each round of the iterative learning process. Consequently, the heterogeneous computation latencies of devices generates “spectrum holes” (unused spectrum-time blocks) that can be scavenged for further improving the energy efficiency. To this end, the scheme of greedy spectrum sharing is designed featuring a novel metric, called energy-bandwidth acceleration rate and defined as the derivative of the energy-bandwidth rate, for selecting an available device to transmit using the spectrum hole. A larger value of the metric, implies steeper energy reduction for a device when it is allocated additional bandwidth. ![A wireless system supporting FEEL over devices capable of CPU-GPU heterogeneous computing.[]{data-label="Fig:system_model"}](system_update_new.pdf){width="95.00000%"} Models and Metrics {#system model} ================== Consider the FEEL system in Fig. \[Fig:system\_model\] comprising a single edge server and $K$ edge devices, denoted by an index set $\mathcal{K}=\{1,\cdots,K\}$. Each iteration between local gradient uploading and global model updating is called a communication round, or round for short. It is assumed that the server has perfect knowledge of the multiuser channel gains and local computation characteristics, which can be obtained by feedback. Using this information, the server determines the energy-efficient strategies for device scheduling and $\text{C}^2$RM. Federated Learning Model ------------------------ A standard federated learning technique (see e.g., [@federatedlearning]) is considered as follows. A global model, represented by the parameter set $\mathbf{w}$, is trained collaboratively across the edge devices by leveraging local labelled datasets. For device $k$, let $\mathcal{D}_k$ denote the local dataset and define the local loss function as $ F_k(\mathbf{w})=\frac{1}{\|\mathcal{D}_k\|}\sum_{(\mathbf{x}_j,y_j)\in\mathcal{D}_k}\ell(\mathbf{w};\mathbf{x}_j,y_j),$ where $\ell(\mathbf{w};\mathbf{x}_j,y_j)$ is the sample-wise loss function quantifying the prediction error of the model $\mathbf{w}$ on the training sample $\mathbf{x}_j$ with regard to its label $y_j$. For convenience, we assume a uniform size for local datasets: $\|\mathcal{D}_k\|\equiv D$, for all $k$. Then the global loss function on all the distributed datasets can be written as $$F(\mathbf{w})=\frac{\sum_{(\mathbf{x}_j,y_j)\in\cup_k\mathcal{D}_k}\ell(\mathbf{w};\mathbf{x}_j,y_j)}{\|\cup_k\mathcal{D}_k\|}=\frac{1}{K}\sum_{k=1}^KF_k(\mathbf{w}).$$ The learning process is to minimize $F(\mathbf{w})$, that is, $\mathbf{w}^=\arg\min F(\mathbf{w})$. For ease of exposition, we focus on the gradient-averaging implementation while the current designs also apply to the model-averaging implementation [@federatedlearning]. In each round, say the $i$-th round, the server broadcasts the current model ${\mathbf{w}}^{(i)}$ as well as selection indicators $\{\rho_k\}$ to all edge devices, where the indicator $\rho_k=1$ if device $k$ is scheduled, or $0$ otherwise. Suppose $M$ devices are scheduled for participation in each round, denoted by an index set $\mathcal{M}_i$ for the $i$-th round. Based on the received model ${\mathbf{w}}^{(i)}$, each scheduled device calculates the gradient $\nabla F_k({\mathbf{w}}^{(i)})$ using its local dataset. Upon completion, the local gradients are transmitted to the server for aggregation: $$\label{gradient-aggregation} {\mathbf{g}}^{(i)}=\frac{1}{M}\sum_{k\in\mathcal{M}_i}\nabla F_k({\mathbf{w}}^{(i)})$$ Synchronized updates by devices are assumed. The global model is then updated by stochastic gradient descent* (SGD) as ${\mathbf{w}}^{(i+1)}={\mathbf{w}}^{(i)}-\eta {\mathbf{g}}^{(i)}$, where $\eta$ is the learning rate. The process iterates until the model converges. Model of Heterogeneous Computing -------------------------------- ### Workload model Adopting a standard model (see e.g., [@zhang2018flops]), the total workload $W$ for a computation task is given as $W=N_{\sf FLOP}\times D$, where $D$ is the local dataset size and $N_{\sf FLOP}$ denotes the number of floating point operations (FLOPs) needed for processing each sample. Furthermore, we define $f_k^{(c)}$ and ${f'}_k^{(c)}$ (in cycle/s) as the clock frequency of the CPU and GPU at device $k$, respectively. It follows that the computing speeds of CPU and GPU can be defined as $f_k = f_k^{(c)}\times n_k$ and $f'_k = {f'}_k^{(c)}\times n_k'$ with $n_k$ and $n_k'$ denoting the number of CPU and GPU FLOPs per cycle, respectively. ### Workload partitioning model For workload partitioning in local gradient computation, we consider input-sample partitioning, also known as data parallelism [@partition; @parallel]. As a result, each processor (CPU or GPU) processes a fraction of input samples. The partitioned workloads for the CPU and GPU at device $k$ are denoted as $W_k$ and ${W'}_k$ respectively, with $$\label{InEq: workload_constrain} \text{(Workload constraint)}\quad W_k+{W}_k'=W,\quad \forall k\in\mathcal K.$$ Given the partitioned workloads and CPU/GPU frequencies, the local computation time for the total workload $W$, denoted as ${t'}_k$, is given as $$\label{InEq: local computation time} \text{(Local computation time)}\quad {t}_k'=\max\left\{\frac{W_k}{f_k},\frac{{W}_k'}{{f}_k'}\right\},\quad \forall k\in\mathcal K.$$ ### DVFS model For a CMOS circuit, the power consumption of a processor can be modeled as a function of clock frequency: $P=\Psi \left[f^{(c)}\right]^3$ with the coefficient $\Psi$ \[in $\text{Watt}/ (\text{cycle/s})^{3}$\] depending on the chip architecture and $f^{(c)}$ being the clock frequency [@liu2012dvfs]. Using this model, the power consumption of CPU and GPU at device $k$ can be written as $$\label{cpu-gpu-power} P_k^{\text{CPU}}=\Psi_k^{\text{CPU}}\l(f^{(c)}_k\r)^3 = C_kf_k^3 \qquad\text{and}\qquad P_k^{\text{GPU}}=\Psi_k^{\text{GPU}}\l({f'}_k^{(c)}\r)^3 = G_k{f'}_k^3,$$ where $C_k=\Psi_k^{\text{CPU}}/n_k^3$ and $G_k=\Psi_k^{\text{GPU}}/{n'}_k^3$. Using these functions, frequency scaling refers to controlling the power of CPU and GPU by adjusting their speeds ${f}_k$ and ${f}_k'$, respectively. (CPU/GPU Computation Efficiency). The coefficient $C_k$ (or $G_k$) in characterizes the computation efficiency* of a CPU (or GPU), defined as the rate of power growth in response to the increase of the cubed computing speeds. In practical data processing based on heterogeneous computing, the GPU tends to play a main role (with $G_k<C_k$) while the CPU contributes supplementary computing resources.* ### Energy consumption model Given the time durations $W_k/f_k$ and $W_k'/f_k'$ for CPU and GPU to complete their tasks with the workloads $W_k$ and $W'_k$, the resultant energy consumption at device $k$ can be written as $$E_k^{\text{CPU}}=C_kW_kf_k^2\qquad\text{and}\qquad E_k^{\text{GPU}}=G_kW_k'{f'}_k^2.$$ Then the total computation energy consumption of device $k$ per round is $$\label{Eq: E_cmp} E_k^{\text{cmp}}=C_kW_kf_k^2+G_k{W}_k'{f'}_k^2,\quad \forall k\in\mathcal{K}.$$ Multiple-Access Model --------------------- Consider the frequency-division multiple access (FDMA) for gradient uploading with the total bandwidth $B$. Let $B_k$ denote the allocated bandwidth for device $k$ in an arbitrary round, which is fixed throughout the round. This assumption is relaxed in Section \[Extension\]. Then we have the following constraint: $$\label{Eq: bandwidth_constraint} \text{(Bandwidth constraint)}\quad \sum_{k=1}^KB_k=B.$$ Channels are assumed to be frequency non-selective. The edge server usually recruits idling devices as workers that either static or at most moving at pedestrian speeds [@federatedlearning]. For this reason, we adopt the model of slow block fading. To be specific, the channel gain of device $k$, denoted as $h_k$, is assumed to remain unchanged within one round but varies independently and identically (i.i.d.) over rounds. Given synchronous updates, a time constraint is set for each round: $$\label{InEq: time_constraint} \text{(Latency constraint)}\quad {t}_k'+t_k\leq T,\quad \forall k\in\mathcal K,$$ where ${t}_k'$ and $t_k$ denote the time for local training/computation and gradient uploading of device $k$, respectively; $T$ is the maximum total time for one round. A prerequisite for a scheduled device is that it can meet the above time constraint. Let $P_k^{\text{TX}}$ denote the transmission power of device $k$. The achievable rate, denoted by $r_k$, can be written as $$\begin{aligned} \label{Eq:shannon} r_k=B_k\log\left(1+\frac{P_k^{\text{TX}}h_k^2}{N_0}\right),\quad \forall k\in\mathcal{K},\end{aligned}$$ where $N_0$ is the spectrum density of the complex white Gaussian channel noise. Let $L=\|\mathbf{g}\|\alpha$ denote the gradient size (in bit) with $\alpha$ denoting a sufficient large number of bits for quantizing each parameter with negligible distortion. Then the data rate is $r_k = L/t_k,\forall k\in\mathcal{K}$. By combining the result and , the communication energy consumption of device $k$ per round is $$\label{Eq:E_upload} E_k^{\text{cmm}}=B_kP_k^{\text{TX}}t_k = \frac{B_kt_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-1\right),\quad \forall k\in\mathcal{K}.$$ Performance Metric ------------------ It is assumed that the data distributed at devices satisfy the condition for model convergence within a finite number of rounds, denoted as $N$ (see e.g., [@signSGD]). As implied by (\[Eq:shannon\]), capacity achieving codes are deployed to ensure reliable transmissions. Consequently, wireless channels have no effects on $N$ though they affect per-round latencies and energy consumption. The objective of energy-efficient RM is to minimize the total energy consumption of all active devices in the $N$-round learning process, namely $\sum_{i=1}^N\sum_{k=1}^K\left(E_{k}^{\text{cmp}}[i]+E_{k}^{\text{cmm}}[i]\right)$ with $E_k^{\text{cmp}}$ and $E_{k}^{\text{cmm}}$ given in (\[Eq: E\_cmp\]) and (\[Eq:E\_upload\]), respectively. Given block fading and the per-round latency constraint in (\[InEq: time\_constraint\]), the objective can be straightforwardly proved to be equivalent to minimize the total energy consumption for each round. This is aligned with a standard approach in adaptive transmission and its proof is similar to those in the literature (see Lemma 1 in [@wen2020joint]) and thus omitted for brevity. It follows that for subsequent designs, it is sufficient to consider an arbitrary round and the corresponding total energy consumption, i.e., $\sum_{k=1}^K\left(E_k^{\text{cmp}}+E_k^{\text{cmm}}\right)$, as the performance metric, is called sum energy hereafter. Energy-Efficient $\text{C}^2$ Resource Management ================================================= In this section, considering the case that all devices are scheduled for uploading, we study: 1) computation RM, 2) communication RM, and 3) joint $\text{C}^2$RM by analyzing and computing the optimal policies. Then, the results are applied to establishing an energy-learning tradeoff. Computation Resource Management ------------------------------- ### Problem Formulation The computation resources distributed at devices can be managed by controlling their workload partitioning and speed scaling to minimize the sum computation energy. The design is formulated as the following optimization problem. $$\label{P:computation}{\bf (P1)}\quad \begin{aligned} \min_{\{W_k,f_k,f_k'\}}\quad&\sum_{k=1}^K\left(C_kW_kf_k^2+G_kW_k'f_k^{'2}\right)\\ \text{s.t.}\quad\quad&W_k+{W'}_k=W,\quad W_k\geq 0,\quad {W'}_k\geq 0,\quad \forall k\in\mathcal{K},\\ &{t}_k'=\max\left\{\frac{W_k}{f_k},\frac{W'_k}{f_k'}\right\}, \quad \forall k\in\mathcal{K},\\ &0\leq{t}_k'\leq {T'}_k, \quad \forall k\in\mathcal{K}. \end{aligned}$$ where the three sets of constraints follow from , , and the time constraint with $T_k'$ being the allowed maximum computation time for device $k$. ### Optimal Policy The optimal policy for computation RM is derived in closed-form below. \[lemma: computation condition\] (Speed Scaling Rule). To minimize the sum computation energy, the computation speeds of CPU and GPU, $f_k$ and $f'_k$, should be scaled by following $$\begin{aligned} \frac{W_k}{f_k}=\frac{W_k'}{f_k'},\quad \forall k\in\mathcal{K},\end{aligned}$$ given an arbitrary workload pair $(W_k, W_k^{'})$ with $W = W_k+W_k'$. See Appendix \[proof: computation condition\]. [The optimal strategy of workload-proportional speed scaling in Lemma 1 is intuitive and results from equalizing CPU and GPU computation time to avoid the slower one becoming the bottleneck of local gradient computation.]{} Then, with Lemma \[lemma: computation condition\], the original Problem [P1]{} can be rewritten as follows: $$\label{P:computation}{\bf (P2)}\quad \begin{aligned} \min_{\{W_k,t_k'\}}\quad&\sum_{k=1}^K \frac{C_kW_k^3+G_k{W}_k^{'3}}{{t}_k^{'2}}\\ \text{s.t.}\quad\quad&W_k+{W}_k'=W,\quad W_k\geq 0,\quad {W}_k'\geq 0,\quad \forall k\in\mathcal{K},\\ &0\leq{t}_k'\leq {T}_k', \quad \forall k\in\mathcal{K}. \end{aligned}$$ Next, it is found that the optimal policy achieves an equilibrium between the CPU and GPU on each device, in terms of energy-workload rate[^2] . \[lemma: energy-load rate\] (Energy-Workload Rate Equilibrium). Consider the energy-workload rates* of CPU and GPU given as $$\begin{aligned} \label{Def: energy-load rate} \frac{\partial E_k^{\text{CPU}}}{\partial W_k}=\frac{3C_kW_k^2}{{t}_k^{'2}} \quad \text{and}\quad \frac{\partial E_k^{\text{GPU}}}{\partial {W}_k'}=\frac{3G_k{W}_k^{'2}}{{t}_k^{'2}},\quad \forall k\in\mathcal{K},\end{aligned}$$ where $E_k^{\text{CPU}} = \frac{C_kW_k^3}{{t}_k^{'2}}$ and $E_k^{\text{GPU}} = \frac{G_k{W}_k^{'3}}{{t}_k^{'2}}$ denote the computation energy of CPU and GPU, respectively. Then, the necessary and sufficient condition for optimal workload allocation is $$\begin{aligned} \label{Equilibrium: energy-load rate} \frac{\partial E_k^{\text{CPU}}}{\partial W_k}=\frac{\partial E_k^{\text{GPU}}}{\partial W_k'},\quad \forall k\in\mathcal{K},\end{aligned}$$ with $W_k+{W}_k'=W$.* See Appendix \[proof: energy-load rate\]. Furthermore, one can observe that the objective function in [P2]{} is a non-increasing function in ${t'}_k,\forall k\in\mathcal{K}$. Therefore, the optimality requires to maximize the computation time of each device, resulting in ${t'}_k^={T'}_k,\forall k\in\mathcal{K}$, which is independent of the workload partitioning. Using the result as well as Lemmas 1 and 2, the optimal computation RM policy is obtained as follows. \[proposition: computation load allocation\] (Optimal Computation RM). The optimal workload allocation is $$\begin{aligned} \label{Eq: Optimal_Workload_Allocation} \text{(Optimal Workload Allocation)}\quad W_k^=\frac{\sqrt{G_k}W}{\sqrt{C_k}+\sqrt{G_k}},\quad W_k'^=\frac{\sqrt{C_k}W}{\sqrt{C_k}+\sqrt{G_k}}, \quad k\in\mathcal{K},\end{aligned}$$ where $C_k$ and $G_k$ are computation coefficients for CPU and GPU, respectively. Moreover, the optimal speed scaling for CPU and GPU is $$\begin{aligned} \label{proposition: frequency scaling} \text{(Optimal Speed Scaling)}\quad f_k^= \frac{W_k^}{{T'}_k} ,\quad f_k'^=\frac{W_k'^}{{T'}_k}, \quad k\in\mathcal{K},~\qquad\qquad\quad\quad\end{aligned}$$ where $T_k'$ is the allowed maximum computation time for device $k$.* (Energy-Efficient CPU-GPU Heterogeneous Computing). According to Proposition \[proposition: computation load allocation\], more workload tends to be allocated to the processor with a smaller computation coefficient indicating a higher computation efficiency (see Remark 1), thereby reducing the devices’ energy consumption. Communication Resource Management --------------------------------- ### Problem Formulation The total radio resources are managed by controlling bandwidth allocation and transmission time to minimize the sum transmission energy. The corresponding optimization problem can be formulated as $$\label{opt_rrm}{\bf (P3)}\quad \begin{aligned} \min_{\{B_k,t_k\}}\quad&\sum_{k=1}^K\frac{B_k t_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-1\right)\\ \text{s.t.}\quad\quad&\sum_{k=1}^KB_k=B,\quad B_k\geq 0,\quad \forall k\in\mathcal{K},\\ &0\leq t_k\leq T_k,\quad \forall k\in\mathcal{K}\nn. \end{aligned}$$ The problem has a standard structure in the literature of energy-efficient communication (see e.g., [@changsheng]). It is convex and can be solved using a standard algorithm such as BCD. In the sequel, we present an alternative solution method yielding an equilibrium property that is useful for low-complexity policy computation faster than conventional methods. ### Properties of Optimal Policies As before, the objective of Problem [P3]{} is observed to be a non-increasing function in $t_k$. Therefore, it is optimal to maximize the transmission time of each device, resulting in $t_k^=T_k,\forall k\in\mathcal{K}$. Next, to derive the optimal bandwidth allocation strategy, a necessary and sufficient condition is given as follows. \[lemma: energy-bandwidth rate\] (Energy-Bandwidth Rate Equilibrium). The energy-bandwidth rate as defined earlier is mathematically obtained as $$\begin{aligned} \label{eqn: define nu_k} \frac{\partial E_k^{\text{cmm}}}{\partial B_k} = \frac{t_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-\frac{L\ln{2}}{B_kt_k}2^{\frac{L}{B_kt_k}}-1\right)<0 ,\quad k\in\mathcal{K}.\end{aligned}$$ The optimal bandwidth allocation equalizes the energy-bandwidth rates as $$\begin{aligned} \label{Eq: energy-bandwidth rate} \frac{\partial E_1^{\text{cmm}}}{\partial B_1}=\frac{\partial E_2^{\text{cmm}}}{\partial B_2}=\cdots=\frac{\partial E_K^{\text{cmm}}}{\partial B_K}=-\nu^,\end{aligned}$$ where $\nu^$ is a constant and $\sum_{k=1}^KB_k = B$.* See Appendix \[proof: energy-bandwidth rate\]. It can be observed from the above lemma that the increase of allocated bandwidth reduces the communication energy consumption of the device. Then, the optimal communication RM policy directly follows from the energy-bandwidth rate equilibrium as stated below. \[lemma: communication bandwidth allocation\] (Optimal Bandwidth Allocation). The optimal policy for bandwidth allocation is $$\begin{aligned} \label{Eq: Optimal Bandwidth Allocation} B_k^=\frac{L\ln{2}}{T_k\left[1+\mathcal{W}_0\!\!\left(\frac{h_k^2\nu^-T_kN_0}{T_kN_0e}\right)\right]},\quad k\in\mathcal{K},\end{aligned}$$ where $T_k$ is the allowed maximum transmission time for device $k$; $\mathcal{W}_0(\cdot)$ is the Lambert $W$ function (principal branch) and $e$ is the Euler’s number. Proposition \[lemma: communication bandwidth allocation\] suggests that $B_k^$ is a non-increasing function with respect to $h_k^2$. It means that more bandwidths should be allocated to devices with weaker channels for the benefit of sum communication energy reduction. Obtaining the optimal bandwidths $\{B^_k\}$ via requires computing the optimal energy-bandwidth rate $\nu^$ by solving the following equation: $$\label{eqn: sum B} \sum_{k=1}^K\frac{L\ln{2}}{T_k\left[1+\mathcal{W}_0\!\!\left(\frac{h_k^2\nu^-T_kN_0}{T_kN_0e}\right)\right]}=B.$$ Due to the intractability of the Lambert function $\mathcal{W}_0$ and the unknown range of $\nu^$, the solution cannot be found efficiently using standard methods such as Newton-Raphson method and bi-section search. An alternative method, derivative-free optimization* (DFO), which requires no derivative, has too high complexity. To address this issue, a fast algorithm for policy computation is designed below. ### Optimal Policy Computation Instead of solving $\nu^$ and $\{B^_k\}$ in a sequential order, the fast algorithm calculates the optimal values iteratively, comprising the following two phases. - [Phase I]{}: (Bandwidth Optimization). Given the energy-bandwidth rate $\nu$, compute the bandwidth allocation $\{B_k\}$ using . As $\nu$ is not optimal, the computed $\{B_k\}$ may not satisfy the bandwidth constraint in . Thus it is necessary to normalize each $B_k$ as $\widetilde{B}_k = \frac{B}{\sum_{k=1}^KB_k}B_k$. - [Phase II]{}: (Energy-Bandwidth Rate Updating). Calculate the respective energy-bandwidth rates $ \nu_k= - \frac{\partial E_k^{\text{cmm}}}{\partial B_k}$, $\forall k\in\mathcal{K}$, by substituting $\{\widetilde{B}_k\}$ and $\{t_k = T_k\}$ into . Then, update the current energy-bandwidth rate $\nu$ using $\{\nu_k\}$ as elaborated in the sequel. The two phases are iterated until convergence as indicated by the energy-bandwidth rate equilibrium in . A key step in the algorithm is the update rule for $\nu$ in Phase II. To derive the rule, a useful property is given as follows. \[Pre:updating\_rule\] See Appendix \[proof: algorithm 1\]. Essentially, by induction, the above Lemma \[Pre:updating\_rule\] simply implies that if the initial point $\nu^{(0)}$ is greater than $\nu^{}$, and the update rule adopted for $\nu$ is $\nu^{(i)} = \max\limits_{k}\{\nu_k^{(i)}\}$, the convergence of $\nu$ to the global optimal $\nu^$ is guaranteed. This is also true for the case of $\nu^{(0)} < \nu^$ by involving the update rule as $\nu^{(i)} = \min\limits_{k}\{\nu_k^{(i)}\}$. It is further noted that the condition of $\nu^{(0)} > \nu^$ or $\nu^{(0)} < \nu^$ can be determined by calculating $\text{sgn}\left(\sum_{k=1}^KB_k^{(1)}-B\right)$, where $B_k^{(1)} = B_k(\nu^{(0)})$ follows from . In summary, we have the following update rule for $\nu$ in Phase II: $$\begin{aligned} \text{(Update rule)} \quad \nu^{(i)} = \left\{ \begin{array}{rcl} \max\limits_{k}\{\nu_k^{(i)}\}, & & \text{sgn}\left(\sum_{k=1}^KB_k^{(1)}-B\right)= -1;\\ \min\limits_{k}\{\nu_k^{(i)}\}, & & \text{otherwise}, \end{array} \right. \end{aligned}$$ where $\{\nu_k^{(i)}\}$ and $\{B_k^{(1)}\}$ are specified in Lemma \[Pre:updating\_rule\]. Based on the above rule, the algorithm for optimal bandwidth allocation is summarized in Algorithm \[Algorithm:solve\_nu\]. \[h\] Input:* initial value of $\nu$.\ Output: optimal $\nu^$ and $\{B^_k\}$.\ Calculate: indicator $y=\text{sgn}\left(\sum_{k=1}^KB_k(\nu)-B\right)$.\ Repeat: - Calculate $\{B_k\}$ by substituting $\nu$ into Proposition \[lemma: communication bandwidth allocation\]; - Normalize $\widetilde{B}_k=\frac{B}{\sum_{k=1}^KB_k}B_k$, $k\in\mathcal{K}$; - Calculate $ \{\nu_k\}$ by substituting $\{\widetilde{B}_k\}$ into , $k\in\mathcal{K}$; - Update $\nu=\frac{1-y}{2}\max\limits_{k}\{\nu_k\}+\frac{1+y}{2}\min\limits_{k}\{\nu_k\}$; Until $\text{var}\left[\{\nu_k\}\right]<\varepsilon$ (to equalize the energy-bandwidth rates). (Low-Complexity and Optimality). The complexity of Algorithm \[Algorithm:solve\_nu\] is $O(\log\frac{1}{\varepsilon})$ with $\varepsilon$ denoting the target accuracy. For comparison, the computation of the DFO method for solving $\nu^$ and $\{B_k^\}$ in a sequential order is $O(\frac{1}{\varepsilon^2})$, and that of the BCD algorithm for directly solving Problem [P3]{} is $O(\frac{K}{\varepsilon}\log\frac{1}{\varepsilon})$, which are much higher than that of Algorithm \[Algorithm:solve\_nu\]. Furthermore, the optimality is guaranteed by Algorithm \[Algorithm:solve\_nu\] as indicated by Lemma \[Pre:updating\_rule\]. Joint $\text{C}^2$ Resource Management -------------------------------------- ### Problem Formulation Building on the preceding results, an energy-efficient $\text{C}^2$RM framework is designed by joint control of workload partitioning, $\text{C}^2$ time division, and bandwidth allocation to minimize sum energy. The optimization problem is formulated as $$\label{P:joint_optimization}{\bf (P4)}\quad \begin{aligned} \min_{\{W_k,{t}_k',B_k,t_k\}}\quad&\sum_{k=1}^K\left[\frac{C_kW_k^3+G_k{W'}_k^3}{{t'}_k^2}+\frac{B_kt_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-1\right)\right]\\ \text{s.t.}\quad\quad&W_k+{W}_k'=W,\quad W_k\geq 0,\quad {W}_k'\geq 0, \quad \forall k\in\mathcal{K},\\ &\sum_{k=1}^KB_k=B,\quad B_k\geq 0, \quad \forall k\in\mathcal{K},\\ &{t}_k'+t_k\leq T,\quad {t}_k'\geq 0,\quad t_k\geq 0, \quad \forall k\in\mathcal{K}\nn. \end{aligned}$$ ### Properties of Optimal Policy It is shown in the sequel that a set of equilibriums exist among devices in terms of energy rates with respect to different types of control variables when $\text{C}^2$RM is optimally energy-efficient. The insights facilitate designing low-complexity policy for solving Problem [P4]{}. First, by the same argument as in Subsections A and B, the latency constraint in Problem [P4]{} should be active for energy minimization: ${t'}_k+t_k=T,\forall k\in\mathcal{K}$. The optimal $\text{C}^2$ time division of $T$ has the following property. \[lemma: energy-time rate\] (Energy-Time Rate Equilibrium). The energy-time rates* as defined earlier can be mathematically obtained as $$\label{def: xi'} \xi_k':=\frac{\partial E_k^{\text{cmp}}}{\partial {t}_k'}=-\frac{2(C_kW_k^3+G_k{W'}_k^3)}{{t'}_k^3},\quad k\in\mathcal{K},$$ and $$\label{def: xi} \xi_k:=\frac{\partial E_k^{\text{cmm}}}{\partial t_k}=\frac{B_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-\frac{L\ln 2}{B_kt_k}2^{\frac{L}{B_kt_k}}-1\right),\quad k\in\mathcal{K}.$$ The optimal $\text{C}^2$ time division for each device requires the energy-time rate equilibrium: $$\frac{\partial E_k^{\text{cmp}}}{\partial {t}_k'}=\frac{\partial E_k^{\text{cmm}}}{\partial t_k},\quad\forall k\in\mathcal{K},$$ with ${t}_k'+t_k= T$.* The proof is similar to that for Lemma \[lemma: energy-load rate\] and thus omitted for brevity. Next, one can observe that Problem [P4]{} integrates [P2]{} and [P3]{} by summing their objectives and combining their constraints. Therefore, [P4]{} is also convex and furthermore the results in Lemmas \[lemma: energy-load rate\] and \[lemma: energy-bandwidth rate\] hold for the current case. Then, combining Lemmas \[lemma: energy-load rate\], \[lemma: energy-bandwidth rate\] and \[lemma: energy-time rate\] yields the following main result. \[theorem: equilibrium condition\] (Equilibrium Based $\text{C}^2$RM). \[Remark: C2-Tradeoff\] (Equalizing $\text{C}^2$ Heterogeneity). In the process of synchronized updates, Theorem 1 suggests that energy-efficient $\text{C}^2$RM should equalize the heterogeneity in communication channels and computation efficiencies using the multi-dimensional control variables. First, the heterogeneity in multiuser channel states and CPU/GPU computation efficiencies at each device is equalized by bandwidth allocation and workload partitioning (with speed scaling) as reflected in the second and third (with fourth) equilibriums. Second, the heterogeneity in $\text{C}^2$ speeds is equalized by adjusting the $\text{C}^2$ time division according to the first equilibrium. It is worth mentioning that $\text{C}^2$ time division represents a $\text{C}^2$ tradeoff* that more communication resources can compensate for the lack of computation resources and vice versa.* ### Optimal Policy Computation Though the optimal solution for Problem [P4]{} can be obtained by finding the equilibriums in Theorem \[theorem: equilibrium condition\] numerically using the standard method of BCD, it has high complexity. To tackle this challenge, we design a more efficient algorithm as follows. First, Problem ${\bf P4}$ can be decomposed into one master problem and two sub-problems as follows: - (Master Problem): The master problem is the optimization of $\text{C}^2$ time division. Denote ${E}^{\text{cmp}}(\{\widetilde T_k\})$ and ${E}^{\text{cmm}}(\{\widetilde T_k\})$ as the computation and communication energy, which are functions of the allowed maximum communication time $\{\widetilde T_k\}$ given optimized resource management strategies output by sub-problem 1 and sub-problem 2. Then, the master problem is cast as $$\label{master_problem}{\bf (MP)}\quad \begin{aligned} \min_{\{\widetilde T_k\}}\quad& {E}^{\text{cmp}}(\{\widetilde T_k\})+ {E}^{\text{cmm}}(\{\widetilde T_k\})\\ \text{s.t.}\quad&0< \widetilde T_k < T,\quad \forall k\in\mathcal{K}. \end{aligned}$$ - (Two Sub-problems): - ([Computation RM]{}): Problem ${\bf P1}$ with $\{{T}_k'=T-\widetilde T_k\}$; - ([Communication RM]{}): Problem ${\bf P3}$ with $\{T_k = \widetilde T_k\}$; As Problem [P4]{} is convex, the optimal solutions can be obtained via iterations between the master problem and two sub-problems. Each iteration comprises two steps: 1) given $\{\widetilde T_k\}$, compute the optimal RM strategies by solving two sub-problems; 2) given allocated bandwidths and partitioned workloads, calculate the sub-gradient of the master problem and apply it to updating $\{\widetilde T_k\}$ via the gradient descent method. The two steps are iterated until convergence. Note that the solution of one sub-problem, namely Problem [P1]{}, can be calculated directly via and while numerically calculation is required for solving the other sub-problem, namely Problem [P3]{}. Given the efficient Algorithm \[Algorithm:solve\_nu\] developed for computing $\{B^_k\}$, in the sequel, we aim at further improving the efficiency by proposing a novel initialization method of $\nu$. To this end, another useful property is introduced in the following lemma. The optimal $\nu^$ under the arbitrary given $\text{C}^2$ time division, namely $T'_k + T_k = T$ with $t'_k \leq T'_k$ and $t_k \leq T_k, \forall k \in \mathcal{K}$, can be calculated as $$\label{nu} \nu^ =-\frac{1}{B}\sum_{k=1}^{K}T_k\frac{\partial E_k^{\text{cmm}}}{\partial t_k}\bigg\|_{T_k} = -\frac{1}{B}\sum_{k=1}^{K}T_k\xi_k(T_k),$$ where $\xi_k(T_k) = \frac{\partial E_k^{\text{cmm}}}{\partial t_k}\big\|_{T_k}$ is the energy-communication time rate under current time division.* The above result can be proved by noting $ T_k\frac{\partial E_k^{\text{cmm}}}{\partial t_k}\big\|_{T_k}=B_k^\frac{\partial E_k^{\text{cmm}}}{\partial B_k}\big\|_{B_k^} = -\nu^* B_k^$ given the bandwidth constraint, namely $\sum_{k=1}^{K}B_k^ = B$. As the values of $\{\xi_k(T_k)\}$ are not attainable, $\nu^$ can not be solved directly. Nevertheless, we can get a good initial point for $\nu$ by taking advantage of . To be specific, due to the fact that $\xi_k'(T'_k)$ and $\xi_k(T_k)$ converge with iterations and $\xi_k'(T'_k)$ can be calculated by solving [P1]{} in closed-form, we propose to initialize $\nu$ as $$\label{eqn: initialize nu} \nu_0:= -\frac{1}{B}\sum_{k=1}^K T_k\xi_k'(T'_k),$$ Such initialization provides an increasingly better initial point that is closer to the optimal solution as the outer iteration proceeds. The algorithm for computing the optimal $\text{C}^2$RM policy is summarized in Algorithm \[Algorithm:joint\_RM\]. \[h\] Input:* initial values of $\{\widetilde T_k\}$.\ Output: optimal $\{B_k^\}$, $\{{t'}_k^ = T-\widetilde T^_k\}$, $\{t_k^ = \widetilde T^_k\}$.\ Repeat:* - Solve the two subproblems with $\{T_k = \widetilde T_k\}$ and $\{T_k' = T-\widetilde T_k\}$ as input: $\triangleright$ Solve Problem ${\bf (P1)}$: : - calculate $\{W_k\}$ and $\{W^{'}_k\}$ using ; : - obtain $\{\xi_k'\}$ by substituting $\{t_k'=T_k'\}$ and $\{W_k, W^{'}_k\}$ into ; $\triangleright$ Solve Problem ${\bf (P3)}$: : - calculate $\{B_k\}$ and solve $\nu$ using Algorithm \[Algorithm:solve\_nu\] by involving : the initialization method ; : - obtain $\{\xi_k\}$ by substituting $\{B_k\}$ and $\{t_k = T_k\}$ into ; - Update the time division: $\widetilde T_k= \widetilde T_k-\eta(\xi_k-\xi_k'),\quad \forall k \in \mathcal{K}$; Until $\\|\boldsymbol{\xi}'-\boldsymbol{\xi}\\|^2\leq\varepsilon$ (to equalize the energy-time rates), where $\boldsymbol{\xi}'\triangleq [\xi_1',\cdots,\xi_K']^{\top}$ and $\boldsymbol{\xi}\triangleq [\xi_1,\cdots,\xi_K]^{\top}$. (Low-Complexity Algorithm). The proposed Algorithm 2 is of low-complexity, which has complexity up to $O(K\log^2\frac{1}{\varepsilon})$ as compared to that of the conventional BCD method for solving ${\bf P4}$, which has complexity $O(\frac{K}{\varepsilon}\log\frac{1}{\varepsilon})$. Discussion on Energy-Learning Tradeoff -------------------------------------- In the literature, the model convergence of federated learning is characterized by either the averaged gradient norm (see e.g., [@signSGD]) or the loss function (see e.g., [@federatedlearning]), as a monotone decreasing function of the number of required rounds, $N$, and participating devices $K$. For example, it is reported in [@signSGD] that with a properly chosen learning rate (step size), the averaged gradient norm over rounds is shown to be proportional to $1/\sqrt{N}$ and $1/\sqrt{K}$. In existing work, the per-round latency, $T$, is assumed constant and thereby can be omitted in the convergence analysis. By considering $T$ or $N$ as a function of $\text{C}^2$ sum energy, denoted as $E_{\varSigma}$, we can discuss a learning-energy tradeoff as follows. (Learning Latency). The learning latency is defined as $T_{\sf total}=N \times T~(\text{in second})$. ### Fixed number of participating devices, $K$ This implies a fixed distributed dataset and thus fixed $N$. On the other hand, it can be inferred from Lemma \[lemma: energy-time rate\] that $T$ is a monotone decreasing function of $E_{\varSigma}$. Thus, the learning latency can be written as $T_{\text{total}}=N\times T(E_{\varSigma})$, which establishes the tradeoff between $T_{\text{total}}$ and $E_{\varSigma}$. Specifically, reducing $E_{\varSigma}$ can result in increased learning latency, and vice versa. Based on Lemma \[lemma: energy-time rate\], one can see that the reduction in the learning latency leads to the increment in sum energy at a rate faster than linear. ### Fixed per-round latency, $T$ For this case, an energy-learning tradeoff also exists. To be specific, it can be easily shown that the number of devices that can satisfy the per-round latency constraint is a monotone increasing function of $E_{\varSigma}$. Thereby, reducing $E_{\varSigma}$ can result in a reduced number of devices participating in learning. This gives rise to a larger required number of rounds for convergence due to the shrinking of distributed dataset. Thus, the learning latency can be written as $T_{\text{total}}=N(E_{\varSigma})\times T$ with $N(E_{\varSigma})$ being a monotone decreasing function. Then the energy-learning tradeoff is one that shortening learning latency needs more energy and vice versa. Particularly, the intermediate parameter, namely the number of participating devices, controls the said tradeoff in this case. This further motivates us to explore $\text{C}^2$ aware device scheduling in the following section. In general, the functions $T(E_{\varSigma})$ in the first case and $N(E_{\varSigma})$ in the second case have no closed form. The analysis of their properties (e.g., scaling laws) is an interesting topic but outside the scope of this work. $\text{C}^2$ Aware Scheduling {#section:device-scheduling} ============================= When there are many devices providing more than sufficient data, it is desirable from the energy-efficiency perspective to select only a subset of devices for participating in model training. In this section, the scheduler design is presented and the effect of scheduling on model convergence is quantified. Scheduler Design ---------------- The mathematical formulation of the design is similar to [P4]{} by modifying the objective as $\sum\limits_{k=1}^K\rho_k\left[\frac{C_kW_k^3+G_k{W'}_k^3}{{t'}_k^2}+\frac{B_kt_kN_0}{h_k^2}\left(2^{\frac{L}{B_kt_k}}-1\right)\right]$ under the additional constraints, namely $\sum\limits_{k=1}^K\rho_k=M,~\rho_k\in\{0,1\}, ~\forall k\in\mathcal{K}$, where $\{\rho_k\}$ are the indicator variables for selecting devices. This mixed integer non-linear problem is NP-hard and classical optimal algorithms (e.g., branch and bound) have too high complexity to be practical when $K$ is large. To tackle this challenge, we propose a novel $\text{C}^2$ aware scheduler design of low-complexity to minimize sum energy. We consider the problem of selecting $M$ out of $K$ devices for FEEL. The data distribution over devices is assumed i.i.d. as commonly make in the literature (see e.g., [@tran2019energy]). Then with $M$ fixed, device scheduling has no effect on the number of rounds $N$ required for learning (whose dependence on $M$ is studied in Section B). This allows scheduling decisions to be based on the $\text{C}^2$ states of devices, giving the name of $\text{C}^2$ aware scheduling. The key element of the scheduler is a scheduling metric designed as follows. Based on the preceding analysis, it is desirable to select devices with good channels or/and good computation efficiencies for energy reduction. To identify such devices, we propose to first perform equal bandwidth allocation over all devices (i.e., $B_k=\bar{B}=B/K,~\forall k\in\mathcal{K}$) and then evaluate the resulting energy consumption of each device. Note that equal bandwidth allocation enables the high energy consumption of a device to indicate: 1) a poor channel, or 2) a low computation efficiency, or 3) both. Therefore, the devices’ energy consumption with equal bandwidth allocation is a suitable scheduling metric given as: $$\label{Eq: Energy_Device_scheduling} E_k = \frac{\bar{B} t_k^N_0}{h_k^2}\left(2^{\frac{L}{\bar{B}t_k^}}-1\right) + \frac{a_kW^3}{(T-t_k^)^2},\quad\forall k\in\mathcal{K},$$ where $a_k\triangleq\frac{C_kG_k}{(\sqrt{C_k}+\sqrt{G_k})^2},\forall k\in\mathcal{K}$. To compute its value, the optimal transmission time $t^_k$ is needed given $B_k=\bar{B}\triangleq B/K$. Based on the energy-time rate equilibrium developed in Lemma \[lemma: energy-time rate\], $t_k^$ solves the equation below: $$\label{eqn: one-shot} f(t_k^) = \frac{\bar{B}N_0}{h_k^2}\left(2^{\frac{L}{\bar{B}t_k^}}-\frac{L\ln{2}}{\bar{B}t_k^}2^{\frac{L}{\bar{B}t_k^}}-1\right) + \frac{2a_kW^3}{(T-t_k^)^3} = 0.$$ A closed-form expression for $t_k^$ is hard to derive but can be numerically computed by a bi-section search since $f(t_k^)$ is a monotonically increasing function. Given the values of $\{E_k\}$, the $\text{C}^2$ aware scheduler selects $M$ devices with the smallest values. The scheduling scheme is summarized in Algorithm \[algorithm: one-shot scheduling\]. \[h\] Initialization: $\forall k\in\mathcal{K},~\rho_k=0,~\bar{B}=B/K$.\ Output: Subset $\mathcal{M}=\{k\in\mathcal{K}\mid \rho_k=1\}$. - Solve for the optimal time divisions $\{t_k^\}$ using and a bi-section search; - Calculate the scheduling metrics $\{E_k\}$ by substituting $\{t_k^\}$ into ; - Select $M$ devices with the smallest $E_k$ and set their $\rho_k=1$. Effect of Scheduling on Convergence ----------------------------------- In this subsection, we quantify the effect of $\text{C}^2$ aware scheduling (without considering data importance) on the convergence rate of FEEL (in round) due to the reduced size of selected dataset for model updating. For tractable analysis, we follow the standard assumptions on the loss function as made in the literature (see e.g., [@sqwang_2018_edge_learning]). \[assumption: convexity and smoothness\] It is assumed that the loss functions has the following two properties: - (Convexity and Smoothness). All functions $\{F_k\}$ are convex and $\beta$-smooth, that is, for all $\mathbf{u},\mathbf{v}$, we have $$\begin{aligned} \langle\nabla F_k(\mathbf{v}),\mathbf{u}-\mathbf{v}\rangle\leq F_k(\mathbf{u})-F_k(\mathbf{v})\leq \langle\nabla F_k(\mathbf{v}),\mathbf{u}-\mathbf{v}\rangle+\frac{\beta}{2}\\|\mathbf{u}-\mathbf{v}\\|^2 \end{aligned}$$ - (Variance Bound). Stochastic gradients $\{\nabla F_k(\mathbf{w})\}$ are unbiased and variance bounded by $\sigma^2$, that is, $$\begin{aligned} \mathbb{E}[\nabla F_k(\mathbf{w})]=\nabla F(\mathbf{w})\qquad \text{and}\qquad \mathbb{E}[\\|\nabla F_k(\mathbf{w})-\nabla F(\mathbf{w})\\|^2]\leq \sigma^2, \end{aligned}$$ where $\nabla F_k(\mathbf{w})$ and $\nabla F(\mathbf{w})$ denote the gradients of a local loss function and the global loss function, respectively, and the expectations are taken over all devices. Under the above assumptions, the upper bound on the convergence rate of FEEL algorithm with device scheduling is given in the following theorem. \[theorem: convergence rate\] (Convergence Rate with Scheduling). Given the learning rate as $\eta=\frac{1}{\sqrt{N}}$, the convergence rate of the FEEL algorithm with device-scheduling can be upper-bounded as $$\label{Learning Convergence Rate} \frac{1}{N}\sum_{i=0}^{N-1}\mathbb{E}_{\mathcal{M}_i}\left[F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right]\leq\frac{1}{\sqrt{N}}\left[\\|\mathbf{w}^{(0)}-\mathbf{w}^\\|^2+\frac{\sigma^2(K-M)}{(K-1)M}\right].$$ where $N$ denotes the number of communication rounds. See Appendix \[proof: convergence rate\]. Compared with the existing results without scheduling (see e.g., [@khaled2019analysis] and [@tran2019energy]), the last term in is new that characterizes the effect of scheduling on learning. One can observe from the term that increasing the number of scheduled devices $M$ leads to the vanishment of the biased term at a rate of $O(\frac{1}{M})$, giving rise to a faster convergence rate, however, at the cost of more sum energy consumption scales faster than linearly with $M$. Extension: Greedy Spectrum Sharing {#Extension} ================================== ![Spectrum holes exist due to heterogeneous $\text{C}^2$ time division.[]{data-label="Fig: Spectrum Holes"}](spectrum_hole_copy.pdf){width="11cm"} In the preceding sections, the bandwidths are allocated at the beginning of each round and then fixed throughout the round. However, as mentioned, the heterogeneity in the computation durations of devices may result in spectrum holes as illustrated in Fig. \[Fig: Spectrum Holes\]. Scavenging them by spectrum sharing among devices can reduce sum energy, which is the theme of this section. While optimizing the spectrum sharing is intractable, we design a practical scheme based on the greedy principle. The scheme allocates the spectrum hole upon its arrival to one of available devices for transmission (in addition to the pre-allocated bandwidth) so as to reduce its energy consumption, called greedy spectrum sharing. To design the scheme, we divide the round into multiple time slots. Furthermore, let $\mathcal{S}$ and $\mathcal{S_{\ell}}$ denote the sets of scheduled devices and a subset of devices that have completed computation at the beginning of a particular time slot, respectively. Then, the resulting spectrum holes have the total bandwidth of $B-\sum_{k\in\mathcal{S}_{\ell}}B_k^$ with $B_k^$ denoting the pre-allocated bandwidth to device $k$, which is determined at the beginning of each round. Intuitively, all the unoccupied bandwidths should be allocated to the device $k\in \mathcal{S_{\ell}}$ that has the largest energy reduction with respect to bandwidth growth. Mathematically, this selection metric is to select the device, denoted as $k^$, with the minimum energy-bandwidth rate at $B_k^$ provided $ \frac{\partial E_k^{\text{cmm}}}{\partial B_k} < 0,\forall k\in\mathcal{K}$, that is $$\begin{aligned} k^* = \arg\min_{k\in \mathcal S_{\ell}}\frac{\partial E_k^{\text{cmm}}}{\partial B_k}\bigg\|_{B_k^}.\end{aligned}$$ However, according to Lemma \[lemma: energy-bandwidth rate\], the energy-bandwidth rates of all scheduled devices are equalized at equilibrium, making this criterion ineffective. To address this issue, we design an alternative metric that can also effectively reduce sum energy. To this end, we introduce the following notion. \[definition: acceleration\] (Acceleration Rate). The acceleration rate is defined as the partial derivative of the energy-bandwidth rate: $$\label{definition: phi} \phi_k(B_k):=\frac{\partial^2E_k^{\text{cmm}}}{\partial B_k^2}=\frac{2^{\frac{L}{B_kt_k}}L^2N_0(\ln 2)^2}{B_k^3t_kh_k^2} > 0,\quad \forall k\in\mathcal{K}.$$* To be mathematical rigor, we note that the notion of acceleration rate can only be defined by the second derivative. However, it can be proved that the second-order partial derivative here is equivalent to the second derivative of energy w.r.t. the allocated bandwidth (see Appendix \[proof: greedy spectrum sharing\]). A device with a small acceleration rate implies a lower energy-bandwidth rate upon being allocated with extra bandwidths, leading to more energy reduction. Therefore, we propose to adopt the criterion of minimum acceleration rate at $\{B_k = B_k^\}$, namely $\{\phi_k(B_k^)\}$, as follows $$\begin{aligned} \label{Eq: device_slection} \text{(Device Selection)}\quad k^=\arg\min_{k\in\mathcal{S}_{\ell}}\phi_k(B_k^).\end{aligned}$$ Based on the above criterion, the scheme is summarized in Algorithm \[algorithm: greedy spectrum sharing\]. \[h\] Initialization: Apply Algorithm \[algorithm: one-shot scheduling\] and \[Algorithm:joint\_RM\] in sequential order to obtain $\{\rho_k^,B_k^\}$.\ Denote $\Delta t$ as the time slot duration and let $t_{\text{count}}=0$.\ For the subset of devices $\mathcal{S}=\{k\in\mathcal{K}\mid \rho_k=1\}$:\ While $t_{\text{count}}<T$: - Denote $\mathcal{S}_{\ell}$ as the set of devices that have completed local computation at time $t_{\text{count}}$; - For $k\notin \mathcal{S}_{\ell}$, no bandwidth will be occupied by them; - For $k\in\mathcal{S}_{\ell}$, each device will first be allocated with bandwidth $B_k^$; - Calculate $\phi_k$ for each $k\in\mathcal{S}_{\ell}$ using ; - Select the device via and allocate all the accessable spectrum $B-\sum_{k\in\mathcal{S}_{\ell}}B_k^$ to it; - $t_{\text{count}}=t_{\text{count}}+\Delta t$. Experimental results {#Simulation} ==================== Experiment Setup ---------------- The simulation settings are as follows unless specified otherwise. In the FEEL system, there are $K=50$ devices and each of them is capable of CPU-GPU heterogeneous computing. The devices’ CPU and GPU coefficients, $\{C_k\}$ and $\{G_k\}$, are uniformly selected from the the set $\{0.020,0.021,\cdots,0.040\}$ and $\{0.001,0.002,\cdots,0.010\}$, respectively. Consider an FDMA system with the uplink bandwidth $B=5$ MHz. The channel gains $\{h_k\}$ are modeled as i.i.d. Rayleigh fading with average path loss set as $10^{-3}$. The noise variance is $N_0=10^{-9}$ W/Hz. The classification task aims at classifying handwritten digits from the well-known MNIST dataset. Each device is randomly assigned $20$ samples. The classifier model is implemented using a 6-layer convolutional neural network (CNN) which consists of two $5\times 5$ convolution layers with ReLU activation, each followed by $2\times 2$ max pooling, a fully connected layer with 50 units and ReLU activation, and a final softmax output layer. The total number of parameters is $21,840$ and the computation workload is $W = 9.75$ MFLOPs. Furthermore, we suppose that each parameter of the training model gradient is quantized into 16 bits, and as a result, the transmission overhead is $L = 3.49\times 10^5$ bits. $\text{C}^2$ Resource Management -------------------------------- ### Energy-efficient RM The performance of the proposed computation RM, communication RM and joint $\text{C}^2$RM policies are evaluated by simulations. To be specific, we consider two settings: a) uniform time division \[see Fig. \[Fig: uniform time division\]\]; and b) optimal time divisions \[see Fig. \[Fig: optimal time division\]\]. Under each setting, we consider four schemes: 1) $\text{C}^2$RM, 2) only computation RM, 3) only communication RM, and 4) without RM. In particular, to distinguish our proposed optimal $\text{C}^2$RM policy, we name the policy with $\text{C}^2$RM under uniform time division setting as “separate $\text{C}^2$RM" while the proposed optimal one is called “joint $\text{C}^2$RM". The curves of the sum energy versus the per-round latency $T$ are illustrated in Fig. \[Fig: C2RM\]. Serveral observations can be made. First, the sum energy reduces as $T$ grows for all cases. This coincides with the results in Lemma \[lemma: energy-time rate\] that the energy consumption is a monotonically decreasing function in computation and communication time. Second, for either setting a) or b), it can be found that the two schemes 2) and 3), namely only computation RM and only communication RM, outperform the scheme without RM but underperform the $\text{C}^2$RM. For example, under setting a), they reduce the sum energy of the policy without RM by $13.8\%$ and $43.5\%$, respectively, for per-round latency equal to $1.0$ s. Meanwhile, the separate $\text{C}^2$RM scheme reduces the sum energy of the policy without RM by $57.3\%$. These results demonstrate the effectiveness of our proposed workload and bandwidth allocation policies for $\text{C}^2$RM. Third, comparing the results in setting a) with b), we find that the strategy of $\text{C}^2$ time division plays a significant role in energy efficiency. For example, the four schemes 1)-4) in b) with optimal time divisions reduce the sum energy of those in a) by $17.2\%$, $47.5\%$, $9.8\%$, and $43.5\%$, respectively, for per-round latency equal to $1.0$ s. This coincides with Remark 4 that the optimal $\text{C}^2$ time division achieves the best energy efficiency by balancing $\text{C}^2$ heterogeneity. Fourth, our proposed joint $\text{C}^2$RM policy outperforms all other schemes, showing its effectiveness. For example, it reduces the sum energy of the schemes 1)-4) in a) by $17.2\%$, $59.0\%$, $37.4\%$, and $64.6\%$, as well as the schemes 2)-4) in b) by $21.9\%$, $31.3\%$, and $37.4\%$, respectively, for per-round latency equal to $1.0$ s. ![Comparison between joint $\text{C}^2$RM with and without greedy spectrum sharing. The lines show sum energy vs. uplink bandwidth with fixed one round time $T=1$ s.[]{data-label="Fig: spectrum sharing"}](spectrum_sharing.pdf){width="55.00000%"} ### Greedy spectrum sharing The performance of the proposed greedy spectrum sharing algorithm in Algorithm \[algorithm: greedy spectrum sharing\] is benchmarked against the optimal $\text{C}^2$RM without spectrum sharing in Algorithm \[Algorithm:joint\_RM\]. Note that the number of devices are set to be $K=20$ with high heterogeneity regarding their computation efficiencies and channels. The curves of sum energy versus the bandwidth $B$ are plotted in Fig. \[Fig: spectrum sharing\]. Two observations can be made. First, the sum energy reduces as $B$ grows as more bandwidths can be traded for lower transmission power. Second, it can be found that the proposed greedy spectrum sharing policy improves the energy efficiency of the $\text{C}^2$RM without spectrum sharing by scavenging unused radio resources. For example, it reduces sum energy of the baseline scheme, namely optimal $\text{C}^2$RM without spectrum sharing by $10.6\%$ for uplink bandwidth equal to $2.5$ MHz. Device Scheduling ----------------- Consider the scenario that the edge server schedules a subset of devices for learning. The communication round number is fixed as $10$ with $T=1$ s for each round. The performance of the proposed $\text{C}^2$-aware scheduling is benchmarked against the random selection scheme. The effects of the number of scheduled devices $M$ on the average learning accuracy of the FEEL algorithm and sum energy consumption are shown in Fig. \[Fig:user\_selection\]. Several observations can be made as follows. First, the average learning accuracy is an increasing function of $M$ as it increases the training data per round. Second, it can be observed that the increase on $M$ leads to the growth of sum energy at a rate faster than linear, which agrees with the preceding discussion. Furthermore, one can observe that the proposed scheduling scheme outperforms the baseline which randomly selects $M$ devices, e.g., achieving $39.8\%$ sum energy reduction for $M=35$. ![The comparison of the energy efficiencies between $\text{C}^2$ aware scheduling and random selection is shown by solid lines. The relationship between test accuracy and the number of scheduled devices per round is illustrated by the dash line.[]{data-label="Fig:user_selection"}](scheduling_update.pdf){width="55.00000%"} Proof of Lemma \[lemma: computation condition\] {#proof: computation condition} ----------------------------------------------- Consider two cases for the second constraint in Problem [P2]{} as follows. First, we consider the case that $\frac{W_k}{f_k}\geq \frac{W_k'}{f_k'}, k\in\mathcal{K}$. Then, we know ${t'}_k=\frac{W_k}{f_k}$. Therefore, we have $f_k=\frac{W_k}{{t'}_k}$ and $f_k'\geq \frac{W_k'}{{t'}_k}$. It follows that the computation energy consumption is $$\begin{aligned} E_k^{\text{cmp}}=C_kW_kf_k^2+G_kW_k'f_k'^2\geq C_kW_k\frac{W_k^2}{{t'}_k^2}+G_kW_k'\frac{W_k'^2}{{t'}_k^2}.\end{aligned}$$ The equality holds if and only if $f_k'=\frac{W_k'}{{t'}_k}$, meaning $\frac{W_k}{f_k}=\frac{W_k'}{f_k'}$. This is also true for the case $\frac{W_k}{f_k}\leq \frac{W_k'}{f_k'}$ due to its symmetric form. Proof of Lemma \[lemma: energy-load rate\] {#proof: energy-load rate} ------------------------------------------ The computation energy of device $k$ can be expressed as $$\begin{aligned} E_k^{\text{cmp}}=E_k^{\text{CPU}}(W_k,{t'}_k)+E_k^{\text{GPU}}(W_k',{t'}_k),\quad k\in\mathcal{K}.\end{aligned}$$ The partial Lagrangian function is defined as $$\begin{aligned} &\mathcal{L}(\{W_k\},\{W'_k\},\{\gamma_k\},\{\lambda_k\},\{\theta_k\})\\ =&\sum_{k=1}^K\left[ E_k^{\text{CPU}}(W_k,{t'}_k)+E_k^{\text{GPU}}(W_k',{t'}_k)+\gamma_k(W_k+{W'}_k-W)-\lambda_kW_k-\theta_k{W'}_k\right], \end{aligned}$$ where $\{\lambda_k\geq 0\}$ and $\{\theta_k\geq 0\}$ are the Lagrange multipliers. Then we have the conditions: $$\begin{aligned} \frac{\partial \mathcal{L}}{\partial W_k}=\frac{\partial E_k^{\text{CPU}}}{\partial W_k}+\gamma_k-\lambda_k=0,\quad \frac{\partial \mathcal{L}}{\partial {W'}_k}=\frac{\partial E_k^{\text{GPU}}}{\partial {W'}_k}+\gamma_k-\theta_k=0,\quad\forall k\in\mathcal{K}.\end{aligned}$$ The complementary slackness conditions give that $$\label{eqn: muW=0 and nuW=0} \lambda_kW_k=0,\qquad \theta_k{W'}_k=0,\qquad \forall k\in\mathcal{K}.$$ Since at least one processor should be active. Assume ${W'}_k>0$ and then we have $\theta_k=0$ from the second condition in . Since the multiplier $\lambda_k$ is required to be $\lambda_k\geq 0$, we know that $$\begin{aligned} \label{RR46} \lambda_k=\frac{\partial E_k^{\text{CPU}}}{\partial W_k}-\frac{\partial E_k^{\text{GPU}}}{\partial {W'}_k}\geq 0,\quad\forall k\in\mathcal{K}.\end{aligned}$$ Substituting results of Lemma \[lemma: energy-load rate\] into (\[RR46\]), we have $W_k\geq\sqrt{\frac{G_k}{C_k}}{W'}_k > 0,~\forall k\in\mathcal{K}.$ Accordingly, we have $W_k>0$ so that $\lambda_k=0$. Therefore, it is optimal for both processors to be active with the energy-workload rate equilibrium as stated in Lemma \[lemma: energy-load rate\]. Proof of Lemma \[lemma: energy-bandwidth rate\] {#proof: energy-bandwidth rate} ----------------------------------------------- Substituting $t_k^* = T_k$ into , the communication energy of device $k$ can thus be expressed as the function of allocated bandwidths, i.e., $E_k^{\text{cmm}}(B_k)$. By introducing Lagrange multipliers $\{\mu_k^\}$ for the inequality constraints $\{B_k\geq 0\}$ and a scalar multiplier $\nu^$ for the equality constraint $\sum_{k=1}^KB_k=B$, the Lagrangian function is defined as $$\mathcal{L}(\{B_k\},\{\mu_k\},\nu)=\sum_{k=1}^K\left[ E_k^{\text{cmm}}(B_k)+\mu_kB_k\right]+\nu\left(\sum_{k=1}^KB_k-B\right),$$ Then we have the following conditions: $$\begin{aligned} \label{eqns:KKT} B_k^\geq 0,\quad \sum_{k=1}^KB_k^=B,\quad \mu_k^\geq 0,\quad \mu_k^B_k^=0, \quad \frac{\partial E_k^{\text{cmm}}}{\partial B_k}\bigg\|_{B_k^}-\mu_k^+\nu^ = 0, \quad \forall k\in\mathcal{K}.\end{aligned}$$ The feasible requirement gives $B_k^>0,\forall k\!\in\!\mathcal{K}$, so that $\mu_k^\!=\!0$. Therefore, $\frac{\partial E_k^{\text{cmm}}}{\partial B_k}\Big\|_{B_k^}\!\!\!=\!-\nu^,~\forall k\!\in\!\mathcal{K}.$ Proof of Lemma \[Pre:updating\_rule\] {#proof: algorithm 1} ------------------------------------- To begin with, we introduce several basic properties: 1) $B_k$ is a decreasing convex function of $\nu$ with the form in Lemma \[lemma: communication bandwidth allocation\]; 2) $\nu_k$ is a decreasing and strictly convex function of $B_k$ with the form in ; 3) $\nu_k(B_k)>0$ for all $k$ and all value of $B_k>0$. The proofs are straightforward and omitted for brevity. Assume the point $\nu^{(i-1)}>\nu^$, it follows from property 1) that $B_k(\nu^{(i-1)})<B_k^$, $\forall k\in\mathcal{K}$. Denote $B_k^{(i)}\triangleq B_k(\nu^{(i-1)})$, and it holds that $\sum_{k=1}^KB_k^{(i)}<\sum_{k=1}^KB_k^=B$. Therefore, $\text{sgn}\left(\sum_{k=1}^KB_k^{(i)}-B\right)=-1$ and $ \widetilde{B}_k^{(i)}=\frac{B}{\sum_{k=1}^KB_k^{(i)}}B_k^{(i)}>B_k^{(i)}.$ By property 2), we know that $\nu_k(\widetilde{B}_k^{(i)})<\nu_k(B_k^{(i)})=\nu^{(i-1)}$, $\forall k\in\mathcal{K}$. Then, we have $\max\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}<\nu^{(i-1)}$. Next, we prove by contradiction that $\max\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}>\nu^$ as follows. First, we assume that $\max\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}<\nu^$ and thus one can have $\nu_k(\widetilde{B}_k^{(i)})<\nu^,~\forall k\in\mathcal{K}$. Accordingly, we have $\tilde{B}_k^{(i)}>B_k^,~\forall k\in\mathcal{K}$, which implies that $\sum_{k=1}^K\widetilde{B}_k^{(i)}>\sum_{k=1}^KB_k^=B$. However, it is invalid as $ \sum_{k=1}^K\widetilde{B}_k^{(i)}=\sum_{k=1}^K\frac{B}{\sum_{k=1}^KB_k^{(i)}}B_k^{(i)}=\frac{B}{\sum_{k=1}^KB_k^{(i)}}\sum_{k=1}^KB_k^{(i)}=B.$ Thus the earlier assumption is false. Thereby, we have $\max\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}>\nu^$. Combining the results above, we have $\text{sgn}\left(\sum_{k=1}^KB_k^{(i)}-B\right)=-1$ and $\nu^<\max\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}<\nu^{(i-1)}$. Following the same procedure, we can derive for the case of $\nu^{(i-1)}<\nu^$, that $y = \text{sgn}\left(\sum_{k=1}^KB_k^{(i)}-B\right)=1$ and $\nu^{(i-1)}<\min\limits_{k}\{\nu_k(\widetilde{B}_k^{(i)})\}<\nu^$. Proof of Theorem \[theorem: convergence rate\] {#proof: convergence rate} ---------------------------------------------- At the beginning of the ($i+1$)-th round, all the devices receive the global model ${\mathbf{w}}^{(i)}$. However, the edge server only schedules $M$ devices for gradient computation, which makes the problem sophisticated to solve. To tackle the challenge, we use a trick that we assume all the devices compute the gradients $\{\nabla F_k({\mathbf{w}}^{(i)})\}$ based on their local datasets while only $M$ of them are aggregated for global model updating. This is equivalent to the learning process in our scenario. Then, the global model update rule can be written as $${\mathbf{w}}^{(i+1)}={\mathbf{w}}^{(i)}-\frac{\eta}{M}\sum_{k\in\mathcal{M}_i}\nabla F_k({\mathbf{w}}^{(i)})=\frac{1}{M}\sum_{k\in\mathcal{M}_i}\left({\mathbf{w}}^{(i)}-\eta\nabla F_k({\mathbf{w}}^{(i)})\right).$$ We define the virtual local updated model at device $k$ as $\mathbf{w}_k^{(i+1)}$ and denote it as $$\begin{aligned} \mathbf{w}_k^{(i+1)}={\mathbf{w}}^{(i)}-\eta\nabla F_k({\mathbf{w}}^{(i)}).\end{aligned}$$ As a result, the global update rule is equivalent to $${\mathbf{w}}^{(i+1)}=\frac{1}{M}\sum_{k\in\mathcal{M}_i}\mathbf{w}_k^{(i+1)}.$$ Next, we introduce another virtual sequence as the average model over all virtual local updates $$\bar{\mathbf{w}}^{(i+1)}=\frac{1}{K}\sum_{k=1}^K\mathbf{w}_k^{(i+1)}.$$ Accordingly, we have $$\label{RR53} \bar{\mathbf{w}}^{(i+1)}={\mathbf{w}}^{(i)}-\frac{\eta}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)}).$$ The averaged virtual model shifts at the end of one round is $$\begin{aligned} &\frac{1}{K}\sum_{k=1}^K\left\\|\mathbf{w}_k^{(i+1)}-\bar{\mathbf{w}}^{(i+1)}\right\\|^2=\frac{\eta^2}{K}\sum_{k=1}^K\left\\|\nabla F_k({\mathbf{w}}^{(i)})-\frac{1}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)})\right\\|^2\\ &=\frac{\eta^2}{K}\sum_{k=1}^K\left\\|\nabla F_k({\mathbf{w}}^{(i)})-\nabla F({\mathbf{w}}^{(i)})\right\\|^2=\eta^2\mathbb{E}\left[\left\\|\nabla F_k({\mathbf{w}}^{(i)})-\nabla F({\mathbf{w}}^{(i)})\right\\|^2\right]\leq \eta^2\sigma^2. \end{aligned}$$ We further notice that $\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2=\\|{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)}+\bar{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2$, and thus we have $$\begin{aligned} &\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2=\\|{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2+\\|\bar{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2+2\langle{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)},\bar{\mathbf{w}}^{(i)}-\mathbf{w}^\rangle\end{aligned}$$ Due to the fact $\mathbb{E}_{\mathcal{M}_i}\langle{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)},\bar{\mathbf{w}}^{(i)}-\mathbf{w}^\rangle=0$, we have $$\begin{aligned} \mathbb{E}_{\mathcal{M}_i}\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2=\mathbb{E}_{\mathcal{M}_i}\\|{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2+\mathbb{E}_{\mathcal{M}_i}\\|\bar{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2.\label{r1:3}\end{aligned}$$ The first term in (\[r1:3\]) can be decomposed into the following two terms. $$\label{r2:1} \begin{aligned} &\\|{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2=\left\\|\frac{1}{M}\sum_{k\in\mathcal{M}_i}\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\right\\|^2=\frac{1}{M^2}\left\\|\sum_{k\in\mathcal{M}_i}\left(\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\right)\right\\|^2\\ &=\frac{1}{M^2}\left(\sum_{k\in\mathcal{M}_i}\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2+\sum_{\substack{k,l\in\mathcal{M}_i\\k\neq l}}\langle \mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)},\mathbf{w}_l^{(i)}-\bar{\mathbf{w}}^{(i)}\rangle\right). \end{aligned}$$ Taking expectation on the first term in (\[r2:1\]), we have $$\label{r2:2} \begin{aligned} &\mathbb{E}_{\mathcal{M}_i}\left[\sum_{k\in\mathcal{M}_i}\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2\right]=\sum_{\substack{\mathcal{M}\subseteq\mathcal{K}\\\|\mathcal{M}\|=M}}{\rm Pr}(\mathcal{M}_i=\mathcal{M})\sum_{k\in\mathcal{M}_i}\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2\\ &=\sum_{\substack{\mathcal{M}\subseteq\mathcal{K}\\\|\mathcal{M}\|=M}}{\rm Pr}(\mathcal{M}_i=\mathcal{M})\sum_{k\in\mathcal{K}}{\rm Pr}(k\in\mathcal{M}_i)\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2\\ &=\frac{\binom{K-1}{M-1}}{\binom{K}{M}}\sum_{k\in\mathcal{K}}\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2=\frac{M}{K}\sum_{k=1}^K\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2. \end{aligned}$$ Taking expectation on the second term in (\[r2:1\]), we have $$\label{r2:3} \begin{aligned} &\mathbb{E}_{\mathcal{M}_i}\left[\sum_{\substack{k,l\in\mathcal{M}_i\\k\neq l}}\langle \mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)},\mathbf{w}_l^{(i)}-\bar{\mathbf{w}}^{(i)}\rangle\right]=\sum_{\substack{\mathcal{M}\subseteq\mathcal{K}\\\|\mathcal{M}\|=M}}{\rm Pr}(\mathcal{M}_i=\mathcal{M})\sum_{\substack{k,l\in\mathcal{M}_i\\k\neq l}}\langle \mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)},\mathbf{w}_l^{(i)}-\bar{\mathbf{w}}^{(i)}\rangle\\ &=\frac{M(M-1)}{K(K-1)}\left(\left\\|\sum_{k=1}^K(\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)})\right\\|^2-\sum_{k=1}^K\left\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\right\\|^2\right)=-\frac{M(M-1)}{K(K-1)}\sum_{k=1}^K\left\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\right\\|^2. \end{aligned}$$ Combining (\[r2:2\]) and (\[r2:3\]), we have the result as follow: $$\label{r7} \mathbb{E}_{\mathcal{M}_i}\\|{\mathbf{w}}^{(i)}-\bar{\mathbf{w}}^{(i)}\\|^2=\frac{K-M}{MK(K-1)}\sum_{k=1}^K\left\\|\mathbf{w}_k^{(i)}-\bar{\mathbf{w}}^{(i)}\right\\|^2\leq \frac{K-M}{M(K-1)}\times \eta^2\sigma^2.$$ Then, we consider the second term in (\[r1:3\]). According to the update rule (\[RR53\]), we know $$\label{r11} \begin{aligned} &\\|\bar{\mathbf{w}}^{(i+1)}-\mathbf{w}^\\|^2 =\left\\|{\mathbf{w}}^{(i)}-\frac{\eta}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)})-\mathbf{w}^\right\\|^2\\ &=\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2+\eta^2\left\\|\frac{1}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)})\right\\|^2-\frac{2\eta}{K}\sum_{k=1}^K\langle \nabla F_k({\mathbf{w}}^{(i)}), {\mathbf{w}}^{(i)}-\mathbf{w}^\rangle. \end{aligned}$$ According to the smoothness property in Assumption \[assumption: convexity and smoothness\], the second term in (\[r11\]) is bounded with $$\begin{aligned} \label{r12} \left\\|\frac{1}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)})\right\\|^2=\left\\|\nabla F({\mathbf{w}}^{(i)})\right\\|^2\leq 2\beta\left(F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right).\end{aligned}$$ According to the convexity property in Assumption \[assumption: convexity and smoothness\], we know that $$\begin{aligned} \langle \nabla F_k({\mathbf{w}}^{(i)}),\mathbf{w}^- {\mathbf{w}}^{(i)}\rangle\leq F_k(\mathbf{w}^)-F_k({\mathbf{w}}^{(i)}).\end{aligned}$$ Therefore, the third term in (\[r11\]) is bounded with $$\label{r15} -\frac{2\eta}{K}\sum_{k=1}^K\langle \nabla F_k({\mathbf{w}}^{(i)}), {\mathbf{w}}^{(i)}-\mathbf{w}^\rangle\leq\frac{2\eta}{K}\sum_{k=1}^K\left(F_k(\mathbf{w}^)-F_k({\mathbf{w}}^{(i)})\right)=2\eta\left(F(\mathbf{w}^)-F({\mathbf{w}}^{(i)})\right).$$ Combining the results in (\[r11\]), (\[r12\]) and (\[r15\]), we obtain that $$\label{r19} \begin{aligned} &\\|\bar{\mathbf{w}}^{(i+1)}-\mathbf{w}^\\|^2=\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2+\eta^2\left\\|\frac{1}{K}\sum_{k=1}^K\nabla F_k({\mathbf{w}}^{(i)})\right\\|^2-\frac{2\eta}{K}\sum_{k=1}^K\langle \nabla F_k({\mathbf{w}}^{(i)}), {\mathbf{w}}^{(i)}-\mathbf{w}^\rangle\\ &\leq \\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2+2\eta^2\beta\left(F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right)+2\eta\left(F(\mathbf{w}^)-F({\mathbf{w}}^{(i)})\right)\\ &=\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2-2\eta(1-\eta\beta)\left(F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right). \end{aligned}$$ Assume $\eta<\frac{1}{\beta}$, so that $1-\eta\beta>0$. In addition, due to $\mathbf{w}^=\arg\min_{\mathbf{w}}F(\mathbf{w})$, it is obvious that $F({\mathbf{w}}^{(i)})- F(\mathbf{w}^)\geq 0$. Combining all the results above, we obtain the upper bound for (\[r1:3\]) as $$\label{Rr64} \begin{aligned} &\mathbb{E}_{\mathcal{M}_{i}}\left[\\|{\mathbf{w}}^{(i+1)}-\mathbf{w}^\\|^2\right]=\mathbb{E}_{\mathcal{M}_{i}}\left[\\|{\mathbf{w}}^{(i+1)}-\bar{\mathbf{w}}^{(i+1)}\\|^2\right]+\mathbb{E}_{\mathcal{M}_{i}}\left[\\|\bar{\mathbf{w}}^{(i+1)}-\mathbf{w}^\\|^2\right]\\ &\leq \frac{\eta^2\sigma^2(K-M)}{M(K-1)}+\mathbb{E}_{\mathcal{M}_i}\left[\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2\right]-2\eta(1-\eta\beta)\mathbb{E}_{\mathcal{M}_i}\left[F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right]. \end{aligned}$$ To ease the notation, denote that $$\begin{aligned} a_{i}=\mathbb{E}_{\mathcal{M}_i}\left[\\|{\mathbf{w}}^{(i)}-\mathbf{w}^\\|^2\right],\quad b=\frac{\sigma^2(K-M)}{M(K-1)},\quad e_i=\mathbb{E}_{\mathcal{M}_i}\left[F({\mathbf{w}}^{(i)})-F(\mathbf{w}^)\right]\end{aligned}$$ Then becomes $$\begin{aligned} a_{i+1}\leq a_i+\eta^2b-2\eta(1-\eta\beta) e_i\label{r23}\end{aligned}$$ Re-write (\[r23\]) and arrange it in the following form: $$\begin{gathered} e_i\leq \frac{a_{i}- a_{i+1}+\eta^2b}{2\eta(1-\eta\beta)}.\end{gathered}$$ Thereby, the result in Theorem \[theorem: convergence rate\] can be derived after taking expectation over rounds: $$\frac{1}{N}\sum_{i=0}^{N-1}e_i \leq\frac{\sum_{i=0}^{N-1} (a_{i}- a_{i+1})}{2\eta(1-\eta\beta)N}+\frac{\eta b}{2(1-\eta\beta)}\leq\frac{a_{0}}{2\eta(1-\eta\beta)N}+\frac{\eta b}{2(1-\eta\beta)}.$$ If we further set the learning rate as $\eta=\frac{1}{\sqrt{N}}\leq\frac{1}{2\beta}$, we have $$\frac{1}{N}\sum_{i=0}^{N-1}e_i \leq\frac{a_{0}}{\eta N}+\eta b=\frac{1}{\sqrt{N}}\left[\\|\mathbf{w}^{(0)}-\mathbf{w}^*\\|^2+\frac{\sigma^2(K-M)}{(K-1)M}\right].$$ Clarification of Definition \[definition: acceleration\] {#proof: greedy spectrum sharing} -------------------------------------------------------- The energy consumption of device $k$ can be expressed as $$E_k=E_k^{\text{cmp}}(W_k,W_k',{t'}_k)+E_k^{\text{cmm}}(B_k,t_k)$$ with $W_k+W_k'=W$ and $t_k'+t_k=T$. Since $W_k$ and $W'_k$ are irrelevant to the allocated bandwidth $B_k$, the derivative of energy with respect to bandwidth can be calculated as follows: $$\begin{aligned} \frac{dE_k}{dB_k}&=\frac{\partial E_k^{\text{cmm}}}{\partial B_k}+\frac{\partial E_k^{\text{cmp}}}{\partial {t'}_k}\frac{d{t'}_k}{dB_k}+\frac{\partial E_k^{\text{cmm}}}{\partial t_k}\frac{dt_k}{dB_k}\\ &=\frac{\partial E_k^{\text{cmm}}}{\partial B_k}+\xi_k\left(\frac{d{t'}_k}{dB_k}+\frac{dt_k}{dB_k}\right)=\frac{\partial E_k^{\text{cmm}}}{\partial B_k}, \end{aligned}$$ where the energy-time rate equilibrium (see Lemma 5) gives $\frac{\partial E_k^{\text{cmp}}}{\partial {t'}_k}=\frac{\partial E_k^{\text{cmm}}}{\partial t_k}\triangleq \xi_k$, and the time constraint, namely ${t'}_k+t_k=T$, gives $\frac{d{t'}_k}{dB_k}+\frac{dt_k}{dB_k}=0$. Then, we can derive the second derivative of energy with respect to bandwidth as follows: $$\begin{aligned} \frac{d^2E_k}{dB_k^2}&=\frac{d}{dB_k}\frac{\partial E_k^{\text{cmm}}}{\partial B_k}=\frac{\partial^2E_k^{\text{cmm}}}{\partial B_k^2}+\frac{\partial^2E_k^{\text{cmp}}}{\partial B_k\partial t'_k}\frac{dt'_k}{dB_k}+\frac{\partial^2E_k^{\text{cmm}}}{\partial B_k\partial t_k}\frac{dt_k}{dB_k}\\ &=\frac{\partial^2E_k^{\text{cmm}}}{\partial B_k^2}+\frac{\partial}{\partial B_k}\left(\frac{\partial E_k^{\text{cmp}}}{\partial t'_k}-\frac{\partial E_k^{\text{cmm}}}{\partial t_k}\right)\frac{dt'_k}{dB_k}=\frac{\partial^2E_k^{\text{cmm}}}{\partial B_k^2}. \end{aligned}$$ [^1]: Q. Zeng, Y. Du, and K. Huang are with The University of Hong Kong, Hong Kong. K. K. Leung is with Imperial College London, UK. Contact: K. Huang (huangkb@eee.hku.hk). [^2]: To be mathematical rigor, we note that the notion of rate can only be defined by the derivative function. However, it can be proved that all the partial derivative functions appeared in this manuscript are equivalent to the derivative ones. Thereby, for consistency of notation, we define the rate with respect to partial derivative hereafter.
--- abstract: 'Barlowite Cu$_4$(OH)$_6$FBr shows three-dimensional (3D) long-range antiferromagnetism, which is fully suppressed in Cu$_3$Zn(OH)$_6$FBr with a kagome quantum spin liquid ground state. Here we report systematic studies on the evolution of magnetism in the Cu$_{4-x}$Zn$_x$(OH)$_{6}$FBr system as a function of $x$ to bridge the two limits of Cu$_4$(OH)$_6$FBr ($x$=0) and Cu$_3$Zn(OH)$_6$FBr ($x$=1). Neutron-diffraction measurements reveal a hexagonal-to-orthorhombic structural change with decreasing temperature in the $x$ = 0 sample. While confirming the 3D antiferromagnetic nature of low-temperature magnetism, the magnetic moments on some Cu$^{2+}$ sites on the kagome planes are found to be vanishingly small, suggesting strong frustration already exists in barlowite. Substitution of interlayer Cu$^{2+}$ with Zn$^{2+}$ with gradually increasing $x$ completely suppresses the bulk magnetic order at around $x$ = 0.4, but leaves a local secondary magnetic order up to $x\sim 0.8$ with a slight decrease in its transition temperature. The high-temperature magnetic susceptibility and specific heat measurements further suggest that the intrinsic magnetic properties of kagome spin liquid planes may already appear from $x>0.3$ samples. Our results reveal that the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr may be the long-thought experimental playground for the systematic investigations of the quantum phase transition from a long-range antiferromagnet to a topologically ordered quantum spin liquid.' author: - Zili Feng - Yuan Wei - Ran Liu - Dayu Yan - 'Yan-Cheng Wang' - Jianlin Luo - Anatoliy Senyshyn - Clarina dela Cruz - Wei Yi - 'Jia-Wei Mei' - Zi Yang Meng - Youguo Shi - Shiliang Li title: 'Effect of Zn doping on the antiferromagnetism in kagome Cu$_{4-x}$Zn$_x$(OH)$_6$FBr' --- introduction ============ A quantum spin liquid (QSL) can be briefly described as a symmetric state without magnetic order emerging from strong quantum fluctuations in frustrated magnetic systems [@BalentsL10; @NormanMR16; @SavaryL17]. The quantum fluctuations are usually enhanced by geometrical frustrations of magnetic ions, which are commonly seen in, e.g., triangle, kagome or pyrochlore lattices. Two-dimensional magnetic kagome lattices have attracted a lot of interests in the search for QSLs [@PLee2008]. Theoretically, it has been shown that the kagome system may exhibit various ordered state and different QSL ground states [@SachdevS92; @JiangHC08; @YanS11; @JiangHC12; @MessioL12; @PunkM14; @BieriS15; @IqbalY15; @KumarK15; @GongSS16; @LiaoHJ17; @MeiJW17; @YCWang2017a; @YCWang2017b], such as chiral and Z$_2$ QSL. These kagome QSLs are usually very close in energy [@MendelsP16] and depend sensitively on the particular form of the superexchange couplings, which render them difficult to be tested experimentally. Experimental progress in finding kagome QSLs has been substantial. Among many kagome magnets, herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ shows several promising properties of a QSL [@NormanMR16]. First of all, it consists of perfect kagome Cu$^{2+}$ ($s=1/2$) planes that show no magnetic order down to at least 20 mK [@ShoresMP05; @BertF07; @MendelsP07; @HeltonJS07]. Inelastic neutron scattering (INS) experiments display broad dispersionless magnetic excitations that are consistent with spinon continuum expected in QSLs [@HanTH12]. Later nuclear magnetic resonance (NMR) and INS experiments suggest that the system may be gapped [@FuM15; @HanTH16]. Interestingly, previous studies have suggested that herbertsmithite may be close to a quantum critical point (QCP) [@HeltonJS10]. However, it is later found that the low-energy spin excitations ($<$ 1 meV) are dominated by the so-called “impurities” of residual interlayer Cu$^{2+}$ ions due to imperfect substitution of inter-kagome Cu by Zn [@NilsenGJ13; @HanTH16]. Moreover, Cu$_4$(OH)$_6$Cl$_2$, the base material that leads to herbertsmithite, has at least four polymorphs with different nuclear structures that are all different from herbertsmithite and have different magnetic orders [@HawthorneFC85; @PariseJB86; @GriceJD96; @JamborJL96; @MalcherekT09; @ZhengXG05; @ZhengXG05b; @ZhengXG05c; @LeeSH07; @KimJH08; @WillsAS08]. INS experiments also do not support the presence of a QCP in the Zn$_x$Cu$_{4-x}$(OD)$_6$Cl$_2$ system since the antiferromagnetic (AF) order in the x = 0 sample becomes spin-glass-like with increasing x before the QSL is established in the x = 1 sample [@LeeSH07]. Recently, a new compound of Cu$_3$Zn(OH)$_6$FBr has been synthesized to exhibit properties that are consistent with a Z$_2$ QSL [@FengZL17; @WenXG17; @WeiY17]. This compound is obtained by substituting interlayer Cu$^{2+}$ in barlowite Cu$_4$(OH)$_6$FBr with nonmagnetic Zn$^{2+}$. The barlowite has perfect Cu$^{2+}$ kagome planes with an AF transition at about 15 K [@HanTH14; @JeschkeHO15; @HanTH16b]. Since the barlowite and Cu$_3$Zn(OH)$_6$FBr have the same space group for the crystal structures at room temperature, it may provide us a rare opportunity to study the quantum phase transition from an AF ordered state to a QSL ground state by continuously tuning Zn substitution level. Therefore, the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system may provide the long-thought experimental playground for the investigation of novel quantum phase transitions between symmetry-breaking phases and symmetric topologically ordered phases. In this paper, we systematically investigate the magnetic properties of the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system. Our results suggest that Zn can indeed be continuously doped into barlowite and suppress the long-range AF order. However, a hexagonal-to-orthorhombic structural change is observed in barlowite, and the low-temperature magnetic structure is directly associated with the orthorhombic structure. According to x-ray diffraction and magnetic susceptibility measurements, we find that the interlayer Cu$^{2+}$ ions result in lattice distortion and a local magnetic order up to $x$ = 0.82. However, the specific-heat measurements suggest that the bulk three-dimensional (3D) antiferromagnetic order should disappear around $x=0.4$ and the spin dynamics of QSL kagome planes may already start to evolve from $x >0.3$, which finally leads to a gapped QSL in Cu$_3$Zn(OH)$_6$FBr. Putting these information together, our comprehensive results suggest that the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system host rich physics of the interplay of frustration, antiferromagnetic order as well as the topologically ordered QSL. experiments =========== Powders of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr were synthesized by a hydrothermal method. We sealed powders of Cu$_2$(OH)$_2$CO$_3$ (1.5 mmol), NH$_4$F (1 mmol), ZnBr$_2$ ($x_{nom}$ mmol) and CuBr$_2$ (1-$x_{nom}$ mmol) in a 50-ml reaction vessel with half water, where $x_{nom}$ is the nominal Zn content. The vessel was slowly heated to 200 $^{\circ}$C and kept for 12 hours before cooling down to room temperature. Powders of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr were obtained by drying the products. The Zn content is determined by the inductively coupled plasma mass spectrometer (Thermo IRIS Intrepid II) with an uncertainty of about 4%. It should be noted for the Cu$_3$Zn(OH)$_6$FBr, the Zn content is determined to be about 0.92. We will use the actual Zn substitution level throughout the paper, so the $x$ = 0.92 sample is equivalent to Cu$_3$Zn(OH)$_6$FBr reported previously [@FengZL17; @WeiY17]. To obtain the deuterated samples, the mixture is changed to CuO, ZnF$_2$, ZnBr$_2$ and CuBr$_2$, and we use heavy water instead. The content of $H$ is less than 2% according to the NMR measurement. The magnetic susceptibility and heat capacity were measured by the magnetic property measurement system (MPMS) and physical property measurement system (PPMS, Quantum Design), respectively. The structures of the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system were measured at room temperature by the x-ray diffractometer (Rigaku Ultima IV) with Cu K$_{\alpha}$ radiation and a scintillation counter detector. The magnetic and nuclear structures of Cu$_4$(OD)$_6$FBr are determined by neutron-diffraction experiments performed on the SPODI diffractometer at FRM-II, Germany and the HB-2A diffractometer at HFIR, USA with wavelengths of 1.5483 Åand 2.4103 Å, respectively. results ======= Nuclear and magnetic structures in Cu$_4$(OH)$_6$FBr ---------------------------------------------------- ![Neutron powder diffraction intensities of Cu$_4$(OD)$_6$FBr (red dots) at (a) 300 K, (b) 20 K, and (c) 3.3 K measured at SPODI. The calculated intensities are shown by the black lines. Short vertical green and red lines represents nuclear and magnetic Bragg peak positions, respectively. The blue line shows the difference between measured and calculated intensities. The weighted profile R-factor ($R_{wp}$) is 4.22%, 4.46%, and 4.29% for (a) - (c), respectively. (d) Comparison between the calculated intensities of the Pnma (red) and Cmcm (blue) structures at 20 K. The labeled peaks are in the Pnma notation and cannot be indexed in the Cmcm structure. ](fig1) ![(a) Nuclear structure at 300 K (left) and 20 K (right) in the view vertical to the kagome plane. The diamond and rectangle are in-plane unit cells at 300 and 20 K, respectively. The subscripts of “H” and “O” denote hexagonal and orthorhombic structures, respectively. We note that the c axis at room temperature becomes the b axis at low temperatures. The three overlapping Cu atoms at 300 K represent three positions that the atom may actually occupy. (b) Detailed low-temperature structure showing only Cu (large blue circles) and O (small red circles) in the view parallel to orthorhombic b (left) and orthorhombic a (right), respectively. (c) Temperature dependence of the (1,0,2)$_O$ structural peak measured at HB-2A. The inset shows 2$\theta$ scans around this peak at 20 and 300 K. ](fig2) Figures 1(a)-1(c) show the neutron-diffraction and refinement results of Cu$_4$(OD)$_6$FBr at 300, 20 and 3.3 K, respectively [@supp1]. The refinement of the data at 3.3 K has to include an AF structure as discussed later in this section. The nuclear structure at room temperature is found to be the same as reported previously [@HanTH14] with parameters shown in Table I(a). At 20 K that is above $T_N$, the data can be well described by an orthorhombic structure (Pnma) as shown in Table I(b). Recently, an orthorhombic structure with a different space group Cmcm was reported for the single-crystal barlowite [@PascoCM18]. Figure 1(d) shows the comparison between the calculated results of these two structures at 20 K on our sample, where the Cmcm structure cannot describe the data. This discrepancy may come from the different methods used to prepare the samples. Figure 2(a) depicts the change in the structure in the view vertical to the kagome planes. According to the low-temperature structure, we label the three Cu$^{2+}$ ions in the kagome planes as Cu(1), Cu(2a), and Cu(2b), the interlayer Cu$^{2+}$ as Cu(3). At room temperature, the first three of them are symmetrically equivalent \[Cu1 in Table I(a)\] and forms an equilateral triangle. Cu(3) \[Cu2 in Table I(a)\] has three equivalent positions with an average position at the center of the triangle, as shown in the left panel of Fig. 2(a). The distribution of Cu(3) ions is presumably random. At 20 K, this random distribution is replaced by a regular pattern as shown in the right panel of Fig. 2(a) and the position of Cu3 in Table I(b). In the meantime, the positions of Cu(2a) and Cu(2b), which are symmetrically equivalent in the Pnma space group and labeled as Cu2 in Table I(b), are distorted both within and out of the kagome plane whereas that of Cu(1) \[Cu1 in Table I(b)\] remains unchanged. The new unit cell is defined according to the positions of the Cu(1) ions. [ccccccc]{} (a) & Site & x & y & z & B (Å$^2$)\ Cu1 & 6g & 0.00000 & 0.50000 & 0.00000 & 1.403(2)\ Cu2 & 6h & 0.36993 & 0.63007(15) & 0.750000 & 1.089(5)\ Br & 2c & 0.66667 & 0.33333 & 0.75000 & 1.788(3)\ F & 2b & 0.00000 & 0.00000 & 0.750000 & 2.137(4)\ O & 12k & 0.20234 & 0.79766(7) & 0.90850(7) & 1.271(2)\ D & 12k & 0.13470 & 0.87530(7) & 0.86677(8) & 2.235(2)\ \ (b) & Site & x & y & z & B (Å$^2$)\ Cu1 & 4a & 0.00000 & 0.00000 & 0.00000 & 0.784(3)\ Cu2 & 8d & 0.25076(21) & 0.51109(17) & 0.24514(28) & 0.728(2)\ Cu3 & 4c & 0.18593(19) & 0.25000 & 0.05766(24) & 0.856(3)\ Br & 4c & 0.33072(23) & 0.25000 & 0.50431(50) & 0.766(2)\ F & 4c & 0.49753(30) & 0.25000 & 0.00719(70) & 1.107(3)\ O1 & 8d & 0.29741(16) & 0.09595(23) & 0.00115(45) & 0.753(3)\ O2 & 8d & 0.10228(24) & 0.09210(28) & 0.19800(41) & 0.772(4)\ O3 & 8d & 0.40066(25) & 0.58780(26) & 0.30230(39) & 0.755(4)\ D1 & 8d & 0.37649(15) & 0.13516(26) & 0.00117(39) & 1.543(4)\ D2 & 8d & 0.06257(25) & 0.13019(26) & 0.31516(39) & 1.619(5)\ ![(a) The three magnetic peaks obtained by subtracting the 20-K data from the 3.3-K data. The solid lines are calculated results of three types of magnetic structures. The inset shows the refined results of raw data measured at SPODI. The subscript “M” denotes the magnetic structure. (b) Temperature dependence of (0,0,1)$_M$ (open red squares) and (1,0,1)$_M$ (filled blue circles) magnetic peaks measured at HB-2A. The solid lines are the fitted results as described in the main text. (c) Magnetic structure with the arrows indicating the sizes and directions of magnetic moments on Cu(3) ions (red and orange balls). The yellow and blue balls are Cu(1) and Cu(2) ions, respectively, whose structures cannot be determined as discussed in the main text. The dashed black lines indicate the magnetic unit cell with the origin shifted. ](fig3){width="\columnwidth"} Figure 2(b) gives the values of the Cu-O bond lengths and Cu-O-Cu angles at 20 K. According to the Goodenough-Kanamori rule [@GOODENOUGH1958; @KANAMORI1959], the nearest-neighbour superexchange changes from positive (antiferromagnetic) to negative (ferromagnetic) when the Cu-O-Cu angle goes through about 95$^{\circ}$ [@MizunoY98]. The Cu-O-Cu angles among Cu(3) and three other Cu$^{2+}$ have values both larger and smaller than 95$^{\circ}$ at 20 K, suggesting very complicated superexchange couplings. Figure 2(c) shows the temperature dependence of the intensity of the (1,0,2)$_O$ structural peak in the orthorhombic notation, which appears below about 270 K and slowly increases with decreasing temperature until about 200 K. As discussed above, the structural change from high temperature to low temperature is associated with a particular position chosen by Cu(3) from three equivalent positions. Since the position of Cu(3) is randomly distributed among these three positions at room temperature, it follows that Cu(3) ions should be able to resonate among them above 200 K so that their positions are not random any more at low temperatures. At the current stage due to the lack of thermodynamic evidence, we are unable to distinguish whether this structural change is a phase transition or rather a crossover resulting from increasing distortion with decreasing temperature. Assuming that there is no structural transition below 20 K, we find that the neutron-diffraction data at 3.3 K \[Fig. 1(c)\] can be refined by introducing an AF order. To be more explicit, the inset of Fig. 3(a) shows the neutron-diffraction data and the refinement results for the first three peaks with lowest angles. Two new peaks at (110)$_{O}$ and (001)$_{O}$ emerge at 3.3 K, which are forbidden in the orthorhombic structure. The intensity of the (101)$_O$ peak significantly increases from 20 to 3.3 K. The main panel of Fig. 3(a) gives the subtracted results, which clearly shows three magnetic peaks. Figure 3(b) shows the temperature dependence of the intensities of two peaks, which is consistent with the AF transition at about 15 K reported previously [@HanTH14], suggesting their magnetic origin. Within the statistics, only one magnetic transition is observed. With $T_N$ fixed at 15.5 K determined by the specific heat measurements as shown in the next section, the intensities of the (0,0,1)$_M$ and (1,0,1)$_M$ AF Bragg peaks can be fitted by $A(1-T/T_N)^{2\beta}$ with $\beta$ as 0.39 $\pm$ 0.06 and 0.33 $\pm$ 0.04, respectively. These values of the order parameter critical exponent are consistent with the 3D Heisenberg universality [@Campostrini2002]. With only three magnetic peaks observed, the magnetic structure cannot be unambiguously determined. Symmetry analysis of the different possible magnetic structures is performed using SARAh[@WillsAS00], which gives the propagation vector $k_{19}$ = (0,0,0). Representational analysis on three copper sites gives eight irreducible representations (IR’s), where IR-7 gives the best refinement results. After trying different models, the following conclusions can be made. First, the magnetic unit cell is the same as the orthorhombic one. Therefore, the positions of the (110)$_O$, (001)$_O$ and (101)$_O$ peaks in the inset of Fig. 3(a) are the same as those of the (110)$_M$, (001)$_M$ and (101)$_M$ peaks in the main panel of Fig. 3(a). Second, the moments on all copper sites are always confined within kagome planes, i.e., the ac plane in the orthorhombic structure. Third, the magnetic structures of the Cu(3) ions are always the same with the moment direction along the orthorhombic a axis, as shown in Fig. 3(c). The magnetic moment is about 0.66(7) $\mu_B$, which changes little in different models. Although the magnetic configurations of the Cu(1) and Cu(2) ions cannot be determined, we give three examples of refinement results, as shown by the solid lines in Fig. 3(a). In the M1 configuration, the moments at the Cu(1) and Cu(2) ions are fixed to be the same and their directions rotate simultaneously. The value of the moments is found to be about 0.182(35) $\mu_B$ with the moment direction about 30$^{\circ}$ away from the orthorhombic a axis. In the M2 configuration, the moment at the Cu(2) ions is set to zero. The value of the moment is about 0.39(22) $\mu_B$, but the direction is rather arbitrary. In the M3 configuration, the directions of the Cu(1) and Cu(2) moments are set to be along the orthorhombic a axis, the same as that of the Cu(3) moment. The values of the moments on the Cu(1) and Cu(2) ions are 0.329(76) $\mu_B$ and 0.079(37) $\mu_B$, respectively. Although M3 gives the best fit, it cannot be really distinguished from the other configurations within the error bars. In any case, the average moment of Cu(1) and Cu(2) is much smaller than that of Cu(3), which is why the magnetic structure of Cu(3) ions can be settled. Evolution of the antiferromagnetic order with Zn substitution ------------------------------------------------------------- ![(a) Room-temperature x-ray diffraction data of selected Cu$_{4-x}$Zn$_x$(OH)$_6$FBr samples. The intensity axis is plotted with the logarithmic scale. (b) Evolution of lattice constants with Zn substitution level $x$ at room temperature. We note that $a_H$ = b$_H$. (c) Doping dependence of the splitting $\Delta$ of Cu(3) from its average position as defined in the inset.](fig4){width="\columnwidth"} Figure 4(a) shows the room-temperature x-ray diffraction data of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr with different $x$’s from 0 to 0.92. Since there is no new peak appearing with Zn substitution, one can conclude that Zn can be continuously doped into barlowite. All the data can be refined by the hexagonal structure with the space group of $P6_3/mmc$. Figure 4(b) gives the substitution evolution of lattice constants $a_H$ and $c_H$, both of which change little with substitution. As discussed in the previous subsection, the interlayer Cu$^{2+}$ in barlowite has to be refined with three equivalent positions. One can define $\Delta$ as the distance from one of these positions to the center of their average positions as shown by the inset of Fig. 4(c). The value of $\Delta$ continuously decreases with increasing $x$ and becomes zero in the $x$ = 0.92 sample, as shown in Fig. 4(c). The Cu-O-Cu angle between Zn$^{2+}$ or Cu$^{2+}$ at the Cu(3) position and other Cu$^{2+}$ ions on kagome planes of the $x$ = 0.92 sample is 94.73$^{\circ}$ at 4 K according to our previous measurements [@WeiY17], which will give nearly zero superexchange couplings [@MizunoY98]. Therefore, it seems that the presence of residual interlayer Cu$^{2+}$ will not affect the spin dynamics of the kagome layers as far as its content is less than 10%. It should be noted the above analysis has assumed that Zn$^{2+}$ only substitutes interlayer Cu$^{2+}$. At the current stage, we cannot exclude the possibility that a few amounts of Zn$^{2+}$ may substitute Cu$^{2+}$ within the kagome planes [@OlariuA08; @FreedmanDE10; @FuM15]. ![(a) Low-temperature specific heat of Cu$_{4-x}$Zn(OH)$_6$FBr. All the samples were measured down to 2 K except for the x = 0.92 sample, which has already been reported previously [@FengZL17]. The arrows indicate the bulk magnetic transition temperature $T_N$, which is determined as the middle point of the drop of $\Delta C/T$ during the transition. The low-temperature hump for the x = 0.92 sample is indicated by the star symbol. (b) Temperature dependence of $\Delta S$ as defined in the main text. The red filled circles are results considering $C/T$ above 2 K, whereas the temperature in obtaining the data of the blue open squares is down to 50 mK. (c) Field dependence of magnetization at 2 K for selected samples at low fields. (d) Doping dependence of retentivity $M_R$. The solid line is an exponential fit. (e) Low-temperature magnetic moments of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr measured by the field-cooling process at 50 Oe. The solid lines are fitted results as described in the main text. The inset shows the data with the logarithmic scale. (f) Doping dependence of $T_N$ from specific heat ($T_N^C$, red squares) and susceptibility ($T_N^M$, blue circles) measurements. The $T_N$ of the $x$ = 0.92 sample is manually set as zero. The dashed lines are guides to the eye. ](fig5){width="\columnwidth"} The AF transition in barlowite can be observed in specific heat as shown in Fig. 5(a), which is similar as reported previously [@HanTH14]. In the x = 0.92 sample, a hump is found in $C/T$ as indicated by the star symbol in Fig. 5(a)[@FengZL17] , which has been attributed to the contribution from residual interlayer Cu$^{2+}$ in herbertsmithite[@HanTH16]. With decreasing Zn content, the hump does not disappear but moves to higher temperature with little change in the integrated area. On the other hand, starting from the x = 0 sample, the AF transition can be still seen in the $x$ = 0.12 sample, but becomes indistinguishable due to the presence of the hump at low temperatures for $x \geq 0.3$. We may estimate the contribution of this hump by analyzing the entropy change $\Delta S$ during the magnetic transition. The high-temperature data are fitted by a simple polynomial function $C_{bg} = \alpha T^2 + \beta T^3$ as performed previously [@HanTH14]. The range has been chosen from 30 K to about 5 K higher than $T_N$ for $x<$ 0.23 or 15 K for others. The magnetic part of the specific-heat $C_M$ associated with the AF order can be obtained by subtracting $C_{bg}$ from the raw data. It should be noted that a substantial magnetic contribution to the specific heat should be present above $T_N$ as discussed in the following subsection and the quadratic term in $C_{bg}$ may be associated with it [@HanTH14]. $\Delta S$ can thus be obtained by integrating $C_M/T$ from 0 to 20 K, assuming that $C_M/T (T = 0 K) = 0$, which is the case for either the AF ordered or the QSL ground state. However, we have only measured the specific heat down to 2 K for most of the samples as shown in Fig. 5(a). The above process will significantly underestimate the contribution below 2 K for samples with large $x$ since the hump temperature becomes lower than 2 K. We have measured the specific heat of the x = 0.92 sample down to 50 mK, which can give us a more precise value of $\Delta S$ as shown by the blue square in Fig. 5(b). A rough linear substitution dependence of $\Delta S$ from the low-substitution samples to the x = 0.92 sample is found, suggesting that the hump contribution of $\Delta S$ is rather independent of substitution. Therefore, the actual entropy change in $\Delta S$ in barlowite is just about 1.38 J/mol/K, which corresponds to 0.06 $k_B \ln2$ per Cu$^{2+}$, only about one-third of that reported previously [@HanTH14]. Figure 5(c) shows low-temperature field dependence of magnetization $M$ at 2 K. Ferromagnetic-like hysteresis can be found in all the samples except for the $x$ = 0.92 one. The substitution dependence of retentivity $M_R$, i.e., the magnetization at the zero field after the magnetic field is removed, is shown in Fig. 5(d), where $M_R$ decreases exponentially with increasing $x$. The presence of ferromagnetic like hysteresis most likely comes from the domains formed at low temperature due to the orthorhombic structure. The exponential decrease in $M_R$ suggests that either the energy required for overcoming the domain walls or the ferromagnetic-like component of the bulk order decreases quickly with increasing $x$. Figure 5(e) shows the temperature dependence of magnetic moment $M$ below 25 K at 50 Oe. The low-temperature signal decreases dramatically with increasing x, but we can still observe magnetic transitions for samples with $x$ up to 0.82. Fitting the data with $A(1-T/T_N)^\beta$ gives the substitution dependence of $T_N^{M}$ in Fig. 5(f), where $T_N^{M}$ only decreases slightly with increasing $x$ and suddenly become zero at $x$ = 0.92. On the other hand, $T_N^{C}$, obtained from the specific heat measurements in Fig. 5(a), decreases quickly with Zn substitution and may becomes zero around $x = 0.4$. It should be pointed out that this value is very subjective and since we cannot observe $T_N^c$ around x = 0.4. Further studies are thus needed to identify the exact substitution level. High-temperature and high-field properties of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr ------------------------------------------------------------------------- ![(a) Temperature dependence of $1/\chi$ for Cu$_{4-x}$Zn$_x$(OH)$_6$FBr. The dashed lines are fitted results between 150 and 300 K. (b) Doing dependence of the Curie temperature $\theta_{CW}$ (left axis) and Curie constant $C_{CW}$ (right axis). (c) Field dependence of magnetization at 2 K for selected samples at high fields. The dashed lines are linear fitted results from 30 to 50 KOe. (d) Doping dependence of slope B fitted from (c). (e) High-temperature specific heat of selected samples. (f) Doping dependence of $C(x)/C(0)$ at several temperatures.](fig6){width="\columnwidth"} Figure 6(a) shows the temperature dependence of $1/\chi$ of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr. The high-temperature data from 150 to 300 K can be fitted by the Curie-Weiss function as $\chi = C_{CW}/(T-\theta_{CW})$. The Curie constant $C_{CW}$ decreases linearly with increasing substitution, as shown in Fig. 6(b), which suggests that it is associated with the content of Cu$^{2+}$ ions. However, the absolute value of the Curie temperature $\theta_{CW}$ only starts increasing above $x$ = 0.3, which is consistent with what is observed in the Zn$_x$Cu$_{4-x}$(OH)$_6$Cl$_2$ system [@ShoresMP05]. This implies there is a substantial change in the nature of magnetic interactions in the system from $x>0.3$, and we believe it is related to the appearance of the magnetic properties of the kagome QSL plane from $x>0.3$ onwards. Figure 6(c) shows high-field magnetization at 2 K. The data between 30 and 50 KOe can be fitted by a linear function as $A + BH$. We note that similar measurements on the single crystal of barlowite have shown large anisotropy for fields parallel and perpendicular to the c axis, but the slopes in the above field range are rather the same [@HanTH16b]. Figure 6(d) shows substitution dependence of $B$, which starts decreasing above $x=0.3$. This is consistent with the strengthening of antiferromagnetic correlations as indicated by the increasing in $\|\theta_{CW}\|$. Figure 6(e) shows the specific heat data up to 150 K. Apart from low-temperature differences due to the presence of the AF order and hump \[Fig. 5(e)\], the high-temperature data also show different behaviors. Figure 6(f) plots the substitution dependence of $C(x)/C(0)$, where $C(x)$ and $C(0)$ are the specific heat with Zn substitution levels of $x$ and zero, respectively. This value roughly decreases monotonically with $x$ for $T >$ 60 K, which suggests that the high-temperature specific heat may be dominated by phonons. However, a dip is found at $x$ = 0.3 for those at lower temperatures, indicating that there is a contribution from the kagome QSL plane as suggested in Fig. 6(b) and 6(d). This is also consistent with the observation that only a small amount of entropy is involved during the magnetic transition in barlowite as shown in the previous section. discussions =========== Our results provide a comprehensive picture of the magnetic order in barlowite. The establishment of the AF order is directly associated with the structural distortion at high temperatures, which is the reason that the magnetic structure in Fig. 3 is different from the incorrectly proposed canted antiferromagnetic order based on the first-principles calculation [@JeschkeHO15] without knowing the structural change. It is interesting to note that three different magnetic structures have been proposed for the clinoatacamite Cu$_2$(OH)$_3$Cl [@LeeSH07; @KimJH08; @WillsAS08]. In our case, the magnetic structure in the M3 configuration is consistent with results from bulk measurements on barlowite. For example, the magnetic entropy associated with the magnetic transition is just about 0.06 $k_B \ln2$ per Cu$^{2+}$. In the simplest model, the magnetic entropy is proportional to $M^2$, i.e., the square of the magnetic moment [@Schwablbook]. Here the sum of the ordered moments of four Cu$^{2+}$ ions in the M3 configuration is 1.076 $\mu_B$, whereas the total moment is $4gS$ = 4.54 $\mu_B$, taking $g$ = 2.27 and $S$ = 1/2 [@HanTH14]. Therefore, the entropy release above the transition is about 5.6% of $k_B \ln2$, which is very close to the experimental value. Moreover, in the magnetic measurements on the single crystal, the hysteresis loop is only observed when H$\bot$c but not for H//c as shown in the single crystal measurements [@HanTH16b], which is consistent with our results that the magnetic/structural domains only present within the kagome planes. When the magnetic field is parallel to the c axis, a saturation moment of 0.29 $\mu_B$ per Cu is found [@HanTH16b]. Although it is attributed to full polarization of the interlayer Cu$^{2+}$, we find that this value is close to the sum of the ordered moments in the M3 configuration. The presence of interlayer Cu$^{2+}$ ions may result in local lattice distortions that will give rise to the magnetic order even when the Zn substitution level is high. This picture is consistent with our observation of magnetic order in magnetic measurements and low-field hysteresis up to $x$ = 0.82, almost substitution independent of $T_N^M$. It also coincides with the nuclear structure refinement results at room temperature, which suggests that the splitting of $\Delta$ of Cu(3) is not zero even when $x$ is as large as 0.82. However, the results from the specific-heat measurements provide another picture where the magnetic order may have already become zero for $x$ larger than 0.4. Since specific heat is a bulk property, it suggests that the magnetic order established at $T_N^M$ is just a secondary phase. This is consistent with our high-temperature and high-magnetic-field results, which suggest that the spin dynamics of the QSL kagome plane may indeed start to appear with Zn substitution level $x >$ 0.3. Based on these analyses, the magnetic properties in the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system can be divided into two parts, the one associated with the kagome planes (and thus bulk) and the one associated with interlayer Cu$^{2+}$ moments (and thus local). In very low substitution samples, the two parts are strongly coupled and cannot be separated, hence the 3D antiferromagnetic order is formed. With increasing Zn substitution, the bulk magnetic order is quickly suppressed and may disappear around $x \sim$ 0.4, but the local magnetic order persists up to $x$ = 0.82 without much change in its $T_N$. It is worth noting that in both the herbertsmithite and the x = 0.92 samples, the spin excitations can be indeed separated into these two independent parts [@NilsenGJ13; @HanTH16; @WeiY17]. The suppression of bulk magnetic order gives rise to two possible scenarios. In the first one, a magnetic QCP is present around $x \sim$ 0.4, which suggests that Cu$_{4-x}$Zn$_x$(OH)$_6$FBr may provide us the long-thought opportunity to study the quantum phase transition from a magnetic ordered state to a QSL state. On the other hand, it is also possible that the $Z_2$ QSL state in Cu$_3$Zn$_x$(OH)$_6$FBr is very robust against interlayer magnetic impurities so that it may persists up to very low Zn substitution. In this case, the disappearance of the bulk magnetic order may be associated with a first-order quantum phase transition or even phase separation between the 3D AF order and the QSL. It should be noted that since the x = 0 and x =0.92 samples have different structures at low temperatures, it will be interesting to see whether the quantum transition around x = 0.4 happens within the same nuclear structure or not. In any case, the physics is rich and interesting, and further studies are needed to clarify the situation. conclusions =========== Our systematical investigation on the Zn substitution effect on the antiferromagnetism in the kagome Cu$_{4-x}$Zn$_x$(OH)$_6$FBr system have revealed three major conclusions. First, the magnetic order in barlowite is associated with a hexagonal-to-orthorhombic structural change. Second, Zn substitution leads to local lattice distortion and may give rise to phase separation and result in a bulk magnetic order and a local magnetic order. Third, an evolution of spin dynamics on the kagome QSL planes may result in a quantum phase transition around $x=0.4$ between 3D AF order and QSL. Our results suggest that Cu$_{4-x}$Zn$_x$(OH)$_6$FBr is an interesting system and an experimental playground to investigate the intriguing physics of kagome antiferromagnets, and possibly realize the long-thought situation where quantum phase transition between symmetry-breaking and topologically ordered phases. Further works are definitely needed to explore the rich physics in these kagome compounds. This work was supported by the National Key R&D Program of China (Grants No. 2016YFA0300502 and No. 2017YFA0302900,2016YFA0300604), the National Natural Science Foundation of China (Grants No. 11874401 and No. 11674406, No. 11374346, No. 11774399, No. 11474330, No. 11421092, No. 11574359 and No. 11674370), the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grants No. XDB25000000 and No. XDB07020000, No. XDB28000000), China Academy of Engineering Physics (Grant No. 2015AB03) and the National Thousand-Young Talents Program of China. Research conducted at ORNL’s High Flux Isotope Reactor was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Z.F. and Y.W. contributed equally to this work. 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--- abstract: \| Several steps of stationary iterative methods serve as preconditioning for Krylov subspace methods for solving singular linear systems. Such an approach is called inner-iteration preconditioning in contrast to the Krylov subspace methods which are outer iterative solvers. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method and present theoretical and practical justifications for using this approach. Numerical experiments show that the generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) inner-iteration preconditioning are more robust and efficient compared to their standard preconditioners for some test problems of large sparse singular linear systems. Keywords: Preconditioner, Inner-outer iteration, GMRES method, Stationary iterative method, Singular linear system. AMS subject classifications: 65F08, 65F10, 65F20, 65F50. author: - 'Keiichi Morikuni[^1] [^2]' bibliography: - 'ref.bib' title: \| Inner-iteration preconditioned GMRES method\ for singular linear systems --- Introduction {#sec:intro} ============ Consider solving linear systems $$\begin{aligned} A \boldsymbol{x} = \boldsymbol{b}, \label{eq:Ax=b}\end{aligned}$$ where $A \in \mathbb{R}^{n \times n}$ may be singular and $\boldsymbol{b} \in \mathbb{R}^n$. For solving large sparse linear systems , iterative methods are preferred to direct methods in terms of efficiency and memory requirement. When the problem is ill-conditioned, the convergence of iterative methods such as Krylov subspace methods tends to be deteriorated and may be accelerated by using preconditioning. However, well-established preconditioners using incomplete matrix factorizations [@Saad1988], [@BaiDuffWathen2001], [@BaiYin2009] require additional memory whose amount is typically comparable to that of the given problem, and may not work in the singular case. Another approach for preconditioning Krylov subspace methods for solving linear systems is to use a splitting matrix of a stationary iterative method such as the successive overrelaxation (SOR) method [@Frankel1950], [@Young1950th]. Instead of one step, several steps of stationary iterative methods can serve as preconditioning for Krylov subspace methods. Such an approach is called the inner-iteration preconditioning. In the singular case, some iterative methods and preconditioners may be infeasible, i.e., they may break down and/or fail to converge. In this paper, we focus on using GMRES with preconditioning since the method is well-established and fairly well understood in the singular case [@Zhang2010], [@HayamiSugihara2011], [@EldenSimoncini2012]. GMRES applied to the linear system with initial iterate $\boldsymbol{x}_0 \in \mathbb{R}^n$ gives the $k$th iterate $\boldsymbol{x}_k$ such that $\\| \boldsymbol{b} - A \boldsymbol{x}_k \\| = \min_{\boldsymbol{x} \in \boldsymbol{x}_0 + \mathcal{K}_k (A, \boldsymbol{r}_0)} \\| \boldsymbol{b} - A \boldsymbol{x} \\|$, where $\\| \cdot \\|$ is the Euclidean norm, $\boldsymbol{r}_0 = \boldsymbol{b} - A \boldsymbol{x}_0$ is the initial residual, and $\mathcal{K}_k (A, \boldsymbol{r}_0) = \operatorname{span}\lbrace \boldsymbol{r}_0, A \boldsymbol{r}_0, \dots, A^{k-1} \boldsymbol{r}_0 \rbrace$ is the Krylov subspace of order $k$. Hereafter, denote $\mathcal{K}_k = \mathcal{K}_k (A, \boldsymbol{r}_0)$ for simplicity. In the singular case, GMRES may fail to determine a solution of . GMRES is said to break down at some step $k$ if $\dim A \mathcal{K}_k < \dim \mathcal{K}_k$ or $\dim \mathcal{K}_k < k$ [@BrownWalker1997 p. 38]. Note that $\dim A \mathcal{K}_k \leq \dim \mathcal{K}_k \leq k$ holds for each $k$. The dimensions of $A \mathcal{K}_k$ and $\mathcal{K}_k$ are related to the uniqueness of the iterate $\boldsymbol{x}_k$, whereas $\dim \mathcal{K}_k$ is related to the degeneracy of the Krylov subspace method. GMRES determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ without breakdown for all $\boldsymbol{b} \in \mathcal{R}(A)$ and for all $\boldsymbol{x}_0 \in \mathbb{R}^n$ if and only if $A$ is a group (GP) matrix $\mathcal{N}(A) \cap \mathcal{R}(A) = \lbrace \boldsymbol{0} \rbrace$ [@BrownWalker1997 Theorem 2.6], [@MorikuniHayami2015 Theorem 2.2], cf. [@Schneider2005th Theorem 4.4.6], where $\mathcal{N}(A)$ is the null space of $A$ and $\mathcal{R}(A)$ is the range space of $A$. The condition that $A$ is a GP matrix is equivalent to that the largest size of the Jordan block of $A$ corresponding to eigenvalue $0$ is not larger than one [@Oldenburger1940 section 3]. Other than Krylov subspace methods, much efforts have been made to study stationary iterative methods for solving singular linear systems (see [@Keller1965], [@MeyerPlemmons1977], [@Dax1990], [@BenziSzyld1997], [@Song1999], [@Yuan2000], [@SongWang2003], [@Cao2004], [@Wang2007a]). Some of modern stationary iterative methods were shown to be powerful used as matrix splitting preconditioning combined with Krylov subspace methods, and can be potentially useful as inner-iteration preconditioning. We shed some light on a preconditioning aspect of stationary iterative methods in the singular case. Consider applying GMRES to the preconditioned linear system $A P^{-1} \boldsymbol{u} = \boldsymbol{b}$, $\boldsymbol{x} = P^{-1} \boldsymbol{u}$, which is equivalent to $A \boldsymbol{x} = \boldsymbol{b}$, where $P$ is nonsingular and a preconditioning matrix given by inner iterations. The right-preconditioned GMRES (RP-GMRES) method with initial iterate $\boldsymbol{x}_0 \in \mathbb{R}^n$ determines the $k$th iterate $\boldsymbol{x}_k$ such that $\\| \boldsymbol{b} - A \boldsymbol{x}_k \\| = \min_{\boldsymbol{x} \in \boldsymbol{x}_0 + \mathcal{K}_k (P^{-1} A, P^{-1} \boldsymbol{r}_0)} \\| \boldsymbol{b} - A \boldsymbol{x} \\|$, where $\boldsymbol{u}_0 \in \mathbb{R}^n$ and $\boldsymbol{r}_0 = \boldsymbol{b} - A P^{-1} \boldsymbol{u}_0 = \boldsymbol{b} - A \boldsymbol{x}_0$. On the other hand, the flexible GMRES (FGMRES) method [@Saad1993] allows to change the preconditioning matrix for each outer iteration. This means that the number of the inner iterations may vary in GMRES. The rest of the paper is organized as follows. In section \[sec:iip\], we give sufficient conditions such that GMRES preconditioned by a fixed number of inner iterations determines a solution without breakdown, a spectral analysis of the inner-iteration preconditioned matrix, and a convergence bound of the method, and discuss a computational complexity of the method. In section \[sec:FGMRES\], we give sufficient conditions such that FGMRES preconditioned by inner iterations determines a solution without breakdowns in the singular case. In section \[sec:nuex\], we show numerical experiment results on test problems comparing the generalized shift-splitting (GSS) and Hermitian and skew-Hermitian (HSS) inner-iteration preconditioning with the GSS and HSS preconditioners, respectively. In section \[sec:conc\], we conclude the paper. GMRES preconditioned by a fixed number of inner iterations {#sec:iip} ========================================================== Consider applying a preconditioner using several steps of a stationary iterative method to RP-GMRES. We give its algorithm as follows (cf. [@DeLongOrtega1995], [@DeLongOrtega1998]). Let $\boldsymbol{x}_0 \in \mathbb{R}^n$ be the initial iterate. $\boldsymbol{r}_0 := \boldsymbol{b} - A \boldsymbol{x}_0$, $\beta := \\| \boldsymbol{r}_0 \\|$, $\boldsymbol{v}_1 := \boldsymbol{r}_0 / \beta$; Apply $\ell$ steps of a stationary iterative method to $A \boldsymbol{z} = \boldsymbol{v}_k$ to obtain $\boldsymbol{z}_k = C^{(\ell)} \boldsymbol{v}_k$; [ [$i = 1, 2, \dots, k$]{} [$h_{i, k} := (\boldsymbol{v}_i, \boldsymbol{w})$, $\boldsymbol{w} := \boldsymbol{w} - h_{i, k} \boldsymbol{v}_i$]{} ]{} [ [$h_{k+1, k} := \\| \boldsymbol{w} \\| = 0$]{} [set $m := k$ and go to line 7]{} [$\boldsymbol{v}_{k+1} := \boldsymbol{w} / h_{k+1, k}$;]{}]{} $\boldsymbol{y}_m := \mathrm{arg\,min}_{\boldsymbol{y} \in \mathbb{R}^m} \\| \beta \boldsymbol{e}_1 - H_{m+1, m} \boldsymbol{y} \\|$, $\boldsymbol{x}_m := \boldsymbol{x}_0 + [\boldsymbol{z}_1, \boldsymbol{z}_2, \dots, \boldsymbol{z}_m] \boldsymbol{y}_m$; Here, $C^{(\ell)}$ is the preconditioning matrix given by a fixed number $\ell$ of inner iterations, $\boldsymbol{e}_i$ is the $i$th column of the identity matrix, and $H_{m+1, m} = \lbrace h_{i, j} \rbrace \in \mathbb{R}^{(m+1) \times m}$. We next give an expression for the preconditioned matrix $A C^{(\ell)}$ for GMRES with $\ell$ inner iterations. Consider the stationary iterative method applied to $A \boldsymbol{z} = \boldsymbol{v}_k$ in lines 3. Note $\boldsymbol{v}_k \in \mathcal{R}(A)$ if $\boldsymbol{b} \in \mathcal{R}(A)$. Let $M$ be a nonsingular matrix such that $A = M - N$. Denote the iteration matrix by $H = M^{-1} N$. Assume that the initial iterate is $\boldsymbol{z}^{(0)} \in \mathcal{N}(H)$, e.g., $\boldsymbol{z}^{(0)} = \boldsymbol{0}$. Then, the $\ell$th iterate of the stationary iterative method is $\boldsymbol{z}^{(\ell)} = H \boldsymbol{z}^{(\ell - 1)} + M^{-1} \boldsymbol{v}_k = \sum_{i = 0}^{\ell - 1} H^i \! M^{-1} \boldsymbol{v}_k$, $\ell \in \mathbb{N}$. Hence, the inner-iteration preconditioning and preconditioned matrices are $C^{(\ell)} = \sum_{i = 0}^{\ell - 1} H^i \! M^{-1}$ and $A C^{(\ell)} = \sum_{i = 0}^{\ell - 1} H^i (\mathrm{I} - H) = M^{-1} \! (\mathrm{I} - H^\ell) M = \sum_{i_0}^{\ell-1} (\mathrm{I} - M^{-1} \! A)^i M^{-1} \! A$, respectively. We give sufficient conditions such that GMRES preconditioned by inner iterations determines a solution of without breakdown so that we prepare the following. \[prop:semiconv\] Let $H$ be a square real matrix. Then, $H$ is semiconvergent, i.e., $\lim_{i \rightarrow \infty} H^i$ exists, if and only if either $\lambda = 1$ is semisimple, i.e., the algebraic and geometric multiplicities corresponding to $\lambda = 1$ are equal, or $\| \lambda \| < 1$ holds for all $\lambda \in \sigma(H) = \lbrace \lambda \mid H \boldsymbol{v} = \lambda \boldsymbol{v}, \boldsymbol{v} \in \mathbb{C}^n \backslash \lbrace \boldsymbol{0} \rbrace \rbrace$ the spectrum of $H$. \[th:nonsing\_C\] If $H$ is semiconvergent, then $\sum_{i = 0}^{\ell - 1} H^i$ is nonsingular for all $\ell \in \mathbb{N}$. Proposition \[prop:semiconv\] shows that there exists a nonsingular matrix $S$ such that $J = S^{-1} \! H S = \tilde{J} \oplus \mathrm{I}$ is the Jordan canonical form (JCF) of $H$ with $\rho(\tilde{J}) < 1$ for $\tilde{J} \in \mathbb{R}^{r \times r}$, where $\oplus$ denotes the Kronecker sum and $\rho (A) = \max \lbrace \|\lambda\|: \lambda \in \sigma (A) \rbrace$ is the spectral radius of $A$. Hence, $\sum_{i = 0}^{\ell - 1} H^i = S \left\lbrace [ (\mathrm{I} - \tilde{J})^{-1} (\mathrm{I} - \tilde{J}^\ell) ] \oplus (\ell \mathrm{I}) \right\rbrace S^{-1}$ holds for all $\ell \in \mathbb{N}$. Since $1- \lambda^\ell \not = 0$ holds for all $\lambda \in \sigma(\tilde{J})$ and for all $\ell \in \mathbb{N}$, $\mathrm{I} - \tilde{J}^\ell$ is nonsingular and hence $\sum_{i = 0}^{\ell - 1} H^i$ is nonsingular for all $\ell \in \mathbb{N}$. \[th:ind1\_C\] If $H$ is semiconvergent, then $\mathrm{I} - H^\ell$ is a GP matrix for all $\ell \in \mathbb{N}$. If $\mathrm{O}$ is the zero matrix, then $\mathrm{I} - H^\ell = S [ (\mathrm{I} - \tilde{J}^\ell) \oplus \mathrm{O} ] S^{-1}$. Since $\mathrm{I} - \tilde{J}^\ell$ is nonsingular, $\mathrm{I} - H^\ell$ is a GP matrix for all $\ell \in \mathbb{N}$. Now we show that GMRES preconditioned by a fixed number of inner iterations determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$. \[th:iGMRES\] Assume that the inner-iteration matrix $H$ is semiconvergent. Then, GMRES preconditioned by inner iterations $C^{(\ell)}$ defined above determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ without breakdown for all $\boldsymbol{b} \in \mathcal{R}(A)$, for all $\boldsymbol{x}_0 \in \mathbb{R}^n$, and for all $\ell \in \mathbb{N}$. Since $\sum_{i = 0}^{\ell - 1} H^i$ is nonsingular for all $\ell \in \mathbb{N}$ from Lemma \[th:nonsing\_C\], $C^{(\ell)} = \sum_{i = 0}^{\ell - 1} H^i M^{-1}$ is nonsingular for all $\ell \in \mathbb{N}$. Hence, the preconditioned linear system $C^{(\ell)} \! A \boldsymbol{x} = C^{(\ell)} \boldsymbol{b}$ is equivalent to $A \boldsymbol{x} = \boldsymbol{b}$. Since $C^{(\ell)} \! A = \mathrm{I} - H^\ell$ is a GP matrix for all $\ell \in \mathbb{N}$ from Lemma \[th:ind1\_C\], the theorem follows from [@MorikuniHayami2015 Theorem 2.2]. This theorem gives [@MorikuniHayami2015 Theorem 4.6] as a corollary if the linear system is a symmetric and positive semidefinite linear system. Theorem \[th:iGMRES\] relies on the property that the preconditioned matrix is GP, which is implies by the semiconvergence of the inner-iteration matrix, irrespective of the property of $A$. Hence, we may extend the class of singular linear systems that GMRES can solve by combining with preconditioners. This means that even though $A$ is not a GP matrix, the inner-iteration preconditioned matrix is a GP matrix for $H$ semiconvergent, and GMRES preconditioned by the inner iterations determines a solution without breakdown (see Theorem \[th:GMRESanyind\] in Appendix). Semiconvergence is a simple and convenient property for deciding if a stationary iterative method is feasible as inner iterations for preconditioning GMRES in the singular case. Indeed, there are many stationary iterative methods whose iteration matrix can be semiconvergent. They are powerful when used as matrix splitting preconditioners for Krylov subspace methods, and potentially useful as inner-iteration preconditioning for GMRES such as the Jacobi, Gauss-Seidel, SOR, and symmetric SOR (SSOR) methods [@Dax1990], extrapolated methods [@Song1999], two-stage methods [@Wang2007a], the GSS method [@CaoMiao2016], and the HSS method and and its variants [@Bai2010], [@LiLiuPeng2012], [@ChenLiu2013], [@WuLi2014], [@YangWuXu2014]. Theorem \[th:iGMRES\] applies to some trivial examples. For example, if $A = L + D + L^\mathsf{T}$ is symmetric and positive semidefinite and the SOR splitting matrix is $M = \omega^{-1} (D + \omega L)$, where $D$ is diagonal, $L$ is strictly lower triangular, and $\omega \in \mathbb{R}$, then the SOR iteration matrix $H = M^{-1} N$ is semiconvergent for $\omega \in (0, 2)$ [@Dax1990 Theorem 13]. On the other hand, if an iteration matrix $H$ is semiconvergent, the extrapolated iteration matrix $(1 - \gamma) \mathrm{I} + \gamma H$ is also semiconvergent for $0 < \gamma < 2 / (1 + \nu (H))$ [@Song1999 Theorem 2.2]. We will recall conditions such that the GSS and HSS iteration matrices are semiconvergent in sections \[sec:GSS\] and \[sec:HSS\], respectively. Hence, these stationary iterative methods can serve as inner-iteration preconditioning for GMRES. Spectral analysis and convergence bound --------------------------------------- Next, consider decomposing GMRES preconditioned by inner iterations into the $\mathcal{R}(A C^{(\ell)}) = \mathcal{R}(A)$ and $\mathcal{R}(A)^\perp$ components to analyze the spectral property of the preconditioned matrix (cf. [@HayamiSugihara2011]). Assume that the inner-iteration matrix $H$ is semiconvergent throughout this subsection and $\boldsymbol{b} \in \mathcal{R}(A)$. Let $r = \operatorname{rank}\! A$, $Q_1 \in \mathbf{R}^{n \times r}$ such that $\mathcal{R}(Q_1) = \mathcal{R}(A)$, $Q_2 \in \mathbf{R}^{n \times (n-r)}$ such that $\mathcal{R}(Q_2) = \mathcal{R}(A)^\perp$, and $Q = \left[ Q_1, Q_2 \right]$ be orthogonal. Then, GMRES applied to $A C^{(\ell)} \boldsymbol{u} = \boldsymbol{b}$ can be seen as GMRES applied to $(Q^\mathsf{T} \! A C^{(\ell)} \! Q) Q^\mathsf{T} \! \boldsymbol{u} = Q^\mathsf{T} \! \boldsymbol{b}$, or $$\begin{aligned} \begin{bmatrix} Q_1^\mathsf{T} \! A C^{(\ell)} Q_1 & Q_1^\mathsf{T} \! A C^{(\ell)} Q_2\\ \mathrm{O} & \mathrm{O} \end{bmatrix} \begin{bmatrix} Q_1^\mathsf{T} \boldsymbol{u} \\ Q_2^\mathsf{T} \boldsymbol{u} \end{bmatrix} \equiv \begin{bmatrix} A_{11} & A_{12} \\ \mathrm{O} & \mathrm{O} \end{bmatrix} \begin{bmatrix} \boldsymbol{u}^1 \\ \boldsymbol{u}^2 \end{bmatrix} = \begin{bmatrix} Q_1^\mathsf{T} \boldsymbol{b} \\ Q_2^\mathsf{T} \! \boldsymbol{b} \end{bmatrix} \equiv \begin{bmatrix} \boldsymbol{b}^1 \\ \boldsymbol{0} \end{bmatrix}. \label{eq:decompeq}\end{aligned}$$ Assume that the initial iterate satisfies $\boldsymbol{u}_0 \in \mathcal{R}(A)$. Then, the $k$th iterate of GMRES applied to is given by $$\begin{aligned} Q^\mathsf{T} \boldsymbol{u}_k \equiv \begin{bmatrix} \boldsymbol{u}_k^1 \\ \boldsymbol{u}_k^2 \end{bmatrix} \in Q^\mathsf{T} \boldsymbol{u}_0 + Q^\mathsf{T} \mathcal{K}_k (A C^{(\ell)}, \boldsymbol{r}_0) = \begin{bmatrix} \boldsymbol{u}_0^1 \\ \boldsymbol{0} \end{bmatrix} + \mathcal{K}_k \left( \begin{bmatrix} A_{1 1} & A_{1 2} \\ \mathrm{O} & \mathrm{O} \end{bmatrix}, \begin{bmatrix} \boldsymbol{r}_0^1 \\ \boldsymbol{0} \end{bmatrix} \right)\end{aligned}$$ which minimizes $\\| Q^\mathsf{T} (\boldsymbol{b} - A C^{(\ell)} \boldsymbol{u}_k) \\|$, or $\boldsymbol{u}_k^1 \in \boldsymbol{u}_0^1 + \mathcal{K}_k (A_{1 1}, \boldsymbol{r}_0^1)$ which minimizes $\\| \boldsymbol{r}_k \\| = \\| \boldsymbol{b}^1 - A_{1 1} \boldsymbol{u}_k^1 \\|$. This means that $\boldsymbol{u}_k^1$ is equal to the $k$th iterate of GMRES applied to $A_{1 1} \boldsymbol{u}^1 = \boldsymbol{b}^1$ for all $k$ (cf. [@HayamiSugihara2011 section 2.5 ]). Now we give the spectrum of the preconditioned matrix $A C^{(\ell)}$ to present a convergence bound on GMRES preconditioned by inner iterations. Let $r = \operatorname{rank}A$. The $r$ nonzero eigenvalues of $A C^{(\ell)}$ are the eigenvalues of $A_{11}$, since $$\begin{aligned} \det \left(A C^{(\ell)} - \lambda \mathrm{I} \right) & = \det \left( \begin{bmatrix} A_{1 1} - \lambda {\operatorname{I}}_r & A_{1 2} \\ \mathrm{O} & \lambda {\operatorname{I}}_{n-r} \end{bmatrix} \right) = (-\lambda)^{n-r} \det (A_{11} - \lambda {\operatorname{I}}_r)\end{aligned}$$ and $A_{11}$ is nonsingular $\Longleftrightarrow$ $A C^{(\ell)}$ is a GP matrix [@HayamiSugihara2011 Theorem 2.3]. If $\mu$ is an eigenvalue of $H$, then $A C^{(\ell)} = M^{-1} (\mathrm{I} - H^\ell) M$ has an eigenvalue $\lambda = 1 - \mu^\ell$. From Proposition \[prop:semiconv\], $H$ has $r$ eigenvalues such that $\| \mu \| < 1$ and $n - r$ eigenvalues such that $\mu = 1$. For $\| \mu \| < 1$, we obtain $\|\lambda - 1\| = \| \mu \|^\ell \leq \nu(H)^\ell < 1$, where $\nu (H) = \max \lbrace \| \lambda \| : \lambda \in \sigma(H) \backslash \lbrace 1 \rbrace \rbrace$ is the pseudo spectral radius of $H$, i.e., the $r$ eigenvalues of $A C^{(\ell)}$ are in a disk with center at $1$ and radius $\nu(H)^{\ell} < 1$. For $\mu = 1$, we have $\lambda = 0$, i.e., the remaining $n - r$ eigenvalues are zero. \[th:bound\] Let $\boldsymbol{r}_k$ be the $k$th residual of GMRES preconditioned by $\ell$ inner iterations $C^{(\ell)}$ and $T$ be the Jordan basis of $A C^{(\ell)}$. Assume that $H$ is semi-convergent. Then, we have $\\| \boldsymbol{r}_k \\| \leq \kappa(T) \sum_{i=0}^{\tau(k, d)} \binom k i \rho(H)^{k \ell - i} \\| \boldsymbol{r}_0 \\|$ for all $\boldsymbol{x}_0 \in \mathcal{R}(C^{(\ell)} A)$ and for all $\boldsymbol{b} \in \mathcal{R}(A)$, where $\kappa(T) = \\| T \\| \\| T^{-1} \\|$, $d$ is the size of the largest Jordan block corresponding to a nonzero eigenvalue of $C^{(\ell)} A$, and $\tau(k, d) = \min(k, d-1)$. Theorem \[th:iGMRES\] ensures that GMRES preconditioned by inner iterations determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ without breakdown for all $\boldsymbol{b} \in \mathcal{R}(A)$ and for all $\boldsymbol{x}_0 \in \mathbf{R}^n$. From [@Bai2000 Theorem 1], we have $\\| \boldsymbol{r}_k \\| = \min_{p \in \mathbb{P}_k, p(0) = 1} \\| p(A C^{(\ell)}) \boldsymbol{r}_0 \\| \leq \kappa(T) \min_{p \in \mathbb{P}_k, p(0) = 1} \max_{1 \leq i \leq s} \\| p(J_i) \\| \\| \boldsymbol{r}_0 \\|$, where $\mathbb{P}_k$ is the set of all polynomials of degree not exceeding $k$ and $J_i$ is a Jordan block of $A C^{(\ell)}$ corresponding to a nonzero eigenvalue, $i = 1, 2, \dots, s$. The second factor is bounded above by $\min_{p \in \mathbb{P}_k, p(0) = 1} \max_{1 \leq i \leq s} \\| p(J_i) \\| \leq \sum_{i=0}^{\tau(k, d)} \binom k i \rho(H)^{k \ell - i}$ [@Bai2000 Theorems 2, 5]. Note that the residual $\\| \boldsymbol{r}_k \\|$ does not necessarily depend only on the eigenvalues of $A C^{(\ell)}$ when $\kappa(T)$ is large (see [@TebbensMeurant2014] and references therein). Computational complexity ------------------------ Compare GMRES for $k \ell$ outer iterations preconditioned by one inner iteration of a splitting method with that for $k$ outer iterations preconditioned by $\ell$ inner iterations of the same splitting method in terms of the Krylov subspaces for the iterate $\boldsymbol{x}_{k \ell}$, $\boldsymbol{x}_k$ and computational complexity, since they use the same total number of inner iterations. \[prop:spacedim\] If $C^{(\ell)}$ and $H$ are as defined above and $H$ is semiconvergent, then we have $$\begin{aligned} \mathcal{K}_k (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) \subseteq \mathcal{K}_{k \ell} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0).\end{aligned}$$ The proof is by induction. Let $\hat{A} = M^{-1} \! A$ and $\hat{\boldsymbol{r}}_0 = M^{-1} \boldsymbol{r}_0$. Consider the case $k = 1$. We have $\mathcal{K}_1 (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) = \operatorname{span}\lbrace C^{(\ell)} \boldsymbol{r}_0 \rbrace$ and $$\begin{aligned} C^{(\ell)} \boldsymbol{r}_0 = \sum_{i=0}^{\ell-1} (\mathrm{I} - M^{-1} \! A)^i M^{-1} \boldsymbol{r}_0 = \sum_{i=0}^{\ell-1} \sum_{j=0}^{i} \binom{i}{j} (- \hat{A})^j \hat{\boldsymbol{r}}_0.\end{aligned}$$ Since $C^{(\ell)} \boldsymbol{r}_0 \in \operatorname{span}\lbrace \hat{\boldsymbol{r}}_0, \hat{A} \hat{\boldsymbol{r}}_0, \dots, \hat{A}^{\ell-1} \hat{\boldsymbol{r}}_0 \rbrace $, we have $$\begin{aligned} \mathcal{K}_1 (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) \subseteq \mathcal{K}_\ell (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0).\end{aligned}$$ Next, assume that $\mathcal{K}_k (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) \subseteq \mathcal{K}_{k \ell} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0) = \mathcal{K}_{k\ell} (\hat{A}, \hat{\boldsymbol{r}}_0)$ holds. Then, $$\begin{aligned} \mathcal{K}_{k+1} (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) & = \mathcal{K}_k (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) + \operatorname{span}\lbrace (C^{(\ell)} \! A)^k C^{(\ell)} \boldsymbol{r}_0 \rbrace, \\ \mathcal{K}_{(k+1)\ell} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0) & = \mathcal{K}_{k\ell} (\hat{A}, \hat{\boldsymbol{r}}_0) + \mathcal{K}_k (\hat{A}, \hat{A}^{k \ell} \hat{\boldsymbol{r}}_0).\end{aligned}$$ From $C^{(\ell)} \! A = \sum_{i=0}^{\ell-1} (\mathrm{I} - \hat{A})^i \hat{A}$, we have $$\begin{aligned} (C^{(\ell)} \! A)^k C^{(\ell)} \boldsymbol{r}_0 & = \left(\sum_{i=0}^{\ell-1} (\mathrm{I} - \hat{A})^i \right)^{k+1} \! \hat{A}^k \hat{\boldsymbol{r}}_0 \in \mathcal{K}_{(k+1)\ell} (\hat{A}, \hat{\boldsymbol{r}}_0) \\ & = \mathcal{K}_{(k+1)\ell} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0).\end{aligned}$$ Hence, $\mathcal{K}_{k+1} (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) \subseteq \mathcal{K}_{k \ell} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0) = \mathcal{K}_{(k+1) \ell} (\hat{A}, \hat{\boldsymbol{r}}_0)$ holds. This proposition shows that GMRES preconditioned by one inner iteration gives an optimal Krylov subspace for the iterate, i.e., any Krylov subspace given by GMRES for $k$ outer iterations preconditioned by $\ell$ inner iterations is not larger than the one given by GMRES for $k \ell$ outer iterations preconditioned by one inner iteration. However, GMRES preconditioned by one inner iteration is not necessarily more efficient than GMRES preconditioned by more than one inner iteration, as will be seen in section \[sec:nuex\]. Indeed, while GMRES for $k \ell$ outer iterations preconditioned by one inner iteration requires $k \ell$ matrix-vector products of $A$ with $\boldsymbol{z}_k$ and $k \ell$ orthogonalizations, GMRES for $k$ outer iterations preconditioned by $\ell$ inner iterations requires $k$ matrix-vector products of $A$ with $\boldsymbol{z}_k$ and $k$ orthogonalizations. Hence, GMRES for $k \ell$ outer iterations preconditioned by one inner iteration needs more computations. Therefore, GMRES preconditioned by more than one inner iteration may be more efficient. Moreover, Proposition \[prop:spacedim\] gives a lower bound of the number of iterations of GMRES preconditioned by inner iterations which is required to determine a solution. Let $s$ be the smallest integer such that $\mathcal{K}_s (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0) < s$. Assume that GMRES preconditioned by $\ell$ inner iterations determines a solution at the $k$th step, where $k$ is the smallest integer such that $\mathcal{K}_k (C^{(\ell)} \! A, C^{(\ell)} \boldsymbol{r}_0) = \mathcal{K}_{s-1} (C^{(1)} \! A, C^{(1)} \boldsymbol{r}_0)$. Then, $k$ is larger than $(s-1) / \ell$. Hence, GMRES preconditioned by $\ell$ inner iterations requires more than $(s - 1) / \ell$ iterations to determine a solution. Flexible GMRES preconditioned by inner iterations. {#sec:FGMRES} ================================================== The preconditioners given in section \[sec:iip\] uses a fixed number of inner iterations. This approach allows a variable number of inner iterations for each outer iteration in line 3, Algorithm \[alg:RiGMRES\] (flexible GMRES (FGMRES) method [@Saad1993]). Let $C^{(\ell_k)}$ be the inner-iteration preconditioning matrix for the $k$th outer iteration. Then, the FGMRES iterate $\boldsymbol{x}_k^\mathrm{F}$ is determined over the space $\boldsymbol{x}_0 + \mathcal{R}(Z_k^\mathrm{F}) = \boldsymbol{x}_0 + \mathcal{R}([C^{(\ell_1)} \boldsymbol{v}_1^\mathrm{F}, C^{(\ell_2)} \boldsymbol{v}_2^\mathrm{F}, \dots, C^{(\ell_k)} \boldsymbol{v}_k^\mathrm{F}])$, which is no longer the Krylov subspace. Quantities denoted with superscript $\mathrm{F}$ are relevant to FGMRES hereafter. Hence, Theorem \[th:iGMRES\] does not apply to FGMRES preconditioned by inner iterations. Similarly to the breakdown of GMRES due to the linear dependence of $\boldsymbol{v}_{k+1}$ on $\boldsymbol{v}_1$, $\boldsymbol{v}_2$, …, $\boldsymbol{v}_k$, FGMRES may break down with $h_{k+1, k}^\mathrm{F} = 0$ due to the matrix-vector product $A C^{(\ell_k)} \boldsymbol{v}_k^\mathrm{F} = \boldsymbol{0}$, i.e., $\boldsymbol{v}_k^\mathrm{F} \in \mathcal{N}(A C^{(\ell_k)})$, in the singular case. If $C^{(\ell_k)}$ is nonsingular, then for $\boldsymbol{b} \in \mathcal{R}(A)$, $\boldsymbol{v}_k^\mathrm{F} \not = \boldsymbol{0}$ $\Longleftrightarrow$ $A C^{(\ell_k)} \boldsymbol{v}_k^\mathrm{F} \not = \boldsymbol{0}$ is equivalent to that $A C^{(\ell_k)}$ is a GP matrix, which is given by the iteration matrix $H$ semiconvergent. Notice that [@Saad1993 Proposition 2.2] holds irrespective of the nonsingularity of $A$: if $\boldsymbol{r}_0 \not = \boldsymbol{0}$, $h_{i+1, i}^\mathrm{F} \not = 0$ for $i = 1, 2, \dots, k-1$, and $H_k^\mathrm{F} = \lbrace h_{i, j}^\mathrm{F} \rbrace \in \mathbb{R}^{k \times k}$ is nonsingular, then $h_{k+1, k}^\mathrm{F} = 0$ is equivalent to that the FGMRES iterate $\boldsymbol{x}_k^\mathrm{F}$ is uniquely determined and is a solution of $A \boldsymbol{x} = \boldsymbol{b}$. Here, the nonsingularity of $H_k^\mathrm{F}$ is ensured by an additional assumption as follows. Let ${Q_k}^\mathsf{T} R_{k+1, k} = H_{k+1, k}^\mathrm{F}$ be the QR factorization of $H_{k+1, k}^\mathrm{F}$, where $Q_k$ is the product of Givens rotations $\Omega_k \Omega_{k-1} \cdots \Omega_1$ such as $\Omega_i = \mathrm{I}_{i-1} \oplus \left[ \begin{smallmatrix} c_i & s_i \\ -s_i & c_i \end{smallmatrix} \right] \oplus \mathrm{I}_{k-i}$ and $R_{k+1, k} \in \mathbb{R}^{(k+1) \times k}$ is upper triangular. The scalars $c_k$ and $s_k$ are chosen to satisfy $c_k^2 + s_k^2 = 1$ and to vanish the $(k+1, k)$ entry of $\Omega_{k-1} \cdots \Omega_1 H_{k+1, k}^\mathrm{F}$. It follows from [@Vuik1995 Lemma 4] that if $\\| \boldsymbol{v}_{k}^\mathrm{F} - A \boldsymbol{z}_k^\mathrm{F} \\| < \|c_{k-1}\|$ for $c_1 \not = 0$, $c_2 \not = 0$, …, $c_{k-1} \not = 0$ and $\boldsymbol{r}_k^\mathrm{F} \not = \boldsymbol{0}$, then $H_k$ is nonsingular. Thus, the following results in. \[th:FGMRES\] If the inner-iteration matrix $H$ defined above is semiconvergent and the inner iterations attain the residual norm $\\| \boldsymbol{v}_k^\mathrm{F} - A \boldsymbol{z}_k^\mathrm{F} \\| < \|c_k\|$ for the $k$th outer iteration, then FGMRES preconditioned by the inner iterations determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ for all $\boldsymbol{b} \in \mathcal{R}(A)$ and for all $\boldsymbol{x}_0 \in \mathbb{R}^n$. Numerical experiments {#sec:nuex} ===================== Numerical experiments on the discretized Stokes problem and artificially generated problems show the feasibility of GMRES and FGMRES preconditioned by inner iterations and the effectiveness of the former. These methods were compared with previous preconditioners in terms of the central processing unit (CPU) time. For instance of inner-iteration preconditioning, we used the generalized shift-splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) and their inexact variants. Although no condition such that GMRES preconditioned by a fixed number of inner iterations of an inexact splitting determines a solution without breakdown is given, we used the method for comparisons. The initial iterates for the inner stationary and outer GMRES and FGMRES iterations were set to zero. No restarts were used for these methods. The inner linear systems in FGMRES were approximately solved to the accuracy on the residual norm $\\| \boldsymbol{v}_k^\mathrm{F} - A \boldsymbol{z}_{k+1}^\mathrm{F} \\| < \|c_k\|$ to ensure that FGMRES determines a solution without breakdown (Theorem \[th:FGMRES\]). The stopping criterion used for (outer) GMRES and FGMRES iterations was in terms of the relative residual norm $\\| \boldsymbol{b} - A \boldsymbol{x}_k \\| \leq 10^{-6} \\| \boldsymbol{r}_0 \\|$. The computations were done on a computer with an Intel Xeon CPU E5-2670 2.50GHz, 256 GB random-access memory (RAM), and Community Enterprise Operating System (CentOS) Version 6.8. All programs for the iterative methods were coded and run in Matlab R2014b for double precision floating point arithmetic with unit roundoff $2^{-53} \simeq 1.1 \cdot 10^{-16}$. Generalized shifted splitting inner-iteration preconditionig. {#sec:GSS} ------------------------------------------------------------- We give numerical experiment results on singular saddle point problems $$\begin{aligned} A \boldsymbol{x} = \begin{bmatrix} C & B^\mathsf{T} \\ - B & \mathrm{O} \end{bmatrix} \boldsymbol{x} = \boldsymbol{b}, \quad B \in \mathbb{R}^{q \times p}, \quad C \in \mathbb{R}^{p \times p}~ \mbox{positive definite}, \label{eq:saddle}\end{aligned}$$ comparing GMRES preconditioned by $\ell$ inner iterations and FGMRES preconditioned by $\ell_k$ inner iterations of the generalized shifted splitting (GSS) $$\begin{aligned} \frac{1}{2} \begin{bmatrix} \alpha \mathrm{I} + C & B^\mathsf{T} \\ - B & \beta \mathrm{I} \end{bmatrix} \boldsymbol{z}^{(i+1)} = \frac{1}{2} \begin{bmatrix} \alpha \mathrm{I} & - B^\mathsf{T} \\ B & \beta \mathrm{I} \end{bmatrix} \boldsymbol{z}^{(i)} + \boldsymbol{d}, \quad i = 1, 2, \dots, \ell ~ \mbox{or} ~ \ell_k \label{eq:GSS}\end{aligned}$$ and its inexact variant (IGSS) with GMRES with the standard GSS and IGSS preconditioning $\ell = 1$ and a sparse direct solver, where $\ell$ is the number of inner GSS and IGSS iterations. Consider test problems of the form given by the Stokes problem $- \mu \mathrm{\Delta} \boldsymbol{u} + \nabla p = \boldsymbol{f}$, $\nabla \cdot \boldsymbol{u} = 0$ in an open domain $\Omega$ in $\mathbb{R}^2$ with the boundary and normalization conditions $\boldsymbol{u} = \boldsymbol{0}$ on $\partial \Omega$ and $\int_\Omega p(x) \mathrm{d} x = 0$, respectively, where $\mu$ is the kinematic viscosity constant, $\mathrm{\Delta}$ is the componentwise Laplace operator, the vector field $\boldsymbol{u}$ denotes the velocity, $\nabla$ and $\nabla \cdot$ denote the gradient and divergence operators, respectively, and the scalar function $p$ denotes the pressure. The Stokes problem was discretized upwind in square domain $\Omega = (0, 1) \times (0, 1)$ on uniform grid. Thus, the matrix representation of the Stokes problem is $C = (\mathrm{I}_q \otimes T + T \otimes \mathrm{I}_q) \oplus (\mathrm{I}_q \otimes T + T \otimes \mathrm{I}_q) \in \mathbb{R}^{2 q^2 \times 2 q^2}, B^\mathsf{T} = [\hat{B}^\mathsf{T}, \boldsymbol{b}_1, \boldsymbol{b}_2] \in \mathbb{R}^{2 q^2 \times (q^2 + 2)}, \hat{B} = [ (\mathrm{I}_q \otimes F)^\mathsf{T}, (F \otimes \mathrm{I}_q)^\mathsf{T} ] \in \mathbb{R}^{q^2 \times 2 q^2}, T = \mu h^{-2} \mathrm{tridiag} (-1, 2, -1) + (2 h)^{-1} \mathrm{tridiag} (-1, 1, 0) \in \mathbb{R}^{q \times q}, F = h^{-1} \mathrm{tridiag} (-1, 1, 0) \in \mathbb{R}^{q \times q}$, where $\otimes$ denotes the Kronecker product, $\boldsymbol{b}_1^\mathsf{T} = [\boldsymbol{e}^\mathsf{T}, \boldsymbol{0}^\mathsf{T}] \hat{B}$, $\boldsymbol{b}_2^\mathsf{T} = [\boldsymbol{0}^\mathsf{T}, \boldsymbol{e}^\mathsf{T}] \hat{B}$, $\boldsymbol{e} = [1, 1, \dots, 1]^\mathsf{T} \in \mathbb{R}^{q^2 / 2}$, and $h = (q+1)^{-1}$ is the discretization meshsize [@BaiGolubPan2004]. The (2,1) and (1,2) blocks were modified to be rank-deficient as done in [@ZhengBaiYang2009 section 5], [@CaoMiao2016 Example 4.1]. We chose three viscosity values $\mu = 10^{-5}$ and $1$ and three kinds of grids $16 \times 16$, $24 \times 24$, and $32 \times 32$. The right-hand side vector for was set to $\boldsymbol{b} = A \boldsymbol{e}$. The GSS inner-iteration matrix is semiconvergent for $\alpha$, $\beta > 0$ [@CaoMiao2016 Theorem 3.2] and GMRES preconditioned by the inner GSS iterations determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ without breakdown for , since $A$ is positive definite (Theorem \[th:iGMRES\]). On the other hand, the GMRES methods preconditioned by IGSS and its inner iterations are not guaranteed to determine a solution without breakdown. The value of $\beta$ for GSS and IGSS was set to $\\| B \\|^2 / \\| C \\|$ [@CaoLiYao2015]. The value of $\alpha$ for GSS and IGSS was experimentally determined to minimize the CPU time. The linear system was solved via [@CaoLiYao2015 Algorithm 2.1] by using the LU factorization for GSS and was solved by using GMRES with the stopping criterion $10^{-1}$ in terms of the relative residual norm for the inexact GSS (IGSS) inner-iterations preconditioning [@BaiGolubNg2008 Section 6]. Tables \[table:Stokes1\] and \[table:Stokes1e-5\] give the number of iterations and the CPU time in seconds for the Stokes problem with $\mu = 1$ and $10^{-5}$, respectively. Iter denotes the number of outer GMRES iterations and Time denotes the CPU time in seconds. GMRES, GSS, IGSS, F-GSS, F-IGSS, and mldivide denote GMRES with no preconditioning, GMRES preconditioned by the inner GSS iterations, its inexact variant, FGMRES preconditioned by the inner GSS iterations, its inexact variant, and the Matlab direct solver function mldivide, respectively. Hence, the CPU time for GMRES preconditioned by inner GSS iterations will improve with a sophisticated choice of the value of $\ell$. The least CPU time for each number of grids among the iterative methods is denoted by bold texts. The number of inner GSS and IGSS iterations was set to three throughout for simplicity, which is not necessarily optimal in terms of the CPU time. For example, GMRES preconditioned by six inner GSS iterations took 76.67 seconds to attain the stopping criterion for the problem with $\mu = 10^{-5}$ and $q = 36$. Table \[table:Stokes1\] shows that for well-conditioned problems $\mu = 1$, IGSS ($\ell = 1$) took the least CPU time to attain the stopping criterion among the iterative methods except for the small problem with grids $16 \times 16$. Table \[table:Stokes1e-5\] shows for ill-conditioned problems $\mu = 10^{-5}$, GSS ($\ell = 3$) took the least CPU time among the iterative methods. For small problems with grids $16 \times 16$ and $24 \times 24$, FGMRES took larger CPU time than other iterative methods. [l\|\[2]{}[r]{}\|\[2]{}[r]{}\|\[2]{}[r]{}]{} & & &\ & & &\ & & & & & &\ GMRES & 145 & 0.279 & 212 & 0.321 & 310 & 2.505\ GSS ($\ell = 1$) & 19 & 0.071 & 20 & 0.315 & 23 & 3.311\ GSS ($\ell = 3$) & 13 & 0.057* & 15 & 0.336 & 17 & 3.311\ IGSS ($\ell = 1$) & 18 & 0.136 & 19 & 0.258 & 21 & 1.088\ IGGS ($\ell = 3$) & 15 & 0.170 & 17 & 0.699 & 19 & 2.966\ F-GSS & 29 & 0.062 & 37 & 0.362 & 40 & 3.238\ F-IGSS & 29 & 0.395 & 38 & 0.739 & 39 & 4.132\ mldivide & & 0.011 & & 0.034 & & 0.066 \[table:Stokes1\] [l\|\[2]{}[r]{}\|\[2]{}[r]{}\|\[2]{}[r]{}]{} & & &\ & & &\ & & & & & &\ GMRES & 766 & 2.915 & 1,723 & 17.51 & 3,861 & 391.3\ GSS ($\ell = 1$) & 740 & 2.693 & 1,585 & 16.32 & 3,036 & 275.2\ GSS ($\ell = 3$) & 561 & 1.885* & 1,130 & 10.64 & 1,549 & 107.2\ IGSS ($\ell = 1$) & 748 & 5.668 & 1,587 & 42.62 & 3,026 & 406.6\ IGSS ($\ell = 3$) & 594 & 7.989 & 1,155 & 59.03 & 1,586 & 276.9\ F-GSS & 32 & 17.17 & 35 & 60.29 & 35 & 451.0\ F-IGSS & 33 & 295.5 & 37 & 500.8 & 37 & 4113.\ mldivide & & 0.011 & & 0.015 & & 0.029 \[table:Stokes1e-5\] Hermitian and skew-Hermitian splitting inner-iteration preconditioning. {#sec:HSS} ----------------------------------------------------------------------- We give numerical experiment results on singular and positive semidefinite linear system for with the generalized saddle point structure $$\begin{aligned} A \boldsymbol{x} = \begin{bmatrix} C & B \\ -B^\mathsf{T} & G \end{bmatrix} \boldsymbol{x} = \boldsymbol{b}, \label{eq:generalizedsaddlepoint}\end{aligned}$$ where $C \in \mathbb{R}^{p \times p}$ and $G \in \mathbb{R}^{q \times q}$ are symmetric and positive semidefinite, and $B \in \mathbb{R}^{p \times q}$. The results compare GMRES preconditioned by $\ell$ inner iterations and FGMRES preconditioned by $\ell_k$ inner iterations of the Hermitian and skew-Hermitian splitting (HSS) [@BaiGolubNg2003] $$\begin{aligned} \begin{cases} (\alpha \mathrm{I} + \mathcal{H}) \boldsymbol{z}^{(i+1/2)} = (\alpha \mathrm{I} - \mathcal{S}) \boldsymbol{z}^{(i)} + \boldsymbol{v}_k, \\ (\alpha \mathrm{I} + \mathcal{S}) \boldsymbol{z}^{(i+1)} = (\alpha \mathrm{I} - \mathcal{H}) \boldsymbol{z}^{(i+1/2)} + \boldsymbol{v}_k, \end{cases} \quad i = 1, 2, \dots, \ell ~ \mbox{or} ~ \ell_k \label{eq:HSS}\end{aligned}$$ and its inexact variant (IHSS) with GMRES with no preconditioning and the standard HSS and IHSS preconditioning $\ell = 1$ for and a sparse direct solver, where $\mathcal{H} = (A + A^\mathsf{T}) / 2$, $\mathcal{S} = (A - A^\mathsf{T}) / 2$, and $\alpha \in \mathbb{R}$. The former and latter systems of were solved by using the Cholesky and LU factorizations, respectively, for HSS, and by using the conjugate gradient (CG) method [@HestenesStiefel1952] and the LSQR method [@PaigeSaunders1982a], respectively, with the maximum number of iterations $n$, with the initial iterate equal to zero, and with the stopping criterion $10^{-1}$ in terms of the relative residual 2-norm for the inner IHSS iterations [@BaiGolubNg2008]. The maximum number of inner IHSS iterations for FGMRES was $n$. We generated test problems with the structure $$\begin{aligned} (U \oplus V)^\mathsf{T} \! A (U \oplus V) = \begin{bmatrix} \hat{C} \oplus \mathrm{O} & \hat{B} \oplus \mathrm{O} \\ - \hat{B}^\mathsf{T} \oplus \mathrm{O} & \hat{G} \oplus \mathrm{O} \end{bmatrix}, \quad \boldsymbol{b} = A [1, 2, \dots, n]^\mathsf{T}, \label{eq:structure}\end{aligned}$$ where $U \in \mathbb{R}^{p \times p}$ and $V \in \mathbb{R}^{q \times q}$ are orthogonal matrices such that $U^\mathsf{T} \! C U = \hat{C} \oplus \mathrm{O}$, $V^\mathsf{T} \! G V = \hat{G} \oplus \mathrm{O}$, and $U^\mathsf{T} B V = \hat{B} \oplus \mathrm{O}$. We set $U^\mathsf{T} \! C U = \operatorname{diag}(\varphi(1), \varphi(2), \dots, \varphi(p-q-1)) \oplus \mathrm{O} \in \mathbb{R}^{p \times p}$ for $\varphi(i) = \kappa^{i / (p - q - 1)}$, $j \in \mathbb{N}$, $V^\mathsf{T} \! G V = \operatorname{diag}(\psi(1), \psi(2), \dots, \psi(q-2)) \oplus \mathrm{O} \in \mathbb{R}^{q \times q}$ for $\psi(i) = \kappa^{i / (q - 1)}$, and $V^\mathsf{T} \! B^\mathsf{T} U = [ V^\mathsf{T} \! G^\mathsf{T} V, \mathrm{O} ] \in \mathbb{R}^{p \times q}$, where $\kappa = 10^{-j}$. We set $q = 16$, $32$, and $64$, $p = q^2$, nonzero density $0.1$% of $A$, and $j = 3$, $6$, and $9$ to show the effect of the condition number on the convergence. The value of $j$ determines the condition number of $A$ such as $\\| A \\| \\| A^\dag \\| = \sqrt{2} \times 10^j$, where $A^\dag$ is the pseudo inverse of $A$. The orthogonal matrices $U \in \mathbb{R}^{p \times p}$ and $V \in \mathbb{R}^{q \times q}$ were the products of random Givens rotations. Hence, the HSS iteration matrix $(\alpha \mathrm{I} + \mathcal{S})^{-1} (\alpha \mathrm{I} - \mathcal{H}) (\alpha \mathrm{I} + \mathcal{H})^{-1} (\alpha \mathrm{I} - \mathcal{S})$ is semiconvergent [@Bai2010 Theorem 3.6], and GMRES preconditioned by inner HSS iterations determines a solution of without breakdown (Theorem \[th:iGMRES\]). On the other hand, the GMRES methods preconditioned by IHSS and its inner iterations are not guaranteed to determine a solution without breakdown. Inner HSS and IHSS iterations involve two parameters: the inner-iteration parameter $\alpha$ and the number of inner HSS iterations $\ell$. Several techniques were proposed for estimating an optimal value of the HSS iteration parameter for the nonsingular case. As pointed out by a referee, parameter estimation techniques proposed in [@BaiGolubLi2006], [@Bai2009], [@Huang2014], [@Chen2015] are developed for the present case. Huang’s technique need not modify for the singular case. In Chen’s technique, the minimum eigenvalue of $\mathcal{H}$ and the minimum singular value of $\mathcal{S}$ were replaced by the nonzero ones. After the value of the inner iteration parameter $\alpha$ was determined, the number of inner iterations $\ell$ was determined by applying the inner HSS iterations alone to and adopted the smallest between 10 and the smallest number of $i$ which satisfies the relative difference norm $\\| \boldsymbol{z}^{(i-1)} - \boldsymbol{z}^{(i)} \\| < 10^{-1} \\| \boldsymbol{z}^{(i)} \\|$. The CPU times required by Huang’s technique to determine the values of the HSS iteration parameter were 0.001 seconds for $q = 16$, 0.002 seconds for $q = 32$, and 0.019 seconds for $q = 64$. Bai et al.’s technique [@BaiGolubLi2006] and Bai’s technique [@Bai2009] did not give more reasonable values of the HSS iteration parameter than Huang’s [@Huang2014] and Chen’s [@Chen2015] techniques for the test problems. Note that Bai’s technique [@Bai2009] is for the saddle-point problem instead of the generalized saddle-point problem , and does not take into account the $(2, 2)$ block of for the estimation. Tables \[tbl:sprad\_optpar3\]–\[tbl:sprad\_optpar9\] give the optimal and estimated values of the HSS iteration parameter and the value of the corresponding pseudo spectral radius of the HSS iteration matrix. The optimal value of the HSS iteration parameter $\alpha_\mathrm{exp}$ was experimentally determined to minimized the pseudo-spectral radius of the HSS iteration matrix. The values of the HSS iteration parameters which were estimated by using Huang’s and Chen’s techniques are denoted with subscript $\mathrm{C}$ and $\mathrm{H}$, respectively. Chen’s technique estimated the values of the parameter close to the optimal one of the parameter which were experimentally determined. Tables \[table:3\]–\[table:9\] give the number of the (outer) iterations and the CPU time in seconds for the test problems with different sizes and condition numbers. HSS, IHSS, HSS$^\prime$, IHSS$^\prime$, F-HSS$^\prime$, and F-IHSS$^\prime$ denote GMRES preconditioned by the HSS preconditioner, its inexact variant, GMRES preconditioned by the HSS inner iterations, its inexact variants, FGMRES preconditioned by inner HSS iterations, and its inexact variant, respectively. $\dag$ means that CG or LSQR for the inner linear systems did not attain the stopping criterion within $n$ iterations or the IHSS inner iterations did not satisfy $\\| \boldsymbol{v}_k^\mathrm{F} - A \boldsymbol{z}_k^\mathrm{F} \\| < \| c_k \|$ within $n$ iterations for the indicated number of outer iterations. $\ddag$ means that the Matlab direct solver mldivide function fails to give a solution, i.e., some of its entries are Not a Number (NaN). [c\|\[3]{}[r]{}]{} $q$ & & &\ $\alpha_\mathrm{exp}$ & `0.03162` & `0.03162` & `0.03162`\ $\nu (H(\alpha_\mathrm{exp}))$ & `0.93869` & `0.93869` & `0.93869`\ $\alpha_\mathrm{H}$ & `0.15678` & `0.08055` & `0.03295`\ $\nu (H(\alpha_\mathrm{H}))$ & `0.98732` & `0.97548` & `0.94109`\ $\alpha_\mathrm{C}$ & `0.03162` & `0.03162` & `0.03162`\ $\nu (H(\alpha_\mathrm{C}))$ & `0.93869` & `0.93869` & `0.93869` \[tbl:sprad\_optpar3\] [c\|\[3]{}[r]{}]{} $q$ & & &\ $\alpha_\mathrm{exp}$ & `0.00100` & `0.00100` & `0.00100`\ $\nu (H(\alpha_\mathrm{exp}))$ & `0.99800` & `0.99800` & `0.99800`\ $\alpha_\mathrm{H}$ & `0.14289` & `0.08068` & `0.03610`\ $\nu (H(\alpha_\mathrm{H}))$ & `0.99999` & `0.99998` & `0.99994`\ $\alpha_\mathrm{C}$ & `0.00100` & `0.00100` & `0.00100`\ $\nu (H(\alpha_\mathrm{C}))$ & `0.99800` & `0.99800` & `0.99800`\ \[tbl:sprad\_optpar6\] [c\|\[3]{}[r]{}]{} $q$ & & &\ $\alpha_\mathrm{exp}$ & `3.16e-5` & `3.16e-5` & `3.16e-5`\ $\nu (H(\alpha_\mathrm{exp}))$ & `0.99993` & `0.99993` & `0.99993`\ $\alpha_\mathrm{H}$ & `0.14102` & `0.13766` & `0.03878`\ $\nu (H(\alpha_\mathrm{H}))$ & `1.00000` & `1.00000` & `1.00000`\ $\alpha_\mathrm{C}$ & `3.16e-5` & `3.16e-5` & `3.16e-5`\ $\nu (H(\alpha_\mathrm{C}))$ & `0.99993` & `0.99993` & `0.99993`\ \[tbl:sprad\_optpar9\] [cl\|\[3]{}[r]{}\|\[3]{}[r]{}\|\[3]{}[r]{}]{} & & & &\ & & & & & & & & & &\ & GMRES & & 112 & 0.061 & & 159 & 0.154 & & 167 & 0.919\ & HSS & & 47 & 0.018 & & 68 & 0.049 & & 81 & 0.493\ & HSS$^\prime$ & 7 & 19 & 0.010 & 10 & 15 & 0.039 & 10 & 15 & 0.405\ & IHSS & & 52 & 0.089 & & 56 & 0.137 & & 70 & 0.834\ & IHSS$^\prime$ & 7 & 29 & $\dag$ & 10 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 19 & 0.014 & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 8 & 0.267 & & 11 & 0.349 & & 11 & 1.442\ & HSS & & 63 & 0.026 & & 62 & 0.043 & & 79 & 0.453\ & HSS$^\prime$ & 5 & 22 & 0.011 & 8 & 15 & 0.033 & 10 & 15 & 0.405\ & IHSS & & 65 & 0.083 & & 63 & 0.129 & & 64 & 0.808\ & IHSS$^\prime$ & 5 & 1 & $\dag$ & 8 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 35 & 0.017 & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 14 & 0.337 & & 15 & 0.425 & & 13 & 1.327\ & mldivide & & & 0.000 & & & 0.001 & & & 0.011 \[table:3\] [cl\|\[3]{}[r]{}\|\[3]{}[r]{}\|\[3]{}[r]{}]{} & & & &\ & & & & & & & & & &\ & GMRES & & 163 & 0.117 & & 464 & 1.136 & & 1,014 & 31.50\ & HSS & & 126 & 0.073 & & 209 & 0.262 & & 373 & 4.634\ & HSS$^\prime$ & 10 & 49 & 0.046* & 10 & 71 & 0.164 & 10 & 109 & 2.139\ & IHSS & & 130 & 0.517 & & 262 & 1.575 & & 738 & 36.84\ & IHSS$^\prime$ & 10 & 1 & $\dag$ & 10 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 1 & $\dag$ & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 4 & $\dag$ & & 2 & $\dag$ & & 1 & $\dag$\ & HSS & & 162 & 0.116 & & 436 & 0.999 & & 326 & 3.397\ & HSS$^\prime$ & 5 & 146 & 0.136 & 7 & 150 & 0.309 & 10 & 76 & 1.520\ & IHSS & & 163 & 0.214 & & 444 & 1.444 & & 362 & 6.001\ & IHSS$^\prime$ & 5 & 1 & $\dag$ & 7 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 32 & $\dag$ & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 6 & $\dag$ & & 11 & $\dag$ & & 12 & $\dag$\ & mldivide & & & 0.000 & & & 0.001 & & & 0.011 \[table:6\] [cl\|\[3]{}[r]{}\|\[3]{}[r]{}\|\[3]{}[r]{}]{} & & & &\ & & & & & & & & & &\ & GMRES & & 117 & 0.062 & & 353 & 0.672 & & 852 & 23.80\ & HSS & & 169 & 0.125 & & 511 & 1.333 & & 1,308 & 49.83\ & HSS$^\prime$ & 10 & 162 & 0.219 & 10 & 317 & 1.063 & 10 & 487 & 13.61\ & IHSS & & 107 & 0.717 & & 297 & 3.455 & & 938 & 76.97\ & IHSS$^\prime$ & 10 & 1 & $\dag$ & 10 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 1 & $\dag$ & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 1 & $\dag$ & & 1 & $\dag$ & & 1 & $\dag$\ & HSS & & 117 & 0.064* & & 348 & 0.659 & & 346 & 4.169\ & HSS$^\prime$ & 5 & 111 & 0.089 & 6 & 190 & 0.389 & 10 & 80 & 1.503\ & IHSS & & 117 & 0.135 & & 350 & 0.895 & & 381 & 6.628\ & IHSS$^\prime$ & 5 & 1 & $\dag$ & 6 & 1 & $\dag$ & 10 & 1 & $\dag$\ & F-HSS$^\prime$ & & 30 & $\dag$ & & 1 & $\dag$ & & 1 & $\dag$\ & F-IHSS$^\prime$ & & 9 & $\dag$ & & 8 & $\dag$ & & 12 & $\dag$\ & mldivide & & & 0.000 & & & 0.001 & & & 0.011 \[table:9\] HSS$^\prime$ took the least CPU time to attain the stopping criterion among the iterative methods except for the case $(j, q) = (9, 16)$. Bai and Chen’s techniques tended to give reasonable values of the HSS iteration parameter for well-conditioned or small problems such as the cases $(j, q) = (3, 16)$, $(3, 64)$, $(6, 16)$, $(6, 32)$, whereas Huang’s technique tended to give reasonable values of the HSS iteration parameter for ill-conditioned or large problems such as the cases $(j, q) = (6, 64)$, $(9, 16)$, $(9, 32)$, $(9, 64)$. Although F-IHSS$^\prime$ took the fewest numbers of iterations, it did not outperform other methods in terms of the CPU time. IHSS$^\prime$ did not converge for all test problems. The Matlab direct solver mldivide function failed to give a solution for the large cases $q = 64$, although it outperformed the iterative methods for the other cases, except for the case $j = 6$. Comparing Tables \[table:3\]–\[table:9\] with Tables \[tbl:sprad\_optpar3\]–\[tbl:sprad\_optpar9\], we see that these estimated optimal values of the HSS iteration parameter in terms of the pseudo spectral radius did not give optimal CPU time for HSS$^\prime$. This implies that a small pseudo-spectral radius does not necessarily gives a fast convergence of HSS, HSS$^\prime$, and IHSS (see also Theorem \[th:bound\]). Conclusions {#sec:conc} =========== We considered applying several steps of a stationary iterative methods as a preconditioner to GMRES and FGMRES for solving singular linear systems. We gave sufficient conditions such that GMRES and FGMRES preconditioned by inner iterations determine a solution without breakdown, and a convergence bound of GMRES preconditioned by inner iterations based on a spectral analysis. We presented a complexity issue of using inner-iteration preconditioning more than one step for GMRES. Numerical experiments showed that GMRES preconditioned by the inner GSS and HSS iterations is efficient compared to previous methods including FGMRES for large and ill-conditioned problems. Appendix {#appendix .unnumbered} ======== If $\operatorname{index}(A) = \min \lbrace d \in \mathbb{N}_0\| \operatorname{rank}A^d = \operatorname{rank}A^{d+1} \rbrace$, where $A^0= \mathrm{I}$ and $\mathrm{I}$ is the identity matrix [@CampbellMeyer1979 Definition 7.2.1], then $d \geq \operatorname{index}(A)$ is equivalent to $\mathcal{R}(A^d) \cap \mathcal{N}(A^d) = \lbrace \boldsymbol{0} \rbrace$ [@CampbellMeyer1979 p. 137]. The following theorem gives conditions such that GMRES determines a solution without breakdown for $\operatorname{index}(A) \geq 1$. \[th:GMRESanyind\] GMRES determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ without breakdown for all $\boldsymbol{b} \in \mathcal{R}(A^d)$ and for all $ \boldsymbol{x}_0 \in \mathcal{R}(A^{d - 1}) + \mathcal{N}(A)$ if and only if $d \geq \operatorname{index}(A)$. Assume $d \geq \operatorname{index}(A)$, or $\mathcal{R}(A^d) \cap \mathcal{N}(A^d) = \lbrace \boldsymbol{0} \rbrace$. Let $\boldsymbol{b} \in \mathcal{R}(A^d)$ and $\boldsymbol{x}_0 \in \mathcal{R}(A^{d - 1}) + \mathcal{N}(A)$. Then, $\boldsymbol{r}_0 \in \mathcal{R}(A^d)$ and $\mathcal{K}_k \subseteq \mathcal{R}(A^d)$. If $k$ is the smallest positive integer such that $\dim \! \mathcal{K}_k < k$, then GMRES determines a solution of $A \boldsymbol{x} = \boldsymbol{b}$ at step $k-1$ (see [@BrownWalker1997 Theorem 2.2]). Now, assume $\dim \! \mathcal{K}_i = i$, $1 \leq i \leq k$. Since $\mathcal{K}_{i+1} = \boldsymbol{r}_0 \cup A \mathcal{K}_i$, we have $\dim A \mathcal{K}_i = \dim \mathcal{K}_i = i$ for $i = 1, 2, \dots, k-1$. Let the columns of $V \in \mathbb{R}^{n \times k}$ form a basis of $\mathcal{K}_k$. If $\dim A \mathcal{K}_k < \dim \mathcal{K}_k$, then there exists $\boldsymbol{c} \not = \boldsymbol{0}$ such that $A V \boldsymbol{c} = \boldsymbol{0}$. Since $V \boldsymbol{c} \not = \boldsymbol{0}$ for $\boldsymbol{c} \not = \boldsymbol{0}$, we have $\mathcal{K}_k \cap \mathcal{N}(A) \not = \lbrace \boldsymbol{0} \rbrace$. From $\mathcal{K}_k \subseteq \mathcal{R}(A^d)$ and $\mathcal{N}(A) \subset \mathcal{N}(A^d)$, we have $\mathcal{R}(A^d) \cap \mathcal{N}(A^d) \not = \lbrace \boldsymbol{0} \rbrace$, which contradicts with $d \geq \operatorname{index}(A)$. Hence, $\dim A \mathcal{K}_k = \dim \mathcal{K}_k$ for all $k \in \mathbb{N}$. Since GMRES does not break down through rank deficiency of the least squares problem $\min_{\boldsymbol{z} \in \mathcal{K}_k} \\| \boldsymbol{r}_0 - A \boldsymbol{z} \\|$, the sufficiency is shown from [@BrownWalker1997 Theorem 2.2]. On the other hand, assume $d < \operatorname{index}(A)$. Then, $\mathcal{N}(A^d) \subset \mathcal{N}(A^{d + 1})$. There exits $\boldsymbol{s} \not = \boldsymbol{0}$ such that $\boldsymbol{s} \not \in \mathcal{N}(A^d)$ and $\boldsymbol{s} \in \mathcal{N}(A^{d + 1})$. Let $\boldsymbol{t} = A^d \boldsymbol{s}$. Then, $\boldsymbol{t} \not = \boldsymbol{0}$ and $A \boldsymbol{t} = A^{d + 1} \boldsymbol{s} = \boldsymbol{0}$. Hence, there exists $\boldsymbol{t} \not = \boldsymbol{0}$ such that $\boldsymbol{t} \in \mathcal{R}(A^d) \cap \mathcal{N}(A)$. Let $\boldsymbol{b} = \boldsymbol{t} + A \boldsymbol{x}_0$ for $\boldsymbol{x}_0 \in \mathcal{R}(A^{d - 1}) + \mathcal{N}(A)$. Then, $\boldsymbol{b} \in \mathcal{R}(A^d)$ and $\boldsymbol{r}_0 = \boldsymbol{b} - A \boldsymbol{x}_0 = \boldsymbol{t} \not = \boldsymbol{0}$. Since $A \boldsymbol{r}_0 = \boldsymbol{0}$, we have $\boldsymbol{r}_1 = \boldsymbol{b} - A \boldsymbol{x}_1 = \boldsymbol{b} - A (\boldsymbol{x}_0 + c \boldsymbol{r}_0) = \boldsymbol{r}_0 - c A \boldsymbol{r}_0 = \boldsymbol{r}_0 \not = \boldsymbol{0}$ for $c \in \mathbb{R}$ and $\dim A \mathcal{K}_1 = 0 < \dim \mathcal{K}_1 = 1$. Hence, GMRES breaks down at step 1 before determining a solution of $A \boldsymbol{x} = \boldsymbol{b}$. This theorem agrees with [@MorikuniHayami2015 Theorem 2.2] for $d = 1$. Although similar results to Theorem \[th:GMRESanyind\] were given in [@WeiWu2000], no attention was paid there in the uniqueness of the GMRES iterate $\boldsymbol{x}_k$, i.e., the dimensions of $A \mathcal{K}_k$ and $\mathcal{K}_k$. acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank Doctor Miroslav Rozložník, Doctor Akira Imakura, and the referees for their valuable comments. [^1]: `morikuni@cs.tsukuba.ac.jp` [^2]: Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba
--- abstract: 'In a neutron polarimetry experiment the mixed state relative phases between spin eigenstates are determined from the maxima and minima of measured intensity oscillations. We consider evolutions leading to purely geometric, purely dynamical and combined phases. It is experimentally demonstrated that the sum of the individually determined geometric and dynamical phases is not equal to the associated total phase which is obtained from a single measurement, unless the system is in a pure state.' author: - 'Jürgen Klepp$^{1}$, Stephan Sponar$^{1}$, Stefan Filipp$^{1}$, Matthias Lettner$^{1}$, Gerald Badurek$^{1}$ and Yuji Hasegawa$^{1,2}$' title: \| Observation of nonadditive mixed state phases with\ polarized neutrons --- Evolving quantum systems acquire two kinds of phase factors: (i) the dynamical phase which depends on the dynamical properties of the system - like energy or time - during a particular evolution, and (ii) the geometric phase which only depends on the path the system takes in state space on its way from the initial to the final state. Since its discovery by Pancharatnam [@Pancharatnam1956] and Berry [@Berry1984] the concept was widely expanded and has undergone several generalizations. Nonadiabatic [@AharanovAnandan1987] and noncyclic [@SamuelBhandari1988] evolutions as well as the off-diagonal case, where initial and final state are mutually orthogonal [@ManiniPistolesi2000], have been considered. Ever since, a great variety of experimental demonstrations has been accomplished [@TomitaChao1986; @SuterEtAl1988], also in neutron optics [@BitterDubbers1987; @AllmanEtAl1997; @HasegawaEtAl2001; @FilippEtAl2005]. Due to its potential robustness against noise [@LeekEtAl2008] the geometric phase is an excellent candidate to be utilized for logic gate operations in quantum information science [@NielsenChuang]. Thus, a rigorous investigation of all its properties is of great importance. In addition to an approach by Uhlmann [@Uhlmann1991] a new concept of phase for mixed input states based on interferometry was developed by Sjöqvist et al. [@SjoeqvistEtAl2000]. Here, each eigenvector of the initial density matrix independently acquires a geometric phase. The total mixed state phase is a weighted average of the individual phase factors. This concept is of great significance for such experimental situations or technical applications where pure state theories may imply strong idealizations. Theoretical predictions have been tested by Du et al. [@DuEtAl2003] and Ericsson et al. [@EricssonEtAl2004] using NMR and single-photon interferometry, respectively. In this letter, we report on measurements of nonadiabatic and noncyclic geometric, dynamical and combined phases. These depend on noise strength in state preparation, defining the degree of polarization, the purity, of the neutron input state. In particular, our experiment demonstrates that the geometric and dynamical mixed state phases $\Phi_{\mbox{\scriptsize{g}}}$ and $\Phi_{\mbox{\scriptsize{d}}}$, resulting from separate measurements, are not additive [@SinghEtAl2003], because the phase resulting from a single, cumulative, measurement differs from $\Phi_{\mbox{\scriptsize{g}}}+\Phi_{\mbox{\scriptsize{d}}}$. This nonadditivity might be of practical importance for applications, e.g. the design of phase gates for quantum computation. A neutron beam propagating in $y$-direction interacting with static magnetic fields $\vec {\mbox{B}}(y)$ is described by the Hamiltonian H$=-\hbar^2/2m\vec\nabla^2-\mu\vec\sigma\vec B(y)$; $m$ and $\mu$ are the mass and the magnetic moment of the neutron, respectively. Zeeman splitting within $\vec {\mbox{B}}(y)$ leads to solutions of the Schrödinger equation $\cos(\theta/2)\|k_+\rangle\|+\rangle+ e^{i\alpha}\sin(\theta/2)\|k_-\rangle\|-\rangle$, where $\|k_{\pm}\rangle$ are the momentum and $\|\pm\rangle$ the spin eigenstates within the field $\vec {\mbox{B}}(y)$. $\theta$ and $\alpha$ denote the polar and azimuthal angles determining the direction of the spin with respect to $\vec {\mbox{B}}(y)$. $k_{\pm}\simeq k_0\mp\Delta k$, where $k_0$ is the momentum of the free particle and $\Delta k=m\mu\|\vec{\mbox{B}}(y)\|/\hbar^2k$ is the field-induced momentum shift. $\Delta k$ can be detected from spinor precession. Omitting the coupling of momentum and spin, we focus on the evolution of superposed spin eigenstates resulting in Larmor precession of the polarization vector $\vec r=\langle\varphi\|\vec\sigma\|\varphi\rangle$, where $\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)$ is the Pauli vector operator. The unitary, unimodular operator $$\begin{aligned} \mbox{U}(\xi',\delta',\zeta')&=&e^{i\delta'}\cos\xi'\|+\rangle\langle +\|-e^{-i\zeta'}\sin\xi'\|+\rangle\langle -\|\\ &+&e^{i\zeta'}\sin\xi'\|-\rangle\langle +\|+e^{-i\delta'}\cos\xi'\|-\rangle\langle -\| %\mbox{U}&=&\cos\xi'[e^{i\delta'}\|k_+\rangle\langle k_+\|+e^{-i\delta'}\|k_-\rangle\langle k_-\|]\otimes \1\nonumber\\ %&+&\sin\xi'[e^{i\zeta'}\|k_-\rangle\langle k_+\|-e^{-i\zeta'}\|k_+\rangle\langle k_-\|]\otimes\sigma_x\end{aligned}$$ describes the evolution of the system within static magnetic fields. The set of SU(2) parameters $(\xi',\delta',\zeta')$ is related to the so-called Cayley-Klein parameters $a,b$ via $a=e^{i\delta'}\cos\xi'$ and $b=-e^{-i\zeta'}\sin\xi'$ (see e.g. [@Sakurai]). Consider the experimental setup shown in Fig.\[fig1\]. A monochromatic neutron beam passes the polarizer P preparing it in the ’up’ state $\|+\rangle$ with respect to a magnetic guide field in $z$-direction (B$_z$). Next, the beam approaches a DC coil with its field B$_x$ pointing to the $x$-direction. B$_x$ is chosen such that it carries out the transformation U$_1$$\equiv$U$(\pi/4,0,-\pi/2)$, corresponding to a $+\pi/2$ rotation around the $+x$-axis. After U$_1$ the resulting state of the system is a coherent superposition of the two orthogonal spin eigenstates: $\|\psi_0\rangle=1/\sqrt{2}(\|+\rangle-i\|-\rangle)$. A subsequent coil, represented by U$(\xi,0,-\pi/2)$, is set to cause a spin rotation around the $+x$-axis by an angle $2\xi$. This second coil and the following propagation distance within B$_z$ – corresponding to a rotation angle $2\delta$ around the $+z$-axis – define an evolution U$_\phi$$\equiv$U$(\xi,\delta,\zeta)$. Undergoing the transformation U$_\phi$ the two spin eigenstates $\|\pm\rangle$ acquire opposite total phase $\pm\phi =\pm\arg\langle\pm\|\mbox{U}_\phi\|\pm\rangle=\pm\delta$. A third coil (U$^{\dagger}_1$) reverses the action of the first one and would therefore transform a state $\|\psi_0\rangle$ back to $\|+\rangle$. The state of the system entering the third coil equals $\|\psi_0\rangle$ only if U$_\phi=\1$. $\phi$ can be extracted by applying an extra dynamical phase shift $\pm \frac{1}{2}\eta$ to $\|\pm\rangle$. It is implemented by adjusting both inter-coil distances from first to second and second to third coil to spin rotation angle equivalents of $2\pi n+\eta$ and $2\pi n'+2\delta-\eta$ ($n,n'$ are integer). By scanning the position of the second coil these rotation angles - referred to as U$_{\eta}$$\equiv$$\mbox{U}(0,\eta/2)$ and U$^{\dagger}_{\eta}$ - are varied to yield intensity oscillations from which $\phi$ is calculated. Note that, because of $\xi'=0$, the parameter $\zeta'$ is undetermined and, therefore, omitted in U$_{\eta}$. After projection on the state $\|+\rangle$ by the analyzer A, the phase $\phi\!=\!\delta$ and its visibility $\nu=\|\cos\xi\|$ can be computed as functions of the maxima and minima of the intensity, I$_{\mbox{\scriptsize max}}$ and I$_{\mbox{\scriptsize min}}$, measured by the detector D [@WaghRakhecha1995]. More general, a neutron beam with incident purity $r_0'\!=\!\|\vec r_0'\|$ along the $+z$-axis ($\vec r_0'\!=\!(0,0,r_0')$) is described by the density operator $\rho_{\mbox{\scriptsize{in}}}(r_0')=1/2(\1+r_0'\sigma_z)$. Calculating $\rho_{\mbox{\scriptsize{out}}}=\mbox{U}^{\dagger}_1\mbox{U}^{\dagger}_{\eta} \mbox{U}_\phi\mbox{U}_{\eta}\mbox{U}_1\rho_{\mbox{\scriptsize{in}}}\mbox{U}^{\dagger}_1\mbox{U}^{\dagger}_{\eta} \mbox{U}^{\dagger}_\phi\mbox{U}_{\eta}\mbox{U}_1 $, we find the intensity $$\begin{aligned} \label{mixedstateintensity} \mbox{I}^{\rho}&\propto& %\mbox{Tr}(\|+\rangle\langle+\|\rho_{\mbox{\scriptsize{out}}})\nonumber\\ \frac{1-r_0'}{2}+r_0'\left(\cos^2\xi\cos^2\delta+\sin^2\xi\cos^2(\zeta-\eta)\right)\end{aligned}$$ after the analyzer A. Considering the maxima and minima $\mbox{I}^{\rho}_{\mbox{\scriptsize max}}$, $\mbox{I}^{\rho}_{\mbox{\scriptsize min}}$ of $\eta$-induced oscillations of $\mbox{I}^{\rho}$ one obtains the mixed state phase [@LarssonSjoeqvist2003] $$\begin{aligned} \label{eq:MixedStatePhInTermsOfImaxImin} \Phi(r_0')&=&%\arg \mbox{Tr}(\mbox{U}_{\phi}\rho_{\mbox{\scriptsize{in}}}) \arccos\sqrt{\frac{[\mbox{I}^{\rho}_{\mbox{\scriptsize min}}/\mbox{I}^{\rho}_{\mbox{\scriptsize n}}-1/2(1-r_0')]/r_0'} {r_0'[1/2(1+r_0')-\mbox{I}^{\rho}_{\mbox{\scriptsize max}}/\mbox{I}^{\rho}_{\mbox{\scriptsize n}}]+ [\mbox{I}^{\rho}_{\mbox{\scriptsize min}}/\mbox{I}^{\rho}_{\mbox{\scriptsize n}}-1/2(1-r_0')]/r_0'}}\end{aligned}$$ with a normalization factor $\mbox{I}^{\rho}_{\mbox{\scriptsize n}}=2\mbox{I}^{\rho}_0/(1+r_0')$. I$^{\rho}_0$ is the intensity measured at U$_\phi=\1$. The noncyclic geometric phase is given by $\phi_g=-\Omega/2$, where $\Omega$ is the solid angle enclosed by a geodesic path and its shortest geodesic closure on the Bloch sphere [@SamuelBhandari1988]: $\phi_{\mbox{\scriptsize{g}}}$ and the total phase $\phi$ are related to the path by the polar and azimuthal angles $2\xi$ and $2\delta$, so that the pure state geometric phase becomes $$\begin{aligned} \label{eq:gamma(delta,xi)} \phi_{\mbox{\scriptsize{g}}} =\phi- \phi_{\mbox{\scriptsize{d}}}=\delta[1-\cos{(2\xi)}],\end{aligned}$$ while its dynamical counterpart is $$\begin{aligned} \label{eq:delta(phi,xi)} \phi_{\mbox{\scriptsize{d}}}=\delta\cos{(2\xi)}.\end{aligned}$$ By proper choice of $2\xi$ and $2\delta$, U$_\phi$ can be set to generate purely geometric, purely dynamical, or arbitrary combinations of both phases as shown in Fig.\[fig2\]. The theoretical prediction for the mixed state phase is [@SjoeqvistEtAl2000; @LarssonSjoeqvist2003] $$\begin{aligned} \label{eq:MixedStatePhase} \Phi(r_0')&=&\arctan\left(r_0'\tan\delta\right)\end{aligned}$$ To access Eq.(\[eq:MixedStatePhase\]) experimentally $r_0'$ needs to be varied. In addition to the DC current, which effects the transformation U$_1$, random noise is applied to the first coil, thereby changing B$_x$ in time. Neutrons, which are part of the ensemble $\rho_{\mbox{\scriptsize{in}}}(r_0')$, arrive at different times at the coil and experience different magnetic field strengths. This is equivalent to applying different unitary operators U$(\pi/4\!+\!\Delta\xi'(t),0,-\pi/2)=\mbox{\~U}_1(\Delta\xi'(t))$. For the whole ensemble we have to take the time integral $$\begin{aligned} \rho=\int\mbox{\~U}_1(\Delta\xi'(t))\|+\rangle\langle +\|\mbox{\~U}_1^{\dagger}(\Delta\xi'(t))dt\nonumber.\end{aligned}$$ Although each separate transformation is unitary, due to the randomness of the signal we end up with a nonunitary evolution that yields a mixed state [@BertlmannEtAl2006]. Note that in this method the purity $r_0'$ of the input state is not affected before Ũ$_1$ creates spin superpositions distributed around $\|\psi_0\rangle$ within the $y,z$-plane. We are left with $\vec r_0\!=\!\left(0,-r_0,0\right)$ where $r_0\!<\!1$, as has been confirmed by a 3D spin analysis of the state $\mbox{\~U}_1\rho_{\mbox{\scriptsize{in}}} \mbox{\~U}^{\dagger}_1$. The experiment was carried out at the research reactor facility of the Vienna University of Technology. A neutron beam – incident from a pyrolytic graphite crystal – with a mean wavelength $\lambda\!\approx\!1.98$Å and spectral width $\Delta\lambda/\lambda\!\approx\!0.015$, was polarized up to $r_0'\!\approx\!99$% by reflection from a bent Co-Ti supermirror array. The analyzing supermirror was slightly de-adjusted to higher incident angles to suppress second order harmonics in the incident beam. The final maximum intensity was about 150counts/s at a beam cross-section of roughly 1cm$^2$. A $^3$He gas detector with high efficiency for neutrons of this energy range was used. Spin rotations around the $+x$-axis were implemented by magnetic fields B$_x$ from DC coils made of anodized aluminium wire (0.6mm in diameter) wound on frames with rectangular profile ($7\!\times\!7\!\times\!2$ cm$^3$). Coil windings in $z$-direction provided for compensation of the guide field B$_z$ at coil positions (see Fig.\[fig1\]) for B$_x$ to be parallel to the $x$-direction. B$_z$ was realized by two rectangular coils of 150cm length in Helmholtz geometry. Maximum coil currents of 2A and guide field strengths of 1mT were sufficient to achieve required spin rotations and prevent unwanted depolarization. The noise from a standard signal generator consisted of random DC offsets varying at a rate of 20kHz. In order to find the coil current values for required spin rotation angles, each coil current was scanned separately. A measured intensity minimum after stepwise increase indicates that a $\pi$ rotation has been achieved. Then, with both coils mounted on translation stages and set to a $\pi/2$ rotation around the $x$-axis, inter-coil distances were varied to search for intensity minima. The distance between two minima (6-7 cm in our case) corresponds to a $2\pi$ rotation around B$_z$ within the $x,y$ plane. For purely geometric phases the parameter sets ($\xi\!=\!\pi/4,\delta,\zeta\!=\!\delta\!-\!\pi/2$) with $\delta\!=\!\phi_{\mbox{\scriptsize{g}}}\! =\!\pi/8,2\pi/8,3\pi/8$ were chosen. For each set the intensity oscillations I$^\rho$ (see Fig.\[fig3\]a, upper graph; data shown for $\delta=\phi_{\mbox{\scriptsize{g}}}=\pi/8$) were measured, scanning the position of the second coil for five values of $r_0$. For the purely dynamical phase - since $2\xi=0$ and Eq.(\[mixedstateintensity\]) reduces to I$^\rho=(1\!-\!r_0')/2\!+\!r_0'\cos^2\delta$ - only one intensity value is needed. With the second coil turned off, the distance between the first and the third coil was chosen such that $\delta=\phi_{\mbox{\scriptsize{d}}}=\pi/8,2\pi/8,3\pi/8$ for five values of $r_0$ (Fig.\[fig3\]b, upper graph). By Eq.(\[eq:MixedStatePhInTermsOfImaxImin\]) the geometric and dynamical mixed state phases $\Phi_{\mbox{\scriptsize{g}}}(r_0)$ and $\Phi_{\mbox{\scriptsize{d}}}(r_0)$ (Fig.\[fig3\], lower graphs) were calculated from intensity values extracted from least square fits for I$^\rho$ (solid lines in Fig.\[fig3\], upper graphs). Vertical error bars in the lower graphs contain two about equal contributions: fitting errors and an estimated uncertainty of 0.5 mm for the reproduction of coil positions. The small horizontal error bars stem from the statistical uncertainty of the purity measurements. The solid lines are theoretical curves according to Eq.(\[eq:MixedStatePhase\]), using the measured value for $\Phi$ without noise as phase reference. The experimental data reproduce well the $r_0$-dependence predicted by Eq.(\[eq:MixedStatePhase\]). We want to emphasize that our investigation focuses on a special property of the mixed state phase: its nonadditivity. Since the Sjöqvist phase is defined as a weighted average of phase factors rather then one of phases (see [@FuChen2004a; @SinghEtAl2003] for a more elaborate discussion) it is true only for pure states that phases of separate measurements can be added up to a total phase. Suppose we carry out two measurements on a pure state system: the state is subjected to a unitary transformation U$_{\mbox{\scriptsize{g}}}$ in the first and to U$_{\mbox{\scriptsize{d}}}$ in the second measurement inducing the phases $\phi_{\mbox{\scriptsize{g}}}$ and $\phi_{\mbox{\scriptsize{d}}}$, respectively. Applying Eqs.(\[eq:gamma(delta,xi)\]) and (\[eq:delta(phi,xi)\]), we can also choose a combination of angles $2\xi$ and $2\delta$ leading to a transformation U$_{\mbox{\scriptsize{tot}}}$ so that we measure the total pure state phase $\phi_{\mbox{\scriptsize{g}}}+\phi_{\mbox{\scriptsize{d}}}$ (note that the three evolution paths induced by U$_{\mbox{\scriptsize{g}}}$, U$_{\mbox{\scriptsize{d}}}$ and U$_{\mbox{\scriptsize{tot}}}$ differ from each other). However, the result of the latter experiment for the system in a mixed state is $\Phi_{\mbox{\scriptsize{tot}}}(r_0) =\arctan{\left[r_0\tan(\phi_{\mbox{\scriptsize{g}}} +\phi_{\mbox{\scriptsize{d}}})\right]}$. The total phase is then not given by $\Phi_{\mbox{\scriptsize{g}}}(r_0) +\Phi_{\mbox{\scriptsize{d}}}(r_0)$, with $\Phi_{\mbox{\scriptsize{g}}}(r_0) =\arctan{(r_0\tan\phi_{\mbox{\scriptsize{g}}})}$ and $\Phi_{\mbox{\scriptsize{d}}}(r_0) =\arctan{(r_0\tan\phi_{\mbox{\scriptsize{d}}})}$. In our experiment we have chosen two examples of U$_{\mbox{\scriptsize{tot}}}$, i.e. two sets of values for B$_x$ in the second coil ($2\xi^{(1)}\!=\!60^{\circ},~2\xi^{(2)}\!=\!48^{\circ}$) and the distance within B$_z$ ($2\delta^{(1)}\!=\!90^{\circ},~2\delta^{(2)}\! =\!135^{\circ}$). According to Eqs.(\[eq:gamma(delta,xi)\]) and (\[eq:delta(phi,xi)\]) the total pure state phases $\phi_{\mbox{\scriptsize{g}}}^{(1)} +\phi_{\mbox{\scriptsize{d}}}^{(1)}$ and $\phi_{\mbox{\scriptsize{g}}}^{(2)} +\phi_{\mbox{\scriptsize{d}}}^{(2)}$ with $\phi^{(1)}_{\mbox{\scriptsize{g}}} =\phi^{(2)}_{\mbox{\scriptsize{g}}}=\pi/8$ and $\phi^{(1,2)}_{\mbox{\scriptsize{d}}}=2\pi/8,\pi/8$ are obtained from intensity oscillations. In Fig.\[fig4\] the resulting mixed state phases $\Phi^{(1,2)}_{\mbox{\scriptsize{tot}}}$ and the sum $\Phi_{\mbox{\scriptsize{g}}}^{(1,2)} +\Phi_{\mbox{\scriptsize{d}}}^{(1,2)}$ are plotted. Note that this nonadditivity of mixed state phases is not due to the nonlinearity of the geometric phase, that occurs – for instance – when the system evolves close to the orthogonal state of the reference state [@Bhandari1997]. Recently there has been a report on NMR experiments [@DuEtAl2007] investigating Uhlmann’s mixed state geometric phase. It is a property of a composite system undergoing a certain non-local evolution of system and ancilla [@EricssonEtAl2003]. Diverse phase definitions, depending on this evolution, are possible. The phase investigated in the present paper is a special case in which the ancilla does not necessarily evolve. While the preconditions for inherent fault tolerance remain intact for the Sjöqvist phase, the question whether other phases offer advantages in terms of robustness remains an exciting issue of discussion. To summarize, we have measured spin-1/2 mixed state phases with polarized neutrons. 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--- abstract: \| Understanding the origin of stellar masses is a key problem in astrophysics. In the solar neighborhood, the mass distribution of stars follows a seemingly universal pattern. In the centre of the Milky Way, however, there are indications for strong deviations and the same may be true for the nuclei of distant starburst galaxies. Here we present the first numerical hydrodynamical calculations of stars formed in a molecular region with chemical and thermodynamic properties similar to those of warm and dusty circum-nuclear starburst regions. The resulting IMF is top-heavy with a peak at $\sim 15$ $M_\odot$, a sharp turn-down below $\sim 7$ $M_\odot$ and a power-law decline at high masses. We find a natural explanation for our results in terms of the temperature dependence of the Jeans mass, with collapse occuring at a temperature of $\sim 100$ K and an H$_2$ density of a few $\times 10^5$ cm$^{-3}$, and discuss possible implications for galaxy formation and evolution. author: - \| Ralf S. Klessen$^{1,2}$, Marco Spaans$^3$, Anne-Katharina Jappsen$^{2,4}$\ [$^1$Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Überle-Str. 2,]{}\ [ 69120 Heidelberg, Germany]{}\ [$^2$Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany]{}\ [$^3$Kapteyn Astronomical Institute, P.O. Box 800, 9700 AV Groningen, The Netherlands]{}\ [$^4$Canadian Institute for Theoretical Astrophysics, McLennan Physics Labs, 60 St. George Street,]{}\ [ University of Toronto, Toronto, ON M5S 3H860, Canada]{} date: 'Received sooner; accepted later' title: ' The Stellar Mass Spectrum in Warm and Dusty Gas: Deviations from Salpeter in the Galactic Centre and in Circum-Nuclear Starburst Regions' --- \[firstpage\] stars: formation – hydrodynamics – turbulence – equation of state – Galaxy: centre – galaxies: starburst Introduction ============ Identifying the physical processes that determine the masses of stars and their statistical distribution, the initial mass function (IMF), is a fundamental problem in star-formation research. It is central to much of modern astrophysics, with implications ranging from cosmic re-ionisation and the formation of the first galaxies, over the evolution and structure of our own Milky Way, down to the build-up of planets and planetary systems. Near the Sun the number density of stars as a function of mass has a peak at a characteristic stellar mass of a few tenths of a solar mass, below which it declines steeply, and for masses above one solar mass it follows a power-law with an exponent $dN/d{\log}m \propto m^{-1.3}$. Within a radius of several kpc this distribution shows surprisingly little variation (Salpeter 1955; Scalo 1998; Kroupa 2001; Kroupa 2002; Chabrier 2003). This has prompted the suggestion that the distribution of stellar masses at birth is a truly universal function, which often is referred to as the Salpeter IMF, although note that the original Salpeter (1955) estimate was a pure power-law fit without characteristic mass scale. On the other hand, there is increasing evidence that the IMF close to the centre of our Milky Way (Stolte et al. 2002, 2005, Nayakshin & Sunyaev 2005, Paumard et al. 2006) and the neighboring Andromeda galaxy (Bender et al. 2005) is dominated by massive rather than low-mass stars. For the circum-nuclear starburst regions in more distant galaxies, very similar IMF deviations are subject to continuing debate (e.g., Scalo 1990, Elmegreen 2005). However, no conclusion has yet been reached, and it appears timely to examine the problem from a theoretical point of view. We approach the problem by means of self-consistent hydrodynamical calculations of fragmentation and star formation in interstellar gas where chemical and thermodynamical properties are described by a realistic equation of state (EOS). We focus on the most extreme environmental conditions such as occur in the nuclear regions of massive star-forming spiral galaxies. There the inferred dust and gas temperatures, gas densities and star formation rates typically exceed the solar-neighborhood values by factors of 3, 10 and $\ge 100$, respectively (e.g. Ott et al. 2005; Israel 2005; Aalto et al. 2002; Spinoglio et al. 2002). Consequently, it has long been speculated that such conditions lead to deviations from the Salpeter IMF (e.g., Scalo 1990, Elmegreen 2005). =0.90 Model ===== Stars and star clusters form through the interplay between self-gravity on the one side and turbulence, magnetic fields, and thermal pressure on the other (for recent reviews see Larson 2003; Mac Low & Klessen 2004; Ballesteros-Paredes et al. 2006). The supersonic turbulence ubiquitously observed in interstellar gas clouds can create strong density fluctuations with gravity taking over in the densest and most massive regions. Collapse sets in to build up stars and star clusters. Turbulence plays a dual role. On global scales it provides support, on local scales it provokes collapse. Stellar birth is thus intimately linked to the dynamic behavior of the parental gas cloud, which governs when and where star formation sets in (as illustrated in Figure \[fig:2D-plot\]). The chemical and thermodynamic properties of interstellar clouds play a key role in this process. In particular, the value of the polytropic exponent $\gamma$, when adopting an EOS of the form $P\propto\rho^\gamma$, strongly influences the compressibility of density condensations as well as the temperature of the gas. The EOS thus determines the amount of clump fragmentation, and so directly relates to the IMF (Vázquez-Semadeni et al. 1996) with values of $\gamma$ larger than unity leading to little fragmentation and high mass cores (Li, Klessen, & Mac Low 2003; Jappsen et al. 2005). The stiffness of the EOS in turn depends strongly on the ambient metallicity, density and infrared background radiation field produced by warm dust grains. The EOS thus varies considerably in different galactic environments (see Spaans & Silk 2000, 2005 for a detailed account). For the circum-nuclear starburst regions that are the subject here we assume a cosmic ray ionisation rate of $3\times 10^{-15}$ s$^{-1}$, solar relative abundances (Asplund et al. 2004; Jenkins 2004) and an overall metallicity of two times solar (Barthel 2005). A velocity dispersion $\Delta V_{\rm tur}=5$ km/s is adopted to take the larger input of kinetic energy (e.g. through supernovae) into account. The dust temperature inside the model clouds is set by a fiducial background star formation rate of $100\,$M$_\odot$ yr$^{-1}$/kpc$^{2}$ which causes dust grains to be at temperatures of about $T_d=30 - 90\,$K, depending on the amount of shielding. Gas temperatures range from $T_g = 40 - 140\,$K, over a density range of $10^4 - 10^7\,$cm$^{-3}$. These values are consistent with gas and dust temperatures determined for circum-nuclear starburst regions (Klaas et al. 1997; Aalto et al. 2002; Spinoglio et al. 2002; Ott et al. 2005; Israel 2005). =0.45 Figure \[fig:starburst-EOS\] shows the resulting polytropic exponent as a function of density. The main feature is the $\gamma > 1$ peak around $n = 10^4\ $cm$^3$. This peak implies that the gas warms up as it is compressed and it is caused mainly by strong photon trapping in opaque H$_2$O and CO lines in the metal-rich nuclear gas. That is, the large optical depth in the cooling lines suppresses the cooling efficiency. Also, warm dust ($T > 40\,$K), heated by the ambient stars, causes H$_2$O collisional de-excitation heating through far-infrared pumping (Takahashi, Hollenbach & Silk, 1983; Spaans & Silk 2005), which adds to the gas-dust heating. Cosmic-ray heating rate is elevated by a high supernova rate, as expected for nuclear starburst regions (Bradford et al. 2003). Adopting this EOS we follow the dynamical evolution of the star-forming gas using smoothed particle hydrodynamics (SPH). This is a Lagrangian method to solve the equations of hydrodynamics, where the fluid is represented by an ensemble of particles, and flow quantities are obtained by averaging over an appropriate subset of SPH particles (Monaghan 2005). The method is able to resolve high density contrasts as particles are free to move, and so the particle concentration increases naturally in high-density regions. The performance and convergence properties of SPH are well understood and tested against analytic models and other numerical schemes in the context of astrophysical flows (see, e.g., Mac Low et al. 1998; Lombardi et al. 1999; Klessen et al. 2000; O’Shea et al. 2005; Ballesteros-Paredes et al. 2006). Artificial fragmentation can be ruled out, as long as the mass within one smoothing volume remains less than half the critical mass for gravitational collapse (Bate & Burkert 1997; Hubber, Goodwin, & Whitworth 2005). We use the publically available parallel code GADGET (Springel et al. 2001). It is modified to replace high-density cores with sink particles (Bate, Bonnell, & Price 1995) that can accrete gas from their surroundings while keeping track of mass and momentum. This enables us to follow the dynamic evolution of the system over many local free-fall timescales. We identify sink particles as the direct progenitors of individual stars. For a more detailed account of the method and a discussion of its convergence properties we refer the reader to Klessen et al. (2000) and Jappsen et al. (2005). =0.45=0.45 We focus on a cubic volume of 11.2$\,$pc in size, which contains $80,000\,$M$_{\odot}$ of gas and has an initial mean particle density $n = 10^3\,$cm$^{_3}$ at a temperature of $21\,$K. Above the characteristic density $n = 10^4\,$cm$^{-3}$ where $\gamma$ is at a maximum, the temperature quickly reaches values of $\sim 100\,$K. This set-up is chosen to describe the typical environment within the central regions of an actively star-forming galaxy such as our own Milky Way or NGC$\,$253. In such galaxies, high-density gas with $n > 10^5\,$cm$^{-3}$, as traced by HCN, typically has a filamentary structure with very low filling factor, while the bulk of the gas is at $n \approx 10^3\,$cm$^{-3}$ (Morris & Serabyn 1996; Hüttemeister et al. 1993; Israel & Baas 2003), exactly as found at the end of our simulation (see Figure \[fig:2D-plot\]). We stop the calculation at a star formation efficiency SFE $\approx 15$%, when roughly 1/6 of the total gas mass has turned into gravitationally collapsed condensations (i.e. sink particles, which we identify as direct progenitors of individual young stars). Throughout the simulation we drive turbulence continuously on large scales, with wave numbers $k$ in the range $1 \le k \le 2$ (see Mac Low 1999) to yield a constant turbulent Mach number $\mathcal{M}_{\mathrm{rms}}\approx 5$. The particle number is $N=8\,000\,000$. This is thus one of the highest-resolution star-formation calculations done with SPH, with a total CPU time of $8 \times 10^4$ hours. The critical density for sink particle formation is $n_c = 10^7\,$cm$^{-3}$, with a sink particle radius of 0.015$\,$pc. The mass of individual SPH particles is $m=0.01\,$M$_{\odot}$, which is sufficient to resolve the minimum Jeans mass in the system $M_{\rm J} \approx 1.5\,$M$_{\odot}$. Except for the EOS and the particle number, the numerical set-up is identical to the study by Jappsen et al. (2005). We have performed a second run for a region of 5.7$\,$pc with 4 times less mass, eight times fewer particles and a sink particle radius of 0.02 pc that has reached a SFE $\sim 36$%. Result and Physical Interpretation ================================== We find that in the considered star-forming region, the mass spectrum of collapsed objects is biased towards high masses. The resulting IMF has a broad peak at $\sim 15\,$M$_{\odot}$ followed by an approximate power-law fall-off with a slope in the range -1.0 to -1.3. Furthermore, there is a clear deficit of stars below $7\,$M$_{\odot}$. This is illustrated in Figure \[fig:starburst-IMF\]a. We contrast this finding with the result from a simulation appropriate for the physical conditions in star forming regions near the Sun (from Jappsen et al. 2005), where $\gamma$ changes from 0.7 to 1.1 at an H$_2$ density of a few $\times 10^5\,$cm$^{-3}$. As expected, Figure \[fig:starburst-IMF\]b shows a mass spectrum that is very similar to the IMF in the solar neighborhood (Kroupa 2002; Chabrier 2003). These striking differences are caused by the very disparate chemical and thermodynamic state of the star forming gas in the two simulations, since all other parameters are very similar. Our results thus support the hypothesis that for extreme environmental conditions as inferred for the centres of most spiral galaxies or more general for IR-luminous circum-nuclear starburst regions the IMF is indeed expected to be top-heavy. There is a natural explanation for our results in terms of the temperature dependence of the Jeans mass $M_{\rm J}$. Compared to a mean temperature of $10\,$K for dense molecular gas in the Milky Way, gravitationally collapsing gas in our simulations has a temperature of $\sim 100\,$K and an H$_2$ density of a few $\times 10^5$ cm$^{-3}$. As the critical mass for gravitational collapse scales as $M_{\rm J} \propto T^{1.5}$, this boosts $M_{\rm J}$ from $0.3\,$M$_\odot$ at $10\,$K to about 10$\,$M$_\odot$ at $100\,$K (see also Klessen & Burkert 2000, Bonnell, Clarke, & Bate 2006). This temperature may seem high, but is quite consistent with molecular cloud observations in the Galactic centre (e.g. Hüttemeister et al. 1993) or with high-density ($n>10^4\,$cm$^{-3}$) NH$_3$ data in the starburst centre of NGC253 (Ott et al. 2005). We also note, that this Jeans mass scaling argument is supported by recent observations in more nearby high-mass star-forming regions. For example, in M17 at a distance of 1.6$\,$kpc from the Sun, the mass spectrum of prestellar cores, which are the direct progenitors of individual stars, peaks at at $\sim 4\,$M$_{\odot}$ at an ambient temperature of $30\,$K (Reid & Wilson 2006). This is well above the corresponding peak in low-mass star-forming regions (e.g. Motte, Andr[é]{}, & Neri 1998). Discussion ========== Our mass spectrum is in good agreement with the IMF estimates in the Galactic centre by Stolte et al. (2002, 2005), Nayakshin et al. (2005), and Paumard et al. (2006). For example, Stolte et al. (2002, 2005) find for the Arches cluster a clear deficit of stars below $7\,$M$_\odot$. This is consistent with our result in the sense that the ambient densities and temperatures found in the Galactic centre are similarly elevated (Helfer & Blitz 1996) as in the circum-nuclear starburst environment we consider. We stress that the turn-down in our model IMF at masses below 10$\,$M$_\odot$ is a direct consequence of the stiff EOS for densities $n$ above a few$\times 10^3\,$cm$^{-3}$ through the Jeans mass temperature dependence, and is not caused by resolution effects. Our two simulations resolve masses down to $\sim 2\,$M$_\odot$ and $\sim 1\,$M$_\odot$, respectively, and our least massive stars (i.e. sink particles) are well above this limit. Rather, the effective Jeans mass at $T\sim 100$K and densities of $\sim 10^5-10^6$ cm$^{-3}$ prevent the formation of low-mass stars. When interpreting our simulation results, there are several caveats that need to be kept in mind. First, our numerical model does not include shear. Strong shear motions may mimic the EOS effects discussed here, as shear adds stability and thus requires larger Jeans masses for collapse to occur. However, the Arches cluster is bound. Thus the Galactic centre shear field cannot play a dominant role in the inner parts of the cluster. Second, our numerical model does not take the effects of magnetic fields into account, which may be of considerable strength in the Galactic centre (Yusef-Zadeh & Morris 1987, but also see Roy 2004 for lower estimates). However, even if there is a rough equipartition between kinetic and magnetic energy, the chemical and thermodynamic properties of the gas are not strongly affected. Our results will still hold at least qualitatively, in the sense that an extreme environment leads to deviatiations from the standard Salpeter IMF. Third, the use of sink particles does not permit us to resolve close binary systems. Massive stars in the solar vicinity are almost always members of a binary or higher-order multiple stellar system (e.g. Vanbeveren et al. 1998). If this trend holds also for starburst environments, then the peak of the stellar IMF will lie below the value reported here. For instance, if each unresolved sink particle in our calculation separates into a binary star, in a statistical sense our mass spectrum needs to be shifted to lower masses by a factor of 0.5. Finally, protostellar feedback may locally affect the accretion onto individual protostars. In this case the mass content of the sink particle may only poorly reflect the mass that ends up in a star. However, even in the extreme case that half the mass is removed by feedback during collapse (for estimates, see Yorke & Sonnhalter 2002; Krumholz, McKee, & Klein 2005), deviations from the standard IMF will still persist. For typical molecular clouds in the Milky less than a few percent of their mass takes part in star formation (e.g. Myers et al. 1986) and this fraction goes up by a factor of a few for cluster-forming cores (e.g. Lada & Lada 2003). A number of observations (Paglione et al. 1997; Mooney & Solomon 1988) indicate that starburst systems like NGC253 and M82, and luminous infrared galaxies in general, have a larger fraction of their interstellar gas mass at high densities (Gao & Solomon 2004). Consequently, their SFE’s are up by as much as an order of magnitude. Our simulations cover this range and the statistics of our mass spectra do not change above a SFE $\sim 10$% in both runs. Hence, the precise SFE that pertains to a starburst environment does not influence our results as long as it is larger than 10%. The computed star formation rate (SFR), defined as the change in mass with time of the sink particles, is typically $860\, M_\odot$ yr$^{-1}/$kpc$^2$ for a SFE $>10$% and when normalised to a surface area of $1\,$kpc$^2$, which is roughly the size scale of the nuclear region inside a starburst galaxy. 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--- abstract: 'In this work, we address the problem of tuning communication libraries by using a deep reinforcement learning approach. Reinforcement learning is a machine learning technique incredibly effective in solving game-like situations. In fact, tuning a set of parameters in a communication library in order to get better performance in a parallel application can be expressed as a game: Find the right combination/path that provides the best reward. Even though AITuning has been designed to be utilized with different run-time libraries, we focused this work on applying it to the OpenCoarrays run-time communication library, built on top of MPI-3. This work not only shows the potential of using a reinforcement learning algorithm for tuning communication libraries, but also demonstrates how the MPI Tool Information Interface, introduced by the MPI-3 standard, can be used effectively by run-time libraries to improve the performance without human intervention.' author: - Alessandro Fanfarillo - Davide Del Vento title: 'AITuning: Machine Learning-based Tuning Tool for Run-Time Communication Libraries' ---
--- abstract: \| We present a method to improve the accuracy of a foot-mounted, zero-velocity-aided inertial navigation system (INS) by varying estimator parameters based on a real-time classification of motion type. We train a support vector machine (SVM) classifier using inertial data recorded by a single foot-mounted sensor to differentiate between six motion types (walking, jogging, running, sprinting, crouch-walking, and ladder-climbing) and report mean test classification accuracy of over 90% on a dataset with five different subjects. From these motion types, we select two of the most common (walking and running), and describe a method to compute optimal zero-velocity detection parameters tailored to both a specific user and motion type by maximizing the detector F-score. By combining the motion classifier with a set of optimal detection parameters, we show how we can reduce INS position error during mixed walking and running motion. We evaluate our adaptive system on a total of 5.9 km of indoor pedestrian navigation performed by five different subjects moving along a 130 m path with surveyed ground truth markers. author: - 'Brandon Wagstaff, Valentin Peretroukhin, and Jonathan Kelly[^1]' bibliography: - 'zupt\_ipin2017.bib' title: \| Improving Foot-Mounted Inertial Navigation\ Through Real-Time Motion Classification --- Introduction ============ Localization within indoor environments can often be challenging because building materials can significantly attenuate or reflect GNSS-based navigation signals. For indoor pedestrian tracking, one potential alternative is body-mounted inertial navigation, a dead-reckoning approach that integrates inertial rates to estimate the position, orientation and velocity of a moving person within a predefined coordinate frame. For low-cost inertial measurement units (IMUs) based on microelectromechanical systems (MEMS), relying on direct integration for any appreciable interval is impractical, as position error grows cubicly with time [@ZVDetect]. To bound this error growth, a zero-velocity-aided inertial navigation system (INS) uses periodic zero-velocity updates (ZUPTs). ZUPTs occur during midstance, a part of the human gait in which the foot is stationary relative to the ground. By mounting an IMU to the foot of the user, a zero-velocity event can be detected through a general likelihood ratio test (LRT) on the inertial data [@zupteval; @ZVDetect], and then incorporated into an extended Kalman filter (EKF) as a pseudo-measurement. [0.48]{} ![(Top) Our system consists of an SVM that classifies motion based on inertial data. We use the motion class to select optimal parameters for a zero-velocity detector. (Bottom) Six different motion types that we use to train and test our SVM-based motion classifier.](figs/system.pdf "fig:"){width="\textwidth"} \[fig:system\] [0.48]{} ![(Top) Our system consists of an SVM that classifies motion based on inertial data. We use the motion class to select optimal parameters for a zero-velocity detector. (Bottom) Six different motion types that we use to train and test our SVM-based motion classifier.](figs/motion_types.pdf "fig:"){width="\textwidth"} In this work we revisit a known drawback of ZUPT-based filters, namely that the optimal zero-velocity detection parameters are dependent on motion type [@ZVDetect]. First, for a fixed motion class, we outline a method to compute an optimal zero-velocity detection threshold that balances both the precision and recall of the detector. Second, we describe a motion classifier based on a support vector machine (SVM) to accurately classify motion in real-time from inertial data. Although we focus on walking and running motion in this paper, we show that our classifier is powerful enough to classify six different motion classes () with accuracies exceeding 90%. Finally, we combine our classifier with a set of optimal zero-velocity thresholds to create a more robust navigation system that can track dynamic, high-speed motions while maintaining high accuracy during walking. illustrates our proposed system. In short, the main contributions of this work are: 1. a procedure to determine optimal zero-velocity detection parameters for a range of motions, 2. a real-time capable classification routine that can reliably distinguish between different motions from a single foot-mounted IMU, and 3. an evaluation of an adaptive, classification-based zero-velocity-aided INS using a substantial 5.9 km indoor navigation dataset with surveyed ground truth markers and involving five different subjects. Related Work ============ Inertial sensors have a rich history in assisting the navigation of ships, airplanes, and spacecraft throughout the twentieth century. While early gyroscopes and accelerometers were large, the availability of lightweight, low-cost, MEMS-based sensors has enabled the production of commercial IMUs for pedestrian navigation. Foxlin et al. [@Foxlin:2005] first introduced a zero-velocity-aided, foot-mounted INS that used ZUPT detection to bound position error.[^2] Midstance detection was studied in detail by Skog et al. [@zupteval; @ZVDetect] who derived four zero-velocity detectors in the general LRT framework. Nilsson et al. [@INS_characterization:2010; @zupteval] demonstrated that the parameters for each LRT (e.g., window size, noise variance, and hypothesis threshold) can be selected to minimize error along a trajectory for a specific user. In further work [@Nilsson:2012], Nilsson showed that the higher hypothesis thresholds required for midstance detection during faster motion lead to increased error during walking. Rantakokko et al. [@Rantakokko:2012] reported that both movement and surface type ‘strongly influenced the performance’ of a zero-velocity-aided INS, and recommended ‘more robust stand-still detection algorithms’ for movements such as sprinting, jogging, sidestepping, ascending and descending stairs, and crawling. Additional work by Nilsson et al. [@Nilsson2:2014] also reported large error accumulation in INS estimates for crawling users (firefighters). Adaptive Thresholding --------------------- To create more robust zero-velocity detectors that work reliably across a range of motion types, several adaptive approaches have been presented in the literature. These techniques vary zero-velocity detection thresholds based on some characterization of the foot state. For instance, Walder and Bernoulli [@context-adaptive] developed a context-adaptive algorithm that aimed to detect midstance during common motion patterns such as walking, running, and crawling by using a velocity-dependent thresholding algorithm. In a similar manner, Ren et al. [@Sensors2016] introduced a velocity-based detector that used a state machine to transition between motion and zero-velocity states, with the transition probability governed by estimated velocity. They argued that their detector performed better than gyroscope-based detection during non-walking motions because angular velocity rates are non-zero at midstance when the user is moving quickly. Li and Wang [@Li:2012] presented a zero-velocity detection algorithm that worked during both walking and running motion by computing two individual zero-velocity detections, each tuned specifically for its motion type. Finally, Tian et al. [@Sensors2016_2] extracted the user’s gait frequency from the IMU signal and used it to adaptively update the zero-velocity threshold. We note that the majority of these methods adapt parameters of a single detector based on foot velocity or gait frequency. While this paradigm may be effective for smooth, continuous motions such as walking and running, it can fail to correctly account for other motion types such as stair-climbing, or crawling (where angular rates, which have been shown to be critical to detect midstance during walking [@ZVDetect], may not play a significant role). Indeed, selecting a single zero-velocity detector that can robustly identify midstance across a range of motions is a difficult task. Motion Classification --------------------- To overcome the challenge of achieving robust midstance detection across a range of motions, our work focuses on classifying motion to facilitate midstance detection. Given a motion type, we can select a detector that is optimized for that particular motion and obviate the need to burden a single detector with multiple motion types. Several machine learning techniques have been used for motion classification in the literature. Mannini and Sabatini [@Mannini2010] compared a variety of classification methods (Naive Bayes, logistic regression, nearest neighbour, and an SVM) to identify motion from several body-mounted sensors. Lau et al. [@svmclass] trained an SVM to predict motion types using data acquired from gyroscopes and accelerometers placed on the shank and foot. Their approach used inertial data collected from test subjects performing five walking motions: level-ground walk, upslope, downslope, stair ascent and stair descent. They reported a classification accuracy of 84.71% for a five-motion classifier, and 100% for two and three-motion classification. Using only foot-mounted inertial sensors, Park et al. [@Park2016] implemented a zero-velocity detection system that used two SVMs: one to classify a user’s motion, and another to identify midstance events. They reported midstance detection accuracies greater than $99\%$, but did not report localization improvements. In our work, we adopt a similar, SVM-based approach to motion classification. However, our system uses classification to modify parameters of an existing LRT-based detector, instead of implementing an entirely new learned mechanism. System Overview {#sec:overview} =============== In the sections that follow, we present our zero-velocity-aided INS, which incorporates a novel motion classifier for improved midstance detection. Zero-Velocity Aided INS with an Extended Kalman Filter ------------------------------------------------------ For our baseline INS, we use an EKF to track the state of a foot-mounted IMU. The filter’s nominal state[^3] consists of the IMU’s position ($\mbf p_k$), velocity ($\mbf v_k$), and orientation in quaternion form ($\mbf q_k$), $$\begin{aligned} \label{eq:state} \mathbf{x}_k &= \bbm \mbf p_k \\ \mbf v_k \\ \mbf q_k \ebm.\end{aligned}$$ To propagate the nominal state, the EKF uses a non-linear motion model $f(\cdot)$ given IMU inputs $\{\mathbf{a}_k^b, \boldsymbol{\omega}_k^b\}$[^4], $$\begin{aligned} \label{eq:naveq} \mbf{x}_k &= f(\mbf{x}_{k-1},\mbf{a}_k^b, \boldsymbol{\omega}_k^b) \nonumber \\ &= \bbm \mbf{p}_{k-1} + \mbf{v}_{k-1}\Delta t \\ \mbf{v}_{k-1} + \left( \mbf{R}(\mbf{q}_{k-1})\mbf{a}_k^b - \mbf{g} \right)\Delta t \\ \mbf{\Omega} (\boldsymbol{\omega}_k^b \Delta t)\mbf{q}_{k-1} \\ \ebm,\end{aligned}$$ where $k$ is a time index, $\Delta t$ is the sampling period, $\mathbf{R}(\cdot) $ is a function that maps quaternions to matrices, $\mathbf{g}$ is the gravity vector, and $\mbf{\Omega}(\boldsymbol{\phi})$ is a 4x4 matrix that updates the quaternion state based on an incremental rotation $\boldsymbol{\phi}$, which we compute through a backward, zeroth order integration of angular-rates (i.e., $\boldsymbol{\phi}=\boldsymbol{\omega}_k^b \Delta t$). Alongside the nominal state, the filter maintains a minimal error state and uses it to apply corrections when a midstance event is detected. During midstance, the filter incorporates a direct observation of velocity and fuses this pseudo-measurement with the motion model according to the standard EKF framework. For a more detailed explanation of an EKF and a zero-velocity-aided INS we refer the reader to [@Foxlin:2005; @Sola:2016]. For midstance detection, we choose to use stance hypothesis optimal detection (SHOE) for its robustness to changes in gait speed and high positional accuracy [@zupteval]. Intuitively, the SHOE detector thresholds the sum of the energy of angular rates and of linear accelerations with gravity ‘removed’, over a window of $W$ measurements. To obviate the need for a global orientation estimate, gravity is ‘removed’ by subtracting a vector of magnitude $g$ in the direction of average acceleration [@ZVDetect]. Concretely, SHOE tracks when the statistic $T_n(\mathbf{a}^b, \boldsymbol{\omega}^b)$, falls below $\gamma$: $$\label{eq:shoe_detector} \begin{split} T_n(\mathbf{a}^b, \boldsymbol{\omega}^b) &= \\ \frac{1}{W} \sum_{k=n}^{n+W-1} & \left( \frac{1}{\sigma_a^2}\norm{ \mathbf{a}_k^b - g\frac{\bar{\mathbf{a}}_n^b}{\norm{\bar{\mathbf{a}}_n^b}}}^2 + \frac{1}{\sigma_\omega^2}\norm{\boldsymbol{\omega}_k^b}^2 \right) < \gamma, \end{split}$$ where $W$ is the window size (the number of sensor readings the detector observes), $\sigma_a^2, \sigma_\omega^2$ are the variances of the specific force and angular rate measurements, $\mbf {\bar{a}}_n^b$ denotes the sample mean over $W$ samples, and $g$ is the magnitude of the local gravitational acceleration. Adaptive Zero-Velocity Detector with Motion Classification {#sec:realtimeINS} ---------------------------------------------------------- In this paper, we introduce an adaptive zero-velocity detector that uses an SVM classifier to determine the user’s motion type. SVMs are linear classifiers that use a hyperplane to maximally separate groups of differently labelled data. Contrary to other linear classifiers, SVMs can typically classify datasets that are not linearly separable by first transforming them to a higher dimensional feature space where they become linearly separable. Our classifier is trained with inertial data that is labelled with the user’s motion type (see ). illustrates the motion classifier’s ability to distinguish walking from running. We post-process the classifier output using a mean filter to remove abrupt motion transitions. Given a motion class, our zero-velocity detector selects an optimal parameter set for the current motion. The entire adaptive system operates in real-time using the Robot Operating System (ROS), displaying the estimated trajectory in RViz, a visualization tool [@rospaper]. ![Unfiltered (top) and filtered (bottom) classification results with associated ground truth.[]{data-label="fig:motionclass"}](figs/oliverwalksubplot.pdf){width="48.00000%"} Experiments =========== We outline and describe the validation of our proposed system in three sections. First, in , we present a method to select optimal parameters of the zero-velocity detector, tailored to a specific motion type and user. In we present our SVM-based motion classification in detail, and evaluate its accuracy with six motion classes. Finally, in we combine the classifier with a set of optimal parameters and evaluate the adaptive INS using an indoor dataset. \[sec:experiments\] Determining Optimal Thresholds for Midstance Detection {#sec:opt_thresh} ------------------------------------------------------ To select optimal parameters for a zero-velocity detector, we generate zero-velocity ground truth by tracking the motion of a subject’s foot using a Vicon infrared motion tracking system. While these parameters can be tuned indirectly by minimizing position error over a trajectory, we optimize zero-velocity events directly to avoid potential dependencies between position error and the geometry of a particular trajectory [@INS_characterization:2010]. In this work, we use the SHOE detector, fixing the parameters $W$ and $\frac{\sigma_a}{\sigma_\omega}$, while focusing our attention on $\gamma$.[^5] ![Our apparatus: an Inertial Elements MIMU22BTP(X) [@Gupta:2015] wireless 4-IMU array mounted to the foot of a test subject. Note that we use both the MIMU22BTP and its X variant (which has a larger lithium-ion battery but identical sensing hardware). []{data-label="fig:foot_imu"}](figs/foot_imu.pdf){width="48.00000%"} ### Data Collection {#sec:param_opt} To record training data, we mounted a wireless IMU to the foot of five different subjects. In all experiments presented in this paper, we used the Inertial Elements Osmium MIMU22BTP(X) (a wireless, Bluetooth-enabled MEMS-based 4-IMU sensor array [@Gupta:2015]). We relied on the internal processing of the sensor to fuse the four sets of inertial measurements into a single 6-axis reading, operating at 125 Hz. To observe the ground truth position of the IMU, we attached a Vicon marker to the sensor itself (see ) and recorded both the inertial data and Vicon motion tracking using ROS. All subjects wore their preferred pair of running shoes, and the IMU was mounted approximately in the centre of the foot using the shoe’s laces. Each user walked or ran for 10 laps within the Vicon tracking volume. A sample trajectory is shown in . In practice, we found that we only needed 1–2 laps of data (approximately 20 seconds) to extract optimal parameters. By numerically differentiating the Vicon position data, we computed foot speeds and applied a threshold to generate ground truth zero velocity events (see ). We used thresholds of 0.1 and 0.25 m/s for walking and running, respectively. The thresholds were empirically selected to ensure the ground truth captured every midstance event without exaggerating their length. ### F-Score Optimization We compared SHOE detector output with zero-velocity ground truth while varying $\gamma$ through the range $[10^2, 10^8]$. At each value of $\gamma$, we computed the precision ($P$) and recall ($R$) to form a precision-recall curve (). To select an optimal operating point, we maximized the $F_\beta$ score (): $$F_\beta = \left(1+\beta^2\right)\frac{P R}{\beta^2 P + R}.$$ In this F-measure, the $\beta$ parameter controls the importance of precision relative to recall. For $\beta < 1$, precision is favoured over recall, and decreasing $\beta$ generally moves the operating point to the left on the precision-recall curve. Empirically, we found that precision was slightly more important for walking compared to running, though both regimes required $\beta < 1$. In this paper, we use $\beta^2$ values of 0.16 and 0.4 for walking and running respectively, and leave a further investigation into potential detector trade-offs for future work. ### Results Table \[optimal\_gammas\] presents the $\gamma$ values that correspond to the maximum $F_\beta$ score for each user. Note that in all cases, the walking threshold is significantly smaller than the running threshold, as expected. [llccccc \| l]{} Subject & & 1 & 2 & 3 & 4 & 5 & Mean ------------------------------------------------------------------------ \ Optimal $\gamma$ & Walk & 0.90 & 1.20 & 1.19 & 0.36 & 1.15 & 0.96\ ($\times 10^5$) & Run & 5.96 & 6.55 & 10.02 & 32.55 & 10.49 & 13.11\ ------------------------------------------------------------------------ [0.49]{} ![image](figs/olivervicwalk.pdf){width="\textwidth"} [0.49]{} ![image](figs/olivervicwalkzv.pdf){width="\textwidth"} \ [0.49]{} ![image](figs/vicf1.pdf){width="\textwidth"} [0.49]{} ![image](figs/vicPrecRec.pdf){width="\textwidth"} Motion Classification {#sec:motionclassification} --------------------- To select which set of optimal parameters to use for zero-velocity detection, we must have some notion of motion type. In our work, we rely on a real-time classification of motion using an SVM classifier. Here, we discuss the training and evaluation of this classifier in more detail. ### Data Collection {#data-collection} We collected foot-mounted inertial data from five people, who each recorded six separate motion trials that consisted of either walking, jogging, running, sprinting, crouch-walking, or ladder-climbing (see ). For each trial, the users moved along a circular trajectory (with a radius of approximately 3 m) for 10 laps.[^6] ### Training To train the classifier, we pre-processed the motion trials in several steps. First, we removed 1000 data points (approximately 8 seconds of inertial data at 125 Hz) from the beginning and end of each trial to ensure that each trial consisted of pure motion data. Next, we normalized the three gyroscope and accelerometer channels of the IMU to ensure that they had similar magnitudes. We separated the normalized IMU samples into training and test sets, using the first half of each motion trial for training, and the second half for evaluation of a test set. Given the IMU specific force ($\mbf a_k$) and angular velocity measurements ($\boldsymbol{\omega}_k$) at timestep $k$, $$\mbf a_k = [a_k^x, a_k^y, a_k^z],$$ $$\boldsymbol{\omega}_k = [\omega_k^x, \omega_k^y, \omega_k^z],$$ we selected $K = 125$ (i.e. 1 second of data) adjacent IMU timesteps to form a training sample ($\mbf d_i$): $$\mbf d_i = [\mbf a_k, \boldsymbol{\omega}_k, \dots, \mbf a_{k+K-1}, \boldsymbol{\omega}_{k+K-1}].$$ Given a sample, $\mbf d_i$, we designed our SVM classifier ($g(\mbf d_i$)) to output a predicted motion type ($y_i$): $$g(\mbf d_i) = y_i \in \{0,1, \dots, 5\},$$ where the integers represent walking, jogging, running, sprinting, crouch-walking, and ladder-climbing respectively. For each motion type, we combined 1000 samples from each of the five users. Our entire dataset consisted of 30,000 training and 30,000 test samples in six motion classes (5000 samples per class). Using the Radial Basis Function (RBF) kernel, we trained the SVM with the six motion classes from all five subjects’ training sets. ### Results The confusion matrix in shows the test set’s classification accuracies for each motion type (averaged over the five subjects). The average motion classification rate was $91.1\%$. When simplifying to a binary classifier (walk vs. sprint) we achieved accuracies above $99.9\%$. ![Confusion matrix depicting SVM Motion classification output.[]{data-label="fig:svm_confusion_matrix"}](figs/confusionmatrix.pdf){width="48.00000%"} Adaptive, Classification-Based INS {#sec:adaptive_ins_eval} ---------------------------------- Combining our classifier with a set of optimal zero-velocity parameters, we evaluated the position accuracy of our adaptive INS to that of a fixed-threshold approach. For this, we limited our motion classification to the binary case of walking and running, and collected an extensive indoor dataset. ### Ground Truth Acquisition We created a ground truth trajectory through our hallway environment at the University of Toronto Institute for Aerospace Studies (UTIAS) by surveying six floor markers spaced at approximately 15 m intervals using a Leica Nova MS50 MultiStation (see ). Each marker consisted of an AprilTag [@apriltag] of 28 cm side length affixed to the floor. Although AprilTags were originally designed as a visual fiducial system, we used only their outline to define the orthogonal coordinate axes of each marker frame, with the bottom left corner of the AprilTag defining the marker itself. We note that the mapped AprilTags can also be used, in principle, to provide high-rate ground truth for a subject wearing a body-mounted camera, but we leave this as future work. Because all of the markers were not visible from a single surveying location, we mapped pairs of consecutive coordinate frames to define frame-to-frame transforms, and compounded them to compute the coordinates of each marker in a single navigation frame. To compute each frame-to-frame transform, we surveyed the coordinates of five locations on each of the two AprilTags in the MultiStation frame: two along one edge, two along an orthogonal edge, and one at the intersection of the two edges (which defined the origin of the frame). We then used an analytic point cloud alignment procedure [@Umeyama1991-ws] to compute the transformation between the MultiStation frame ($\cframe{m}$) and each of the two AprilTag frames ($\cframe{i}$ and $\cframe{i+1}$). We obtained the final transform between each of the AprilTag coordinate frames by evaluating $$\Matrix{T}_{i+1, i} = \Matrix{T}_{i+1, m} \Matrix{T}^{-1}_{i, m},$$ where $\Matrix{T} \in \text{SE(3)}$. We repeated this process for every set of adjacent AprilTags, and then compounded the transforms to compute all six marker positions in a global navigation frame. To estimate the accuracy of the ground truth markers, we applied this procedure in the forward and backward directions to determine the loop closure error. Our method achieved a loop closure error of 0.31 m over a path length of 130.6 m (0.24$\%$ error). For the final ground truth map, we used the position of the markers from the forward direction only, although we note that it is also possible to use pose-graph relaxation to incorporate both sets of measurements into one consistent map. Our accurate mapping procedure allows us to evaluate INS position error at intermediate points along the trajectory (as opposed to computing only a loop closure error based on the start and finish). Due to symmetries within a particular trajectory, loop-closure errors can be deceptively low compared to errors at other points along the trajectory. For example, if the zero-velocity detector is set to a higher-than-optimal value, zero-velocity detection will occur slightly before and after midstance. The INS will then underestimate the user’s step length [@Nilsson:2012]. However, if the user returns to the origin, the underestimation occurs in both directions along the trajectory, effectively removing its effect from the final loop closure error. By observing the position error at the furthest point from the origin, we can obtain a better quantification of error accumulation for these types of trajectories. ### Data Collection {#data-collection-1} We collected inertial data from five individuals who each recorded three walking, running, and combined running/walking motion trials along the trajectory with the surveyed ground truth. For each trial, users started at the origin, walked through a hallway (approximately 50 m), made one $90^o$ right-handed turn, walked a further 20 m to the furthest marker, and then turned around and retraced their steps to the origin (see ). At each ground truth marker, the subject pressed a handheld trigger that recorded a timestamp to facilitate temporal alignment with ground truth. For the combined trial, we instructed the users to alternate motions between every consecutive ground truth marker (beginning the trial with walking, and ending with running). ### Motion Classification {#motion-classification-1} We trained user-specific SVM motion classifiers with approximately one minute of walking and running data that we collected for each subject prior to evaluation. We filtered the SVM output in order to reduce artifacts around motion transitions by applying a mean filter, $$\bar{y}_{i} = \left\{ \begin{array}{ll} 1 & \frac{1}{W_s} \sum_{i}^{i+W_s} y_i \geq 0.2 \\ 0 & \frac{1}{W_s} \sum_{i}^{i+W_s} y_i \leq 0.2, \\ \end{array} \right.$$ where, for this work, we use $W_s = 15$. depicts an example of the unfiltered and filtered binary motion classifier while a user alternated between walking and running. Filtering with a threshold less than 0.5 causes the classifier to identify running rather than walking during motion transitions, reducing the likelihood that the walking threshold is applied to the first running steps when a user abruptly increases their movement speed (a case that often leads to missed midstance detection, and increased error). We note the classification requires 1 second of data (125 IMU samples) and therefore there is a short lag behind the true motion being performed. We compared the SVM motion classification with known ground truth (based on either the known motion class for that trial, or the handheld trigger signal for the combined trial). For the combined motion case, we note that our ground truth motion may differ from the user’s true motion because users cannot instantaneously transition between walking and running. Rather, there is a transition period, which we choose not to model in this work. [0.48]{} ![image](figs/hallway_topdown.pdf){width="\textwidth"} [0.48]{} ![image](figs/trevorwalktraj.pdf){width="\textwidth"} \ [0.48]{} ![image](figs/trevorruntraj.pdf){width="\textwidth"} [0.48]{} ![image](figs/trevorcombtraj.pdf){width="\textwidth"} ### Results For each trial, we determined pose estimates along the trajectory using the EKF-based INS described in . We used each of the user-specific optimized static thresholds ($\gamma_{walk}$ and $\gamma_{run}$), and also the proposed adaptive technique that applied the appropriate threshold (identified as $\gamma_{adapt}$) for the user’s current motion type $$\gamma_{adapt} = \left\{ \begin{array}{ll} \gamma_{walk} & \bar{y}_i=0 \\ \gamma_{run} & \bar{y}_i=1. \\ \end{array} \right.$$ To evaluate the accuracy of each thresholding method, we computed the Euclidean norm of 2D x-y position error at the furthest point of the trajectory. Figure \[fig:halltraj\] depicts three characteristic motion trials from subject 4, with trajectories produced by each of the three thresholding methods. Table \[tab:endRMSE\] summarizes the results of each thresholding method. First, we observe that, as expected, using the optimized threshold for a given motion results in the lowest error during single-motion trials: $\gamma_{walk}$ results in a lower position error than $\gamma_{run}$ for every user during walking, and $\gamma_{run}$ results in a lower error than $\gamma_{walk}$ for every user during running. Note that the difference in the running trials is more apparent because $\gamma_{walk}$ is too low to detect the majority of zero-velocity events while running. Owing to the high accuracy of our classifier, using the adaptive threshold for the pure motion cases results in position errors approximately equal to that for the optimized fixed thresholds. In the combined case, our adaptive approach ($\gamma_{adapt}$) resulted in more accurate position estimates than $\gamma_{walk}$ or $\gamma_{run}$, with an average end-point error of 2.68 m compared to 7.35 and 3.30 m for $\gamma_{walk}$ and $\gamma_{run}$, respectively. We note that the error reduction using $\gamma_{adapt}$ relies on the SVM being able to accurately classify motion type. shows that our SVM classifier achieved accuracies greater than 95% for pure walking and pure running trials, but a slightly lower accuracy (81% or better) for all combined motion trials. This reduction in accuracy can be explained, in part, by the fact that our ground truth motion types switch instantaneously between markers, while our subjects often needed a few steps to make the transition. Modelling this transition is left as future work. [cccccc]{} & & &\ & & & $\gamma_{walk}$ & $\gamma_{run}$ & $\gamma_{adapt}$\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ & 1 & 99.01 & 2.17 & 3.68 & 2.24\ & 2 & 97.93 & 1.33 & 2.32 & 1.26\ & 3 & 96.80 & 0.60 & 1.49 & 0.63\ & 4 & 98.61 & 0.62 & 4.97 & 0.60\ & 5 & 98.53 & 1.84 & 3.87 & 1.81\ & Mean & 98.18 & 1.31 & 3.27 & 1.31\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ & 1 & 95.56 & 18.37 & 3.93 & 3.54$^1$\ & 2 & 98.41 & 109.26 & 4.28 & 4.23\ & 3 & 98.06 & 98.11 & 2.90 & 2.91\ & 4 & 95.73 & 248.63 & 7.75 & 7.74\ & 5 & 96.63 & 165.57 & 2.65 & 2.67\ & Mean & 96.88 & 127.99 & 4.30 & 4.22\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ & 1 & 87.12 & 7.87 & 4.77 & 4.37\ & 2 & 81.23 & 5.05 & 3.91 & 3.51\ & 3 & 87.77 & 3.14 & 1.33 & 0.96\ & 4 & 82.39 & 20.04 & 3.69 & 2.69\ & 5 & 81.97 & 0.64$^2$ & 2.79 & 1.86\ & Mean & 84.10 & 7.35 & 3.30 & 2.68\ 1\. The mean error for the running trials is slightly lower for our adaptive approach, likely due to subject 1’s slow running pace. 2\. During the run/walk trial, subject 5 did not increase their running speed to the point where midstance detection using $\gamma_{walk}$ failed, resulting in a lower error than $\gamma_{adapt}$. Conclusion & Future Work ======================== We have presented an adaptive zero-velocity-aided INS that uses a motion classifier to improve tracking during sundry motion types. We evaluated our SVM-based classifier and reported classification accuracies exceeding 90% on a dataset of five subjects performing six different motion types. For a particular motion type, we described a method to compute optimal zero-velocity detector thresholds by maximizing an F-score given training data collected within a motion capture room. Combining the classifier with the optimal thresholds, we evaluated our final adaptive INS on a substantial indoor navigation dataset with 5.9 km of walking and running data from five different subjects. During pure walking or running motion, our system achieved localization accuracy that was equivalent to that achieved using a fixed, optimized threshold for that particular motion. In combined walking and running activities, our adaptive approach resulted in lower position errors than with either the optimized running or walking threshold alone. These results are a proof-of-concept demonstration of the capability of an adaptive, classification-based INS. In future work, we hope to analyze the effect of optimizing additional parameters, extend the real-time classification to other motion types and motion transitions, and to incorporate other zero-velocity detection paradigms (trained or hand crafted) that may work better for motions like crawling. [^1]: All authors are with the Space & Terrestrial Autonomous Robotic Systems (STARS) Laboratory at the University of Toronto Institute for Aerospace Studies (UTIAS), Canada [{brandon.wagstaff, valentin.peretroukhin}@robotics.utias.utoronto.ca, jkelly@utias.utoronto.ca]{} [^2]: Zero-velocity measurements can correct for position, velocity, accelerometer biases, pitch, roll, and the pitch and roll gyro biases. The yaw and yaw gyro bias remain unobservable during the zero-velocity update. [^3]: We do not incorporate any sensor biases into the state, as Nilsson et al. [@Nilsson:2012] note that doing so will not improve the accuracy of the system since the modelling error associated with zero-velocity measurements significantly outweighs error due to sensor bias. [^4]: $\mathbf{a}_k^b$ and $\boldsymbol{\omega}_k^b$ are the three-axis acceleration and angular velocity expressed in the IMU (body) frame. [^5]: We note that the threshold affects motion-specific detection more than the other tuning parameters do. While tuning other parameters may improve detection in non-walking motions, we leave this as future work. Our fixed parameters were: $W = 5$, $\sigma_a=0.01$, and $\sigma_\omega =0.00174$. [^6]: For ladder climbing, each subject ascended and descended a step ladder ten times.
--- abstract: \| Study of exoplanets is one of the main goals of present research in planetary sciences and astrobiology. Analysis of huge planetary data from space missions such as CoRoT and Kepler is directed ultimately at finding a planet similar to Earth – the Earth’s twin, and answering the question of potential exo-habitability. The Earth Similarity Index (ESI) is a first step in this quest, ranging from 1 (Earth) to 0 (totally dissimilar to Earth). It was defined for the four physical parameters of a planet: radius, density, escape velocity and surface temperature. The ESI is further sub-divided into interior ESI (geometrical mean of radius and density) and surface ESI (geometrical mean of escape velocity and surface temperature). The challenge here is to determine which exoplanet parameter(s) is important in finding this similarity; how exactly the individual parameters entering the interior ESI and surface ESI are contributing to the global ESI. Since the surface temperature entering surface ESI is a non-observable quantity, it is difficult to determine its value. Using the known data for the Solar System objects, we established the calibration relation between surface and equilibrium temperatures to devise an effective way to estimate the value of the surface temperature of exoplanets for further analysis with our graphic methodology. ESI is a first step in determining potential exo-habitability that may not be very similar to a terrestrial life. A new approach, called Mars Similarity Index (MSI), is introduced to identify planets that may be habitable to the extreme forms of life. MSI is defined in the range between 1 (present Mars) and 0 (dissimilar to present Mars) and uses the same physical parameters as ESI. We are interested in Mars-like planets to search for planets that may host the extreme life forms, such as the ones living in extreme environments on Earth; for example, methane on Mars may be a product of the methane-specific extremophile life form metabolism. author: - 'J. M. Kashyap' - 'S. B. Gudennavar' - Urmi Doshi - 'M. Safonova' title: 'Similarity indexing of exoplanets in search for potential habitability: application to Mars-like worlds.' --- Introduction ============ The search for life elsewhere outside the Earth has been a very fascinating area in recent years. A lot of efforts have been channeled in this direction in the form of space missions looking for potential habitable planets [e.g. @sea]. Analysis of huge volume of collected planetary data from space missions such as CoRoT and Kepler is directed at finding a planet similar to Earth – the Earth’s twin, and answering the question of exo-habitability. A full assessment of the habitability of any planet requires very detailed information about it, which we still cannot achieve. Currently, the best we can do is to compare the properties we can measure (or infer), in other words determine the potential habitability. Here, we consider habitability in the Earth way, or in some modified but still recognizable by us way (e.g. mars, or Titan, or Europa/Enciladus). The key parameters that are typically available from observations are stellar flux that can help to judge the temperature of the planet, and the planet’s size, derived either from transit observation – radius, or from RV measurements – low limit on mass. In the latter case, from the statistical point of view the true mass differs from the low limit by only a factor of $4/\pi \sim 1.2$ (e.g. Maruyana et al. 2013). With the developed theoretical models, it is already possible to exclude planets from the potential habitability list based on mass/radius values. Thus, since we prefer to start the search only on the rocky worlds, only planets of up to $1.6\,R_{\rm E}$ or $2\,M_{\rm E}$ can be rocky (e.g. Grosset, Mocqut&Sotin, 2007; Chen & Kipping, 2016). With the basic criteria and data already available, we can start gauging the potential habitability of exoplanets. The challenge here is to determine which exoplanet parameter(s) are important in finding this similarity. To address this challenge, the Earth Similarity Index (ESI), a parametric index to analyze the exoplanets’ data, was introduced to assess the Earth-likeness of exoplanets [@schu; @mascaro]. This multi-parameter ESI scale depends on radius, density, escape velocity and surface temperature of a planet. The ESI index is used by the Planetary Habitability Laboratory (PHL), University of Puerto Rico, Arecibo, to estimate the potential habitability of all discovered to date exoplanets. Here, the total ESI ranges from 0 (totally dissimilar to Earth) to 1 (identical to Earth). A planet with ESI $\geq 0.8$ is considered an Earth-like, but even planets with ESI $\geq 0.73$ (e.g. Mars) are optimistically called potentially habitable planets (PHPs). As of now (January 2017), there are 44 assumed PHPs[^1]. We focus on indexing the exoplanet data taken from the online Exoplanets Catalog (PHL-EC)[^2] maintained by the PHL. The ESI is subdivided into the interior ESI — an estimate of probability of a planet to have a rocky interior (calculated from radius and density), and exterior ESI – a surface similarity, an estimate of probability of a surface temperature to be within a habitable range (calculated from surface temperature and escape velocity); the total (global) ESI is their geometric mean. This way, the scale is useful for the overall concept of planet similarity for both interior and surface properties. One of our objectives here is to establish how the individual parameters, entering the interior and surface ESI, are contributing to the global ESI, using graphic analysis approach. Presently, one of the ESI parameters, the mean surface temperature, is estimated for the rocky planets in PHL-EC by following a correction factor of 30-33 K based on the Earth’s greenhouse effect. Another objective of our work in calculations of the global ESI is to try to introduce a better estimate of the surface temperature of rocky exoplanets, because it may be a crucial element in the search for habitable planets (Carone et al. 2016). From graphic analysis of the known data for some of the Solar System objects, we establish the calibrated relation between surface and equilibrium temperatures. This relation is extended to estimate the surface temperatures of exoplanets from their equilibrium temperatures. Though Earth is currently the only place where life as we know it exists, there is a good probability that life could have existed on Mars in the past (Abramov & Mojzsis, 2016). Indeed, Mars is technically inside the Sun’s habitable zone (HZ), though its ESI is only 0.73 (PHL). However, Mars-like conditions may be suitable for the extremophile life forms, such as the ones living in extreme environments on Earth (numerous experiments have proven several organisms to survive the simulated martian conditions, e.g. Onofri et al. 2015). Reports on the presence of methane in the martian atmosphere (Webster et al. 2015) raise the possibility of the existence of a methane-specific extremophile life forms; in fact, on Earth methanogens thrive in conditions similar to martian environments: in dry desert soils and in 3-km deep glacial ice (see Hu et al. 2016, and references therein). Therefore, we are interested in Mars similarity to look for planets potentially habitable by extremophiles, say extreme PHPs. For that, we introduce the new indexing formulation, called Mars Similarity Index (MSI), where the range is 0 (no similarity to present Mars) and 1 (present Mars). Mars here represents a sort of test-bed that can indicate the potential habitability of small rocky planets. A planet with MSI $> 0.63$ is recognized to be Mars-like, e.g. Earth, which is a habitable planet. We find from our data analysis six Mars-like planets (Mars, Earth, Kepler-186 f, Kepler-442 b, Kepler-438 b, Kepler-62 f, GJ 667C c, GJ 667C f), which are also included in the PHP list. The structure of this paper is as follows. In Section 2, we give the detailed derivation of the ESI and MSI scales. Data analysis is presented in Section 3, which includes ESI analysis, surface temperature analysis, and MSI analysis as subsections. Finally, discussion and conclusions are given in Section 4. Mathematical formulation of the ESI and MSI =========================================== Distance/similarity measurements are widely used in, for example, classification of objects, clustering and retrieval problems, or to compute the overlaps between quantitative data. Here, the distance $d$ is represented as dissimilarity, and proximity is equivalent to similarity $s$. Mathematically, the concept of distance is a metric one – a measure of a true distance in Euclidean space ${\bf R}^n$. This problem can be addressed by using Minkowski’s space of $L_p$ form [@Sung], in which $p$-norm stands for finite $n$-dimensional vector space, $$d_{\rm } =\left( \sum_{i=1}^{n}\|p_i-q_i\|^p \right)^{1/p}\,, \label{eq:Lp}$$ where the city-block, or Manhattan, $L_1$ distance $$d_{\rm CB} = \sum_{i=1}^{n}\|p_i-q_i\| \,, \label{SIIeq1}$$ and Euclidean $L_2$ distance $$d_{\rm Euc} =\left( \sum_{i=1}^{n}(p_i-q_i)^2\right)^{1/2}\,, \label{eq:euc}$$ are the special cases. Here, $p_i$ and $q_i$ are the coordinates of $p$ and $q$ in dimension $i$, and $i= 1, 2, 3,\ldots, n$. $L_1$ form has an advantage that it can be decomposed into contributions made by each variable being the sum of absolute differences (e.g. for the $L_2$ form, it would be the decomposition of the squared distance). However, in comparing multivariate data sets, some distance measures can be applied that do not conform to the usual definition of a metric (i.e. metric axioms, e.g. Deza and Deza 2016), but are still very useful as a measure of difference (or similarity) between samples. Here, we are interested in finding similarities between different planets based on their various characteristics, in other words, a multivariate data sets. In such cases, especially abundant in ecological and environmental studies to quantify the differences between samples collected at different sampling locations [e.g @12], the Bray-Curtis distance is the most used scale [@BrayCurtis]; it is also sometimes called an ecological distance [@TreeBook]. The advantages in using the ecological distance is that differences between datasets can be expressed by a single statistic. Bray-Curtis is a modified Manhattan distance[^3], where the summed differences between the variables are standardized by the summed variables of the objects, $$d_{\rm BC} = {\sum_{i=1}^{n}\|p_i-q_i\| \over \sum_{i=1}^{n}\left(p_i+q_i\right)}\,. \label{SIIeq2}$$ Here $p_i$ and $q_i$ are two different precisely measurable quantities between which the distance is to be measured, and $n$ is the total number of variables. The assumption in the Bray-Curtis scale is that samples are taken from same physical measure, e.g. mass, or volume. It is because the distance is found from the raw counts, so that if there is a higher abundance in one sample comparing to the other, it is a part of the difference between the two samples. The advantage of the Bray-Curtis scale is the simplicity in interpretation: $0$ means the samples are exactly the same, and $1$ means they are completely disjoint. It shall be kept in mind that Bray-Curtis distance is not the true metric distance since it violates the triangular inequality – one of the axioms of a true metric. Though it is still called a semi-metric in some earlier works, in modern mathematics it is defined as a non-metric [@Deza]. The intersection between two distributions is a more widely used form of similarity (as opposed to distance). Most similarity measures for intersection can be transformed from the distance measure by the transformation technique (see Bloom 1981, but not exclusively), $$s_{\rm BC} = 1- d_{\rm BC} = 1- {\sum_{i=1}^{n}\|p_i-q_i\| \over \sum_{i=1}^{n}\left(p_i+q_i\right)}\,. \label{eq:similarity}$$ Here, the value of $0$ means complete absence of relationships, and the value of $1$ shows a complete matching of the two data records in the $n$-dimensional space [@JanSchulz]. Distances/similarities based on heterogeneous data can be found after a process of standardization — balancing of the contribution of different types of variables in an equitable way [@Book]. One way to do that is to calculate the similarity for each set of homogeneous variables and then combine them using various methods (see Sec. \[sec:ESI CALCULATION\]). Higher values in one set may impact the result of the Bray-Curtis similarity more dominantly and imply that these variables are more likely to discriminate between sets. Therefore, user-defined weighting is a convenient (though subjective) method for down-weighing the differences for a set of variables. In our case, we would like to compare sets of different variables of one planet with that of the reference value, for example, Earth, to find planets that are similar to Earth. We rewrite Eq. (\[eq:similarity\]) as $$s=\left[1- \frac{\|x-x_0\|}{(x+x_0)}\right]^{w_x}\,, \label{last_sim}$$ where $x$ is the physical property of the exoplanet, $w_x$ is the weight for this property, $x_0$ is the reference value, and the dimension $n=1$, since we are constructing the index separately for each physical property. We find the weights by defining the threshold value ($V$) in the similarity scale for each quantity, $$V = \left[1-\Big\|\frac{x_0-x}{x_0+x}\Big\| \right]^{w_x}\,. \label{SIIeq6}$$ Traditionally, the similarity indices are subdivided into equal 0.2 intervals [@Bloom], defining very low, low, moderate, high and very high similarity regions. Therefore, the threshold can be defined on this grounds, for example considering only very high similarity region with the threshold $V=0.8$. Defining the physical limits $x_a$ and $x_b$ of the permissible variation of a variable with respect to $x_0$ (i.e. $x_a<x_0<x_b$), we calculate the weight exponents for the lower $w_a$ and upper $w_a$ limits, $$w_a = \frac {\ln{V}} {\ln\left[1-\left\|\frac{x_0-x_a} {x_0+x_a}\right\|\right]} \,,\quad w_b = \frac {\ln{V}} {\ln\left[1-\left\|\frac{x_b-x_0}{x_b+x_0}\right\|\right]} \,,\quad \label{eq:weight_exponent}$$ The average weight is found by the geometric mean, $$w_x=\sqrt{{w_a}\times {w_b}}\,. \label{eq:geom_mean}$$ In this paper, we use Eq. \[eq:similarity\] to define the Earth similarity index $$ESI_x = {\left[1-\Big\| \frac{x-x_0}{x+x_0}\Big\| \right]^{w_x}}\,, \label{eq:esi}$$ and Mars similarity index $$MSI_x = {\left[1-\Big\| \frac{x-x_0}{x+x_0}\Big\| \right]^{w_x}}\,, \label{msi}$$ where $x$ is the physical property of the exoplanet (for example, radius or density), and $x_0$ is the reference to Earth in ESI, and to Mars in MSI. We focus on indexing the exoplanet data taken from the online PHL Exoplanets Catalog (PHL-EC)[^4]. The catalog contains more than 60 observed and derived stellar and planetary parameters for all currently confirmed exoplanets. The input parameters for similarity scales are radius $R$, density $\rho$, surface temperature $T_S$ and escape velocity ${V_{e}}$. These parameters, except the surface temperature, are expressed in Earth Units (EU) in the ESI calculations. In addition, we normalized the mean radius, bulk density and escape velocity to Mars Units (MU) in the calculations of the Mars Similarity Index. The corresponding weight exponents for both ESI and MSI scales were found using the threshold value $V= 0.8$, indicating very high similarity region. The weight exponents for the upper and lower limits of parameters were calculated for the Earth-like parameter range [@schu]: radius 0.5 to 1.9 EU, mass 0.1 to 10 EU, density 0.7 to 1.5 EU, surface temperature 273 to 323 K, and escape velocity 0.4 to 1.4 EU, through Eqs. (\[eq:weight\_exponent\]) and (\[eq:geom\_mean\]). Similarly, the weight exponents for the lower and upper limits of parameters were defined for the Mars-like conditions: radius range 0.72 to 1.88 MU, mass range 0.514 to 9.30 MU, density range 0.89 to 1.402 MU, surface temperature range 233 to 418 K, and escape velocity range 0.85 to 2.23 MU. Here, MU are Mars Units, where radius is 3390 km, density is 3.93 g/cm$^3$, escape velocity is 5.03 km/s, and the mean surface temperature 240 K [@MarsBook]. The reason behind the limits definitions were to have a rocky planet, with lower limit in comparison to Mars (mass and radius are chosen as for Mercury, the smallest planet in our Solar System, and density as for Io), and with Earth as the upper limit. The temperature range is chosen on the basis of the range known to be suitable for extremophile life forms on our planet, between $-40^{\circ}$ and $+145^{\circ}$C [@Tung2005]. The corresponding weight exponents were calculated using the same method as for the ESI, and are given in Table \[table:3.1\] along with Earth and Mars unit parameters. [l\[4]{}[c]{}r]{} Planetary Property &Ref. Value & Ref. Value & Weight Exponents & Weight Exponents\ & for ESI & for MSI & for ESI & for MSI\ Mean Radius & 1EU & 1MU & 0.57 & 0.86\ Bulk Density & 1EU &1MU & 1.07 & 2.10\ Escape Velocity &1EU& 1MU& 0.70 & 1.09\ Surface Temperature & 288K& 240K&5.58 & 3.23\ \ [EU = Earth Units, with Earth radius 6371 km, density 5.51 g/cm$^3$, escape velocity 11.19 km/s.\ MU = Mars Units, with Mars radius 3390 km, density 3.93 g/cm$^3$, escape velocity 5.03 km/s.]{} Data analysis ============= The PHL-EC contains 3635 confirmed exoplanets (as of January 2017). However, some of them do not have all the required input parameters for calculating the global ESI. Wherever we could, we supplemented the missing data after the additional search through the following catalogs: Habitable Zone Gallery, Open Exoplanet Catalogue, Extrasolar Planets Encyclopaedia, Exoplanets Data Explorer and Nasa Exoplanet Archive. In addition, we discarded the entries with unrealistic values. For example, some of the planets had the equilibrium/surface temperatures of less than 3.2 K, and some had the density values of around 500 EU. In such cases, we have done extensive search through all available exoplanet catalogs and discovery papers, and supplemented those values that we could find. As an example, there is a lot of confusion with the available data on Kepler-53c, Kepler-57c and Kepler-59b planets. Their densities are listed in the PHL-EC as 162, 573.18 and 492 Earth Units, respectively. In such cases, the density of, say, Kepler 57c, becomes 21 times the density of the Sun’s core. There is obviously a mistake in the retrieved data. We have searched through the catalogs and found that for Kepler 53c, the mass of 5007.56 EU used in calculating that density was, in fact, an upper limit from the stability analysis (Steffen et al. 2013), and it was subsequently updated (Haden et al. 2014) to $35.4^{+19.}_{-14.8}$ EU with nearly the same value for the radius. Using this number, the density becomes 1.169 EU. Similarly, for Kepler 57c, the density of 573.18 EU was obtained using the upper limit on mass of 2208.83 EU which, after updating the mass to $7.4^{+9.4}_{-6.3}$ EU (Haden et al. 2014) with essentially the same radius, fell within the normal range, 1.139 EU. Correspondingly, we have corrected the data for these planets in our catalog. Some of the entries had to be removed owing to the absence of available data (or very conflicting values), which left us with 3566 exoplanets for our analysis (1650 of them are rocky). SURFACE TEMPERATURE ANALYSIS ---------------------------- Surface temperature of the planet enters the exterior ESI index – a measure of surface similarity. Usually, the extrasolar planets temperatures are estimated from the calculated temperature of the parent star and other observational data (e.g. distance to the star, etc.) [@Nature], in the assumption that the planet does not have an atmosphere e.g. [e.g. @williams], which gives a so-called radiative equilibrium temperature. It is essentially an effective temperature attained by an isothermal planet that is in complete radiation equilibrium with the parent star. If a planet has no atmosphere, this temperature equals the surface temperature of the planet. Actual temperatures of the gas giants are usually higher because there is an internal heat source. If the planet has a substantial atmosphere, then this is the temperature near the tropopause of the planet – approximately the level where the radiation from the planet is emitted to space. Terrestrial planets with atmospheres, such as Earth or Venus, for example, have higher temperatures due to the greenhouse effect. So, though equilibrium temperature may be different from the actual surface temperature of the planet, it is a useful parameter in comparing planets. It has been used to estimate the boundaries of Habitable Zones around stars (e.g. Kaltenegger and Sasselov 2011; Schulze-Makuch et al. 2011), and is an essential quantity for exoplanets when it is not possible to measure (or calculate) the surface T. The albedo entering the surface temperature equation is generally not known and has to be assumed. The albedo depends on many factors, some of which are geometry, composition, and atmospheric properties. In addition, there is also the emissivity index which represents atmospheric effects, and depends on the type of surface and many climate models. In calculations of the surface temperatures of rocky planets by the PHL, a correction factor of $30-33$ K is usually used, based on the Earth’s green-house effect . However, we shall not ignore the good quality data available for other planets in our Solar System, because unlike with the exoplanets, many parameters are actually been measured. We have used the known data of our Solar System objects (only rocky bodies), using several available sources, such as NASA planetary fact sheet [@williams], and extrapolated the resulting relation to obtain the surface temperatures of rocky exoplanets in our dataset. The relation between the surface temperature $(T_s)$ and the equilibrium temperature $(T_e)$ (Fig. \[figure: 3.4\]) is $$T_s=9.65+1.0956\times T_e\,, \label{eq:regression}$$ ![Calibration of surface temperature. Venus is masked as red dot due its very high surface temperature.\[figure: 3.4\]](SurfaceT.jpg){width="9cm"} In Table \[Table:3.3\], we present the equilibrium and surface temperatures of several Solar System objects, and the sample of our results for a few potentially habitable exoplanets. [l\[4]{}[c]{}r]{} Planet & Equilibrium Temp & Surface Temp & $T_s=9.65+1.0966 \, T_e$\ & (K) & (K) & (K)\ Earth & 255 & 288 & 289.28\ Mars & 217 & 240 & 247.61\ Moon & 157 & 197 & 181.81\ Venus & 227 & 730 & 258.57\ Titan & 82 & 94 & 99.57\ Triton & 34.2 & 38 & 47.15\ GJ 667Cc & 246.5 & 277.4 & 279.96\ Kepler-442 b& 233 & unknown & 265.16\ Kepler-438 b& 276 & unknown & 312.31\ GJ 667 C f & 220.7 & unknown & 251.67\ \[Table:3.3\] Surface temperature of the Moon varies from Farside 120 K to the 350 K of a subsolar point, hence we adopted the average surface temperature for the Moon of 197 K as the most physically robust estimate [@Vol]. This empirical correlation is, of course, very simplified. However, such regression analysis was already successfully used to predict the mean global temperatures of rocky planets in @Vol1 (see their Figs. 1 – 4). The objective of our analysis was to evaluate how well the resulting new relation may predict the observed mean surface temperatures; in Table \[Table:3.3\] we show the measured and calculated by Eq. 12 surface temperatures, and observe that calculated values are pretty close to the measured ones. ESI CALCULATION {#sec:ESI CALCULATION} --------------- First we converted the input parameters to Earth Units (EU), except the surface temperature, which is left in Kelvin. Some of the planets in the PHL-EC do not have estimates for the surface temperature. To mitigate this, we have calculated the surface temperature of all rocky planets using Eq. (\[eq:regression\]). The corresponding Earth Similarity Index for each parameter was calculated using Eq. (\[eq:esi\]). Then these indices were separately combined to form an interior similarity and surface similarity. The interior ESI is thus $$ESI_I = \sqrt{{ESI_R} \times {ESI_\rho}}\,, \label{eq:interiorESI}$$ and surface ESI is: $$ESI_S = \sqrt{ESI_T \times ESI_{V_e}}\,, \label{eq:surfaceESI}$$ where $ESI_R$, $ESI_\rho$, $ESI_T$ & $ESI_{V_e}$ are Earth Similarity Indices, calculated for radius, density, surface temperature and escape velocity, respectively. The global ESI is their geometric mean: $$ESI = \sqrt{{ESI_I} \times {ESI_S}}\,. \label{eq:globalESI}$$ The results of the ESI calculations for all 3566 currently confirmed exoplanets are presented in Table \[Table:3.2\] (only few planets are shown as an example). The full dataset is available online (Kashyap et al. 2017). The sample calculation of ESI for Mars is given in the Appendix. [l\[8]{}[c]{}r]{} Names & Radius & Density &Temp & E. Vel & $ESI_S$ & $ESI_I$ & ESI\ & (EU) & (EU) & (K) & (EU) & & &\ Earth & 1.00 & 1.00 & 288 &1.00 & 1.00 &1.00 &1.00\ Mars & 0.53 & 0.73 & 240 &0.45 & 0.65 &0.82 &0.73\ Kepler-438 b & 1.12 & 0.90 & 312 & 1.06& 0.88 &0.95 & 0.91\ Proxima Cen b & 1.12 & 0.90 & 259 & 1.06& 0.85 &0.95 & 0.90\ GJ 667C c & 1.54 & 1.05 & 280 & 1.57& 0.88 & 0.92 & 0.90\ Kepler-296 e & 1.48 & 1.03 & 302 & 1.5& 0.86 &0.93 & 0.89\ \[Table:3.2\] ESI ANALYSIS ------------ Results of Section 3.1 are presented as a histogram of global ESI (Fig. \[figure: 3.2\]). According to the PHL project, surface ESI is dominating the interior ESI, because the surface temperature weight exponent value is much higher than that of the interior parameters. We found, however, that this is only true for the giant planets. For the rocky planets, we found that the interior ESI is a predominant factor in the global ESI, where the real values of interior and surface ESI play a larger role than the weight exponent. The 3-D histogram (Fig. \[figure: 3.1\]) is the result of overplotting interior and surface ESI for all the rocky exoplanets. ![Histogram of the global ESI values of 3566 exoplanets[]{data-label="figure: 3.2"}](ESI.jpg){width="9cm"} ![3-D histogram of interior and surface ESI for 1650 rocky planets.[]{data-label="figure: 3.1"}](3D_ESI.jpg){width="9cm"} In Fig. \[fig:ESIplot\], we present a scatter plot of interior ESI versus surface ESI. Blue dots are the giant planets, red dots are the rocky planets, and cyan circles are the Solar System objects. The dashed curves are the isolines of constant global ESI, with values shown in the plot. Planets above ESI=0.8 are conservatively considered Earth-like, and planets with ESI $\gtrsim 0.73$ are optimistically potentially habitable planets (PHL). We see that there are 29 Earth-like planets (ESI > 0.8) in 3566 planets that we have considered. In this plot we also see the predominant nature of the interior ESI. However, due to the geometrical mean nature of the global ESI formula, we need to consider all the four parameters to check the habitability of the planet. We also find from the plot that there seem to be a definite division between gaseous and rocky planets, at approximately $ESI_I=0.67$ (interior ESI of the Moon). It is interesting to note that this division separates Moon and Io, rocky satellites, (especially Io, which is closer in bulk composition to the terrestrial planets) and, say Pluto and Europa, which are composed of water ice–rock. ![Plot of interior ESI versus surface ESI. Blue dots are the giant planets, red dots are the rocky planets, and cyan circles are the Solar System objects (Table 3). The dashed curves are the isolines of constant global ESI, with values shown in the plot. The giant planets above 0.67 dotted lines are of water-gas composition, and the planets to the right of 0.67 dotted line are rocky planets. Planets with ESI$\gtrsim 0.8$ are considered Earth-like. However, the optimistic limit is $\sim 0.67$.[]{data-label="fig:ESIplot"}](ESIplot2017.png){width="13cm"} When the ESI index was proposed, it was accepted that even planets with ESI between 0.6 and 0.8 could be potentially habitable, or at least similar to Earth. Thus, we propose to extend the optimistic limit from 0.73 to 0.67. There are now we about 100 rocky exoplanets with ESI $> 0.67$. For example, the ESI of Kepler-445 d is 0.76. It is located in the HZ, and has an estimated surface temperature of 305 K, which would make it suitable for life. MSI CALCULATION --------------- In our Solar System, we have discovered many planetary bodies with conditions similar to some of the terrestrial environments where we know that Earth extremophiles live. Particularly notable here is Mars. It is now believed that Mars had a much wetter and warmer environment in its early history (Grotzinger et al. 2015; Wray et al. 2016). The discovery of a desert varnish on Mars (Krinsley et al. 2012), believed to be the product of the specific bacteria on Earth (Dorn and Oberlander 1981), has gotten us even further down the road of whether life existed/exists on Mars with its super-extreme conditions for habitation. We introduce here the Mars Similarity Index (MSI), calculated using Eq. (\[msi\]), to look for Mars-like planets as potential planets to host extremophile life forms. We are interested in this study to compare extreme environments similar to Mars. Interior MSI is $$MSI_I = \sqrt{{MSI_R} \times {MSI_\rho}}\,,$$ and the surface MSI is $$MSI_S = \sqrt{MSI_{T_S} \times MSI_{V_e}}\,,$$ where $MSI_R$, $MSI_\rho$, $MSI_{T_S}$ & $MSI_{V_{e}}$ are Mars similarity indices calculated for radius, density, surface temperature and escape velocity, respectively. The global MSI is given by: $$MSI = \sqrt{{MSI_I} \times {MSI_S}}$$ A result of MSI calculations is presented in Table \[Table:msi\] (only few entries are shown). The full dataset is available online (Kashyap et al. 2017). For the graphic analysis part we have used all 3566 confirmed exoplanets as in the ESI case. [l\[8]{}[c]{}r]{} Names & Radius & Density &Temp & E. Vel & $MSI_S$ & $MSI_I$ & MSI\ & (MU) & (MU) & (K) & (MU) & & &\ Mars & 1.00 & 1.00 & 240 & 1.00 & 1.00 & 1.00 & 1.00\ Earth & 1.88 & 1.407 & 288 & 2.23 & 0.659 & 0.70 & 0.68\ Moon & 0.513 & 0.853 & 197 & 0.48 & 0.66 & 0.77 & 0.71\ Kepler-42 d & 1.07 & 1.05 & 503& 1.11 & 0.58 & 0.95 & 0.74\ Kepler-378 c & 1.29 & 1.08 & 462& 1.37 & 0.59 & 0.90 & 0.73\ Kepler-438 b & 2.10 & 1.23 & 312& 2.35 & 0.70 & 0.73 & 0.72\ Proxima Cen b& 2.10 & 1.23 & 259& 2.35 & 0.69 & 0.73 & 0.71\ \[Table:msi\] ANALYSIS OF THE MSI ------------------- Results of Section 3.3 are represented as a histogram of the global MSI (Fig. \[fig:MSI\]) for 3566 confirmed exoplanets. In Fig. \[fig:MSI\_two\], we show the 3-D histogram of the interior and surface MSI. As with the ESI, we can see that interior MSI is more dominant factor than surface MSI for the rocky exoplanets in the global MSI. ![Histogram of the global MSI values for 3566 confirmed exoplanets.[]{data-label="fig:MSI"}](MSI.jpg){width="10cm"} ![Interior and Surface MSI for 1650 rocky planets.[]{data-label="fig:MSI_two"}](3D_MSI.jpg){width="10cm"} In Fig. \[fig:MSIplot\], we present a scatter plot of interior MSI versus surface MSI for 3566 confirmed exoplanets. The dashed curves are the isolines of constant global MSI, with values shown in the plot (the value for the Earth is 0.68). Planets above MSI $\sim 0.63$ are considered Mars-like. For example, Kepler-186 f has MSI$\gtrsim 0.69$ is potentially habitable for extremophile life forms. There is a noticeable difference of similarity to Mars planets (29) compared with the Earth-like planets, where we find 99 of the Earth-like PHPs. The reason is that, probably due to the selection bias, smaller planets are under-represented in the catalog of detected planets. In Fig. \[fig:density\], we present mass-radius plots for small size planets. Left plot is in the Earth units for planets of $\leq 20$ Earth masses, and the right plot in Mars units for planets $\leq 2$ Earth masses, or $\sim 20$ Mars masses. The regular features seen on this plot, as well as on Figs. 4 and 7, are due to the use of modelled values, inferring mass from radius or radius from mass. We have plotted the model curves for the mass-radius relation adopted by the PHL on both plots. It is obvious that data for the most of the rocky planets was estimated from these relations. Planets that are noticeably not following the relations are those whose mass and radius were estimated independently, such as, e.g. Kepler-138 b (Jontof-Hutter et al. 2015). ![Plot of interior MSI versus surface MSI. Blue dots are the giant planets, red dots are the rocky planets, and cyan circles are the Solar System objects. The dashed curves are the isolines of constant global MSI, with values shown in the plot. Planets with MSI$\gtrsim 0.63$ are assumed Mars-like; we obtain 29 planets.[]{data-label="figure: 3.7"}](MSIplot2017.png){width="13cm"} \[fig:MSIplot\] ![[Left]{}: Mass-radius diagram for exoplanets with measured masses less than 20 EU along with model curves for different mass-radius relation: black line is $R=M^{0.3}$ for $M_{\rm E} < 1$; blue-dotted line is $R=M^{0.5}$ for $1< M_{\rm E}< 200$. Red crosses indicate rocky planets, blue crosses are gas giants, cyan squares are our Solar System objects. In 2016, the data in the catalog suggested only two rocky exoplanets smaller than Earth. In the present data, there are many more smaller planets. [Right]{}: Blow-up of the previous plot for small-size planets, in terms of the Mars units. Line of same mass-radius relation are marked on the plot, along with the isolines of constant density. Some interesting planets are marked by names.[]{data-label="fig:density"}](MassRadius.png "fig:"){width="8.7cm"} ![[Left]{}: Mass-radius diagram for exoplanets with measured masses less than 20 EU along with model curves for different mass-radius relation: black line is $R=M^{0.3}$ for $M_{\rm E} < 1$; blue-dotted line is $R=M^{0.5}$ for $1< M_{\rm E}< 200$. Red crosses indicate rocky planets, blue crosses are gas giants, cyan squares are our Solar System objects. In 2016, the data in the catalog suggested only two rocky exoplanets smaller than Earth. In the present data, there are many more smaller planets. [Right]{}: Blow-up of the previous plot for small-size planets, in terms of the Mars units. Line of same mass-radius relation are marked on the plot, along with the isolines of constant density. Some interesting planets are marked by names.[]{data-label="fig:density"}](small_MassRadius.png "fig:"){width="8.7cm"} Discussion and Conclusion ========================= The current definition of the habitable planets as those that are to be found in habitable zones (HZ) of the hosts has a caveat. A habitable planet may not necessarily harbour life (e.g. Venus), although from a distance its biosignatures may show positive signs; while Earth-like planets (similar atmosphere, size, mass, at a similar distance and orbiting a solar-type star) with high probability may actually have life on them (e.g. Jheeta 2013). Other factors can come in next, such as, for example, the age of the planet [e.g. @safonova] since life requires time to change its environment to become noticeable. The search for habitable exoplanets has essentially two goals, both of which (if fulfilled) will have profound implications to our civilization. One is to seek life elsewhere outside the Earth. Another one is to seek a twin-Earth, preferably nearby. The second goal, in principle, is to have a planet habitable for our kind of life, but uninhabited, so that we can shift there in the far away future, if needed. Another aspect of the second goal is that it is easier to search for the biosignatures of life as we know it on a planet which looks just like Earth. It is estimated that one in five solar-type stars and approximately half of all M-dwarf stars may host an Earth-like planet in the HZ. Extrapolation of Kepler’s data shows that in our Galaxy alone there could be as many as 40 billion such planets (e.g. Borucki et al. 2010; Batalha et al. 2013; Petigura et al. 2013). And it is quite possible that quite soon we may actually observe most of them. With the ultimate goal of a discovery of life, astronomers do not have millennia to quietly sit and sift through more information than even pentabytes of data. In addition, obtaining the spectra of an Earth-like planet around a Sun-like star is difficult, and would require a large-scale expensive space mission (such as e.g. JWST), which still may be able to observe only about a hundred stars over its lifetime (e.g., Turnbull et al. 2012). Thus, it is necessary to prioritise the planets to look at, to introduce a quick estimate of whether a planet can be habitable from the measured properties of the star and the planet. The Earth Similarity Index (ESI), a parametric index to analyze the exoplanets data, was introduced to access precisely that; to evaluate the potential habitability (Earth-likeness) of all discovered to date exoplanets. Since our search for habitable exoplanets (aka Earth-like life, which is clearly favoured by Earth-like conditions) is by necessity anthropocentric, as all we know for sure is only the Earth-based habitability, any such indexing has to be centred around finding Earth-like life, at least initially. Here, we have shown how the ESI can be derived from the initial mathematical concept of similarity. Out of the four parameters, entering the global ESI, only one (radius) is a direct observable, while the remaining three parameters, surface temperature, escape velocity and density, are generally calculated. According to the PHL project, surface ESI is dominating the interior ESI, because the weight exponent value for the surface temperature is much higher than that of the interior parameters. We found, however, that the interior ESI is a predominant factor in the global ESI for the rocky exoplanets, where the real values of interior and surface ESI play a larger role than the weight exponents (Fig. 2). However, even though evaluation of only radius and density parameters may be enough to suggest a rocky nature of an exoplanet, due to the geometrical mean nature of the ESI formulation, we need to consider the surface temperature to verify the Earth-likeness. For example, if we consider surface temperature values as 10 K, 100 K and 2500 K, and keep interior ESI the same as for the Earth, the corresponding global ESI values will be 0.02, 0.40 and 0.11, respectively; clearly not habitable. Thus, the surface temperature and plays a key role in balancing the global ESI equation. However, there is always an observational difficulty in estimating the surface temperature value of the exoplanet. We introduced the calibration technique in Section 3.2 to try to mitigate this difficulty for the case of rocky planets. We have used the existing observational data from Solar System rocky objects and performed the linear regression to extrapolate to all rocky exoplanets. It is now believed that in its early history, Mars had a much wetter and warmer environment, just at the time when life on Earth is now known to have originated (this date was recently moved back to 4.1 Ga (Gigayears ago) (Bell et al. 2015). Curiosity data indicates early ($\sim 3.8$ Ga) Martian climate with stable water lakes on the surface for thousands to millions of years at a time (Grotzinger et al. 2015), and a recently discovered evidence of carbonate-rich ($\sim 3.8$ Ga) bedrock (Wray et al. 2016) suggested the habitable warm environment. It is possible that after the presumed catastrophic impact-caused loss of most of the atmosphere (e.g., Melosh et al., 1989; Webster et al. 2013), only the toughest life forms had survived, the ones we call here on Earth as extremophiles. They would have adapted to the currently existing conditions and just like the terrestrial extremophiles would need such conditions for the survival; for example, terrestrial methanogens have developed biological mechanism that allows them to repair DNA and protein damage to survive at temperatures from $-40^{\circ}$C to $145^{\circ}$C [@Tung2005]. The usual conditions for habitability would be different for such life forms. Carbon and water have the dominant role as the backbone molecule and a solvent of biochemistry for Earth life. However, the abundance of carbon may not be a useful indication of the habitability of an exoplanet. The Earth is actually significantly depleted in carbon compared with the outer Solar System. Here, on Earth, we have examples of life, both microbial and animal, that do not require large amounts of water either. For example, both bacteria and archaea are found thriving in the hot asphalt lakes (Schulze-Makuch et al. 2011b) with no oxygen and virtually no water present. They respire with the aid of metals, perhaps iron or manganese, and create their own water by breaking down hydrocarbons, just like [E. coli]{} gut bacteria that can generate most of their own water from light hydrocarbons (Kreuzer-Martin et al. 2005). We have introduced the Mars similarity index to study the Mars-like planets as potential planets to host extremophile life forms. In this scale, Moon has the MSI of $\sim 0.75$, Earth has the MSI of $\sim 0.68$, and the next closest exoplanet is Kepler-186 f (MSI=0.69), which is listed as potentially habitable planet in the HEC. Mars-like planets can tell us about the habitability of small worlds rather than planets that are far from their star. For example, Earth at Mars distance would most probably still be habitable [@MarsClimate]. Given constant exchange of impact ejecta between the planets, it is possible that biota from the Earth reached and survived on Mars, which thus could have been ‘extremophile’-habitable throughout all its history. It is interesting to note that when we started this work, only two small (less than Earth in size or mass) exoplanets were known. In this year, many new small planets were discovered, with resulting 29 planets that we can call Mars-like. So can such similarity scale be useful? It actually might be developed into a sort of same scale as the stellar types in astronomy, such as classifying stars based on information about size, temperature, and brightness. It can be used as a quick tool of screening planets in important characteristics in Earth-likeness. Different ranking scales for evaluating habitability perspectives for follow-up targets have been already proposed (e.g. habitability index for transiting exoplanets (HITE, Barnes et al. 2015), or Cobb-Douglas Habitability Index (CDHI, Bora et al. 2016). We conclude that it is necessary to arrive at the a multiparameter calculator that, though based on a current similarity scale – ESI, may include more input parameters (e.g. orbital properties, temperature, escape velocity, radius, density, activation energy and so on), and extended to other planet-likeness, such as similarity to Mars – MSI. If we find habitable possibilities on Titan for example, the scale can be modified for the Titan-like habitability. We would like to call this future calculator a Life Information Score (LIS), which shall be used as an overall calculator to detect life itself. The LIS is almost similar to the anthropic selection, which basically deals with the preconditions for the emergence of life and, ultimately, intelligent observers [@Wal]. But the expected outcome of this LIS is to accurately measure the possibility of a planet to host any form of life using only the parametric data. Calculation of Mars ESI as an example {#calculation-of-mars-esi-as-an-example .unnumbered} ===================================== $ESI_x$ calculations for Mars are performed using Eq. (\[eq:esi\]), with weight exponents from Table \[table:3.1\], by using the following values for the input parameters, $$\begin{aligned} &R=0.53 \times 6371\,km = 3376.63\,km\,,\nonumber\\ & \rho=0.71 \times 5.51\, g/cm^3 = 3.9121 \,g/cm^3\,,\nonumber\\ &V_{e}= 0.45 \times 11.19\, km/s = 5.0355\, km/s \,,\nonumber\\ &T_{s}= 240 \, K\,. \nonumber\end{aligned}$$ The ESI for each parameter are, accordingly, $$\begin{aligned} &ESI_R=\left(1-\left\|3376.63\,km-6371\,km\right\|\Big/ \left\|3376.63\,km +6371\,km\right\|\right)^{0.57} = 0.8124\,,\nonumber\\ & ESI_{\rho} =\left(1-\|3.9121\, g/cm^3 - 5.51 \,g/cm^3\|\Big/\|3.9121 \,g/cm^3+5.51 g/cm^3\|\right)^{1.07} = 0.8218 \,,\nonumber \\ & ESI_{v_{e}} =\left(1-\|5.0355\, km/s-11.19\, km/s\|\Big/\|5.0355\, km/s+11.19\, km/s\|\right)^{0.7} = 0.7162\,,\nonumber\\ &ESI_{T_{s}} =\left(1-\|240\,K-288\,K\|\Big/\|240\,K+288\,K\|\right)^{5.58} = 0.5875\,.\nonumber\end{aligned}$$ Interior ESI from Eq. (\[eq:interiorESI\]) is:\ $ESI_I = \sqrt{0.8124\times0.8218} \approx 0.8171$.\ Surface ESI from Eq. (\[eq:surfaceESI\]) is:\ $ESI_S = \sqrt{0.7162 \times 0.5875} \approx 0.6487$.\ And the global ESI for Mars (Eq. \[eq:globalESI\]) is:\ $ESI = \sqrt{0.8171 \times 0.6487} \approx 0.728$. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Jayant Murthy (Indian Institute of Astrophysics, Bangalore) and Yuri Shchekinov (Lebedev Institute, Moscow) for the fruitful discussions. 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--- abstract: 'State-of-the-art approaches for image captioning require supervised training data consisting of captions with paired image data. These methods are typically unable to use unsupervised data such as textual data with no corresponding images, which is a much more abundant commodity. We here propose a novel way of using such textual data by artificially generating missing visual information. We evaluate this learning approach on a newly designed model that detects visual concepts present in an image and feed them to a reviewer-decoder architecture with an attention mechanism. Unlike previous approaches that encode visual concepts using word embeddings, we instead suggest using regional image features which capture more intrinsic information. The main benefit of this architecture is that it synthesizes meaningful thought vectors that capture salient image properties and then applies a soft attentive decoder to decode the thought vectors and generate image captions. We evaluate our model on both Microsoft COCO and Flickr30K datasets and demonstrate that this model combined with our semi-supervised learning method can largely improve performance and help the model to generate more accurate and diverse captions.' author: - \| Wenhu Chen\ RWTH Aachen\ [hustchenwenhu@gmail.com]{} - \| Aurelien Lucchi\ ETH Zürich\ [aurelien.lucchi@inf.ethz.ch]{} - \| Thomas Hofmann\ ETH Zürich\ [thomas.hofmann@inf.ethz.ch]{} - '\' bibliography: - 'egbib.bib' title: 'A Semi-supervised Framework for Image Captioning' ---
--- abstract: 'Cherenkov telescopes have the capability of detecting high energy tau neutrinos in the energy range of 1–1000PeV by searching for very inclined showers. If a tau lepton, produced by a tau neutrino, escapes from the Earth or a mountain, it will decay and initiate a shower in the air which can be detected by an air shower fluorescence or Cherenkov telescope. In this paper, we present detailed Monte Carlo simulations of corresponding event rates for the VERITAS and two proposed Cherenkov Telescope Array sites: Meteor Crater and Yavapai Ranch, which use representative AGN neutrino flux models and take into account topographic conditions of the detector sites. The calculated neutrino sensitivities depend on the observation time and the shape of the energy spectrum, but in some cases are comparable or even better than corresponding neutrino sensitivities of the IceCube detector. For VERITAS and the considered Cherenkov Telescope Array sites the expected neutrino sensitivities are up to factor 3 higher than for the MAGIC site because of the presence of surrounding mountains.' address: - 'Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erwin-Rommel-Str. 1, D-91058 Erlangen, Germany' - 'DESY, Platanenallee 6, D-15738 Zeuthen, Germany' - 'Institute of Nuclear Physics PAN, Radzikowskiego 152, Kraków, Poland' author: - 'D. Góra' - 'E. Bernardini' - 'A. Kappes' title: Searching for tau neutrinos with Cherenkov telescopes --- Introduction ============ Neutrinos have long been anticipated to help answering some fundamental questions in astrophysics like the mystery of the source of the cosmic rays (for a general discussion see [@uhecr]). For neutrinos in the TeV range, prime source candidates are Galactic supernova remnants [@remnats]. Neutrinos in the PeV range and above are suspected to be produced by Active Galactic Nuclei (AGN) and Gamma Ray Bursts (GRB) with many AGN models predicting a significant neutrino flux [@Atoyan:2004pb; @PhysRevD.80.083008; @Mucke2003593]. Recently, the IceCube Collaboration has reported the very first observation of a cosmic diffuse neutrino flux which lies in the 100 TeV to PeV range [@science_icecube]. Individual sources, however, could not be identified up to now. While many astrophysical sources of origin have been suggested [@icecube_inter], there is yet not enough information to narrow down the possibilities to any particular source. Due to the low interaction probability of neutrinos, a large amount of matter is needed in order to detect them. One of the detection techniques is based on the observation of inclined extensive air showers (EAS) induced by taus from tau neutrino interactions deep in the atmosphere. As these showers are initiated close to the surface of the Earth, they are still very young when reaching the detector and hence have a significant electromagnetic component leading to a broad time structure of the detected signal. In contrast, showers from cosmic-ray nuclei are induced in the upper atmosphere and therefore have a strongly reduced electromagnetic component when reaching the detector. However, because of its low density, neutrino interactions are not very likely to happen inside the atmosphere. A solution to this problem is to look for so-called Earth skimming (up-going) tau neutrinos [@fargion0; @fargion1; @zas; @Letessier:2001; @feng; @tseng; @aramo] which interact within the Earth or a mountain and produce a tau. For neutrino energies of about a EeV, the charged leptons have a range of a few kilometers and hence may emerge from the Earth or mountain, decay shortly above the ground and produce EAS detectable by a surface detector. In some cases, two consecutive EASs might be observable, one coming from a tau neutrino interaction close to the surface and one from the decay of the resulting tau lepton. These two showers, coming from the same direction in a time interval corresponding to the tau decay time, can generate a unique signature in the detector called a Double-Bang event [@learne]. The detection of such a Double-Bang event would be very important both from the astrophysical and the particle physics point of view, as it would be an unambiguous sign for an ultra-high energy (UHE) tau-neutrino. Up to now, there has been no clear identification of tau neutrinos at high energies. The detection of PeV tau neutrinos through optical signals also seems possible. A combination of fluorescence and Cherenkov light detectors in the shadow of steep cliffs could achieve this goal [ [@fargion0; @fargion1; @doi:10.1142/S0217732304014458]]{}. Recently, it has also been shown by the All-sky Survey High Resolution Air-shower detector (Ashra) experiment, that such kind of experiments could be sensitive to tau neutrinos from fast transient objects such as nearby GRBs [@Asaoka:2012em].We note, that the recent IceCube results do not show any neutrino events at or above the Glashow resonance at 6.3 PeV [@science_icecube]. This likely means that there is either a cutoff in the astrophysical neutrino flux below $\sim 6$ PeV or the neutrino spectrum is steeper than the usually assumed $E^{-2}$ spectrum. In principle also existing Imaging Air Cherenkov Telescopes (IACTs) such as MAGIC [@magic], VERITAS [@veritas] and H.E.S.S. [@hess] have the capability to detect PeV tau neutrinos by searching for very inclined showers [@fargion]. In order to do that, the Cherenkov telescopes need to be pointed in the direction of the taus escaping from the Earth crust, i.e. at or a few degree below the horizon. This is because the trajectory of the tau lepton has to be parallel to the pointing direction of the telescope within a few degrees as the Cherenkov light is very much beamed in the forward direction. For example, the MAGIC telescope is placed on top of a mountain on La Palma at an altitude of about 2200m a.s.l. Thus, it can look down to the Sea and monitor a large volume within its field of view (FOV). In [@upgoing_magic], the effective area for up-going tau neutrino observations with the MAGIC telescope was calculated analytically with the maximum sensitivity in the range from 100TeV to $\sim$ 1EeV. However, the calculated sensitivity for diffuse neutrinos was very low because of the limited FOV, the short observation time and the low expected neutrino flux. On the other hand, if flaring or disrupting point sources such as GRBs are observed, one can expect an observable number of events even from a single GRB if close by. In the case of MAGIC, however, the topographic conditions allow only for a small window of about 1 degree width in zenith and azimuth to point the telescope downhill. In case of other IACT sites with different topographic conditions, the acceptance for up-going tau neutrinos will be increased by the presence of mountains. Mountains can work as an additional target and will lead to an enhancement in the flux of emerging tau leptons. A target mountain can also shield against cosmic rays and star light. For Cherenkov telescope sites, very often nights with high clouds prevent the observation of gamma-ray sources. In such conditions, pointing the telescopes to the horizon could significantly increase the observation time and the acceptance for up-going tau neutrinos. Next-generation Cherenkov telescopes, i.e. the Cherenkov Telescope Array (CTA) [@cta], can in addition exploit their much larger FOV (in extended observation mode), a higher effective area and a lower energy threshold. In this work, we present an update of the work in [@ouricrc], where a detailed Monte Carlo simulation of event rates induced by Earth skimming tau neutrinos was performed for an ideal Cherenkov detector. Neutrino and lepton propagation was simulated taking into account the local topographic conditions at the MAGIC site and at four possible locations of Cherenkov instruments: two in Argentina (San Antonio, El Leoncito), one in Namibia (Kuibis) and one on the Canary Islands (Tenerife). In this work, similar simulations have been performed for the location of the VERITAS telescopes and for two sites located close to VERITAS: Meteor Crater and Yavapai Ranch. These two sites were recently also considered as possible locations for CTA. Results are shown for a few representative neutrino fluxes expected from giant AGN flares. We would like to stress that in this work we are exploring the effect of different topographic conditions rather than providing a comprehensive survey of potential sites. Method ====== The propagation of a given neutrino flux through the Earth and the atmosphere is simulated using an extended version of the ANIS code [@gora:2007]. For fixed neutrino energies, $10^{6}$ events are generated on top of the atmosphere with zenith angles ($\theta$) in the range $90^{\circ}$–$105^{\circ}$ (up-going showers) and with azimuth angles in the range $0^{\circ}$–$360^{\circ}$. Neutrinos are propagated along their trajectories of length $ \Delta L$ from the generation point on top of the atmosphere to the interaction volume, defined as the volume which can contribute to the expected event rate, in steps of $\Delta L$/1000 ($\Delta L/1000 \geq 6$ km). At each step of propagation, the $\nu$–nucleon interaction probability is calculated according to parametrization of its cross section based on the chosen parton distribution function (PDF). In particular, the propagation of tau leptons through the Earth is simulated. All computations are done using digital elevation maps (DEM) [@dem] to model the surrounding mass distribution of each site under consideration. The flux of the leptons emerging from the ground as well as their energy and the decay vertex positions are calculated inside an interaction volume, modeled by a cylinder with radius of 35km and height 10km. The detector acceptance for an initial neutrino energy $E_{\nu_\tau}$ is given by: $$\begin{aligned} A(E_{\nu_\tau}) =N_{\mathrm{gen}}^{-1} \times \sum_{i=1}^{N_{k}} P_{i}(E_{\nu_\tau},E_{\tau},\theta) \nonumber \\ \times T_{\mathrm{eff},i}(E_{\tau},x,y,h,\theta) \times A_i(\theta)\times \Delta \Omega, \label{aperture}\end{aligned}$$ where $N_{\mathrm{gen}}$ is the number of generated neutrino events. $N_k$ is the number of $\tau$ leptons with energies $E_{\tau}$ larger than the threshold energy $E_{\mathrm{th}}=1$PeV and a decay vertex position inside the interaction volume. $P(E_{\nu_\tau},E_{\tau},\theta)$ is the probability that a neutrino with energy $E_{\nu_\tau}$ and zenith angle $\theta$ produces a lepton with energy $E_{\tau}$ (this probability was used as “weight” of the event). $A_i(\theta)$ is the physical cross-section of the interaction volume seen by the neutrino and $\Delta\Omega$ is the solid angle. $T_{\mathrm{eff}}(E_{\tau},x,y,h,\theta)$ is the trigger efficiency for tau-lepton induced showers with the decay vertex position at ($x$, $y$) and height $h$ above the ground. The trigger efficiency depends on the response of a given detector and is usually estimated based on Monte-Carlo simulations. In this work, we used an average trigger efficiency extracted from [@Asaoka:2012em], namely $\langle T_{\mathrm{eff}} \rangle =10$%, which is comparable to what was calculated for up-going tau neutrino showers studied in [@doi:10.1142/S0217732304014458]. This is a qualitative estimation and as such it is the major source of uncertainty on the results presented hereafter. ![\[fig::spectrum2\] [A sample of representative neutrino fluxes from photo-hadronic interactions in AGNs. See text for more details. Flux-1 and Flux-2 are calculations for $\gamma$-ray flare of 3C 279 [@2009IJMPD]. Flux-3 and Flux-4 represent predictions for PKS 2155-304 [@Becker2011269]. Flux-5 corresponds to a prediction for 3C 279 calculated in [@PhysRevLett.87.221102]. ]{}](Fig1.eps){width="\columnwidth"} Equation (\[aperture\]) gives the acceptance for diffuse neutrinos. The acceptance for a point source can be estimated as the ratio between the diffuse acceptance, defined in Eq. (\[aperture\]), and the solid angle covered by the diffuse analysis, multiplied by the fraction of time the source is visible $f_{\mathrm{vis}}(\delta_{s},\phi_{\mathrm{site}})$. This fraction depends on the source declination ($\delta_{s}$) and the latitude of the observation site ($\phi$). In this work, the point source acceptance is calculated as $ A^{\mathrm{PS}}(E_{\nu_\tau})\simeq A(E_{\nu_\tau}) / \Delta \Omega \times f_{\mathrm{vis}}(\delta_{s},\phi_{\mathrm{site}})$. For energies above 1PeV, neutrinos no longer efficiently penetrate the Earth and are preferentially observed near the horizon where they traverse a reduced chord of the Earth. Thus the main contribution to the acceptance comes from a few degrees below horizon. In this work, we simulated propagation of tau neutrinos with zenith angles in the range $90^{\circ} - 105^{\circ}$ corresponding to $\Delta \Omega=1.62$sr. ![image](Fig2A.eps){width="\columnwidth" height="6.1cm"} ![image](Fig2B.eps){width="\columnwidth" height="6.1cm"} ![image](Fig2C.eps){width="\columnwidth" height="6.1cm"} ![image](Fig2D.eps){width="\columnwidth" height="6.1cm"} ![image](Fig2E.eps){width="\columnwidth" height="6.1cm"} ![image](Fig2F.eps){width="\columnwidth" height="6.1cm"} Flux-1 Flux-2 Flux-3 Flux-4 Flux-5 --------------------------------------------------------- -------- -------- -------- -------- -------- -- -- $N_{\mathrm{La Palma}}$ 2.8 1.5 0.86 8.6 2.6 $N_{\mathrm{VERITAS}}$ 8.2 3.9 1.6 16 6.9 $N_{\mathrm{Meteor Crater}}$ 9.0 4.2 1.7 17 7.5 $N_{\mathrm{Yavapai Ranch}}$ 7.7 3.7 1.5 15 6.5 $N^{\mathrm{Northern \mbox{ } Sky}}_{\mathrm{IceCube}}$ 6.8 2.5 0.46 4.6 8.8 $N^{\mathrm{Southern \mbox{ } Sky}}_{\mathrm{IceCube}}$ 11.0 3.2 0.76 7.6 8.8 In Fig. \[fig::spectrum2\], a compilation of fluxes expected from AGN flares are shown. Flux-1 and Flux-2 are calculations for Feb 23, 2006 $\gamma$-ray flare of 3C 279 [@2009IJMPD]. Flux-3 and Flux-4 represent predictions for PKS 2155-304 in low-state and high-state, respectively [@Becker2011269]. Flux-5 corresponds to a prediction for 3C 279 calculated in [@PhysRevLett.87.221102]. The total observable rates (number of expected events) were calculated as $N=\Delta T \times \int_{E_{\mathrm{th}}}^{E_{\mathrm{max}}} A^{\mathrm{PS}}(E_{\nu_\tau})\times\Phi(E_{\nu_\tau})\times dE_{\nu_\tau}$, where $\Phi(E_{\nu_\tau})$ is the neutrino flux and $\Delta T$ the observation time (3 hours in Table \[tab::rate222\]). Figure \[fig::magic\] shows the expected event rates for a detector with an average trigger efficiency of 10% located at the VERITAS site, Meteor Crater and Yavapai Ranch, respectively, as a function of the incoming neutrino direction (defined by the zenith and azimuth angles). Correlations can be observed between the expected rate and the local topography. For the VERITAS site, the expected number of events from South-West and South-East is larger than from other directions due to the larger amount of matter encountered by incoming neutrinos from these directions. For tau lepton energies between $10^{16}$eV and $10^{17}$eV, the decay length is a few kilometers, hence, detectable events should mainly come from local hills not further away than about 50km. For Meteor Crater the largest mass distribution is seen in the South-West, thus for this direction the calculated event rate is the largest. In case of the Yavapai Ranch site, the topography map shows many hills around the site leading to a rather complicated dependence of the event rate on azimuth and zenith. It is also worth to remark, that the presence of too many local hills close to the detector can in fact lead to a decrease of the up-going tau lepton flux and hence to a smaller event rate, as showers induced by up-going tau neutrinos are attenuated in the hills. Results {#sec:results} ======= In Fig. \[fig::acccc\] we show as a function of the neutrino energy the estimated point source acceptance for the three studied sites in comparison to the La Palma site (MAGIC) and other possible locations of Cherenkov instruments. Due to the lack of results from IceCube in the tau-neutrino channel, we use IceCube’s muon neutrino acceptance [@IC-80-acc] for a sensitivity comparison. This is motivated by the fact that at Earth we expect an equal flavor flux from cosmic neutrino sources due to full mixing [@mixing]. In [@up-icecube; @diff-icecube] it is also shown that for neutrino energies between 1PeV and 1000PeV, the muon-neutrino acceptance is only slightly larger than that for tau neutrinos. The IceCube acceptance shows an increase for energies between $10^{6}$GeV and $10^{9}$GeV, and is on average about $~2\times10^{-3}$ km$^{2}$. A potential detector with an average trigger efficiency of 10% located at the VERITAS, Meteor Crater or Yavapai Ranch site can have an acceptance as large as a factor 10 greater than IceCube in the northern sky at energies larger than $\sim 5\times 10^{7}$GeV. For Flux-3 and Flux-4, this results in an event rate that is about a factor 3–4 larger than the rate calculated for IceCube in the northern sky assuming three hours of observation time (see Table \[tab::rate222\]). For neutrino fluxes covering the energy range below $\sim 5\times10^{7}$GeV (Flux-1, Flux-2, Flux-5), the number of expected events for these sites is comparable to that estimated for IceCube. Thus, Cherenkov telescopes pointing to the horizon, which is not their typical operating mode, could have a sensitivity comparable or even larger than that of neutrino telescopes such as IceCube in case of short neutrino flares (i.e. with a duration of about a few hours). For longer durations, the advantage of neutrino telescopes as full-sky 100%-duty cycle instruments will be relevant. This can partially be compensated by Cherenkov telescopes with observations during nights with high clouds. ![\[fig::acccc\] Acceptance for point source, $A^{\mathrm{PS}}(E_{\nu_\tau})$ to earth-skimming tau neutrinos as estimated for the La Palma site and a sample selection of few future locations of Cherenkov instruments (with a trigger efficiency of 10%) and IceCube (as extracted from [@IC-80-acc]). ](Fig3.eps){width="\columnwidth"} Table \[tab::rate222\] also shows that in case of sites surrounded by mountains (VERITAS, Meteor Crater and Yavapai Ranch) event rates are at least a factor two higher than for sites without surrounding mountains (La Palma). The influence of systematic uncertainties is evaluated in terms of final event rates. We studied the influence on the expected event rate arising from uncertainties on the tau-lepton energy loss and different neutrino-nucleon cross-sections. The average energy loss of taus per distance travelled (unit depth $X$ in gcm$^{-2}$) can be described as $\left\langle dE/dX \right \rangle = \alpha(E) + \beta(E) E$. The factor $\alpha(E)$, which is nearly constant, is due to ionization. $\beta(E)$ is the sum of $e^+e^-$-pair production and bremsstrahlung (both well understood) and photonuclear scattering, which is not only the dominant contribution at high energies but at the same time subject to relatively large uncertainties. In this work, the factor $\beta_{\tau}$ is calculated using the following models describing contribution of photonuclear scattering: ALLM [@allm], BB/BS [@bbbs] and CMKT [@ckmt], and different neutrino-nucleon cross-sections: GRV98lo [@GRVlo], CTEQ66c [@cteq], HP [@hp], ASSS [@CooperSarkar:2007cv], ASW [@Albacete:2005ef]. Results are listed in Table \[tab::rate\] for Flux-1 and Flux-3. model PDF $\beta_{\tau}$ sum -------- ------------------- ------------------- -------------------- -- Flux-1 $^{+14\%}_{-2\%}$ $^{+2\%}_{-7\%}$ $^{+14\%}_{-7\%}$ Flux-3 $^{+42\%}_{-7\%}$ $^{+7\%}_{-14\%}$ $^{+43\%}_{-16\%}$ : \[tab::rate\] [Relative contributions to the systematic uncertainties on the up-going tau neutrino rate. As a reference value the expected event rate for the La Palma site calculated for Flux-1 and Flux-3 was used.]{} Definitive conclusions on the capability of Cherenkov telescopes for detecting tau neutrinos, however, require accurate neutrino-trigger and background simulations which go beyond the scope of this paper. The acceptance/sensitivity to point sources also depends on the source declination ($\delta_s$), which defines the fraction of a day for which a source is visible in the sky at zenith angles between $90^\circ$ and $105^\circ$. Figure \[fig::visibility\] shows this fraction as a function of source declination for the different detector locations. As an example, 3C 279 ($\delta_s = - 5.8^{\circ}$) and PKS 2155-305 ($\delta_s = - 30.22^{\circ}$) are observed for $\sim 11$% of a sidereal day (for about 2.4 hours per day), while for point sources with declination $\delta_s =-60^{\circ}$ the observation time is at least two times longer. The plot also shows that for the different detector locations the visibility has a quite similar behavior. In conclusion, even though the sensitivity window of the Earth-skimming analysis in zenith angle is small ($15^{\circ}$), it permits a point source survey in a broad range in declination angles spanning more than $100^{\circ}$ in the sky. ![\[fig::visibility\] Fraction of a sidereal day in percent having a point-like source at declination $\delta$ detectable with zenith angle $ 90^{\circ} <\theta<105^{\circ}$ i.e. for the simulated zenith range of Earth-skimming (up-going) neutrinos . The fraction of time is plotted as a function of source declination for a few selected sites.](Fig4.eps){width="\columnwidth"} Summary ======= In this paper, detailed Monte Carlo simulations have been employed to evaluate the sensitivity of air Cherenkov telescopes to high-energy cosmic tau neutrinos in the PeV to EeV energy range coming from the horizon. 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--- abstract: \| We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a generalization of a result of Kazdan-Kramer about existence of solutions of certain quasi-linear elliptic equations. [Mathematics Subject Classification:]{} (2000): 58C40, 53C42 [Key words:]{} Barta’s theorem, Cheng’s Eigenvalue Comparison Theorem, spectrum of Nadirashvili minimal surfaces, stability of minimal hypersurfaces. author: - 'G. Pacelli Bessa' - 'J. Fábio Montenegro[^1]' title: \| An Extension of Barta’s Theorem\ and Geometric Applications --- Introduction ============ The fundamental tone $\lambda^{\ast}(M)$ of a smooth Riemannian manifold $M$ is defined by $$\lambda^{\ast}(M)=\inf\{\frac{\smallint_{M}\vert {{{\rm{grad}\,}}}f \vert^{2}}{\smallint_{M} f^{2}};\, f\in {L^{2}_{1,0}(M )\setminus\{0\}}\}$$ where $L^{2}_{1,\,0}(M )$ is the completion of $C^{\infty}_{0}(M )$ with respect to the norm $ \Vert\varphi \Vert^2=\int_{M}\varphi^{2}+\int_{M} \vert\nabla \varphi\vert^2$. When $M$ is an open Riemannian manifold, the fundamental tone $\lambda^{\ast}(M)$ coincides with the greatest lower bound $\inf \Sigma$ of the spectrum $\Sigma \subset [0,\infty)$ of the unique self-adjoint extension of the Laplacian $\triangle$ acting on $C_{0}^{\infty}(M)$ also denoted by $\triangle$. When $M$ is compact with piecewise smooth boundary $\partial M$ (possibly empty) then $\lambda^{\ast}(M)$ is the first eigenvalue $\lambda_{1}(M)$ of $M$ (Dirichlet boundary data if $\partial M\neq \emptyset$). A natural question is what bounds can one give for $\lambda^{\ast}(M)$ in terms of Riemannian invariants? Or if $M$ is an open manifold, for what geometries does $M$ have $\lambda^{\ast}(M)>0$? See [@kn:berger-gauduchon-mazet], [@kn:chavel], [@kn:G] or [@kn:schoen-yau]. A simple method for giving bounds the first Dirichlet eigenvalue $\lambda_{1}(M)$ of a compact smooth Riemannian manifold $M$ with piecewise smooth boundary[^2] $\partial M\neq \emptyset$ is Barta’s Theorem. All Riemannian manifolds in this paper are smooth and connected. \[barta\]Let $M$ be a bounded Riemannian manifold with piecewise smooth non-empty boundary $\partial M$ and $f\in C^{2}(M)\cup C^{0}(\overline{M})$ with $f\vert M>0$ and $f\vert \partial M=0$ and $\lambda_{1}(M)$ be the first Dirichlet eigenvalue of $M$. Then $$\label{eqBarta1}\sup_{M} ( -\triangle f/f) \geq \lambda_{1}(M ) \geq \inf_{M} (-\triangle f/f).$$With equality in (\[eqBarta1\]) if and only if $f$ is a first eigenfunction of $M$. \[remarkBarta\]To obtain the lower bound for $\lambda_{1}(M )$ we may suppose that $f\vert\partial M \geq 0$. Cheng applied Barta’s Theorem in a beautiful result known as Cheng’s eigenvalue comparison theorem. Let $N$ be a Riemannian $n$-manifold and $B_{N}(p,r)$ be a geodesic ball centered at $p$ with radius $r<inj (p)$. Let $c$ be the least upper bound for all sectional curvatures at $B_{N}(p,r)$ and let $\mathbb{N}^{n}(c)$ be the simply connected $n$-space form of constant sectional curvature $c$. Then $$\label{eqThmcheng1}\lambda_{1}(B_{N}(p, r))\geq \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r)).$$ In particular, when $c=-1$ and $inj (p)=\infty$, Cheng’s inequality (\[eqThmcheng1\]) becomes $\lambda^{\ast}(N)\geq \lambda^{\ast}(\mathbb{H}^{n}(-1))$ which is McKean’s inequality, see [@kn:mckean]. Our main result gives lower bound for the fundamental tone $\lambda^{\ast}(M)$ of an arbitrary smooth Riemannian manifolds $M$ in terms of the divergence of certain vector fields regardless the smoothness degree of its boundary. Let $M$ be a Riemannian manifold and a vector field $X\in L^{1}_{loc}(M)$ $($meaning that $\vert X\vert \in L_{loc}^{1}(M))$. A function $g\in L_{loc}^{1}(M)$ is a weak divergence of $X$ if $$\int_{M} \phi \,g =-\int_{M}\langle {{{\rm{grad}\,}}}\phi,\,X\rangle,\,\, \forall \,\phi \in C^{\infty}_{0}(M).$$\[defDiv1\] It is clear that there exists at most one weak divergence $g \in L^{1}_{loc}(M)$ for a given $X\in L^{1}_{loc}(M)$ and we may write $g={{{\rm{Div}\,}}}X$. For $C^{1}$ vector fields $X$ the classical divergence ${{{\rm{div}\,}}}X$ and the weak divergence ${{{\rm{Div}\,}}}X$ coincide. \[defDiv2\]Let ${\cal W}^{1,1}(M)$ denote the Sobolev space of all vector fields $ X \in L^{1}_{loc}(M)$ possessing weak divergence $ {{{\rm{Div}\,}}}X $. \[remarkW11\] If $X \in {\cal W}^{1,1}(M)$ and $f\in C^{1}(M)$ then $fX\in {\cal W}^{1,1}(M)$ with ${{{\rm{Div}\,}}}(f X) = \langle {{{\rm{grad}\,}}}f,\,X\rangle + f \,{{{\rm{Div}\,}}}X$. In particular for $f \in C^{\infty}_{0}(M)$ we have that $$\label{eqDivergence1}\int_{M}{{{\rm{Div}\,}}}(fX)=\int_{M} \langle {{{\rm{grad}\,}}}f,\,X\rangle - \langle {{{\rm{grad}\,}}}f,\,X\rangle=0.$$Conversely, if $fX\in {\cal W}^{1,1}(M)$ for all $f\in C_{0}^{\infty}(M)$ then $X\in {\cal W}^{1,1}(M)$. Our main result is the following theorem. \[teoremaPrincipal\]Let $M$ be a Riemannian manifold. Then $$\label{eqThmP1} \lambda^{\ast}(M)\geq \sup_{{\cal W}^{1,1} }\{\inf_{ M} ({{{\rm{Div}\,}}}X- \vert X \vert^{2})\}.$$ If $M$ is a compact Riemannian manifold with smooth boundary then $$\label{eqThmP2}\lambda_{1}(M)= \sup_{{\cal W}^{1,1} }\{\inf_{ M} ({{{\rm{Div}\,}}}X- \vert X \vert^{2})\} .$$ Our first geometric application of Theorem (\[teoremaPrincipal\]) is an extension of Cheng’s lower eigenvalue estimates. We show that inequality (\[eqThmcheng1\]) is valid for arbitrary geodesic balls $B_{N}(p,r)$ provided the $(n-1)$-Hausdorff measure ${\cal H}^{n-1}({\rm Cut} (p)\cap B_{N}(p,r))=0$, where ${\rm Cut}(p)$ is the cut locus of $p$. Moreover, we show that equality of the eigenvalues occurs if and only if $B_{N}(p,r)$ and $B_{\mathbb{N}^n(c)}(r)$ are isometric. \[Cheng\] Let $N$ be a Riemannian $n$-manifold with radial sectional curvature $K(x)(\partial t, v)\leq c$, $x\in B_{N}(p,r)\setminus {\rm Cut}(p)$, $v\perp \partial t$ and $\vert v\vert \leq 1$. Let $\mathbb{N}^{n}(c)$ be the simply connected $n$-space form of constant sectional curvature $c$ and suppose that ${\cal H}^{n-1}({\rm Cut} (p)\cap B_{N}(p,r))=0$. Then $$\label{eqCheng}\lambda^{\ast}(B_{N}(p,r))\geq \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r)).$$Equality in (\[eqCheng\]) holds iff $B_{N}(p,r)$ and $B_{\mathbb{N}^{n}(c)}(r)$ are isometric. Our second geometric application (Theorem \[submanifold\]) is a generalization of the following Cheng-Li-Yau’s result proved in [@kn:cheng-li-yau]. Let $M^{m}\subset \mathbb{N}^{n}(c)$ be an immersed $m$-dimensional minimal submanifold of the $n$-dimensional space form of constant sectional curvature $c\in\{-1,0,1\}$ and $D\subset M^{m}$ a $C^{\,2}$ compact domain. Let $a=\inf_{p\in D}\sup_{z\in D} dist_{\mathbb{N}^{n}(c)}(p,z)>0$ be the outer radius of $D$. If $c=1$ suppose that $a\leq \pi/2$. Then $$\label{eqCLY1}\lambda_{1}(D)\geq \lambda_{1}(B_{\mathbb{N}^{m}(c)}(a))$$ Equality in (\[eqCLY1\]) holds iff $M$ is totally geodesic in $\mathbb{N}^{n}(c)$ and $D=B_{\mathbb{N}^{m}(c)}(a)$. \[submanifold\]Let $N$ be a Riemannian $n$-manifold with radial sectional curvature $K(x)(\partial t, v)\leq c$, for all $x\in B_{N}(p,r)\setminus {\rm Cut}(p)$, and all $v\perp \partial t $ with $\vert v\vert \leq 1$. Let $M\subset N$ be an $m$-dimensional minimal submanifold and $\Omega \subset M\cap B_{N}(p,r)$ be a connected component. Suppose that the $(m-1)$-Hausdorff measure ${\cal H}^{m-1}(\Omega \cap {\rm Cut}_{N}(p))=0$. If $c>0 $, suppose in addition that $r< \pi /2\sqrt c$. Then $$\label{eqSubm1}\lambda^{\ast}(\Omega) \geq \lambda_{1}(B_{\mathbb{N}^{m}(c)}(r)),$$where $B_{\mathbb{N}^{m}(c)}(r)$ is the geodesic ball with radius $r$ in the simply connected space form $\mathbb{N}^{n}(c)$ of constant sectional curvature $c$. If $\Omega $ is bounded then equality in (\[eqSubm1\]) holds iff $\Omega=B _{\mathbb{N}^{m}(c)}(r)$ and $M=\mathbb{N}^{m}(c)$. After Nadirashvili ’s bounded minimal surfaces in $\mathbb{R}^{3}$, (see [@kn:Nadirashvili]), Yau in [@kn:yau] asked if the spectrum of a Nadirashvili minimal surface was discrete. A more basic question is if the lower bound of the spectrum of a Nadirashvili minimal surface is positive. The following corollary shows that this is the case. \[nadirashvili\]Let $M\subset B_{\mathbb{R}^{3}}(r)$ be a complete bounded minimal surface in $\mathbb{R}^{3}$. Then $$\lambda^{\ast}(M)\geq \lambda_{1}(\mathbb{D}(r))=c/r^{2}.$$ Where $c>0$ is an absolute constant. Let $M\subset\mathbb{R}^{3}$ be a minimal surface with second fundamental form $A$ and $B_{M}(p,r)$ be a stable geodesic ball with radius $r$. Schoen, [@kn:schoen] showed that $\Vert A \Vert^{2}(p) \leq c/r^{2}$ for an absolute constant $c>0$. We have a converse of Schoen’s result for minimal hypersurfaces of the Euclidean space. \[stability\]Let $M\subset \mathbb{R}^{n+1}$ be a minimal hypersurface with second fundamental form $A$ and $B_{M}(r)$ be a geodesic ball with radius $r$. If $$\sup_{B_{M}(r)} \Vert A \Vert^{2} \leq \lambda_{1}(B_{\mathbb{R}^{n}}(0,r))=c(n)/r^2,$$ then $B_{M}(r)$ is stable. Here $B_{\mathbb{R}^{n}}(0,r)$ is a geodesic ball of radius $r$ in the Euclidean space $\mathbb{R}^{n}$ and $c(n)>0$ is a constant depending on $n$. Finally we apply Barta’s Theorem and Theorem (\[teoremaPrincipal\]) to the theory of quasi-linear elliptic equations. \[Elliptic1\] Let $M$ be a bounded Riemannian manifold with smooth boundary and $F\in C^{0}(\overline{M})$. Consider this problem, $$\label{eqElliptic1}\left\{\begin{array}{rccrl} \triangle u - \vert {{{\rm{grad}\,}}}u\vert^{2} &=& F & in& M\\ u&=& +\infty & on& \partial M . \end{array}\right.$$If (\[eqElliptic1\]) has a smooth solution then $\inf_{M} F\leq \lambda_{1}(M) \leq \sup_{M} F $. If either $\inf_{M} F=\lambda_{1}(M)$ or $\lambda_{1}(M)= \sup_{M} F$ then $F=\lambda_{1}(M)$. On the other hand if $F=\lambda$ is a constant then the problem (\[eqElliptic1\]) has solution if and only if $\lambda=\lambda_{1}(M)$. Now if we allow continuous boundary data on problem (\[eqElliptic1\]) we have the following generalization of a result of Kazdan-Kramer [@kn:kazdan-kramer]. \[Elliptic2\] Let $M$ be a bounded Riemannian manifold with smooth boundary and $F\in C^{0}(\overline{M})$ and $\psi \in C^{0}(\partial M)$. Consider the problem $$\label{eqElliptic2}\left\{\begin{array}{rclll} \triangle u - \vert {{{\rm{grad}\,}}}u\vert^{2} &=& F & in& M\\ u&=& \psi & on& \partial M. \end{array}\right.$$then if $\sup_{M}F < \lambda_{1}(M)$ then (\[eqElliptic2\]) has solution. Moreover if (\[eqElliptic2\]) has solution then $\inf_{M}F< \lambda_{1}(M)$. \[remarkkazdan\] - If we set $f=e^{-u}$ then (\[eqElliptic1\]) becomes $$\label{eqremark2}\left\{\begin{array}{rclll}\triangle \,f + F\,f&= &0 & in& M\\ f&=& 0 & on& \partial M . \end{array}\right.$$ Kazdan-Warner in [@kn:kazdan-warner] studied this problem (\[eqremark2\]) and they showed that if $F\leq \lambda_{1}(M)$, then (\[eqremark2\]) has solution, with $\sup F=\lambda_{1}(M)$. Thus Theorem (\[Elliptic1\]) is a complementary result to Kazdan-Warner result. - If we impose Dirichlet boundary data ($u=0$ on $\partial M$) on problem (\[eqElliptic1\]) then there is a solution if $F\leq \lambda_{1}(M)$ with strict inequality in a positive measure subset of $M$. This was proved by Kazdan-Kramer in [@kn:kazdan-kramer]. [Acknowledgement:]{}[The beginning of this work was greatly motivated by a paper of Kazdan & Kramer [@kn:kazdan-kramer] for which we thank Professor Djairo de Figueiredo for bringing it to our attention. We also should mention a paper of Cheung-Leung [@kn:cheung-leung] which contains the germ of Theorem (\[teoremaPrincipal\]). We are grateful to the referee for pointing out a gap in the proof of Theorem (\[Cheng\]).]{} An extension of Barta’s theorem =============================== We can consider Theorem (\[teoremaPrincipal\]) as an extension of Barta’s theorem for if $M$ is a bounded Riemannian manifold with piecewise smooth boundary $\partial M\neq \emptyset$ and $f\in C^{\,2}(M)\cup C^{\,0}(\overline{M})$ is a positive function on $M$ then for the vector field $X=-{{{\rm{grad}\,}}}\log f$ we obtain that ${{{\rm{div}\,}}}X-\vert X \vert^{2}=-\triangle f/f$. [Theorem \[teoremaPrincipal\]]{} [ Let $M$ be a Riemannian manifold. Then $$\label{eqThmP3} \lambda^{\ast}(M)\geq \sup_{{\cal W}^{1,1}(M ) }\{\inf_{ M} ({{{\rm{Div}\,}}}X- \vert X \vert^{2})\}.$$ If $M$ is compact with smooth non-empty boundary then $$\label{eqThmP4}\lambda_{1}(M)= \sup_{{\cal W}^{1,1}(M ) }\{\inf_{ M} ({{{\rm{Div}\,}}}X- \vert X \vert^{2})\}.$$ ]{} [Proof:]{} Let $X\in {\cal W}^{1,1}(M) $ and $f\in C_{0}^{\infty}(M)$. As observed in the Remark (\[remarkW11\]) we have that $\smallint_{M}{{{\rm{Div}\,}}}(f^{2} X)=0$. On the other hand we have that $$\begin{aligned} 0=\int_{M}{{{\rm{Div}\,}}}(f^{2}X) & = & \int_{M} \langle grad\,f^{2}, X\rangle + \int_{M} f^{2}\,{{{\rm{Div}\,}}}(X) \nonumber \\ & & \nonumber\\ &\geq & -\int_{M}\vert grad\,f^{2}\vert\cdot \vert X \vert+ \, \int_{M}f^{2}\,{{{\rm{Div}\,}}}\,X \nonumber\\ & & \nonumber \\ &= & -\int_{M}2 \cdot \vert f\vert\ \cdot \,\vert X \vert \cdot \vert grad\,f\vert + \, \int_{M}f^{2}\,{{{\rm{Div}\,}}}\,X \nonumber\\ & & \nonumber \\ & \geq & -\int_{M} f^{2}\cdot \vert X \vert^{2} - \int_{M}\vert {{{\rm{grad}\,}}}f \vert^{2} +\int_{M}f^{2}\,{{{\rm{Div}\,}}}\,X \nonumber\\ && \nonumber\\ & = & \int_{M}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2})\cdot f^{2} - \int_{M}\vert {{{\rm{grad}\,}}}f \vert^{2} \nonumber\\ && \nonumber \\ &\geq & \inf_{ M}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2})\int_{M}f^{2} - \int_{M}\vert {{{\rm{grad}\,}}}f \vert^{2}. \nonumber\end{aligned}$$ Then $$\int_{M}\vert {{{\rm{grad}\,}}}f \vert^{2} \geq \inf_{ M}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2})\int_{M}f^{2},$$ and thus $$\int_{M}\vert {{{\rm{grad}\,}}}f \vert^{2} \geq \sup_{{\cal W}^{1,1} }\inf_{ M}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2})\int_{M}f^{2}.$$Therefore $$\lambda^{\ast}(M) \geq \sup_{{\cal W}^{1,1} }\inf_{ M}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2}).$$ This proves (\[eqThmP3\]). Suppose that $M$ is compact with smooth non-empty boundary and let $v$ be its first eigenfunction. If we set $X_{0}=-{{{\rm{grad}\,}}}(\log v)$ then we have that ${{{\rm{div}\,}}}X_{0} -\vert X_{0} \vert^{2}= -\triangle v/v=\lambda_{1}(M)$ and (\[eqThmP4\]) is proven. \[remarkBartaGen\]This same proof above shows that $$\lambda^{\ast}(M) \geq \sup_{{\cal W}^{1,1} }\inf_{ M\setminus G}({{{\rm{Div}\,}}}\,X -\vert X \vert^{2}),$$ where $G$ has zero Lebesgue measure. Geometric Applications ====================== In the geometric applications of the Theorem (\[teoremaPrincipal\]) we need to know when a given vector field $X$ is in the Sobolev space ${\cal W}^{1,1}(M)$. The following lemma give sufficient conditions. \[divergence\] Let $\Omega\subset M$ be a bounded domain with piecewise smooth boundary $\partial \Omega$ in a smooth Riemannian manifold $M$. Let $G\subset \Omega$ be a closed subset with $(n-1)$-Hausdorff measure ${\cal H}^{n-1}(G)=0$. Let $X$ be a vector field of class $ C^{1}(\Omega\setminus G)\cap L^{\infty}(\Omega)$ such that ${{{\rm{div}\,}}}(X)\in L^{1}(\Omega)$. Then $X \in {\cal W}^{1,1}(\Omega)$. [Proof:]{} In fact we are going to show that $$\label{eqDivergence2}\int_{\Omega}{{{\rm{div}\,}}}(X) = \int_{\partial \Omega}\langle X,\nu\rangle,$$where $\nu$ is the outward unit normal vector field on $\partial \Omega \setminus Q$ and $Q\subset \partial \Omega$ is a closed subset with ${\cal H}^{n-1}(Q)=0$, see the footnote $(\ref{foot1})$ on page $2$. The equation (\[eqDivergence2\]) implies that $X\in {\cal W}^{1,1}(\Omega)$ since for all $\phi \in C^{\infty}_{0}$ we have that $$\int_{\Omega}{{{\rm{div}\,}}}(\phi\,X)=\int_{\Omega}\phi\,{{{\rm{div}\,}}}(X)+\int_{\Omega}\langle {{{\rm{grad}\,}}}\phi ,X \rangle$$ and $\int_{\Omega}{{{\rm{div}\,}}}(\phi\,X)=0$ by (\[eqDivergence2\]). Suppose first that $M=\mathbb{R}^{n}$. We may assume that $G$ is connected otherwise we work with each connected component. By Whitney’s Theorem there exists a smooth function (we may suppose to be non-negative) $f\hspace{-.7mm}:\hspace{-.7mm}\mathbb{R}^{n}\hspace{-.7mm} \rightarrow \mathbb{R}$ such that $G\hspace{-.8mm}=f^{-1}(0)$. By Sard’s theorem we can pick a sequence of regular values $\epsilon_{i}\rightarrow 0$ such that $f^{-1}(\epsilon_{i})$ is a smooth $(n-1)$-dimensional submanifold $\partial N_{i}$ bounding a connected set $N_{i}=f^{-1}([0,\epsilon_{i}])$ containing $G$, moreover, $N_{i}\subset \Omega$ for $\epsilon_{i}$ sufficiently small. Set $\Omega_{i}=\Omega \setminus N_{i}$ and let $\chi_{i}$ its characteristic function. Then $ \chi_{i}\cdot {{{\rm{div}\,}}}X\hspace{-.1cm} \to {{{\rm{div}\,}}}X$ a.e. in $\Omega$ and $\chi_{i}\cdot {{{\rm{div}\,}}}X \leq{{{\rm{div}\,}}}X\in L^{1}(\Omega)$. By the Lebesgue Convergence Theorem $\int_{\Omega_{i}}{{{\rm{div}\,}}}X=\int_{\Omega}\chi_{i}\cdot {{{\rm{div}\,}}}X \to \int_{\Omega} {{{\rm{div}\,}}}X $. On the other hand applying the divergence theorem to $X$ on $\Omega_{i}$ we obtain $$\begin{array}{lcl} \int_{\Omega_{i}}{{{\rm{div}\,}}}X & = & \int_{\partial \Omega_{i}}\langle X, \nu \rangle\\ && \\ &=& \int_{\partial \Omega}\langle X, \nu \rangle -\int_{\partial N_{i}}\langle X, \nu_{i} \rangle , \end{array}$$ $\nu_{i}$ is the outward (pointing toward $G$) unit vector field normal to $\partial N_{i} $. But $\left \vert \int_{\partial N_{i}}\langle X, \nu_{i} \rangle\right\vert \leq Vol_{n-1}(\partial N_{i}) \Vert X\Vert_{\infty}=Vol_{n-1}(f^{-1}(\epsilon_{i})) \Vert X\Vert_{\infty}$ and $Vol_{n-1}(f^{-1}(\epsilon_{i}))\to {\cal H}^{n-1}(G)=0$ as we will show later. Passing to the limit we have $$\begin{array}{lcl}\int_{\Omega} {{{\rm{div}\,}}}X& = & \lim_{i\to \infty}\int_{\Omega_{i}}{{{\rm{div}\,}}}X\\ && \\& =& \int_{\partial \Omega}\langle X, \nu \rangle - \lim_{i\to \infty}\int_{\partial N_{i}}\langle X, \nu_{i} \rangle\\ && \\ &=& \int_{\partial \Omega}\langle X, \nu \rangle\end{array}$$ To show $Vol_{n-1}(f^{-1}(\epsilon_{i}))\to {\cal H}^{n-1}(G)=0$ recall the $(n-1)$-dimensional spherical measure of $A\subset \mathbb{R}^{n}$ is defined by ${\cal S}^{n-1}(A)=\sup_{\delta>0}{\cal S}^{n-1}_{\delta}(A)$ $=\lim_{\delta\to 0}{\cal S}^{n-1}_{\delta}(A)$, where ${\cal S}^{n-1}_{\delta}(A)=\inf\sum {\rm diam} (B_{j})^{n-1}$ the infimum taken over all coverings of $A$ by balls $B_{j}$ with diameter ${\rm diam} (B_{j})\leq \delta$. The $(n-1)$-spherical measure is related to $(n-1)$-Hausdorff measure by ${\cal H}^{n-1}(A)\leq {\cal S}^{n-1}(A)\leq 2^{n-1}{\cal H}^{n-1}(A)$, $\forall \,A\subset \mathbb{R}^{n}$, see [@kn:matilla] for more details. Since ${\cal S}^{n-1}(G)=0$ we have that ${\cal S}^{n-1}_{\delta}(G)=0$ for all $\delta>0$. Then for $\delta>0$ and all $k>0 $ there exists a (finite) covering of $G$, ($G$ is compact), by closed balls $\{B_{kj}\}$ of diameter ${\rm diam}(B_{kj})\leq\delta$ such that $\sum_{j} {\rm diam} (B_{kj})^{n-1} \leq 1/k$. For $\epsilon_{i}$ sufficiently small, say $\epsilon_{i}\leq\epsilon_{i_{0}}$, the balls $B_{kj}$ are also a covering for the submanifold $f^{-1}(\epsilon_{i})$. This means that ${\cal S}^{n-1}_{\delta}(f^{-1}(\epsilon_{i}))\leq 1/k$. Taking a sequence $\delta_{l}\to 0$ we can find a sequence of pairs $(k_{l},\epsilon_{l})$, $k_{l}\rightarrow \infty$, $\epsilon_{l}\to 0$ such that ${\cal S}_{\delta_{l}}^{n-1}(f^{-1}(\epsilon_{i}))\leq 1/k_{l}$ for all $\epsilon_{i}\leq \epsilon_{l}$. Thus we have that $\lim_{l\to \infty}{\cal S}^{n-1}_{\delta_{l}} (f^{-1}(\epsilon_{l}))=0$ and since ${\cal S}^{n-1} (f^{-1}(\epsilon_{l}))\leq {\cal S}^{n-1}_{\delta_{l}} (f^{-1}(\epsilon_{l}))$ we have that $\lim_{l\to \infty}{\cal S}^{n-1} (f^{-1}(\epsilon_{l}))=0$. On the other ${\cal S}^{n-1} (f^{-1}(\epsilon_{l}))\geq {\cal H}^{n-1}(f^{-1}(\epsilon_{l}))=c(n)\cdot vol_{n-1}(f^{-1}(\epsilon_{l}))$. This shows that $vol_{n-1}(f^{-1}(\epsilon_{i}))\to 0$. This completes the proof of Lemma (\[divergence\]) for $M=\mathbb{R}^{n}$. The general case is done similarly using partition of unit. Geodesic coordinates -------------------- Let $M$ be a Riemannian manifold and a point $p\in M$. For each vector $\xi \in T_{p}M$, let $\gamma_{\xi}$ the unique geodesic satisfying $\gamma_{\xi}(0)=p$, $\gamma_{\xi}'(0)=\xi$ and $d(\xi)=\sup\{t>0: {\rm dist}_{M}(p,\gamma_{\xi}(t))=t\}$. Consider the largest open subset ${\cal D}_{p}=\{t\,\xi\in T_{p}M: 0\le t<d(\xi), \,\vert \xi\vert=1\}$ of $T_{p}M$ such that for any $\xi \in {\cal D}_{p}$ the geodesic $\gamma_{\xi}(t)=\exp_{p}(t\,\xi)$ minimizes the distance from $p$ to $\gamma_{\xi}(t)$ for all $t\in [0,d(\xi)]$. The cut locus of $p$ is given by ${\rm Cut }(p)=\{ \exp_{p}(d(\xi)\,\xi),\, \xi \in T_{p}M,\, \vert \xi \vert =1\}$ and $M=\exp_{p}({\cal D}_{p})\cup {\rm Cut }(p)$. The exponential map $\exp_{p}:{\cal D}_{p}\to \exp_{p}({\cal D}_{p})$ is a diffeomorphism and is called the geodesic coordinates of $M\setminus {\rm Cut}(p)$. Fix a vector $\xi \in T_{p}M$, $\vert \xi \vert =1$ and denote by $\xi^{\perp}$ the orthogonal complement of $\{\mathbb{R}\xi\}$ in $T_{p}M$ and let $\tau_{t}:T_{p}M\to T_{\exp_{p}(t\,\xi)}M$ be the parallel translation along $\gamma_{\xi}$. Define the path of linear transformations $${\cal A}(t,\xi):\xi^{\perp}\to\xi^{\perp}$$ by $${\cal A}(t,\xi)\eta=(\tau_{t})^{-1}Y(t)$$ where $Y(t)$ is the Jacobi field along $\gamma_{\xi}$ determined by the initial data $Y(0)=0$, $(\nabla_{\gamma_{\xi}'}Y)(0)=\eta$. Define the map $${\cal R}(t):\xi^{\perp}\to \xi^{\perp}$$ by $${\cal R}(t)\eta=(\tau_{t})^{-1}\, {\rm R}(\gamma_{\xi}'(t),\tau_{t}\,\,\eta)\gamma_{\xi}'(t),$$ where ${\rm R}$ is the Riemann curvature tensor of $M$. It turns out that the map ${\cal R}(t)$ is a self adjoint map and the path of linear transformations ${\cal A}(t,\xi)$ satisfies the Jacobi equation ${\cal A}''+{\cal R}{\cal A}=0$ with initial conditions ${\cal A}(0,\xi)=0$, ${\cal A}'(0,\xi)=I$. On the set $\exp_{p}({\cal D}_{p})$ the Riemannian metric of $M$ can be expressed by $$\label{eqmetricGeoCoord}ds^{2}(\exp_{p}(t\,\xi))=dt^{2}+\vert {\cal A}(t,\xi)d\xi\vert^{2}.$$ Setting $\sqrt{g(t,\xi)}=\det{\cal A}(t,\xi)$ we have by Rauch comparison theorem this following comparison theorem due to R. Bishop [@kn:bishop-crittenden], see also [@kn:chavel]. \[thmBishop\]If the radial sectional curvatures along $\gamma_{\xi}$ satisfies $\langle {\rm R}(\gamma_{\xi}'(t),v)\gamma_{\xi}'(t),v\rangle \leq c\vert v\vert^{2}$, $\forall t\in (0,r)$ and if $S_{c}(t)$ does not vanishes on $(0,r)$ then $$\begin{aligned} \label{eqRauch} [\sqrt{g(t,\xi)}/S_{c}^{n-1}(t)]'&\geq &0\,\,on \,\,(0,r)\nonumber \\ && \\ \sqrt{g(t,\xi)}-S_{c}^{n-1}(t)&\geq&0\,\,on \,\,(0,r]\nonumber \end{aligned}$$ Moreover equality occurs in one of these two inequalities (\[eqRauch\]) at a point $t_{0}\in (0,r)$ iff ${\cal R}=c\cdot I$ and ${\cal A}=S_{c}\cdot I $ on all $[0,t_{0}]$. Where $C_{c}(t)=S_{c}'(t)$ and $S_{c}(t)$ is given by $$\label{eqSc}S_{c}(t)=\left\{\begin{array}{clll}\displaystyle\frac{1}{\sqrt c}\sin (\sqrt c \,t) , & if& c>0 &\\ t & if & c=0\\ \displaystyle\frac{1}{\sqrt-c} \sinh ( \sqrt-c\, t) & if & c<0 \end{array}\right.$$ Proof of Theorem \[Cheng\] -------------------------- [Theorem \[Cheng\]]{} [Let $N$ be a Riemannian $n$-manifold with radial sectional curvature $K(x)(\partial t, v)\leq c$, $x\in B_{N}(p,r)\setminus {\rm Cut}(p)$ and $v\in T_{x}N\cap (\partial t)^{\perp}$ with $\vert v\vert \leq 1$. Let $\mathbb{N}^{n}(c)$ be the simply connected $n$-space form of constant sectional curvature $c$ and suppose that ${\cal H}^{n-1}({\rm Cut} (p)\cap B_{N}(p,r))=0$. Then $$\label{eqCheng2}\lambda^{\ast}(B_{N}(p,r))\geq \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r)).$$Equality in (\[eqCheng2\]) holds iff $B_{N}(p,r)$ and $B_{\mathbb{N}^{n}(c)}(r)$ are isometric.]{} [Proof:]{} Observe that if $c>0$ and $r>\pi/\sqrt{c}$ there is nothing to prove. Because in this case $\mathbb{N}^{n}(c)= \mathbb{S}^{n}(c)=B_{\mathbb{N}^{n}(c)}(r)$ and $\lambda_{1}(B_{\mathbb{N}^{n}(c)}(r))=0$. Hence, we may assume that $r<\pi/\sqrt{c}\,$ if $c>0$. Let $v$ be a positive first eigenfunction of $B_{\mathbb{N}^{n}(c)}(r)$. It is well known that $v$ is a radial function satisfying the differential equation $$v''(t) + (n-1)\frac{C_{c}(t)}{S_{c}(t)}v'(t)+\lambda_{1}(B_{\mathbb{N}^{n}(c)}(r))v(t)=0, \,\, \forall \,\,t\in[0,r] \label{eqCheng3}$$ with $v'(t)\leq 0$ and with $v'(t)=0$ iff $t=0$. Define $u :B_{N}(p,r) \rightarrow [0,\infty)$ by $$u(x)=\left \{ \begin{array}{cll} v(t) & if & x=\exp_{p}(t\,\xi),\;t\in [0,d(\xi))\cap[0,r]\\ &&\\ 0 &if &x=\exp_{p}(d(\xi)\,\xi)\in {\rm Cut} (p) \end{array}\right.$$ Set $X(x)=-{{{\rm{grad}\,}}}\log u (x)$ if $x\in B_{N}(p,r)\setminus(\{p\}\cup {\rm Cut}(p))$ and $X(x)=0$ if $x\in B_{N}(p,r)\cap (\{p\}\cup {\rm Cut}(p))$. The vector field $X$ is expressed in geodesic coordinates by $$X(x)=\left\{\begin{array}{cll}\displaystyle -\frac{v'(t)}{v(t)}\cdot \partial t & if &x=\exp_{p}(t\,\xi),\;t\in (0,d(\xi))\cap (0,r]\\ & &\\ 0&if & x=p,\,\;\;{\rm or}\;\; x=\exp_{p}(d(\xi)\,\xi). \end{array}\right.$$ The vector field $X\in {\cal W}^{1,1}(B_{N}(p,r))$ as we will prove it later. Setting $B\setminus G=B_{N}(p,r)\setminus (\{p\}\cup {\rm Cut}(p))$ for simplicity of notation we have by Theorem (\[teoremaPrincipal\]) and Remark (\[remarkBartaGen\]) we have that $$\lambda^{\ast}(B_{N}(p,r)) \geq \inf_{B\setminus G} \{{{{\rm{Div}\,}}}X - \vert X \vert^{2}\}=\inf_{B\setminus G} \{{{{\rm{div}\,}}}X - \vert X \vert^{2}\}=\inf_{B\setminus G}-\frac{\triangle u}{u}$$ By (\[eqRauch\]) and (\[eqCheng3\]) we have that for $0<t<d(\xi)$ $$\begin{aligned} -\frac{\triangle u}{u}(\exp_{p}(t\,\xi))&=&-\displaystyle\frac{1}{v(t)}\left\{v''(t) + \frac{\sqrt{g(t,\xi)}'}{\sqrt{g(t,\xi)}} v'(t)\right\}\nonumber \\ && \nonumber\\ &\geq& -\frac{1}{v(t)}\left\{v''(t) +(n-1) \frac{S'_{c}(t)}{S_{c}(t)}v'(t) \right\}\label{eqCheng4} \\ \nonumber \\ & =& \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r))\nonumber\end{aligned}$$ Thus $$\lambda^{\ast}(B_{N}(p,r))\geq \inf_{B\setminus G}[-\displaystyle\frac{\triangle u}{u}]\geq \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r))$$ This proves (\[eqCheng2\]). To handle the equality case in (\[eqCheng2\]) we observe that the set of non-smooth points of the boundary $\partial B_{N}(p,r)$ is exactly the intersection $ {\rm Cut}(p)\cap\partial B_{N}(p,r)$ but since ${\cal H}^{n-1}( {\rm Cut}(p)\cap B_{N}(p,r))=0$ the Hausdorff measure ${\cal H}^{n-1}({\rm Cut}(p)\cap \partial B_{N}(p,r))=0$. Thus $B_{N}(p,r)$ has piecewise smooth boundary (see [@kn:whitney] pages 99-100). Hence there exists a positive Dirichlet eigenfunction for $B_{N}(p,r)$ with eigenvalue $\lambda_{1}(B_{N}(p,r))=\lambda^{\ast}(B_{N}(p,r))$. The equality $\lambda^{\ast}(B_{N}(p,r))= \lambda_{1}(B_{\mathbb{N}^{n}(c)}(r))$ implies that $u$ is a first eigenfunction of $B_{N}(p,r)$, see Barta’s Theorem (\[barta\]). Looking at (\[eqCheng4\]) we see that the equality implies that $$\frac{\sqrt{g(t,\xi)}'}{\sqrt{g(t,\xi)}}= (n-1) \frac{S'_{c}(t)}{S_{c}(t)}$$ on all $[0,r]$. Thus by Bishop’s Theorem (\[thmBishop\]) we have that ${\cal R}=c\cdot I$ and ${\cal A}=S_{c}\cdot I $. This is saying that $B_{N}(p,r)$ is isometric to $ B_{\mathbb{N}^{n}(c)}(r)$. To finish the proof of Theorem (\[Cheng\]) we need to show that the vector field $X\in {\cal W}^{1,1}(B_{N}(p,r)) $. As observed in Remark (\[remarkW11\]) it suffices to show that $f^{2}X\in {\cal W}^{1,1}(B_{N}(p,r))$ for all $f\in C_{0}^{\infty}(B_{N}(p,r))$. Since the $(n-1)$-Hausdorff measure ${\cal H}^{n-1}(\{p\}\cup {\rm Cut}(p))=0$ by the Lemma (\[divergence\]) it is sufficient to show that - ${{{\rm{div}\,}}}(f^{2}X ) \in L^{1}(B_{N}(p,r))$ - $f^{2}X\in \ C^{1}\left(B_{N}(p,r)\setminus \{p\}\cup {\rm Cut}(p)\right)\cap L^{\infty}(B_{N}(p,r))$. The vector field $X$ is clearly smooth on $B_{N}(p,r)\setminus \{p\}\cup {\rm Cut}(p)$ thus on this set ${{{\rm{div}\,}}}X = {{{\rm{Div}\,}}}X$ and ${{{\rm{div}\,}}}(f^{2}X)=\langle {{{\rm{grad}\,}}}f^{2},X\rangle + f^{2}{{{\rm{div}\,}}}X.$ Integrating over $B_{N}(p,r)\cap {\rm Supp}(f)$ we have $$\begin{aligned} \label{eqDiv(fX)}\int_{B_{N}(p,r)\cap {\rm Supp}(f)}\vert {{{\rm{div}\,}}}(f^{2}X)\vert & \leq & \int_{B_{N}(p,r)\cap {\rm Supp}(f)}\vert {{{\rm{grad}\,}}}f^{2}\vert \vert X\vert \nonumber \\ && \\ &+ & \int_{B_{N}(p,r)\cap {\rm Supp}(f)}\vert f^{2}\vert \vert{{{\rm{div}\,}}}X\vert\nonumber\end{aligned}$$ The first term of (\[eqDiv(fX)\]) is finite $$\begin{aligned} \int_{B_{N}(p,r)\cap {\rm Supp}(f)}\vert {{{\rm{grad}\,}}}f^{2}\vert \vert X\vert & \leq &\sup_{ {\rm Supp}(f)}\{\vert {{{\rm{grad}\,}}}f^{2}\vert\cdot \vert X\vert\}\cdot vol (B_{N}(p,r))\nonumber \\ && \nonumber \\ & < & \infty \nonumber\end{aligned}$$ since $\vert X (x) \vert \leq \vert (v'/v)(t)\vert <\infty $ for $x=\exp_{p}(t\,\xi)\in B_{N}(p,r)\cap {\rm Supp}(f)$. We have that $$(f^{2}{{{\rm{div}\,}}}X)(x)=f^{2}(x)[\displaystyle-\frac{v''}{v}(t)+\frac{v'^{2}}{v^{2}}(t)-\frac{v'}{v}(t)\frac{\sqrt{g(t,\xi)}\,'}{\sqrt{g(t,\xi)}}].$$ Integrating over $B_{N}(p,r)\cap {\rm Supp}(f)$ we have $$\begin{aligned} \int_{B_{N}(p,r)\cap {\rm Supp}(f)}\vert f^{2}\vert \vert{{{\rm{div}\,}}}X\vert& \leq & \int_{\xi}\int_{0}^{t (\xi)}(\displaystyle\vert\frac{v''}{v}\vert+\vert\frac{v'^{2}}{v^{2}}\vert)\vert f^{2} \vert \sqrt{g(t,\xi)}dtd\xi \nonumber\\ && \nonumber \\ & &+ \int_{\xi}\int_{0}^{t (\xi)}\displaystyle\vert\frac{v'}{v}\vert \vert f^{2}\vert\sqrt{g(t,\xi)} \,'dtd\xi<\infty \nonumber\end{aligned}$$ that the second term on the right hand side of (\[eqDiv(fX)\]) is also finite. Where $t(\xi)$ is the largest $t<\min\{d(\xi),\,r \}$ such that $\exp_{p}(t\,\xi)\in {\rm Supp}(f)$. This shows that ${{{\rm{div}\,}}}(f^{2}X ) \in L^{1}(B_{N}(p,r))$. Showing the item (ii) is a trivial task. Proof of Theorem ----------------- [Theorem \[submanifold\]]{} [Let $N$ be a Riemannian $n$-manifold with radial sectional curvature $K(x)(\partial t, v)\leq c$, for all $x\in B_{N}(p,r)\setminus {\rm Cut}(p)$, and all $v\perp \partial t $ with $\vert v\vert \leq 1$. Let $M\subset N$ be an $m$-dimensional minimal submanifold and $\Omega \subset M\cap B_{N}(p,r)$ be a connected component. Suppose that the $(m-1)$-Hausdorff measure ${\cal H}^{m-1}(\Omega \cap {\rm Cut}_{N}(p))=0$. If $c>0 $, suppose in addition that $r< \pi /2\sqrt c$. Then $$\label{eqSubm2}\lambda^{\ast}(\Omega) \geq \lambda_{1}(B_{\mathbb{N}^{m}(c)}(r)),$$where $B_{\mathbb{N}^{m}(c)}(r)$ is the geodesic ball with radius $r$ in the simply connected space form $\mathbb{N}^{n}(c)$ of constant sectional curvature $c$. If $\Omega $ is bounded then equality in (\[eqSubm1\]) holds iff $\Omega=B _{\mathbb{N}^{m}(c)}(r)$ and $M=\mathbb{N}^{m}(c)$]{}. [Proof:]{} Let $v:B_{\mathbb{N}^{m}(c)}(r)\to \mathbb{R}$ be a positive first Dirichlet eigenfunction of $B_{\mathbb{N}^{m}(c)}(r)$. It is known that $v$ is radial with $v'(t)\leq 0$ and $v'(t)=0$ iff $t=0$. We can normalize $v$ such that $v(0)=1$. The differential equation $\triangle_{\mathbb{N}^{m}(c)}v(t)+\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))v(t)=0$ is expressed in geodesic coordinates by $$v''(t) + (m-1)\frac{C_{c}(t)}{S_{c}(t)}v'(t)+\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))v(t)=0, \,\, \forall \,\,t\in[0,r]. \label{eqSubm3}$$Recall that for each $\xi \in T_{p}N$, $\vert \xi \vert=1$, $d(\xi)>0$ is the largest real number (possibly $\infty$) such that geodesic $\gamma_{\xi}(t)=\exp_{p}(t\,\xi)$ minimizes the distance from $\gamma_{\xi}(0)=p$ to $\gamma_{\xi}(t)$ for all $t\in [0,d(\xi)]$. We have that $B_{N}(p,r)\setminus {\rm Cut}(p)=\exp_{p}(\{t\,\xi\in T_{p}N: 0\le t<\min\{r, d(\xi)\}, \,\vert \xi\vert=1 \})$. Define $u:B_{N}(p,r)\to\mathbb{R}$ by $u(\exp_{p}(t\xi))=v(t)$ if $t<\min\{r, d(\xi)\}$ and $u(r\xi)=u(d(\xi)\xi)=0$. Let $\Omega \subset M\cap B_{N}(p,r)$ be a connected component and $\psi:\Omega \rightarrow \mathbb{R}$ defined by $\psi=u\circ \varphi$, where $\varphi$ is the minimal immersion $\varphi : M \subset N$. The vector field $X=-{{{\rm{grad}\,}}}\log \psi$ identified with $d\varphi (X)$ is not smooth at $G= \Omega \cap (\{p\}\cup {\rm Cut}(p))$. By hypothesis ${\cal H}^{m-1}(G)=0$ and it can be shown that the vector field $X\in C^{1}(\Omega\setminus G)\cap L^{\infty}(\Omega)$ and ${{{\rm{div}\,}}}X\in L^{1}(\Omega)$ thus $X\in {\cal W}^{1,1}$ and satisfies (\[eqDivergence2\]) and by Theorem (\[teoremaPrincipal\]) we have that $$\lambda^{\ast}(\Omega)\geq \inf_{\Omega\setminus G} [{{{\rm{Div}\,}}}X -\vert X\vert^{2}]=\inf_{\Omega\setminus G} [{{{\rm{div}\,}}}X -\vert X\vert^{2}]=\inf_{\Omega\setminus G}[-\triangle \psi/\psi].$$ Where $\triangle \psi $ is given by the following formula, (see [@kn:bessa-montenegro], [@kn:cheung-leung], [@kn:jorge-koutrofiotis]), $$\begin{aligned} \triangle \,\psi (x) & = & \sum_{i=1}^{m}{{{\rm Hess}\,}}\,u(\varphi (x)) \,(e_{i},e_{i})+ \langle {{{\rm{grad}\,}}}u\,,\, \stackrel{\rightarrow}{H}\rangle\label{eqlaplacianpsi}\\ & =&\sum_{i=1}^{m}{{{\rm Hess}\,}}\,u(\varphi (x)) \,(e_{i},e_{i})\nonumber\end{aligned}$$where $\varphi(x)=\exp_{p}(t\xi) $, $\stackrel{\rightarrow}{H}=0$ is the mean curvature vector of $\Omega$ at $\varphi (x)$ and $\{ e_{1},\ldots e_{m}\}$ is an orthonormal basis for $T_{\varphi (x)}\,\Omega$. Choose this basis such that $e_{2},\ldots e_{m}$ are tangent to the distance sphere $\partial B_{N}(p,t)\subset N$ and $e_{1}= \cos (\beta(x) )\,\partial/\partial t + \sin (\beta(x) )\,\partial/\partial \theta $, where $ \partial/\partial \theta \in [[e_{2}, \ldots e_{m}]]$, $\vert \partial/\partial \theta\vert=1$. From (\[eqlaplacianpsi\]) we have for $\varphi (x)\in \Omega\setminus G$ that $$\begin{aligned} \triangle \,\psi (x)& =& \sum_{i=1}^{m} {{{\rm Hess}\,}}u (\varphi (x))(e_{i},e_{i})\nonumber \\ && \nonumber \\ & = & v''(t)(1-\sin^{2}\beta(x) ) \nonumber \\ &&\label{eqlaplacianpsi2} \\ & +& v'(t)\,\sin^{2}\beta(x)\,{\rm Hess}(t)(\partial /\partial\theta ,\partial/\partial \theta ) \nonumber \\ &&\nonumber \\ & +& v'(t)\,\sum_{i=2}^{m}{\rm Hess}(t)(e_{i},e_{i}) \nonumber\end{aligned}$$ where $t={\rm dist}_{N}(p,x)$. Adding and subtracting $(C_{c}/S_{c})(t)\,v'(t)\sin^{2}\beta(x) $ and $ (m-1)(C_{c}/S_{c})(t)\,v'(t)$ in (\[eqlaplacianpsi2\]) we have $$\begin{aligned} \label{eqlaplacianpsi3} \triangle \,\psi (x)&=&v''(t)+ \,(m-1)\frac{C_{c}}{S_{c}}(t) \,v'(t)\nonumber \\ &&\nonumber \\ & +& \left({\rm Hess}(t)(\partial /\partial\theta ,\partial/\partial \theta )-\frac{C_{c}(t)}{S_{c}(t)}\right) \,v'(t)\sin^{2}\beta(x) \nonumber \\ && \\ &+& \sum_{i=2}^{m}[{\rm Hess }(t)(e_{i},e_{i})-\frac{C_{c}}{S_{c}}(t)]\,v'(t)\nonumber \\ &&\nonumber \\ & +& \nonumber\left( \frac{C_{c}(t)}{S_{c}(t)}v'(t)-v''(t)\right)\sin^{2}\beta(x)\end{aligned}$$ From (\[eqSubm3\]) and (\[eqlaplacianpsi3\]) we have that $$\begin{aligned} \displaystyle-\frac{\triangle \,\psi}{\psi} (x)& =& \label{eqlaplacianpsi4} \lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))\nonumber \\ & -& \left({\rm Hess}(t)(\partial /\partial\theta ,\partial/\partial \theta )-\frac{C_{c}(t)}{S_{c}(t)}\right) \,\frac{v'(t)}{v(t)}\sin^{2}\beta(x) \nonumber \\ && \\ &-& \sum_{i=2}^{m}[{\rm Hess }(t)(e_{i},e_{i})-\frac{C_{c}}{S_{c}}(t)]\,\frac{v'(t)}{v(t)}\nonumber \\ &&\nonumber \\ & -&\frac{1}{v(t)} \left( \frac{C_{c}(t)}{S_{c}(t)}v'(t)-v''(t)\right)\sin^{2}\beta(x) \nonumber\end{aligned}$$Since the radial curvature $K(x)(\partial t, v)\leq c$ for all $x\in B_{N}(p,r)\setminus {\rm Cut}(p)$ and all $v\perp \partial t $ with $\vert v\vert \leq 1$ then by the Hessian Comparison Theorem (see [@kn:schoen-yau]) we have that ${{{\rm Hess}\,}}(t(x))(v,v) \geq (C_{c}/S_{c})(t)$ for all $v\perp \partial t$, $t(x)=t$, $x=\exp_{p}(t\xi)$. But $v'(t)\leq 0$ then we have that the second and third terms of (\[eqlaplacianpsi4\]) are non-negative. If the fourth term of (\[eqlaplacianpsi4\]) is non-negative then we would have that $$-\frac{\triangle \,\psi }{\psi} (x) \geq \lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))$$ By Theorem (\[teoremaPrincipal\]) we have that $$\label{eqlaplacianpsi5}\lambda^{\ast}(\Omega )\geq \inf (-\frac{\triangle \,\psi }{\psi} )\geq \lambda_{1}(B_{\mathbb{N}^{m}(c)}(r)).$$ This proves (\[eqSubm2\]). We can see that $\displaystyle -\left( \frac{C_{c}(t)}{S_{c}(t)}\frac{v'(t)}{v(t)}-\frac{v''(t)}{v(t)}\right)\sin^{2}\beta(x)\geq 0$ is equivalent to $$\label{eqSubm4}m \frac{C_{c}(t)}{S_{c}(t)}v'(t)+\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))v(t) < 0, \,\, t\in (0,r).$$ To prove (\[eqSubm4\]) we will assume without loss of generality that $c=-1,0,1$. Let us consider first the case $c =0$ that presents the idea of the proof. The other two remaining cases ($c=-1 $ and $c=1$) we are going to treat (quickly) with the same idea. When $c=0$ the inequality (\[eqSubm4\]) becomes $$\label{eqSubm5}\frac{mv'(t)}{t}+ \lambda_{1}v(t) < 0, \,\, t\in (0,r),$$where $\lambda_{1}:=\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))$. Let $\mu (t):=\exp\{-\displaystyle\frac{\lambda_{1}t^{2}}{2m}\}$. The functions $v$ and $\mu$ satisfy the following identities, $$\label{eqSubm6}\begin{array}{lcl} (t^{m-1}v'(t))' + \lambda_{1}t^{m-1}v(t)& =& 0\\ && \\ (t^{m-1}\mu'(t))' + \lambda_{1}t^{m-1}(1- \displaystyle\frac{\lambda_{1}\,t^{2}}{m^{2}})\mu(t)& =& 0 \end{array}$$In (\[eqSubm6\]) we multiply the first identity by $\mu$ and the second by $-v$ adding them and integrating from $0$ to $t$ the resulting identity we obtain, $$t^{m-1}\,v'(t)\,\mu(t) -t^{m-1}\,v(t)\,\mu'(t)=-\frac{\lambda_{1}^{2}}{m^{2}}\int_{0}^{t}\mu(t)\,v(t) <0,\,\,\ \forall t\in (0,r).$$ Then $\mu(t)v'(t) < \mu'(t)v(t) $ and this proves (\[eqSubm5\]). Assume that now that $c=-1$. The inequality (\[eqSubm4\]) becomes $$\label{eqSubm7}m\displaystyle\frac{C_{-1} (t)}{S_{-1}(t)}v'(t) + \lambda_{1}v(t)<0$$ Set $\mu(t):=C_{-1} (t)^{-\lambda_{1}/m}$. The functions $v$ and $\mu$ satisfy the the following identities $$\label{eqSubm8}\displaystyle\begin{array}{lll}(S_{-1}^{m-1}v')' + \lambda_{1}S_{-1}^{m-1}v& =& 0 \\ && \\ (S_{-1} ^{m-1}\mu')' + \lambda_{1}S_{-1}^{m-1}\left(\displaystyle \frac{m-1}{m}+\displaystyle \frac{1}{m C_{-1}^{2} } - \displaystyle\frac{\lambda_{1}}{m^{2}}\displaystyle\frac{S_{-1}^{2}}{C_{-1}^{2}}\right)\mu&=&0 \end{array}$$In (\[eqSubm8\]) we multiply the first identity by $\mu$ and the second by $-v$ adding them and integrating from $0$ to $t$ the resulting identity we obtain $$S_{-1}^{m-1}\left(v'\mu-\mu'v\right)(t)+ \int_{0}^{t}\lambda_{1}S_{-1}^{m-1}\left[\frac{1}{m}-\frac{1}{mC_{-1}^{2}}+ \frac{\lambda_{1}}{m^{2}}\frac{S_{-1}^{2}}{C_{-1}^{2}}\right]\mu v=0$$ The term $ S_{-1}^{m-1}\left[\displaystyle\frac{1}{m}-\frac{1}{mC_{-1}^{2}}+ \frac{\lambda_{1}}{m^{2}}\frac{S_{-1}^{2}}{C_{-1}^{2}}\right]\mu v $ is positive (one can easily check) therefore we have that $(v'\mu-\mu'v)(t)<0$ for all $t\in (0,r)$. This proves (\[eqSubm7\]). For $c=1$ the inequality (\[eqSubm4\]) becomes the following inequality $$\label{eqSubm9}m\frac{C_{1}}{S_{1}}v'(t)+\lambda_{1}v(t)<0,\,\,\,0<t<\pi/2$$Set $ \mu(t):=C_{1} (t)^{-\lambda_{1}/m},\,\,0<t<\pi/2$. The functions $v$ and $\mu$ satisfy the the following identities $$\label{eqSubm10}\begin{array}{lll}(S_{1}^{m-1}v')' + \lambda_{1}S_{1}^{m-1}v&=& 0\\ && \\ (S_{1} ^{m-1}\mu')' - \lambda_{1}S_{1}^{m-1}\left(\displaystyle \frac{m-1}{m}+ \displaystyle\frac{1}{m C_{1}^{2} } + \displaystyle\frac{\lambda_{1}}{m^{2}}\displaystyle\frac{S_{1}^{2}}{C_{1}^{2}}\right)\mu&=&0 \end{array}$$In (\[eqSubm10\]) we multiply the first identity by $\mu$ and the second by $-v$ adding them and integrating from $0$ to $t$ the resulting identity we obtain$$S_{1}^{m-1}\left(v'\mu-\mu'v\right)(t)+ \int_{0}^{t}\lambda_{1}S_{1}^{m-1}\left[2-\frac{1}{m}+\frac{1}{mC_{1}^{2}}+ \frac{\lambda_{1}}{m^{2}}\frac{S_{1}^{2}}{C_{1}^{2}}\right]\mu v=0$$ The term $\displaystyle S_{1}^{m-1}\left[2-\frac{1}{m}+\frac{1}{mC_{1}^{2}}+ \frac{\lambda_{1}}{m^{2}}\frac{S_{1}^{2}}{C_{1}^{2}}\right]\mu v$ is positive therefore we have that $(v'\mu-\mu'v)(t)<0$ for all $t\in (0,r)$. This proves (\[eqSubm9\]) and thus the fourth term in (\[eqlaplacianpsi4\]) is non-negative. To finishes the proof of the Theorem (\[submanifold\]) we need to consider the equality case in (\[eqSubm2\]) when $\Omega $ is bounded. From the spectral theory it is known that for a given bounded domain in a Riemannian manifold there is $u\in C^{\infty}(\Omega)\cap H_{0}^{1}(\Omega)$, positive in $\Omega$ satisfying $\triangle u+\lambda_{1}(\Omega)u=0$, where $\lambda_{1}(\Omega)=\lambda^{\ast}(\Omega)$. This function is also called an [eigenfunction]{}. Observe that $u\vert \partial \Omega =0$ only a.e. thus $u$ is not considered a solution for the Dirichlet eigenvalue problem. The proof of existence of $u$ is the same proof of existence of Dirichlet eigenvalues for a smooth domain, since it (the proof) does need smoothness of the boundary but the boundedness of the domain. With an approximation argument Barta’s Theorem can be extend to arbitrary bounded open sets. \[3.1\]Let $\Omega$ be a bounded domain in a smooth Riemannian manifold. Let $v\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$, $v>0$ in $\Omega$ and $v\vert \partial \Omega =0$. Then $$\lambda^{\ast}(\Omega)\geq \inf_{\Omega}(-\frac{\triangle v}{v}).\label{eqProp1}$$Moreover, $\lambda^{\ast}(\Omega)= \inf_{\Omega}(-\displaystyle\frac{\triangle v}{v})$ if and only if $v=u$, where $u$ is a positive eigenfunction of $\Omega$, i.e. $\triangle u+\lambda^{\ast}(\Omega)u=0$. [Proof:]{} Let $\epsilon_{i}\to 0$ be a sequence of positive regular values of $v$ and let $\Omega_{\epsilon_{i}}^{v}=\{x\in \Omega;\, v(x)>\epsilon_{i}\}$. Applying Barta’s Theorem we have that $$\label{eqProp2}\lambda^{\ast}(\Omega_{\epsilon_{i}}^{v})= \lambda_{1}(\Omega_{\epsilon_{i}}^{v})\geq \inf_{\Omega_{\epsilon_{i}}^{v}}(-\displaystyle\frac{\triangle v}{v})\geq \inf_{\Omega}(-\displaystyle\frac{\triangle v}{v})$$But $\lim_{\epsilon_{i}\to 0}\lambda^{\ast}(\Omega_{\epsilon_{i}}^{v})=\lambda^{\ast}(\Omega)$, see in [@kn:chavel], page 23. Let $u\in C^{\infty}\cap H_{0}^{1}(\Omega)$ be a positive eigenfunction of $\Omega$. Then $$\label{eqProp3}\lambda^{\ast}(\Omega)= \displaystyle -\frac{\triangle u}{u} =-\frac{\triangle v}{v}+\frac{u\triangle v-v\triangle u}{uv}$$ [Claim:]{} $\displaystyle\int_{\Omega}(u\triangle v-v\triangle u)=0$.\ [Proof:]{} Let $\epsilon_{i}\to 0$ be a sequence of positive regular values of $u$ and let $\Omega_{\epsilon_{i}}^{u}=\{x\in \Omega;\, u(x)>\epsilon_{i}\}$. Set $$\label{eqProp4}u_{\epsilon_{i}}=\left\{\begin{array}{lll} u-\epsilon_{i} & {\rm on} & \Omega_{\epsilon_{i}}^{u} \\ && \\ 0 &{\rm on} & \Omega \setminus\Omega_{\epsilon_{i}}^{u}\end{array}\right.$$One can show that $u_{\epsilon_{i}}\to u$ in $H^{1}(\Omega)$ using the Lebesgue Convergence Theorem. Therefore $$\begin{aligned} \displaystyle\int_{\Omega_{\epsilon_{i}}^{u}}(u_{\epsilon_{i}}\triangle v-v\triangle u_{\epsilon_{i}}) &=&-\displaystyle\int_{\Omega_{\epsilon_{i}}^{u}}\langle {{{\rm{grad}\,}}}u_{\epsilon_{i}}, {{{\rm{grad}\,}}}v\rangle + \lambda^{\ast}(\Omega)\int_{\partial\Omega_{\epsilon_{i}}^{u}}u v\nonumber \\&& \\ & \to &\displaystyle-\int_{\Omega}\langle {{{\rm{grad}\,}}}u, {{{\rm{grad}\,}}}v\rangle + \lambda^{\ast}(\Omega) \int_{\partial\Omega}u v=0\nonumber\end{aligned}$$Since $v\in H_{0}^{1}(\Omega)$ and $u$ is a weak solution of $\triangle u+\lambda^{\ast}(\Omega)u=0$. On the other hand $ \displaystyle\int_{\Omega_{\epsilon_{i}}^{u}}(u_{\epsilon_{i}}\triangle v-v\triangle u_{\epsilon_{i}})\to \displaystyle\int_{\Omega}(u\triangle v-v\triangle u) $. If $\Omega$ is bounded we have that $\partial \varphi (\Omega)\subset \partial B_{N}(p,r)$. This implies that the function $\psi= u\circ \varphi\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ is such that $\psi \,\vert \partial \Omega =0$. Suppose that $ \lambda^{\ast}(\Omega)=\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))$. Then by Proposition (\[3.1\]), $\psi:\Omega\to \mathbb{R}$ is an eigenfunction of $\Omega$ and we have that $ \lambda^{\ast}(\Omega)=-\triangle \psi/\psi$. From (\[eqlaplacianpsi4\]) we have that $$\begin{array}{lll} \left({\rm Hess}(t)(\partial /\partial\theta ,\partial/\partial \theta )-\displaystyle\frac{C_{c}(t)}{S_{c}(t)}\right) \,\displaystyle\frac{v'(t)}{v(t)}\sin^{2}\beta(x)&=&0 \\ && \\ \displaystyle\sum_{i=2}^{m}[{\rm Hess }(t)(e_{i},e_{i})-\displaystyle\frac{C_{c}}{S_{c}}(t)]\,\displaystyle\frac{v'(t)}{v(t)}&=&0 \\ &&\nonumber \\ \displaystyle\frac{1}{v(t)} \left( \frac{C_{c}(t)}{S_{c}(t)}v'(t)-v''(t)\right)\sin^{2}\beta(x) &=&0, \end{array}$$for all $t$ such that $\varphi (x)=\exp_{p}(t\xi)\in \Omega$. This implies $\sin^{2}\beta (x)=0$ for all $x\in \Omega$ and we have that $e_{1}(\varphi (x))=\partial /\partial t $. Integrating the vector field $\partial/\partial t$ we have a minimal geodesic (in $N\cap \varphi ( \Omega)$) joining $\varphi (x)$ to the center $p$. This imply that $\Omega$ is the geodesic ball in $M$ centered at $\varphi^{-1}(p)$ with radius $r$ i.e. $\Omega =B_{M}(\varphi^{-1}(p),r)$. Since $\psi$ is an eigenfunction with the same eigenvalue $\lambda_{1}(B_{\mathbb{N}^{m}(c)}(r))$ we have that $$\label{eq29}\triangle_{M}\,v (t)=\triangle_{\mathbb{N}^{m}(c)}\,v(t),\,\,t=dist_{N}(p,\varphi (q)), \forall q\in \Omega .$$Rewriting this identity (\[eq29\]) in geodesic coordinates we have that $$\frac{\sqrt{g(t,\xi)}'}{\sqrt{g(t,\xi)}} (t,\theta)v'(t) + v''(t)= (m-1)\frac{C_{c}(t)}{S_{c}(t)}v'(t) + v''(t)$$ This imply that by Bishop Theorem $\Omega= B_{M}(\varphi^{-1}(p)$ and $B_{\mathbb{N}^{m}(c)}(r)$ are isometric. By analytic continuation $M=\mathbb{N}^m(c)$. Proof of Corollary \[nadirashvili\] ----------------------------------- Let $M\subset B_{\mathbb{R}^{3}}(r)$ be a complete bounded minimal surface in $\mathbb{R}^{3}$. Then $$\lambda^{\ast}(M)\geq \lambda_{1}(\mathbb{D}(r))=c/r^{2}.$$ Where $c>0$ is an absolute constant. The proof of this result follows directly from Theorem (\[submanifold\]). The theorem says that the fundamental tone $\lambda^{\ast}(\Omega)\geq \lambda_{1}(B_{\mathbb{R}^{3}}(r))$ for any connected component of $M\cap B_{\mathbb{R}^{3}}(r)$. In particular, for $\Omega= M$ we have that $\lambda^{\ast}(M)\geq \lambda_{1}(B_{\mathbb{R}^{3}}(r))=c(2)/r^{2}$, where $c(2)$ is the first zero of the Bessel function $J_{0}$, see [@kn:chavel], page 46. Proof of Corollary ------------------- Let $\varphi :M \hookrightarrow N^{n+1}$ be a complete orientated minimal hypersurface and $A(X) =- \nabla_{X}\, \eta$ its second fundamental form, where $\eta$ is globally defined unit vector field normal to $\varphi (M)$. A normal domain $D\subset M$ is said to be stable if the first Dirichlet eigenvalue $\lambda_{1}^{L}=\inf\{-\smallint_{D} u\,Lu/\smallint_{D} u^2,\,u\in C_{0}^{\infty}(D)\}$ of the operator $L=\triangle + Ric(\eta)+ \Vert A\Vert^{2}$ is positive. On the other hand we have that $$\begin{array}{lll}-\smallint_{D}u\,Lu & = & \smallint_{D} \left[\,\vert {{{\rm{grad}\,}}}u \vert^{2}-\left( Ric (\eta)+\Vert A\Vert^{2}\right)u^{2}\right]\\ && \\ & \geq &\smallint_{D}\lambda_{1}^{\triangle}(D)- \left( Ric (\eta)+\Vert A\Vert^{2}\right)u^{2}. \end{array}$$ Therefore if $\lambda_{1}^{\triangle}(D)\geq \sup_{x\in D}\{ Ric (\eta)+\Vert A\Vert^{2}(x)\}$ then $D$ is stable. In [@kn:bessa-montenegro] we give estimates for $\lambda_{1}^{\triangle}(D)$ in submanifolds with locally bounded mean curvature, in particular minimal hypersurfaces. With those estimates we have obvious statements for stability theorems. If $D=B_{M}(p,r)$ is a ball in $M^{n}\hookrightarrow \mathbb{R}^{n+1} $ obviously that $\varphi (D)\subset B_{n+1}(\varphi (p),r)$. And we have that $\lambda_{1}(D)\geq \lambda^{\ast}(\varphi^{-1}(B_{n+1}(\varphi (p),r))\geq \lambda_{1}(B_{n}(0,r))$. This proves the Corollary (\[stability\]). Quasilinear elliptic equations ------------------------------ In this section we want to apply Barta’s Theorem to study the existence of solutions to certain quasi-linear elliptic equations. Let $M$ be a bounded Riemannian manifold with smooth non-empty boundary $\partial M$ and $f\in C^{2}(M)$, $f>0$. If we set $u=-\log f$[^3], $f>0$ then the problem $ \triangle\,f + F f=0$ becomes $\triangle u - \vert {{{\rm{grad}\,}}}u\vert^{2}= F $. Hence Barta’s Theorem (\[barta\]) can be translated as [Theorem \[Elliptic1\] ]{} [Let $M$ be a bounded Riemannian manifold with smooth boundary and $F\in C^{0}(\overline{M})$. Consider this problem, $$\label{eqElliptic3}\left\{\begin{array}{rccrl} \triangle u - \vert {{{\rm{grad}\,}}}u\vert^{2} &=& F & in& M\\ u&=& +\infty & on& \partial M . \end{array}\right.$$If (\[eqElliptic3\]) has a smooth solution then $\inf_{M} F\leq \lambda_{1}(M) \leq \sup_{M} F $. If either $\inf_{M} F=\lambda_{1}(M)$ or $\lambda_{1}(M)= \sup_{M} F$ then $F=\lambda_{1}(M)$. On the other hand if $F=\lambda$ is a constant the problem (\[eqElliptic3\]) has solution if and only if $\lambda=\lambda_{1}(M)$.]{} Likewise Theorem (\[teoremaPrincipal\]) can be translated into language of quasi-linear elliptic equations. [Theorem \[Elliptic2\]]{} Let $M$ be a bounded Riemannian manifold with smooth boundary and $F\in C^{0}(\overline{M})$ and $\psi \in C^{0}(\partial M)$. Consider the problem $$\label{eqElliptic4}\left\{\begin{array}{rclll} \triangle u - \vert {{{\rm{grad}\,}}}u\vert^{2} &=& F & in& M\\ u&=& \psi & on& \partial M. \end{array}\right.$$then 1. If $\sup_{M}F < \lambda_{1}(M)$ then (\[eqElliptic4\]) has solution. 2. If (\[eqElliptic4\]) has solution then $\inf_{M}F< \lambda_{1}(M)$. [Proof:]{} The operator $L=-\triangle -F $ is compact thus its spectrum is a sequence of eigenvalues $\lambda_{1}^{L}< \lambda_{2}^{L}\leq \lambda_{3}^{L}\leq \cdots \nearrow +\ \infty$. Suppose that $\sup_{M}F<\lambda_{1}(M)$ then we have that $$\begin{aligned} \lambda_{1}^{L}& = & \inf\,\{ \int_{M} \vert {{{\rm{grad}\,}}}u\vert^{2} -F\,u^{2};\, u\in H_{0}^{1}(M),\,\smallint_{M}u^{2}=1\}\nonumber \\ & \geq & \lambda_{1}(M) - \sup_{M}F>0.\nonumber\end{aligned}$$ Therefore $L$ is invertible. On the other hand, $u=-\log f $ is a solution of (\[eqElliptic4\]) if and only if $f$ is solution of $$\label{eqElliptic5}\left\{\begin{array}{lcl} L f & = & 0 \,\,\,\, in \,\, M \\ f &= & e^{-\psi}\,\, on \, \, \partial M \end{array}\right.$$ Consider the harmonic extension $v$ of $e^{-\psi}$ on $\partial M$, ($\triangle \,v=0$ in $M $ and $v=e^{-\psi}$ on $\partial M$), and $h= f-v$. We have that $L \,h = F\, v$ in $M $ and $h=0 $ on $\partial M$. Then $h = L^{-1}(F\,v)$ and $f= -v + L^{-1} (F \,v)$ is solution of (\[eqElliptic5\]). Item b. Suppose that the equation (\[eqElliptic4\]) has a smooth solution $u$. Let $\varphi$ be the first eigenfunction of the operator $-\triangle $, ($\triangle\, \varphi + \lambda_{1}(M ) \varphi =0$ in $M $ and $\varphi\vert\partial M =0$), and $f$ a solution of (\[eqElliptic5\]). By Green we have that $$\int_{ M }f \triangle \varphi - \varphi \triangle f= \int_{\partial M }e^{-\psi}\frac{\partial \varphi}{\partial \eta}- \varphi \frac{\partial f}{\partial \eta}= \int_{\partial M}e^{-\psi}\frac{\partial \varphi}{\partial \eta}\;.$$ Thus $$\int_{\partial M}e^{-\psi}\frac{\partial \varphi}{\partial \eta} = \int_{M}(F-\lambda_1)\varphi f$$ if $v$ is the harmonic extension $v$ of $e^{-\psi}$ then we have $$\int_{M}-\lambda_1f\varphi =\int_{ M }v \triangle \varphi - \varphi \triangle v= \int_{\partial M }v\frac{\partial \varphi}{\partial \eta}- \varphi \frac{\partial v}{\partial \eta}=\int_{\partial M}e^{-\psi}\frac{\partial \varphi}{\partial \eta}<0$$ since $\inf_{M}v=\inf_{\partial M}v=\inf_{\partial M}e^{-\psi}>0$. Therefore $\smallint_{M}(F-\lambda_1)\varphi f <0$ and $\inf F<\lambda_1(M)$. [abcd]{} Barta, J.: [Sur la vibration fundamentale d’une membrane.]{} C. R. Acad. Sci. 204, 1937, 472-473. Berger, M., Gauduchon, P. and Mazet, E.: [Le Spectre d’une Variété Riemannienes]{}. Lect. Notes Math. 194, 1974, Springer-Verlag. Bessa, G. P., Montenegro, J. F.: [Eigenvalue estimates for submanifolds with locally bounded mean curvature.]{} Ann. Global Anal. and Geom. 24, 2003, 279-290. Bishop, R., Crittenden, R.: [Geometry of Manifolds]{}. Academic Press, New York, 1964. Chavel, I.: [Eigenvalues in Riemannian Geometry.]{} Pure and Applied Mathematics, 1984, Academic Press, INC. Cheng, S. Y., [Eigenfunctions and eigenvalues of the Laplacian.]{} Am. Math. Soc Proc. Symp. Pure Math. 27, part II, 1975, 185-193. Cheung, Leung-Fu and Leung, Pui-Fai, [Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space.]{} Math. Z. 236, 2001, 525-530. Cheng, S. Y., Li, P. and Yau, S. T., [Heat equations on minimal submanifolds and their applications.]{} Amer. J. Math. 106, 1984, 1033-1065. Fisher-Colbrie, D. and Schoen, R,. [The structure of complete stable minimal surfaces in $3$-manifolds of non negative scalar curvature.]{} Comm. Pure and Appl. Math. 33, 1980, 199-211. Grigor’yan, A.: [Analytic and geometric background of recurrence and non-explosion of the brownian motion on Riemannian manifolds]{}. Bull. Amer. Math. Soc. 36, 2, 1999, 135-249. Kazdan, J. and Kramer, R., [Invariant Criteria for existence of solutions to second-order quasilinear elliptic equations.]{} Comm. Pure Appl. Math. 31, 5, 1978, 619-645. Kazdan, J. and Warner, F. W., [Remarks on some quasilinear elliptic equations.]{} Comm. Pure Appl. Math. 28, 1975, 567-597. Jorge, L. and Koutrofiotis, D., [An estimate for the curvature of bounded submanifolds.]{} Amer. J. Math., 103, 4, 1980, 711-725. McKean, H. P.: [An upper bound for the spectrum of $\triangle $ on a manifold of negative curvature.]{} J. Differ. Geom. 4, 1970, 359-366. Matilla, P., [Geometry of Sets and Measures in Euclidean Spaces.]{} Cambridge University Press, 1995. Nadirashvili, N., [Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces.]{} Invent. Math., 126, 1996, 457-465. Schoen, R., [Stable minimal surfaces in three manifolds.]{} Seminar on Minimal Submanifolds. Annals of Math Studies. Princenton University Press. Schoen, R. and Yau, S. T., [Lectures on Differential Geometry.]{} Conference Proceedings and Lecture Notes in Geometry and Topology, [vol. 1]{}, 1994. Yau, S. T., [Review of Geometry and Analysis.]{} Asian J. Math. vol.4, 1, 2000, 235-278. Whitney, H., [Geometric Integration Theory.]{} Princenton Mathematical Series, 1957, Princenton University Press. [^1]: Both author were partially supported by CNPq Grant. [^2]: \[foot1\]Piecewise smooth boundary here means that there is a closed set $Q\subset \partial M$ of $(n-1)$-Hausdorff measure zero such that for each point $q \in \partial M\setminus Q$ there is a neighborhood of $q$ in $\partial M$ that is a graph of a smooth function over the tangent space $T_{q}\partial M$, see Whitney [@kn:whitney] pages 99-100. [^3]: This transformation $u=-\log f$ we learned from Kazdan & Kramer [@kn:kazdan-kramer] but it also appears in [@kn:fischer-colbrie-schoen]
--- abstract: 'Every system in physics is described in terms of interacting elementary particles characterized by modulated spacetime recurrences. These intrinsic periodicities, implicit in undulatory mechanics, imply that every free particle is a reference clock linking time to the particle’s mass, and every system is formalizable by means of modulated elementary spacetime cycles. We propose a novel consistent relativistic formalism based on intrinsically cyclic spacetime dimensions, encoding the quantum recurrences of elementary particles into spacetime geometrodynamics. The advantage of the resulting theory is a formal derivation of quantum behaviors from relativistic mechanics, in which the constraint of intrinsic periodicity turns out to quantize the elementary particles; as well as a geometrodynamical description of gauge interaction which, similarly to gravity, turns out to be represented by relativistic modulations of the internal clocks of the elementary particles. The characteristic classical to quantum correspondence of the theory brings novel conceptual and formal elements to address fundamental open questions of modern physics.' author: - Donatello Dolce bibliography: - '../cycles.bib' title: Cyclic spacetime dimensions --- #### Introduction {#introduction .unnumbered} — According to the Standard Model, natural phenomena are described in terms elementary particles and their relativistic interactions (including gravity). In addition to this, in agreement with our empirical observations of nature, undulatory mechanics tell us that to every particle of local four-momentum $\bar k_{\mu}=\{\bar \omega, -\mathbf{\bar{k}}\}$ corresponds an instantaneous spacetime recurrence $T^{\mu}=\{T, \vec \lambda\}$ of fundamental topology $\mathbb S^{1}$. This is, in fact, the Lorentz projection $\Lambda \rightarrow \gamma T - \gamma \vec \beta \cdot \vec \lambda$ of the Compton proper-time periodicity $\Lambda = 2 \pi / \bar m $ of a rest particle of mass $\bar m$ — in natural units ($\hbar = c = 1$). In modern physics the recurrence of elementary particles are indeed represented as “periodic phenomena” [@Broglie:1924; @1996FoPhL], such as waves or phasors (wave-particle duality), in which spacetime enters as angular variables: their periodicities are locally fixed through the Planck constant by the kinematical states, according to the phase harmony condition $\bar m \Lambda \equiv 2 \pi \rightarrow {\bar{k}}_{\mu} T^{\mu}\equiv 2 \pi $, where $\bar \omega = \gamma \bar m$ and $\bar k_{i} = \gamma \beta_{i} \bar m$. Thus, every system in nature is formalizable in terms of elementary spacetime cycles and their modulations [@Dolce:2009ce; @Dolce:tune; @Dolce:AdSCFT]. As also noted for instance by R. Penrose, under such an assumption of Intrinsic Periodicity (IP) every isolated particle is a relativistic reference clock [@Dolce:2009ce; @Penrose:cycles], as “clocks directly linking time to a particle’s mass” [@Lan01022013]. In fact, “[a relativistic clock \[is\] a phenomenon passing periodically through identical phases]{}” [@Einstein:1910]. Actually, these “ticks” of the internal clocks of the elementary particles (Compton clocks) can be indirectly observable experimentally [@2008FoPh...38..659C; @Lan01022013], and in principle used to define our time axis similarly to the atomic clocks. This suggests a relational interpretation of the flow of time. An isolated system (universe) composed by a single particle has a perfectly homogeneous cyclic evolution (persistent “ticks”), as a pendulum in the vacuum. Nevertheless, a system composed by more particles is ergodic as the elementary periodicities are in general not rational each others; if we also consider interactions and the consequent modulations of periodicities, we find that the evolution of this non elementary system of elementary cycles is chaotic, according to our physical observations. Every instant in time is therefore characterized by a unique combination of the phases of the elementary cycles associated to the particles constituting the system, similarly to the reference temporal cycles of a stopwatch or of a calendar. The infinite spacetime periodicities of very low energy massless particles (photons and gravitons have infinite Compton worldline periodicities) set the underlying non-compact reference spacetime framework for the ordinary causal relational description of retarded events in relativity. The simple question, never considered before, that we want to address is: what happens if we impose to a particle its natural local modulated spacetime recurrence as a constraint?[^1] Remarkably, the result is a consistent unified description of fundamental aspects of modern physics based on cyclic spacetime dimensions. Similarly to Bohr’s atom or a ’particle in a box’, such a constraint of IP is in fact a quantization condition. We will see that the resulting cyclic dynamics formally match ordinary relativistic Quantum Mechanics (QM) in both the canonical and Feynman formulations [@Dolce:2009ce; @Dolce:tune; @Dolce:AdSCFT]. The local retarded variations of four-momentum of avery particle interacting in a given interaction scheme can be equivalently described, through the Planck constant, in terms of corresponding local retarded modulations of elementary spacetime cycles. In this way we will see that gauge interactions can be derived from spacetime geometrodynamics similarly to gravity and Weyl’s original proposal [@Weyl:1918ib; @Blagojevic:2002du]. The assumption of IP is also an interesting, novel interpretational key of the fundamental properties of modern integrable theories. #### Cyclic spacetime geometrodynamics {#cyclic-spacetime-geometrodynamics .unnumbered} — We represent mathematically the elementary spacetime cycles of a bosonic particle (“periodic phenomenon”) in terms of compact spacetime dimensions of length $T^{\mu}$ with Periodic Boundary Conditions (PBCs). In this way, a free elementary bosonic system of fundamental mass $\bar m$, in the rest or in a generic reference frame, is represented, respectively, by the equivalent (upon Lorentz transformations) actions $${\mathcal{S}}^{\Lambda} = \int^{\Lambda}\!\!\!\! d s {\mathcal{L}}(\partial_s \Phi(s),\Phi(s)) \rightarrow \int^{T^\mu} \!\!\!\!\!\! d^4 x {\mathcal{L}}(\partial_\mu \Phi(x),\Phi(x)) \label{action:compact}$$ This formalism is manifestly covariant (the action is a scalar) as the PBCs (or anti-PBCs) minimize the action at the boundary [@Dolce:2009ce; @Dolce:tune]. A global Lorentz transformation $x^\mu \rightarrow x'^\mu = \Lambda^{\;\mu}_{\nu} x^\nu $ in the free action (\[action:compact\]) implies a transformation of the boundary $T^\mu \rightarrow {T'}^\mu = \Lambda^{\;\mu}_\nu T^\nu$. The resulting recurrence ${T'}^\mu $ actually describes the four-momentum $\bar k_\mu \rightarrow {{\bar k}'}_\mu = \Lambda_{\;\mu}^\nu \bar k_\nu$ of the free particle in the new frame, according to $\bar k'_{\mu} T'^{\mu} = 2 \pi$. The geometric quantity $T^{\mu}$ is therefore a contravariant tangent four-vector [@Kenyon:1990fx] satisfying the reciprocal of the relativistic dispersion relation: $\bar m^{2}= \bar k_{\mu} \bar k^{\mu} \Leftrightarrow \frac{1}{\Lambda^{2}} = \frac{1}{T^{\mu}} \frac{1}{T_{\mu}}$. Thus, as long as modulations of periodicities are considered, the resulting description of elementary spacetime cycles are fully consistent with special relativity. In this section we only consider the non-quantum limit of a bosonic massive particle. This corresponds to the fundamental solutions of (\[action:compact\]), denoted by the bar sign, $\bar \Phi(s) \propto \exp[-i \bar m s] \rightarrow \bar \Phi(x) \propto \exp[-i \bar k_{\mu} x^{\mu}]$. Indeed, the boundary at $\Lambda$ and $T^{\mu}$ select the kinematical state of the particle, the BCs select a single Klein Gordon (KG) mode describing the free boson at rest or in a generic frame, respectively. Thus, in a generic frame, we see that in (\[action:compact\]) such a free fundamental solution $\bar \Phi(x)$ of persistent periodicity $T^{\mu}$ is described by the ordinary KG lagrangian ${\mathcal{L}} \Rightarrow \bar{\mathcal{L}}_{KG} = \frac{1}{2} [\partial_\mu \bar \Phi\partial^\mu\bar\Phi-\bar{m}^{2}\bar{\Phi}^{2}]$. Generic interactions are introduced by considering the corresponding periodicity modulations, as prescribed by undulatory mechanics. That is, in every point $x=X$, a generic interaction scheme (a potential) can be described by the corresponding relativistic variations of four-momentum with respect to the free case $\bar{k}_{\mu}\rightarrow\bar{k}'_{\mu}(X)=e_{\;\mu}^{a}(x)\|_{x=X}\bar{k}_{a}$. This implies relativistic modulations of instantaneous four-periodicity $T^{\mu}\rightarrow T'^{\mu}(X)\sim e_{a}^{\;\mu}(x)\|_{x=X}T^{a}$. Under the assumption of IP, every free particle is an inertial reference clock. Generic interactions are therefore modulations of these elementary clocks and in turns can be encoded in corresponding deformations of the spacetime metric [@Birrell:1982ix], similarly to General Relativity (GR). In our formalism, modulations of periodicity are in fact realized through PBCs by local deformations of compact spacetime dimensions. Indeed, in the free action (\[action:compact\]), the local transformations of reference frame $x^{\mu}\rightarrow x'^{\mu}(x)=x^{a}\Lambda_{a}^{\;\mu}(x)$, where $e^{\;\mu}_{a}(x) = [\partial x'^{\mu}(x)/\partial x^{a}]$ (for simplicity’s sake we work in the linear approximation, neglecting self-interactions) leads to the transformed action with locally deformed metric $\eta_{\mu\nu}\rightarrow g_{\mu\nu}(X)=[e_{\;\mu}^{a}(x)e_{\nu}^{\; b}(x)]_{x=X}\eta_{ab}$ and local boundary $T^{\mu} \rightarrow \Sigma'^{\mu}(X) =\Lambda^{\;\mu}_{a}(x)\|_{x=X}T^{a}$: $$\mathcal{S}^{\Lambda}\simeq\int^{\Sigma'^{\mu}(X)}d^{4}x\sqrt{-g(x)}\mathcal{L}(e_{a}^{\;\mu}(x)\partial_{\mu}\Phi'(x),\Phi'(x))\,. \label{eq:defom:action:generic:int}$$ The resulting fundamental solution has the typical form of a modulated wave $\bar\Phi'(x)\propto\exp[-i\int^{x_{\mu}}dy^{\mu}\bar {k}'_{\mu}(y)]$ of instantaneous periodicity $T'^{\mu}(x)$. Indeed, it actually describes the local four-momentum $\bar{k}'_{\mu}(x)$ of the interacting bosonic particle. It is easy to check that this description of interaction matches the ordinary geometrodynamical description of gravitational interaction of GR. A weak Newtonian interaction in fact corresponds to the local variations of energy $\bar{\omega}\rightarrow\bar{\omega}(x)'\sim\left(1+{GM_{\odot}}/{\|\mathbf{x}\|}\right)\bar{\omega}$, implying through the Planck constant the corresponding modulations of time periodicity $T_{t}\rightarrow T_{t}'(x)\sim\left(1-{GM_{\odot}}/{\|\mathbf{x}\|}\right)T_{t}$. That is, elementary cycles applied to Newtonian interaction immediately yields two fundamental aspects of GR: redshift and time dilatation, see [@Ohanian:1995uu]. By also considering the variations of momentum and the corresponding modulations of wavelength we find that, in (\[eq:defom:action:generic:int\]), the Newtonian interaction is actually encoded by the linearized Schwarzschild metric. As known, ordinary Einstein’s equation follows from this linearized description by considering self-interaction [@Ohanian:1995uu]. Note that relativity, if rigorously interpreted, only fixes the differential structure of spacetime (metric), whereas the only requirement for the BCs comes from the variational principle, the BCs must minimize the action at boundary of a relativistic theory. Einstein’s equation is defined modulo boundary terms: we do not know “what \[and where\] is the boundary of GR” [@springerlink:10.1007/BF01889475]. We also note that our approach fulfills the holographic principle [@'tHooft:1993gx]: the boundary $\Sigma'^{\mu}(x)$ of (\[eq:defom:action:generic:int\]) is directly related to the local spacetime periodicity $T'^{\mu}(x)$ and therefore, through $\hbar$, it explicitly encodes the kinematics $\bar{k}'_{\mu}(x)$ of the interaction. The assumption that every particle is a reference clock yields a geometrodynamical description of gauge interaction, similar to GR as in Weyl’s original proposal. Intuitively we want to describe the motion of a particle interacting electromagnetically in terms of the corresponding local transformations of flat reference frame: $dx^{\mu}(x)\rightarrow dx'^{\mu}\sim dx^{\mu} - e dx^{a} \zeta^{\;\mu}_{a}(x)$, ($g_{\mu\nu}(x) \propto \eta_{\mu\nu}$). This can be parametrized by the vectorial field $\bar{A}_{\mu}(x) \equiv \zeta_{\;\mu}^{a}(x)\bar{k}_{a}$ so that, according to our geometrodynamical description, the resulting interaction scheme is the minimal substitution $\bar{k}'_{\mu}(x) \sim \bar{k}_{\mu}- e\bar{A}_{\mu}(x)$ [@Dolce:tune]. The locally rotated boundary of (\[eq:defom:action:generic:int\]) leads to modulated solutions of the type $\bar \Phi'(x) \propto \exp[{{-i \bar k_{ \mu} x^{\mu}} + i e \int^{x^{\mu}} d y^{\mu} \bar A_{ \mu} (y) }]$. The gauge connection (Wilson line) $\bar U(x) = \exp[{ i e \int^{x^{\mu}} d y^{\mu} \bar A_{ \mu}(y) }]$ (postulated in ordinary gauge theory) here is derived from local transformations of flat reference frame. Indeed, through the gauge transformed terms $ \bar \Phi(x)=\bar U^{-1}(x) \bar \Phi'(x)$ and $\partial_{\mu} \bar \Phi(x) =\bar U^{-1}(x)D_{\mu} \bar \Phi'(x) $, where $D_{\mu}=\partial_{\mu}-ie\bar{A}_{\mu}(x)$, the modulated periodicity $T'^{\mu}(x)$ of the interacting particle $ \bar \Phi'(x)$ is “tuned” to the persistent periodicity $T^{\mu}(x)$ of the free case $\bar \Phi(x)$. By substituting $\bar \Phi(x)$ in ${\bar\mathcal{L}_{KG}}$, we see that the modulated solution $\bar \Phi'(x)$ of the interacting action with locally deformed boundary (\[eq:defom:action:generic:int\]) is equivalently obtained from the “tuned” fundamental lagrangian ${\mathcal{L}} \Rightarrow {\bar\mathcal{L}_{tuned}} = \frac{1}{2} [D_\mu \bar \Phi' D^\mu \bar \Phi' - \bar m^{2} \bar {\Phi'}^{2}]$ in the action (\[action:compact\]) with persistent boundary $T^{\mu}$. In general [@Dolce:tune], gauge invariant terms are “tunable” to fulfill the variational principle at a common boundary $T^{\mu}$. In this way we also justify the kinematic term $-\frac{1}{4}\bar F_{\mu\nu} \bar F^{\mu\nu}$ for $\bar A_{\mu}$, formally obtaining a Yang-Mills theory (this constrains $\zeta^{\;\mu}_{a}(x)$ to be “polarized”). We have inferred that local transformations of flat reference frame, inducing local rotations of the boundary, correspond to transformations of the fundamental solution of (\[eq:defom:action:generic:int\]) formally equivalent to the internal transformations of ordinary gauge theory (in a sort of parametrization invariance applied to the boundary). Though relativity fixes the differential structure of spacetime, it allows us to “play” with spacetime boundaries as long as the variational principle is fulfilled. Nevertheless, BCs (playing a central role in “old” QM) have a “marginal” role in Quantum Field Theory (QFT). For instance, the KG field used for mathematical computations is the more general solution of the KG equation, so that variations of the boundaries and BCs are not considered in the ordinary KG fields. We have introduced a formalism in which the boundary selects the kinematical state of the particle. Interactions can be therefore described as corresponding local retarded modulations of the spacetime periodicity of a KG mode from the initial state to the final state, as suggested by undulatory mechanics, instead of creating and annihilating modes with different spacetime periodicities (four-momenta) as in ordinary QFT. #### Correspondence to quantum mechanics {#correspondence-to-quantum-mechanics .unnumbered} — By playing with BCs in a consistent way in a relativistic wave theory, it is possible to derive at a classical level the fundamental aspects of relativistic QM [@Dolce:2009ce; @Dolce:tune; @Dolce:AdSCFT]. The most general bosonic solution of the free action (\[action:compact\]) is a vibrating “cord”[^2] with all the harmonics allowed by the PBCs at $T^{\mu}$: $\Phi(x) = \sum_{n} \phi_n(x) = \sum_{n} \mathcal N {a^}_{n}(\bar \mathbf k) \exp[{-i k_{n \mu} x^{\mu}}]$; the KG mode $\bar \Phi$ considered so far is the fundamental harmonic, $\mathcal N$ and $a^{}_{n}(\bar \mathbf k) $ are the normalization and population coefficients of the harmonics, respectively. Homogeneous spacetime periodicity $T^{\mu}$ (isolated particle) implies a harmonic quantization of the conjugate energy-momentum spectrum $k_{n \mu} T^{\mu} = n \bar k_{\mu} T^{\mu} = 2 \pi n$; that is, time periodicity $T$ implies the quantized harmonic energy spectrum $\omega_{n} = n \bar \omega = 2 \pi n / T$. Since $T^{\mu}$ transforms as a contravariant four-vector, $T$ varies with $\bar \mathbf k$ (relativistic Doppler effect). In a generic reference frame, the resulting energy spectrum of our harmonic system is therefore $\omega_{n} (\bar \mathbf k) = 2 \pi n / T (\bar \mathbf k) = n \sqrt{\bar \mathbf k^{2} + \bar m^{2}}$. This is the same energy spectrum prescribed for free bosons by ordinary second quantization in QFT (after normal ordering[^3]). The modulated IP $T'^{\mu}(X)$ of an interacting particle $ \exp[- i\int^{\Sigma'^{\mu}(X)} d x^{\mu} \bar {k}'_{n \mu}(x) ]\equiv \exp[-i2\pi n]$ implies the Bohr-Sommerfeld quantization $\oint_{X} d x^{\mu} \bar {k}'_{n \mu}(x) = 2 \pi n$, from which it is possible to solve semiclassically fundamental quantum systems [@refId0]. This modulated harmonic system is the typical classical system representable locally in a Hilbert space. The modulated harmonics of our vibrating “cord” form locally a complete set with inner product $\left\langle \phi'_{n}(X,t_{f})\|\phi'_{n'}(X,t_{i})\right\rangle \equiv\int_{V({\mathbf{X})}}\frac{{d\mathbf{x}}}{V(\mathbf{x})} {\phi'_{n'}}^{}(\mathbf{x},t_{f})\phi'_{n}(\mathbf{x},t_{i})$ and Hilbert eigenstates $\left\langle X \| \phi'_{n}\right\rangle \equiv \phi'_{n}(x)$ ($V(\mathbf{x})$ corresponds to a sufficiently large or infinite number of spatial periods to contain the interaction region, so that $V(\mathbf{x})\simeq V$). Thus, $\bar \Phi'(x)$ is represented by a corresponding Hilbert state $ \left\| \Phi' \right\rangle = \sum a_{n} \left\| \phi'_{n} \right\rangle$. The composition of more harmonic systems is represented by the tensor product of the Hilbert spaces with a quantum number for every fundamental periodicity; $\{n\}$ corresponds to the $\mathbb S^{1}$ spacetime recurrence[^4]. The energy-momentum spectrum of the locally modulate harmonic system defines the non-homogeneous Hamiltonian and momentum operators $\mathcal H'$ and $\mathcal P'_{i}$ as $\mathcal P'_{\mu}\left \|\phi'_{n} \right\rangle \equiv k'_{n \mu} \left\|\phi'_{n} \right\rangle$, where $\mathcal{P}'_\mu = \{\mathcal H', - \mathcal{P}'_{i}\}$. The “square root” of the modulated wave equation leads to the spacetime evolution of every harmonic $i \partial_{\mu} \phi'_{n}(x) = k'_{n}(x) \phi'_{n}(x)$ with infinitesimal unitary evolution operator $\mathcal U(x+d x) = \exp[-i \mathcal P'_{\mu}(x) dx^{\mu}]$. Thus, the resulting time evolution of our modulated vibrating “cord” in Hilbert notation is described by the ordinary Schrödinger equation $i \partial_{t} \left\|\Phi' \right\rangle = \mathcal H' \left\|\Phi' \right\rangle$. Moreover, the ordinary commutation relations of QM are directly derived* from the constraint of IP. In analogy with Feynman’s demonstration [@Feynman:1942us], the expectation value of a total derivative $\partial_{i} \mathcal F(x)$ for generic Hilbert states, after integration by parts and considering that the boundary terms cancel owing IP, yields the same commutators of ordinary QM: $ [\mathcal F(x),\mathcal{P}'_{i}] = i \partial_{i} \mathcal F(x) $, and $ [x_{j},\mathcal{P}'_{i}] = i \delta_{i j} $ for $\mathcal F(x)=x_{j}$ [@Dolce:2009ce; @Dolce:tune]. This correspondence to ordinary relativistic QM is also proved by the fact that the classical evolution of our modulated harmonic system is mathematically described by the ordinary Feynman path integral [@Dolce:2009ce; @Dolce:tune] $$\mathcal{Z}=\int_{V} {\mathcal{D}\mathbf{x}} e^{i \mathcal{S}'(t_{f},t_{i})}\,. \label{eq:Feynman:Path:Integral}$$ The resulting lagrangian $\mathcal{L}' = {\mathcal P}'_{i} x_{i}- \mathcal H'$ associated to $\mathcal S'$ is, by construction, the classical action of the corresponding interaction scheme, according to the ordinary formulation. This remarkable result has an intuitive, purely classical interpretation. Without relaxing the classical variational principle, in our cylindric geometry of topology $\mathbb S^{1}$ the classical evolution of $\Phi'(x)$ is the result of the interference of all the (potentially infinite) possible classical paths with different windings numbers between its two extremal configurations. By extending all the volume integrals to an infinite number of periods ($V\rightarrow \infty$) we obtain formally the same equations of relativistic QM, and thus of our modern description of nature. It is straightforward to check that our formalism provides a correct classical limit for $\hbar \rightarrow 0$. The harmonics of $\Phi'(x)$ are interpreted as quantum excitations of the system. The corpuscular limit corresponds to $T \rightarrow 0$, so that only the fundamental energy level (KG mode) is populated (infinite energy gap). IP applied to fermions represents a natural realization of the Zitterbewegung model, providing a semi-classical description of the spin and the Dirac equation [@Hestenes:zbw:1990; @Trzetrzelewski:2013xia], as well as interesting reconsiderations of supersymmetry foundational aspects [@CasalbuoniZitterSusy]. The modulated classical evolution (\[eq:Feynman:Path:Integral\]) in the case of gauge interaction turns out to be described by the ordinary Feynman path integral of scalar QED (similarly, the scattering matrix describes the modulations of the harmonic modes in Hilbert notation) [@Dolce:tune]. Such a semiclassical description is confirmed by recent progresses in integrable theories with fundamental analogy to our theory, such as light-front-quantization [@Zhao:2011ct] (where PBCs are the quantization condition), Twistor theory [@ArkaniHamed:2009si] (“cyclic coordinates” attached to every spacetime point) and AdS/CFT (discussed below). The modulated PBCs applied locally to the modulo of the gauge transformation (Wilson loop with generic winding number) $ \exp[ i e \int^{\Sigma'^{\mu}(X)} d x^{\mu} A_{ \mu}(x) ] = \exp[ i \pi n]$ implies the Dirac quantization for magnetic monopoles $ \oint_{X} d x^{\mu} e A_{ \mu}(x) = \pi n$. For example, as described in a forthcoming paper, IP in gauge transformations $A_{ \mu}(x) \rightarrow A_{ \mu}(x) - e \partial_{\mu} \Theta(x)$ implies that the corresponding Goldstone mode $\Theta(x)$, being a phase of a “periodic phenomenon”, is periodic and can only vary by discrete amounts: $ e \Theta(x^{\mu}) = e \Theta(x^{\mu} + T^{\mu}) + \pi n \Rightarrow \delta \Theta(x) = n /2 e$. Thus, through Stokes’ theorem, this means that the magnetic flux along an orbit $\Sigma$ characterized by pure gauge is quantized and the current cannot decay smoothly: $\int_{\mathcal{S}_{\Sigma}}\mathbf{B}(\mathbf{x},t)\cdot d\mathbf{S} = \oint_{\Sigma_{\mathcal{}}}\mathbf{A}(\mathbf{x},t)\cdot d\mathbf{x} = \oint_{\Sigma}\mathbf{\nabla}{{\theta}}(\mathbf{x})\cdot d\mathbf{x}= n / 2 e$. We have therefore obtained the description of superconductivity given in [@Weinberg:1996kr]. Superconductivity has historically originated the Higgs mechanism, in our description the corresponding gauge symmetry breaking mechanism can be interpreted as a quantum-geometrodynamical effect, see [@Dolce:tune; @Dolce:SuperC]. We propose the following interpretation of the mathematic results above. This formal correspondence suggests a possible statistical origin of QM associated to the fact that, typically, the Compton periodicity of elementary particles is extremely fast (except neutrinos) with respect to the present experimental time resolution $\sim 10^{-17} s$. For instance, the recurrence of a simple electrodynamical system is fixed by the electron Compton time: $\Lambda_{e}= 8.09329972 \times 10^{-21} s$. As also noted by ’t Hooft [@'tHooft:2001ar], “there is a deep relationship between a particle moving \[very fast\] along a circle and the quantum harmonic oscillator” (the basic element of QFT). Indeed, such a relativistic particle moving very fast along a circle can only be described statistically (Hilbert notation) as a fluid (continuity equation) of total density one (Born rule)[^5]. Since in our statistical (Hilbert) description of a “periodic phenomenon” (reference clock) the phase cannot be observed, to determine the energy and define time with good accuracy we must count a large number of “ticks”, according to $\|\exp[-i{\bar \omega t}]\|= \exp[-i({\bar \omega t}+ \pi)] = \exp[-i(\bar \omega + \Delta\bar{\omega}) t ] = \exp[-i \bar \omega ( t + \Delta t)]$. This corresponds to Heisenberg’s relation $\Delta\bar{\omega}\Delta t\ge 1/2$ [@Dolce:2009ce; @Dolce:tune]. Note that we have not introduced any local-hidden-variable in the theory. The assumption of IP is the only quantization condition and represents an element of non-locality (namely “complete coherence”[^6], consistent with relativistic causality). The formal correspondence to ordinary QM suggests that the theory could violate Bell’s inequalities as ordinary QM, and thus a possible deterministic nature at time scales smaller than the Compton time of the system. #### Applications to modern physics {#applications-to-modern-physics .unnumbered} — Elementary spacetime cycles represent a new interpretational key of fundamental aspects of modern physics, yielding elements of the mathematical beauties of extra-dimensional or string theories. The Compton periodicity of the worldline parameter enters into the equations of motion of the free field $\Phi(x)$ in perfect analogy with the cyclic eXtra-Dimension (XD) of a Kaluza-Klein (KK) theory, [@Klein:1926tv; @Kaluza:1921tu]. If we have a KK massless theory $dS^{2} = dx_{\mu} d x^{\mu} - d s^{2} \equiv 0$ with cyclic XD $s$ and compactification length $\Lambda$, and we identify the cyclic XD with the Minkowskian worldline parameter, $d s^{2} = dx_{\mu} d x^{\mu} $, we obtain exactly our purely 4D theory with Compton worldline periodicity $s \in (0,\Lambda]$ and thus, through Lorentz transformations, elementary spacetime cycles. We address this identification by saying that the XD is Virtual (VXD) [@Dolce:AdSCFT]. For instance, the rest energy spectrum of our harmonic system, or equivalently of a second quantized KG field, is the analogous of the KK mass tower $m_{n} = n \bar m = 2 \pi n / \Lambda \equiv \omega_{n} (0) = n\bar \omega (0) = 2 \pi n / T(0)$[^7]. In this case the KK modes are virtual: they are not independent KK particles of different masses, they describes the energy excitations of the same 4D harmonic system, the quantum excitations of the same system. The holographic approach in XD theories actually corresponds to the same collective[^8] description of the KK modes, but in addition to this the heavy modes are integrated out in terms of the source field $\phi_{\Sigma}(x) \sim \bar \Phi(x)$. The result of the holographic approach is thus equivalent to an effective description of the VXD: $ \mathcal{S}'^{Holo}_{\Phi\|_{\Sigma}=e\phi_{\Sigma}}(s_{f},s_{i}) \sim {\mathcal{S}}^{VXD}(s_{f},s_{i})+\mathcal{O}(E^{eff}/\bar{m})$ (with implicit source term) [@Dolce:AdSCFT; @Casalbuoni:2007xn]. We note that, similarly to light-front-quantization [@Honkanen:2010nt], Hilbert notation with Schrödinger evolution $i \partial_{s} \left\|\Phi'\right\rangle = \mathcal M \left\|\Phi' \right\rangle$ and implicit commutation relation $[s,\mathcal{M}] = i $ between mass operator $\mathcal M \left \|\phi_{n} \right\rangle \equiv m_n \left\|\phi_{n} \right\rangle$ and XD $s$ can also be applied to KK theories [@Dolce:AdSCFT]. An example [@Dolce:AdSCFT; @Dolce:graphene] of this geometric interpretation of the particle’s mass as Compton proper-time periodicity, as well as of the analogy between XD geometry and quantum behavior, comes from graphene physics. In graphene the electrons behave as 2D massless particles. However, as a dimension is curled up to form a nanotube, the electrons at rest with respect to the axial direction (rest frame) have residual cyclic motions along the radial direction: the residual proper-time periodicity of corresponds to the Compton time defining the effective mass of the electrons in nanotubes, [@deWoul:2012ed; @RevModPhys.79.677; @Dolce:AdSCFT; @Dolce:graphene]. A similar example is given by graphene bilayer, in which the effective mass is originated by the residual proper-time periodicity associated to the cyclic motion of the electrons between the two layers at different potentials [@2011arXiv1103.1663Z]. By means of the duality to XD theories, the generic interaction scheme $\bar{k}'_{\mu}(x)$ is equivalently encoded in a corresponding deformed[^9] VXD metric $d S^{2} \simeq g_{\mu\nu} d x^{\mu} d x^{\nu} - d s^{2}$. As confirmation of our description of gauge interaction, it can be shown that the corresponding spacetime geometrodynamics discussed above are equivalently encoded in a virtual Kaluza-like metric (Kaluza’s miracle) [@Kaluza:1921tu]. By combining the correspondence with relativistic QM [@Dolce:2009ce], the geometrodynamical formulation of interactions [@Dolce:tune] and the dualism to XD theories, we obtain that elementary cycles pinpoints the fundamental classical to quantum correspondence of Maldacena’s conjecture and holography [@Witten:1998zw; @Gherghetta:2010cj]. This implies that the holographic representation of the classical configurations of $\Phi'$ in a curved XD background describes the low energy quantum behavior of the corresponding interaction scheme [@Dolce:AdSCFT] $$\int_{V} {\mathcal{D}\mathbf{x}} e^{ i \mathcal{S}'(t _{f},t_{i})} \leftrightsquigarrow e^{ i \mathcal{S}'^{Holo}_{\Phi\|_{\Sigma}=e\phi_{\Sigma}}(s_{f},s_{i})}\,.\label{VXD:QFT:corr}$$ Another example of this classical to quantum correspondence is given by the Quark-Gluon-Plasma (QGP) freeze-out. We assume that the QGP is described by the classical Bjorken hydrodynamical model [@Magas:2003yp]. This means that during the freeze-out the energy decays exponentially with the proper-time $s$. Thus, neglecting masses ($\omega \sim k$), the four-momentum has exponential conformal decay $\bar{k}_{\mu}\rightarrow\bar{k}'_{\mu}(s)\simeq e^{-K s}\bar{k}_{\mu}$[^10]. This equivalently means that the spacetime periodicity has exponential conformal dilatation $T^{\mu}\rightarrow T'^{\mu}(s)\simeq e^{K s }T^{\mu}$. According to our geometrodynamical description, this interaction scheme is encoded in the warped metric $ds^{2}=e^{-2K s}dx_{\mu}dx^{\mu}$, resulting from the deformation $dx_{\mu}\rightarrow dx'_{\mu}(s)\simeq e^{-ks}dx_{\mu}$, and in turns in the virtual AdS metric $dS^{2}\simeq e^{-2 K s }dx_{\mu}dx^{\mu}-ds^{2}\equiv0$. The time periodicity of the fields during the QGP freeze-out turns out to be the conformal parameter $T(s) = e^{K s }/K = 2 \pi / E(s)$. In agreement with the AdS/CFT dictionary, this naturally describes, by means of the Planck constant, the inverse of the energy during the freeze-out. Infinite VXD means that the system has infinite Compton recurrence. Indeed, according to our classical to quantum correspondence, the modulated classical configurations of a harmonic system in such a unbounded warped metric encode the quantum behavior of a massless quantum system, of a conformal theory. In particular the warped modulations of the harmonic solution leads to asymptotic freedom, as shown in [@Pomarol:2000hp; @ArkaniHamed:2000ds]. A massive system has finite Compton periodicity, compact VXD. Indeed, this breaks the conformal invariance. It is known from AdS/QCD that the collective modulated harmonics in this compact warped configuration qualitatively matches the quantum behaviors and mass spectrum of the hadrons (improvements of this model are mentioned in [@Dolce:AdSCFT]). Similarly to Veneziano’s original idea, in our description the hadrons are indeed collective energy (quantum) excitations, virtual KK modes, of the same fundamental vibrating “string” [@FoundString]. This allows us to introduce another remarkable property of the theory. From the free rest action in (\[action:compact\]) we see that the Compton wordline recurrence of elementary particles allow us the possibility to define of a novel, minimal string theory based on a single compact world-parameter and a purely 4D target spacetime. From a mathematical point of view, the compact worldline parameter of the theory plays a role similar to both the compact worldsheet parameter of ordinary string theory and of the XD of the KK theory. For this reason the theory inherits fundamental aspects of both string and XD theories, avoiding the introduction of any unobserved XD. As we will show in forthcoming papers, this yields interesting analogies to Veneziano’s amplitude [@Afonin:2011hk], Virasoro’s algebra [@Dolce:tune], phenomenological properties of Randall-Sundrum models and implications to quantum gravity. The compact 4D target spacetime has also important justifications in QCD confinement [@PhysRevD.9.3471]. #### Conclusions {#conclusions .unnumbered} — Pure quantum systems are characterized by IP, so that isolated elementary particles can be regarded as reference clocks “linking time to the particle’s mass” [@Lan01022013]. We infer that a formulation of physics in terms of intrinsically cyclic spacetime dimensions is fully consistent with the theory of relativity [@Dolce:2009ce; @Dolce:tune; @Dolce:AdSCFT]. In fact, relativity fixes the spacetime differential structure (metric) leaving the freedom to play with BCs, as long as the variational principle is fulfilled. This allows us to encode undulatory mechanics directly into the spacetime geometry, enforcing the wave-particle duality and the local nature of relativistic spacetime. We have shown that ordinary quantum behaviors are consistently derived from corresponding classical cyclic dynamics. This provides an intriguing unified description of fundamental aspects of modern physics, including a geometrodynamical description of gauge interaction analogous to gravity, and pinpoints the fundamental classical to quantum correspondence characteristic of important integrable theories. #### Acknowledgments: {#acknowledgments .unnumbered} Parts of this work have been presented in recent international conferences [@Dolce:cycle; @Dolce:ICHEP2012; @Dolce:Dice2012; @Dolce:TM2012]. [^1]: This means to promote IP to a general principle of physics. [^2]: We use the term “cord” to avoid confusion with string theory [^3]: Another equivalently allowed possibility is to assume anti-PBCs instead of PBCs, obtaining $\omega_{n}(\bar \mathbf k) = (n + \frac{1}{2}) \frac{2 \pi}{ T(\bar \mathbf k)} = (n + \frac{1}{2}) \bar \omega (\bar \mathbf k) $. [^4]: E.g. spatial spherical symmetry $\mathbb S^{2}$ implies ordinary angular momentum quantization with two additional quantum numbers $\{l,m\}$), with related harmonic expansion and implicit commutation relations. [^5]: It is interesting to note that this is analogous to the statistical description of the outcomes of a die rolling very fast. In a die the outcomes can be predicted by observing the motion with a sufficient resolution in time (an imaginary observer with infinite time resolution has no fun playing dice, “God does not play dice”, A. Einstein.) [^6]: The quantum recurrence is destroyed by the dissipative thermal noise (chaotic interactions) [@2013arXiv1304.6295F]. This correctly describe the quantum to classical transition. Cyclic dynamics suggests a physical interpretation of the “mathematical trick” of Euclidean time periodicity imposed to quantize statistical systems and of Wick’s rotation [@Dolce:SuperC]: the assumption in IP is in general a quantization condition for a system at the equilibrium. [^7]: This rest spectrum is observed in quantum phenomena such as superconductivity and nanotubes [@deWoul:2012ed]. [^8]: In the holographic propagator $\Pi^{Holo}(\bar k^{2})$ the effective propagation of the all KK modes is in fact described in a collective way in terms of the same fundamental $\bar k_{\mu}$ fixed by the source field. [^9]: In the case of finite Compton periodicity, dilatons or softwalls should be included as well as mixing terms $\mathcal O(dx^{\mu} ds)$. [^10]: In QCD thermodynamics [@Satz:2008kb], $K$ is the cooling gradient (Newton’s law).
--- abstract: 'The main objective of this work is to accelerate the Maximum-Likelihood (ML) estimation procedure in radio interferometric calibration. We introduce the OS-LS and the OS-SAGE radio interferometric calibration methods, as a combination of the Ordered-Subsets (OS) method with the Least-Squares (LS) and Space Alternating Generalized Expectation maximization (SAGE) calibration techniques, respectively. The OS algorithm speeds up the ML estimation and achieves nearly the same level of accuracy of solutions as the one obtained by the non-OS methods. We apply the OS-LS and OS-SAGE calibration methods to simulated observations and show that these methods have a much higher convergence rate relative to the conventional LS and SAGE techniques. Moreover, the obtained results show that the OS-SAGE calibration technique has a superior performance compared to the OS-LS calibration method in the sense of achieving more accurate results while having significantly less computational cost.' author: - \| S. Kazemi$^{1}$[^1], S. Yatawatta$^{2}$, S. Zaroubi$^{1}$\ $^{1}$Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, the Netherlands\ $^{2}$ASTRON, Postbus 2, 7990 AA Dwingeloo, the Netherlands bibliography: - 'references.bib' title: 'Radio Interferometric Calibration via Ordered-Subsets Algorithms: OS-LS and OS-SAGE calibrations' --- =-.8in \[firstpage\] methods: statistical, methods: numerical, techniques: interferometric Introduction {#sec:intro} ============ Radio interferometry is the technique of combining and correlating signals from two or more separate antennas to observe the target astronomical object with a resolution determined not by the size of a single antenna but by the area covered with all the incorporated antennas. Therefore, a much better angular resolution can be achieved using radio interferometers with multiple antennas instead of single dishes [@A.R.1; @burke]. The main objective of designing the new generation of radio interferometers, such as the Square Kilometer Array (SKA)[^2], the Murchison Widefield Array (MWA)[^3], the Precision Array to Probe Epoch of Reionization (PAPER)[^4], the 21-cm Array (21CMA)[^5], the Hydrogen Epoch of Reionization Array (HERA)[^6], the Long Wavelength Array (LWA)[^7] and the LOw Frequency ARray (LOFAR)[^8], with the ability to collect enormous amounts of data, is improving the sensitivity, resolution and frequency coverage of observations. Therefore, to deliver their scientific goals, there is a need for processing a large amount of data and for upgrading the accuracy as well as the processing time of the existing calibration techniques. Propagation medium and the receivers’ effect in radio interferometric data are initially unknown and have to be calibrated and corrected before imaging. Self-calibration [@selfcal] estimates the Maximum-Likelihood (ML) estimate of the unknowns utilizing only the measurements, and due to its high accuracy, it has become the method of choice, as in this paper, for calibrating the new generation of radio synthesis arrays. In the presence of additive Gaussian noise, calibration is performed as a non-linear Least-Squares (LS) optimization that calculates the ML estimation using iterative gradient based methods such as Levenberg-Marquardt (LM) method [@A.L.1; @K.L1]. However, the LS calibration suffers from a very low convergence rate because the parameters must be updated simultaneously on a complete data space. Solving for a large number of unknowns, the Jacobian computation corresponding to the applied gradient based method is considerably costly. This makes the LS calibration impractical for calibration of giant radio telescopes like SKA with thousands of receivers. The convergence rate and computational efficiency of calibration is significantly improved by the recently proposed Space Alternating Generalized Expectation maximization (SAGE) calibration technique [@S.2************; @S.K]. SAGE method [@J.A.1] is a specific version of the Expectation Maximization (EM) algorithm [@M.2] which converges even faster than the conventional EM [@FA]. The SAGE algorithm partitions the data space to smaller “hidden” data spaces and at every iteration, it alternates between updating parameters on some or all of them. Obtaining the ML estimate for the parameters of these small data spaces, which carry less information compared to the complete data space, provides SAGE algorithm with a superior accuracy as well as lower computational cost compared to the LS technique. Nevertheless, there is still a need for improving the speed of calibration process especially for radio synthesis arrays such as LOFAR and SKA. In this paper, we utilize the Ordered-subsets (OS) algorithm for accelerating the speed of calibration. The well known OS algorithm accelerates the convergence rate of iterative ML estimations and has been widely used in medical imaging [@OS1; @OS2]. This method decomposes the objective (likelihood) function to several sub-objective functions and updates the parameters by using the gradient of one, or some, of the sub-objective functions as an approximation to the original objective function’s gradient. At the initial iterations, when the parameters are far from the optimum point, these approximations are quite reasonable since the gradient is only an approximation at those stages. Thus, they can be efficient substitutions for the gradient of the original cost function and considerably accelerate the computations of the OS algorithm. However, it must be taken into account that the highest accuracy that OS methods can achieve is the same as the one which could be obtained by the conventional (non-OS) techniques. Close to the optimal solution, OS methods generally do not converge but rather become stuck at a sub-optimal limit cycle of as many points as there are sub-objective functions. Therefore, if OS method becomes globally convergent [@Ahn; @OS1], it maintains exactly the same accuracy of the convergent non-OS methods. This paper is organized as follows: In sections \[Calibration Data Model\] and \[algoritms\], we present the general data model of radio interferometric calibration and the classical LS and SAGE calibration techniques. In section \[pro\], we introduce the OS-LS and OS-SAGE calibration techniques in order to cut down the processing time of the conventional LS and SAGE calibration methods. The computational advantages of applying the OS type calibrations instead of the conventional methods are also shown. For the ML estimations, the LM method is applied. At the end of section \[pro\], we show an application of OS calibration to accelerate computations when calibrating for an individual data sample. The technique is based on partitioning data over baselines and hence could also be useful in speeding up the calibration procedure of radio telescopes with a large number of receivers. In section \[Results\], we give results based on simulations to demonstrate the superior convergence rate of the OS calibration schemes compared to the non-OS ones. Finally, we draw our conclusions in section \[Conclusions\]. The following notations are used in this paper: Bold, lowercase letters refer to column vectors, e.g., [y]{}. Upper case bold letters refer to matrices, e.g., [C*]{}. The transpose, Hermitian transpose, and conjugation of a matrix are presented by $(.)^T$, $(.)^H$, and $(.)^$, respectively. The matrix Kronecker product is denoted by $\otimes$. $\mathbb{R}$ is the set of Real numbers. $\operatorname{E}\{\}$ denotes the statistical expectation operator. The real and imaginary parts of complex quantities are shown by $\mathfrak{Re}$ and $\mathfrak{Im}$, respectively. Calibration Data Model {#Calibration Data Model} ====================== In this section, the general measurement equation of a polarimetric radio interferometer is presented. For some introduction to radio polarimetry and calibration the reader is referred to @J.P.1 and @J.P.2. Consider a radio interferometer with $N$ antennas which observes $K$ uncorrelated sources. The induced voltage at antenna $p$, ${\tilde{\bf{v}}}_{pl}$, due to radiation of the $l$-th source, ${\bf{e}}_l$, is given by ${\tilde{\bf{v}}}_{pl}={\tilde{\bf{J}}}_{pl}{{\bf{e}}}_{l}$ where ${\tilde{\bf{J}}}_{pl}$ is the complex ${2\times 2}$ Jones matrix [@J.P.1] corresponding to the sky and instrumental corruptions of the signal.\ The total signal obtained at antenna $p$, ${\bf v}_p$, is a linear superposition of $K$ such signals plus the antenna’s thermal noise. After correcting for geometric delays and the instrumental effects, the $p$-th antenna voltage is correlated with the other $N-1$ antennas voltages. The correlated voltages $\operatorname{E}\{{\bf v}_p{\bf v}_q^H\}$, referred to as [[visibility]{}]{} [@J.P.1] of baseline $p-q$ is given by $${\bf V}_{pq}={\bf G}_p \left( \sum_{l=1}^K {{\bf{J}}}_{pl}{{\bf{C}}}_{l}{{\bf{J}}}^H_{ql} \right) {\bf G}_q^H +{\bf{N}}_{pq},\label{s2}$$ where ${\bf N}_{pq}$ is the baseline’s additive noise and ${{\bf{C}}}_{l}=\operatorname{E}\{{{\bf{e}}}_{l}{{\bf{e}}}_{l}^H\}$ is the $l$-th source [[coherency]{}]{} matrix [@bornwolf; @J.P.1]. The errors common to all directions (mainly the receiver delay and amplitude errors) are given by ${\bf G}_p$ and ${\bf G}_q$. We assume that an initial calibration, at a finer time and frequency resolution, is performed to estimate ${\bf G}_p$-s (direction independent effects). Then, the corrected data is obtained as $$\widetilde{\bf V}_{pq}={\bf G}_p^{-1} {\bf V}_{pq} {\bf G}_q^{-H},$$ where $\widetilde{\bf V}_{pq}$ are the visibilites after correction for effects common to all directions. The remaining errors are unique to a given direction, but residual errors in ${\bf G}_p$-s are also absorbed into these errors, which are denoted by ${\bf{J}}_{pl}$ in the usual notation. The vectorized form of corrected visibilities are given by $${\bf v}_{pq}\equiv\mbox{vec}(\widetilde{\bf V}_{pq})=\sum_{l=1}^K {\bf s}_{pql}+{\bf n}_{pq},\label{s3}$$ where ${\bf s}_{pql}={\bf J}^_{ql}\otimes{\bf J}_{pl}\mbox{vec}({\bf C}_{l})$ and ${\bf n}_{pq}=\mbox{vec}({\bf G}_p^{-1} {\bf N}_{pq} {\bf G}_q^{-H})$. The unknowns of the calibration problem are the real and imaginary parts of the Jones matrices complex elements $$\pmb{\theta}=[\mbox{vec}(\mathfrak{Re}\{{\bf J}_{11}\})^T\ \mbox{vec}(\mathfrak{Im}\{{\bf J}_{11}\})^T\ \mbox{vec}(\mathfrak{Re}\{{\bf J}_{12}\})^T\ldots]^T, \label{s333}$$ and therefore, ${\pmb{\theta}}\in \mathbb{R}^{8KN\times 1}$. Consider a dataset of $\tau$ time and frequency samples that form a small enough time and frequency interval over which $\pmb{\theta}$ is invariant. Stacking up the real and imaginary parts of the instrument’s visibilities and noise vectors in ${\bf y}=[\mathfrak{Re}\{{\bf v}^T_{12}\}\ \mathfrak{Im}\{{\bf v}^T_{12}\}\ \mathfrak{Re}\{{\bf v}^T_{13}\}\ \ldots]^T$ and ${\bf n}=[\mathfrak{Re}\{{\bf n}^T_{12}\}\ \mathfrak{Im}\{{\bf n}^T_{12}\}\ \mathfrak{Re}\{{\bf n}^T_{13}\}\ \ldots]^T,$ respectively, the general measurement equation becomes $${\bf y}=\sum_{l=1}^K {\bf s}_l({\pmb{\theta}})+{\bf n}.\label{s4}$$ In (\[s4\]), ${\bf s}_l({\pmb{\theta}})=[\mathfrak{Re}\{{\bf s}^T_{12l}\}\ \mathfrak{Im}\{{\bf s}^T_{12l}\}\ \mathfrak{Re}\{{\bf s}^T_{13l}\}\ \ldots]^T$. ${\bf y},\ {\bf n},$ and ${\bf s}_l$ are vectors of size $4\tau N(N-1)$, and the noise vector ${\bf n}$ is assumed to be white Gaussian. Calibration is the ML estimation of the unknown parameter vector ${\pmb{\theta}}$ from (\[s4\]). Note that calibration methods could also be applied to the uncorrected visibilities of (\[s2\]) to estimate ${\bf G}_p$ and ${\bf G}_q$ errors as well. Moreover, having a large enough $N$ and small enough $K$, there will be enough constrains to solve for the $8KN$ unknown parameters of ${\pmb{\theta}}$ using the $4\tau N(N-1)$ measurements of ${\bf y}$. The LS and SAGE Calibration Methods {#algoritms} =================================== In this section, both the LS and SAGE calibration algorithms are briefly outlined. The OS scheme is applied to both methods. LS calibration {#Normalcal} -------------- Since the noise vector $\bf{n}$ in the calibration data model (\[s4\]) is assumed to be white Gaussian, LS calibration method estimates the ML estimate of ${\pmb{\theta}}\in \mathbb{R}^{8KN\times 1}$ by minimizing the sum of squared errors: $$\begin{array}{c} \widehat{{\pmb{\theta}}}=\mbox{arg}\ \mbox{min}\ \|\|{\bf y}-\sum_{l=1}^K {\bf s}_l({\pmb{\theta}})\|\|^2.\\ {\pmb{\theta}}\quad\quad\quad\quad\quad\quad\quad\label{s5} \vspace{-2mm}\end{array}$$ Gradient-based optimization techniques are used for solving (\[s5\]). Among those, the LM method [@A.L.1; @K.L1] is one of the most robust in the sense that it mostly converges to a global optimum. Defining the cost function $\phi({\pmb{\theta}})={\bf y}-\sum_{l=1}^K {\bf s}_l({\pmb{\theta}})$, where $\phi({\pmb{\theta}})\in\mathbb{R}^{4\tau N(N-1)\times 1}$, and initializing the starting point $\widehat{\pmb{\theta}}^1$, the LS calibration method via LM algorithm is outlined as follows:\ [for]{} every iteration $k=1,2,\ldots$ until an upper limit or convergence of $\widehat{\pmb{\theta}}^k$, [Calculate $\widehat{\pmb{\theta}}^{k+1}\in \mathbb{R}^{8KN\times 1}$ using LM algorithm as $$\widehat{\pmb{\theta}}^{k+1}=\widehat{\pmb{\theta}}^k-({\pmb{\bigtriangledown}}^T_{\pmb{\theta}} {\pmb{\bigtriangledown}}_{\pmb{\theta}} +\lambda {\bf H})^{-1}{\pmb{\bigtriangledown}}^T_{\pmb{\theta}} \phi({\pmb{\theta}})\|_{\widehat{\pmb{\theta}}^k}.\label{s6}$$ ]{} [endfor]{} In (\[s6\]), ${\pmb{\bigtriangledown}}_{\pmb{\theta}}=\frac{\partial }{\partial {\pmb{\theta}}} \phi({\pmb{\theta}})$, $\lambda$ is the damping factor [@Dampingterm], and ${\bf H}=\mbox{diag}({\pmb{\bigtriangledown}}^T_{\pmb{\theta}} {\pmb{\bigtriangledown}}_{\pmb{\theta}})$ is the diagonal of the Hessian matrix. The sizes of the Jacobian ${\pmb{\bigtriangledown}}_{{\pmb{\theta}}}$ and the linear system solved in (\[s6\]) are $4\tau N(N-1)\times 8KN$ and $8KN$, respectively. Consequently, the cost of computing ${\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}} {\pmb{\bigtriangledown}}_{{\pmb{\theta}}}$ is $\mathcal{O}((8KN)^2\times4\tau N(N-1))$. Therefore, since at every iteration all the $8KN$ parameters of ${\pmb{\theta}}$ are simultaneously updated, LS calibration has a very low speed of convergence. Furthermore, estimating a large number of unknowns, the Jacobian computation also becomes considerably costly. SAGE calibration {#SAGE} ---------------- In the case of solving for multiple sources in the sky, the SAGE calibration algorithm [@S.K; @S.2************] has a significantly improved computational cost and convergence rate compared to the LS calibration. The key point is that, in general, the SAGE algorithm [@J.A.1] partitions the complete data space to smaller “hidden” data spaces and estimates parameters in them rather than in the complete data space. Applying the SAGE algorithm to the calibration problem, the contribution of every $l$-th source in the observation is assumed to depend only on a subset of parameters, ${\pmb{\theta}}_l \in \mathbb{R}^{8N\times 1}$. Therefore, the parameter vector ${\pmb{\theta}}\in \mathbb{R}^{8KN\times 1}$ could be partitioned for different directions (sources) in the sky as $${\pmb{\theta}}=[{\pmb{\theta}}^T_1\ {\pmb{\theta}}^T_2\ldots{\pmb{\theta}}^T_K]^T.$$ This partitioning is justifiable when the sources are sufficiently separated from each other. Initializing a starting parameter vector $\widehat{\pmb{\theta}}^1$, where $\widehat{\pmb{\theta}}^k$ denotes the estimate of ${\pmb{\theta}}$ obtained at the $k$-th iteration, SAGE calibration algorithm is executed as follows:\ [for]{} every iteration $k=1,2,\ldots$ until an upper limit for $k$ or convergence of $\widehat{\pmb{\theta}}^k$: [for*]{} all or some $l\in\{1,2,\ldots,K\}$, update the $l$-th source parameters ${\pmb{\theta}}_l \in \mathbb{R}^{8N\times 1}$: 1. Define the hidden data space as $${\bf x}_{l}={\bf s}_l({\pmb{\theta}}_l)+{\bf n}\in\mathbb{R}^{4\tau N(N-1)\times 1}.\label{n17}$$ Thus, the observed data ${\bf y}\in\mathbb{R}^{4\tau N(N-1)\times 1}$ is given by $${\bf y}={\bf x}_l+\sum_{\substack{z=1\\z\neq l}}^K{\bf s}_z({\pmb{\theta}}_z).\label{n18} \vspace{-2mm}$$ 2. [[SAGE E Step]{}]{}: Calculate the conditional mean $\widehat{{\bf x}}_l^k=\operatorname{E}\{{\bf x}_l\|{\bf y},{\widehat{\pmb{\theta}}}^k\}$ as $$\widehat{{\bf x}}_l^k={\bf s}_l(\widehat{\pmb{\theta}}_l^k)+({\bf y}-\sum_{z=1}^K{\bf s}_z(\widehat{\pmb{\theta}}_z^k))={\bf y}-\sum_{\substack{z=1\\z\neq l}}^K{\bf s}_z(\widehat{\pmb{\theta}}_z^k).\label{n19} \vspace{-2mm}$$ 3. [[SAGE M Step]{}]{}: Estimate $$\begin{array}{c} \widehat{\pmb{\theta}}_l^{k+1}=\mbox{arg}\ \mbox{min}\ \|\|[\widehat{{\bf x}}_l^k-{\bf s}_l({\pmb{\theta}}_l)]\|\|^2,\\ {\pmb{\theta}}_l\quad\quad\quad\quad \end{array}\vspace{-3mm}$$\ by the LM method as $$\widehat{\pmb{\theta}}_l^{k+1}=\widehat{\pmb{\theta}}_l^k-({\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} {\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l} +\lambda {\bf H})^{-1}{\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} \phi({\pmb{\theta}}_l)\|_{\widehat{\pmb{\theta}}_l^k},\label{nn1}$$ where $\phi({\pmb{\theta}}_l)=[\widehat{{\bf x}}_l^k-{\bf s}_l({\pmb{\theta}}_l)]\in \mathbb{R}^{4\tau N(N-1)\times 1}$. [endfor]{} [endfor]{}\ Based on the above, at every $k$-th iteration, SAGE method alternates between updating parameters of some or all the sources, $l\in\{1,2,\ldots,K\}$. Calculating the ML estimate of ${\pmb{\theta}}_l \in \mathbb{R}^{8N\times 1}$ in (\[nn1\]), instead of the ML estimate of all parameters ${\pmb{\theta}}\in \mathbb{R}^{8KN\times 1}$ as in (\[s6\]), it has been proved that the SAGE algorithm benefits from an accelerated convergence rate [@J.A.1] compared to the LS method. The sizes of the Jacobian ${\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l}$ and the linear system solved in (\[nn1\]) are $4\tau N(N-1)\times 8N$ and $8N$, respectively. In addition, the cost of computing ${\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} {\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l}$ is $\mathcal{O}((8N)^2\times4\tau N(N-1))$. Thus, applying LM algorithm for estimating ${\pmb{\theta}}_l$ from (\[nn1\]), the computational expense of the SAGE calibration is much cheaper compared to the LS calibration. Note that in the SAGE calibration, instead of partitioning the parameters of the individual sources, one could also make partitions including more than a single source sharing common parameters [@S.K.3]. This is more efficient when some sources have a small angular separation from each other in the sky and hence share some parameters. The OS-LS and OS-SAGE Calibration Methods {#pro} ========================================= In this section, OS-LS [@LL.] and OS-SAGE [@H.Z.] calibration algorithms, combinations of Ordered-Subsets (OS) algorithm with LS and SAGE calibration methods, are introduced to speed up the conventional LS and SAGE calibration procedures. Ordered-Subsets (OS) algorithm is applied to those optimization problems with a cost function that can be expressed as a sum of several other cost functions for accelerating the convergence rate. The solutions obtained by the OS method attain almost the same accuracy as those obtained by the non-OS optimization methods in a fraction of the time [@OS1]. The key idea is to consider the Jacobian of one, or some, sub-cost functions as an approximate gradient of the original cost function. These approximations are quite reasonable when one is far from the optimal point, and provide OS method with a very fast convergence rate. However, at later iterations and when the parameters are close to the global optimum, the approximations restrict the OS method to a sub-optimal limit cycle (the optima of the individual sub-observations which are processed in OS iterations). Therefore, the OS method does not converge globally [@Ahn]. Denote the visibility vectors of the $\tau$ time and frequency samples that have the fixed gain errors $\pmb{\theta}\in \mathbb{R}^{8KN\times 1}$ by ${\bf y}_1, {\bf y}_2, \ldots, {\bf y}_{\tau}$, where ${\bf y}_t\in\mathbb{R}^{4N(N-1)\times 1}$, for $t\in\{1,2,\ldots,\tau\}$. Since the noise is statistically independent, calibration problem could be restated as $$\begin{aligned} \widehat{{\pmb{\theta}}}=\mbox{arg}\ \mbox{max}\prod_{t=1}^{\tau}f_t({{\bf y}_t};{\pmb{\theta}})=\mbox{arg}\ \mbox{max}\sum_{t=1}^{\tau} \mathcal{L}_t({\pmb{\theta}}\|{\bf y_t}),\\[-3mm] {\pmb{\theta}}\quad\quad\quad\quad\quad\quad\quad\quad\quad{\pmb{\theta}}\quad\quad\quad\quad\quad\quad\quad\nonumber\label{s55}\end{aligned}$$ where $f_t$ and $\mathcal{L}_t$ are the probability density and the log-likelihood functions for the visibility vector ${\bf y}_t$, respectively. OS algorithm is applied for accelerating the maximization of this sum of log-likelihood functions. Supposing that the following Jacobian equivalence conditions hold $${\pmb{\bigtriangledown}}_{\pmb{\theta}} \mathcal{L}_1\cong {\pmb{\bigtriangledown}}_{\pmb{\theta}} \mathcal{L}_2 \cong \ldots \cong {\pmb{\bigtriangledown}}_{\pmb{\theta}} \mathcal{L}_{\tau},\label{mman}$$ then the OS method sequentially updates the parameters ${\pmb{\theta}}$ for one or some visibility vectors ${\bf y}_t$ (sub-observations). The solution of every sub-observation is used as the starting point of the next sub-observation. Since each sub-cost function $\mathcal{L}_t$ involves a subset of data, ${\bf y}_t$, which is independent from the others, the method is named “ordered subsets”. Sub-observations might be ordered for updating by some scheme that gives preferences to the data items, or, as in this work, in random. An introduction to the OS algorithm is presented by @Ahn. In the following, the OS-LS and OS-SAGE methods are outlined. Note that the size of sub-observations ${\bf y}_t$-s must be grater than or equal to the number of unknown parameters in ${\pmb{\theta}}$. OS-LS calibration {#mehm1} ----------------- In the presented OS-LS calibration, the LM method is selected as the gradient-based ML estimation algorithm of the LS calibration. Starting with an initial suggestion $\widehat{\pmb{\theta}}^1 \in \mathbb{R}^{8KN\times 1}$, OS-LS is executed as:\ [for]{} every iteration $k=1,2,\ldots$ until an upper limit or convergence of $\widehat{\pmb{\theta}}^k$, run $m$ [OS iterations]{}: [for]{} some or all sub-observation $\{{\bf y}_t\|t=1,\ldots,m\leq \tau\}$: [Select ${\pmb{\theta}}^k=\widehat{\pmb{\theta}}^t$, and calculate]{} [ $${\pmb{\theta}}^{k+1}={\pmb{\theta}}^k-({\pmb{\bigtriangledown}}^T_{\pmb{\theta}} {\pmb{\bigtriangledown}}_{\pmb{\theta}}+\lambda {\bf H})^{-1}{\pmb{\bigtriangledown}}^T_{\pmb{\theta}} \phi({\pmb{\theta}})\|_{{\pmb{\theta}}^k},\label{mano}$$]{} [where $\phi({\pmb{\theta}})=[{\bf y}_t-\sum_{l=1}^K {\bf s}_l({\pmb{\theta}})]\in\mathbb{R}^{4N(N-1)\times 1}$. ]{} [Select $\widehat{\pmb{\theta}}^{(t \bmod m)+1}={\pmb{\theta}}^{k+1}$ for the next sub-observation.]{} [endfor]{} [endfor]{} As given above, at every LM iteration, parameters are sequentially updated for some or all sub-observations. The sizes of the Jacobian ${\pmb{\bigtriangledown}}_{{\pmb{\theta}}}$ and the linear system solved in (\[mano\]) are $4 N(N-1)\times 8KN$ and $8KN$, respectively. Moreover, the cost of computing ${\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}}{\pmb{\bigtriangledown}}_{{\pmb{\theta}}}$ is $\mathcal{O}((8KN)^2\times4 N(N-1))$. When (\[mman\]) holds, the Jacobian is calculated only for one, or a few, number of sub-observations per iteration and hence, the OS-LS method’s convergence rate is considerably increased compared to the LS method. OS-SAGE calibration {#mehm2} ------------------- In this section, the OS-SAGE calibration method is introduced. A similar OS-SAGE technique is used for positron emission tomography (PET) by @H.Z.. Initializing $\widehat{\pmb{\theta}}^1 \in \mathbb{R}^{8N\times 1}$, OS-SAGE is outlined as follows:\ [for]{} every $k=1,2,\ldots$ until an upper limit for $k$ or convergence of $\widehat{\pmb{\theta}}^k$, execute $m$ [OS iterations]{}: [for]{} some or all sub-observations $\{{\bf y}_t\|t=1,\ldots,m\leq \tau\}$: [Select ${\pmb{\theta}}^k=\widehat{\pmb{\theta}}^t$.]{} all or some $l\in\{1,2,\ldots,K\}$, update the $l$-th source\ parameters ${\pmb{\theta}}_l \in \mathbb{R}^{8N\times 1}$: 1. Define $${\bf y}_t={\bf x}_l+\sum_{\substack{z=1\\z\neq l}}^K{\bf s}_z({\pmb{\theta}}_z),\quad{\bf x}_{l}={\bf s}_l({\pmb{\theta}}_l)+{\bf n}.\label{n1117}$$ 2. [[SAGE E Step]{}]{}: Calculate $\widehat{{\bf x}}_l^k=\operatorname{E}\{{\bf x}_l\|{\bf y}_t,{{\pmb{\theta}}}^k\}$ as $$\widehat{{\bf x}}_l^k={\bf y}_t-\sum_{\substack{z=1\\z\neq l}}^K{\bf s}_z({\pmb{\theta}}_z^k),\ {\bf y}_t\in\mathbb{R}^{4N(N-1)\times 1}.\label{n1119}$$ 3. [[SAGE M Step]{}]{}: Similar to (\[nn1\]), estimate $$\begin{array}{c} {\pmb{\theta}}_l^{k+1}=\mbox{arg}\ \mbox{min}\ \|\|[\widehat{{\bf x}}_l^k-{\bf s}_l({\pmb{\theta}}_l)]\|\|^2,\\ {\pmb{\theta}}_l\quad\quad\quad\quad \end{array}\vspace{-3mm}$$\ using the LM method, by $${\pmb{\theta}}_l^{k+1}={\pmb{\theta}}_l^k-({\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} {\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l} +\lambda {\bf H})^{-1}{\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} \phi({\pmb{\theta}}_l)\|_{{\pmb{\theta}}_l^k}\label{nn111}$$ [endfor]{} Select $\widehat{\pmb{\theta}}^{(t \bmod m)+1}={\pmb{\theta}}^{k+1}$ for the next sub-observation. [endfor]{} [endfor]{} OS method reduces the data size from $4\tau N(N-1)$ to $4N(N-1)$, since it calculates the partial gradients for sub-observations ${\bf y}_t\in\mathbb{R}^{4N(N-1)\times 1}$, $t\in\{1,2,\ldots,\tau \}$, instead of the whole observed data ${\bf y}\in\mathbb{R}^{4\tau N(N-1)\times 1}$. Thus, the size of the Jacobian ${\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l}$, where $\phi({\pmb{\theta}}_l)=[\widehat{{\bf x}}_l^k-{\bf s}_l({\pmb{\theta}}_l)]\in \mathbb{R}^{4N(N-1)\times 1}$, calculated by LM method for every OS iteration of the OS-SAGE calibration at (\[nn111\]), is $4N(N-1)\times 8N$. The size of the linear system solved in (\[nn111\]) is $8N$ and the cost of computing ${\pmb{\bigtriangledown}}^T_{{\pmb{\theta}}_l} {\pmb{\bigtriangledown}}_{{\pmb{\theta}}_l}$ is $\mathcal{O}((8N)^2\times4 N(N-1))$. When $m\ll \tau$, the OS-SAGE method converges much faster than the conventional SAGE algorithm for which the Jacobian size is $4\tau N(N-1)\times 8N$. On the other hand, for every $t$-th OS iteration, the updated result of the $(t-1)$-th sub-observation is used as the starting point. Every OS-SAGE iteration includes $m$ number of SAGE iterations. Therefore, at initial iterations when (\[mman\]) holds, OS-SAGE algorithm increases the likelihood function as equivalent to SAGE method with $m$ iterations. Thus, the convergence of OS-SAGE compared with SAGE is accelerated. Partitioning the baselines {#bp} -------------------------- So far, we have divided the data into sub-observations only based on their integration time and frequency. However, there are cases in which we need to calibrate for a single time and frequency interval. For instance, consider calibrating only for the $i$-th time and frequency interval when $1\leq i\leq \tau$. To apply OS calibration to such a case, one can define sub-observations by partitioning the data vector ${\bf y}_i$ over the instrument’s baselines as, $${\bf y}_i=[{\bf y}_{i1}^T\ {\bf y}_{i2}^T\ \ldots {\bf y}_{iB}^T]^T, \quad B\ll \frac{N(N-1)}{2}.$$ Then, similar to (\[s55\]), the calibration problem becomes $$\begin{aligned} \widehat{{\pmb{\theta}}}=\mbox{arg}\ \mbox{max}\sum_{b=1}^{B} \mathcal{L}_b({\pmb{\theta}}\|{\bf y}_{ib}),\\[-3mm] \quad\quad\quad\quad\quad\quad\quad\quad\quad{\pmb{\theta}}\quad\quad\quad\quad\quad\quad\quad\nonumber\label{s5555}\end{aligned}$$ for which OS methods presented by sections \[mehm1\] and \[mehm2\] are applicable, and where [OS iterations]{} are executed over $\{{\bf y}_{ib}\|b=1,\ldots,m\leq B\}$. Utilizing such an OS calibration could also be beneficial in cutting down the computational expense of calibration of interferometers with a large number of receivers. The only points that should be taken into account are: - [Every partition of data (sub-observation) ${\bf y}_{ib}$, for $b\in\{1,2,\ldots,B\}$, must have visibilities from different baselines such that the baselines cover all the receivers of the instrument (or all the parameters).]{} - [The number of visibilities of every sub-observation must be equal to, or larger than, the number of calibration unknowns, $$\|\|{\bf y}_{ib}\|\|_1\geq8KN.$$]{} Discussion ---------- To wrap up all the discussed calibration algorithms, we present a general overview in Fig. \[modiosls\]. Fig. \[modiosls\] illustrates LS, SAGE, OS-LS, and OS-SAGE calibrations algorithms. $\begin{array}{cc} { \begin{array}{l} \hspace{-1cm}\epsfig{file=ff1.eps, bb= 95 95 650 500,clip=,width=7cm,scale=0.4} \\[5mm] \hspace{-1cm}\epsfig{file=ff2.eps, bb= 95 95 650 500,clip=,width=7cm,scale=0.4} \\ \end{array} } & {\vspace{-5mm}\begin{array}{l} \hspace{1cm}\epsfig{file=ff3.eps, bb= 95 95 650 500,clip=,width=7cm,scale=0.4} \\[5mm] \hspace{1cm}\epsfig{file=ff4.eps, bb= 95 95 650 500,clip=,width=7cm,scale=0.4} \\ \end{array}} \end{array}$ Note that: - [As it is discussed at the beginning of this section \[pro\], the OS algorithms do not necessarily converge. Nevertheless, there exist two major approaches in dealing with the convergence problem of the OS method: (i) using relaxation parameters (stepsizes) [@Ahn]. Calculating suitable relaxation parameters per every iteration is considerably costly. That makes the approach of progressively decreasing the number of sub-observations in OS method to be preferable. (ii) Reducing the number of subsets with increasing iterations until the complete dataset estimate is reconstructed [@OS1]. In the OS method, one can incrementally combine some sub-observations together until there are no individual sub-observations remaining. Therefore, at the final iteration, the OS method is in fact changed to the non-OS technique which is used for the ML approximations, solving for the complete dataset. This approach guarantees global convergence as long as the non-OS ML estimation techniques (LS, SAGE, etc.) converge. However, it must be taken into account that the highest accuracy achievable by the proposed scheme is equal to any non-OS optimization methods. Modifying OS calibration in order to achieve an accuracy superior to the ones obtained by non-OS calibrations is addressed in future work. ]{} - When the Signal to Noise Ration (SNR) is poor, shifting to non-OS calibrations after running a few number of OS iterations is recommended.. Moreover, instead of running the OS method on every individual time and frequency sub-observation ${\bf y}_t$, for $t\in\{1,2,\ldots,\tau\}$, one could also apply the method to combinations of two or more sub-observations to improve the SNR. Fig. \[subsets\] shows examples of having incrementally ordered datasets of size two, randomly chosen datasets of the same size, and randomly chosen datasets from different sizes, from left to right, respectively. The datasets could be arranged in different orders depending on the characteristics of specific observations. Similarly, subsets of frequency ordered sub-observations could be introduced. \ - [In the calibration data model presented by (\[s4\]), we consider a very general form of the Jones matrices $\bf{J}$, as complex $2\times 2$ matrices, and then search for the real and imaginary parts of their elements which are collected in $\pmb{\theta}$. However, one can use a more detailed presentation of the Jones matrices in the data model, for instance, when the elements of the Jones matrices are functions of time $\zeta$ and frequency $\xi$, $${\bf J}= \left[ \begin{array}{cc} \eta_1(\zeta,\xi) & \eta_2(\zeta,\xi) \\ \eta_3(\zeta,\xi) & \eta_4(\zeta,\xi) \end{array} \right].\label{gek}$$ Then, calibration is estimation of these functions, denoted by $\eta$ in (\[gek\]). But, this leads again to estimation of some constant parameters which define the functions. Therefore, OS calibration is also useful for such a case as well and its partitioning of data to time and frequency sub-observations would not cause any degradation of the accuracy of calibration.]{} Results {#Results} ======= In this section, simulated data are used to compare the performance of LS and SAGE calibrations with OS-LS and OS-SAGE ones. Note that $n$ in this section denotes the number of iterations of the conventional LS and SAGE methods. The implementation of the calibration algorithms are done using MATLAB software. The unit of color bars of all the images are in Jansky (Jy). Simulations {#sim} ----------- A 12 hour observation of Westerbork Synthesis Radio Telescope (WSRT), including 14 receivers observing a sky with 50 sources, is simulated. Three sources are very bright with intensities 160, 107, and 108 Jy, and forty seven other sources are faint with intensities below 15 Jy. The source positions are following a uniform distribution. The Jones matrices are generated as multiplications of different linear combinations of $sin$ and $cos$ functions. Their gradients vary slowly (coherence time about three minutes) as a function of time such that on a few seconds time intervals the variation could be negligible. We keep the $\mbox{SNR}=80$. The simulated single channel image at 355 MHz is shown in Fig. \[FigureA1\] in which the background faint sources are almost invisible. We partition the simulated data to ten seconds time intervals, $\tau=10$, including sub-observations obtained from ten individual seconds, for which the gain errors are assumed to be the same. Then, we calibrate the data partitions only for the three brightest sources via the LS and SAGE calibration methods. The residual images, obtained after $n=9$ iterations, are presented in Fig. \[FigureA2\]. As Fig. \[FigureA2\] shows, among those three subtracted bright sources, the central one is the best removed (slightly underestimated) by both SAGE and the LS calibration methods. The unsolved forty seven faint sources are also visible in both residual images. But, the two other bright sources are not subtracted perfectly (overestimated in the left and right sides and underestimated in the central parts). This problematic pattern is expected to be improved by increasing the number of iterations. There is no significant difference between the residual images produced by the LS and SAGE methods in Fig. \[FigureA2\]. However, as it is shown in Table \[table1\], the noise level in the residual image of the SAGE calibration is lower than the one of the LS method. Therefore, SAGE calibration reveals a superior performance compared to the LS calibration since it achieves more accurate results with a considerably less computational complexity [@S.K]. $ \begin{array}{cc} \multicolumn{2}{c}{\hspace{-6mm}\epsfig{file=colormap.eps, bb= -98 393 637 438, scale=0.71}}\vspace{-10cm}\\ \hspace{-1mm}\epsfig{file=fls9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4} & \hspace{-3mm}\epsfig{file=fsage9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4}\vspace{-5mm}\\ \mbox{\color{white}\large{\bf (a)}}&\mbox{\color{white}\large{\bf (b)}}\vspace{1cm}\\ \end{array}$ The data is also calibrated by the OS-LS and OS-SAGE methods using $n=9$ iterations. OS iterations are executed for $m=1,2$ number of sub-observations which are randomly chosen. The residual images after subtracting the three brightest sources are presented in Fig. \[FigureA3\]. As Fig. \[FigureA3\] shows, the central source becomes problematic in the results of the OS calibrations and it was much better removed by the conventional LS and SAGE calibrations in Fig. \[FigureA2\]. Except for this source, the OS calibrations have a similar quality in the residual images to the conventional LS and SAGE calibrations. The two other subtracted sources are not perfectly removed and the other forty seven faint sources are visible in the images, similar to Fig. \[FigureA2\]. The residual images obtained for m=1 and m=2 OS iterations look almost the same. There is no significant improvement in the residual noise level when using $m=2$ OS iterations instead of $m=1$, as it is evident in Table \[table1\]. In this case, the OS calibration with m=1 OS iteration is preferable in comparison with m=2 since it carries a lower computational cost. $ \begin{array}{cc} \multicolumn{2}{c}{\hspace{-6mm}\epsfig{file=colormap.eps, bb= -98 393 637 438, scale=0.71}}\vspace{-19cm}\\ \hspace{-1mm}\epsfig{file=fos1ls9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4} & \hspace{-3mm}\epsfig{file=fos1sage9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4}\vspace{-5mm}\\ \mbox{\color{white}\large{\bf (a)}}&\mbox{\color{white}\large{\bf (b)}}\vspace{2mm}\\ \hspace{-1mm}\epsfig{file=fos2ls9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4} & \hspace{-3mm}\epsfig{file=fos2sage9.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4}\vspace{-5mm}\\ \mbox{\color{white}\large{\bf (c)}}&\mbox{\color{white}\large{\bf (d)}}\vspace{1cm}\\ \end{array}$ The calibrations execution times, in minutes, and the residual noise levels, in milliJansky (mJy), are presented in Table \[table1\]. Table \[table1\] shows that the OS calibrations have a much faster processing speed compared to the conventional LS and SAGE calibrations. Among OS calibrations, the ones with a smaller number of OS iterations always have faster execution, as it is the case comparing the processing times for $m=1$ to $m=2$. The fastest execution speed of the calibration method belongs to the OS calibrations with $m=1$ OS iteration. On the other hand, the OS calibrations including a large number of OS iterations usually produce more accurate solutions since they use a higher level of information in their computations. As the results of Table \[table1\] demonstrate, the accuracy obtained by $m=2$ number of OS iterations is slightly higher than the one achieved by $m=1$. However, the use of $m=1$ number of OS iterations is still preferred compared to $m=2$ since it has a considerably lower processing time. Note that the use of the SAGE type calibration methods are always preferred compared to the LS ones, providing more accurate results in a lower processing time.\ [\|l@\|l@\|l@\|l@\|l\|]{}\ m= number of OS iterations &&\ LS or SAGE iterations $n=9$ &Time & Noise & Time & Noise\ & \[minutes\] & \[mJy\] & \[minutes\] & \[mJy\]\ OS, $m=1$ & 41.3 & 234.2 & 9.7 & 226.1\ OS, $m=2$ & 75.5 & 232.9 & 20.4 & 225.7\ Conventional methods & 103.9 & 180.1 & 86.3 & 179.2\ Fig. \[FigureA6\] illustrates the residual noise level achieved by the calibration procedures versus the number of iterations of the LS and SAGE methods, when it varies between one to nine, $n\in\{1,\ldots,9\}$. The number of OS iterations are denoted by $m$. In the plots of Fig. \[FigureA6\], the residual noise levels of the OS calibrations are higher than the ones of the non-OS calibrations. However, it must be taken into account that these results are obtained by using a comparably less computational cost compared to the classical LS and SAGE calibrations. By increasing $n$, the result of SAGE calibrations are always better than the one of LS calibrations. Moreover, the accuracy of OS calibration using $m=2$ OS iterations are also always superior to the results obtained by $m=1$. ![image](probeer5.eps){width="20cm" height="12cm"} As we have seen so far in this simulation, among the OS calibrations, the ones with a smaller number of OS iterations (smaller $m$) have a lower execution time. On the other hand, the OS calibrations including a large number of OS iterations usually produce more accurate solutions since they use a higher level of information in their computations of the Jacobian. However, the use of a small number of OS iterations is still preferable since it is considerably faster and applying a high enough number of calibration iterations, we would achieve the same accuracy as with large $m$. In this section, we also demonstrate the applicability of the OS calibration in calibrating for a single time and frequency data sample, as it is discussed in section \[bp\], where the data must be partitioned over the instrument’s baselines. There are various ways of such a partitioning of visibilities among which we use the most efficient one for this specific simulation. 1. The first question is “[what is the maximum number of partitions of data over the baselines that we can define such that the baselines of every single partition cover all the receivers of the interferometer?]{}”. The reason of searching the maximum is to get the highest level of information at every calibration’s sub-observation later on. To answer this question, we use some well-known definitions of graph theory [@graph] .\ Consider the interferometer as a complete graph of order $N$ [^9] where the receivers and the baselines are the nodes and edges of the graph, respectively. Therefore, since in this simulation $N$ is even, the answer to our question is the chromatic index of this graph which is qual to $N-1$. This means we can color the $\frac{N(N-1)}{2}$ edges of the graph by $N-1$ colors where every color is covering all the $N$ nodes and $\frac{N}{2}$ number of edges. For instance, Fig. \[FigureB3\] shows a complete graph of order eight, colored by $8-1=7$ colors, where every color covers all the nodes by $\frac{8}{2}=4$ number of edges. We partition the visibilities based on the color of their corresponding baselines in the graph. Thus, at every partition, we have $\frac{N}{2}$ number of visibility matrices. 2. [The second question is “[how many partitions should be collected at every OS calibration’s sub-observation to ensure that (\[s4\]) is not an under-determined system?’]{}’. Every partition has $\frac{N}{2}$ of baselines and we are trying to estimate $KN$ Jones matrices. Therefore, we must have at least $x$ partitions at every OS calibration’s sub-dataset where $$x \frac{N}{2} > KN.$$ Thus, $$x \geq 2K+1.$$ ]{} We have $N=14$ number of receivers in WSRT. Thus, $\frac{N(N-1)}{2}=91$ number of baselines, providing $2 \times 2$ visibility matrices, at every time and frequency sample. According to (i) we can make thirteen partitions of baselines so that every partition includes $\frac{N}{2}=7$ number of visibilities covering all the receivers. Since we calibrate for $K=3$ bright sources A, B, and C, using (ii), $x \geq 7$. This means at every OS sub-observation we must collect at least seven number of those partitions. Thus, at every sub-observation we have $x\times \frac{N}{2} =49$ number of visibility matrices and that is enough for estimating $KN=42$ number of Jones matrices. Indeed better accuracy of OS calibration is expected to be obtained by increasing $x$ till $x\leq N-1$. This approach of defining sub-observations of the OS calibration is demonstrated in Fig. \[FigureB5\]. As this figure shows, there are no overlaps between the baselines of the thirteen different partitions. Therefore, the maximum information level, achievable by using $x\times\frac{N}{2}=49$ number of visibilities, is provided for every sub-observation of the OS calibration. OS-SAGE calibration is executed, using $m=2$ number of time samples at every iteration (two number of OS iterations), for $x=7$ and $x=10$. The residual images are shown in Fig. \[FigureB4\]. We can see that by increasing the number of visibilities in the sub-observations from forty nine ($x=7$) to seventy ($x=10$), the calibration accuracy is highly improved. We also can see that the two images of Fig. \[FigureB4\] have a higher residual noise and artifacts compared to the result obtained for $x=N-1=13$, which is presented by Fig. \[FigureA3\] as image (d). This shows that better accuracy of the OS calibration is achieved when the number of visibilities in every sub-observation is large. However, the calibration’s processing times for $x=7$ and $x=10$ are $73.5$ and $92.8$ minutes, respectively, while for $x=13$ it is $108.8$ minutes (Table 1). Remember that the whole point of partitioning the baselines was to cut down the computations. We also can benefit from this approach to speed up the initial calibration iterations for the telescope with a large number of baselines such as SKA. $ \begin{array}{cc} \multicolumn{2}{c}{\hspace{-7mm}\epsfig{file=colormap.eps, bb= -98 393 637 438, scale=0.72}}\vspace{-10cm}\\ \hspace{-2mm}\epsfig{file=47baseossage0t10m2.eps, bb= -87 12 700 800,clip=,width=8.8cm,scale=0.4} & \hspace{-3mm}\epsfig{file=70baseossage0t10m2.eps, bb= -87 12 700 800,clip=,width=8.8cm,scale=0.4}\vspace{-5mm}\\ \mbox{\color{white}\large{\bf (a)}}&\mbox{\color{white}\large{\bf (b)}}\vspace{1cm}\\ \end{array}$ As a final remark, for partiting baselines of a telescope with an odd number of receivers $N$, an alternative would be: (i) first partitioning baselines for $N-1$ number of receivers, as it is already explained in this section, and (ii) assigning the remained baselines to these $N-1$ partitions. Averaging of visibilities ------------------------- The OS calibration method divides the data into sub-observations and alternates. The use of fewer data samples in each iteration is the principle cause of the speedup. So far, we have used segments of data consisting of multiple integrations in time and have considered the individual integrations as the sub-observations. This is reasonable for the use of OS calibrations. However, for the non-OS type calibrations all of these integrations are explicitly considered to be equivalent. Therefore, one could ask if it is easier to average the data before calibration to decrease the computational cost.\ To answer this question, consider the case of calibrating data for a point source far away from the phase center of an observation. Based on (\[s3\]), the visibilities of baseline $p-q$ at every sub-observation are formulated as $${\bf v}_{pq}={\bf J}^_{q}\otimes{\bf J}_{p}\mbox{vec}({\bf C})+{\bf n}_{pq},\label{kh2}$$ where, $${{\bf{C}}}=e^{\frac{-2\pi j \xi}{c} (ul+vm+w(\sqrt{1-l^2-m^2}-1))} \left[ \begin{array}{cc} \frac{{{I}}}{2}&0\\ 0&\frac{{{I}}}{2} \end{array}\right].\label{kh22}$$ In (\[kh22\]), $j^2=-1$, $\xi$ is the frequency of the observation, $c$ is the speed of light, $(l,m)$ are the source direction components corresponding to the observation phase center, $(u,v,w)$ are the geometric components of baseline $p-q$, and $I$ is the intensity of the source.\ Since the source is far away from the phase center, $(l,m)$ in (\[kh22\]) are large. Therefore, even very small variation of the baselines $(u,v,w)$ on different sub-observations cause huge differences in the phase terms of (\[kh22\]). Subsequently, averaging the visibilities of (\[kh2\]) causes de-correlation (losing amplitude) and smearing effects in the calibration residuals. To illustrate this, we simulate a 12 hour observation of WSRT from a very bright source with 130 Jy intensity is simulated. The source is about four degrees away from the phase center. In the center of the field we also put twenty three faint sources with intensities below 9 Jy. The Jones matrices for the faint sources are considered as identity matrices. For the bright source, they are multiplications of different linear combinations of $sin$ and $cos$ functions which are invariant on twenty five seconds time intervals. That provides time samples of size $\tau=25$ including sub-observations, from every individual second, for which the gain errors are exactly the same. White Gaussian noise is also added to the simulated data. It is expected that traditional calibration after averaging data performs as equivalent as the OS calibration which iterates on the individual sub-observations. The reason is that the simulated corruptions in the signals on twenty five seconds time intervals are invariant. However, the results, illustrated by Fig \[FigureB2\], is completely the opposite. $ \begin{array}{cc} \multicolumn{2}{c}{\hspace{-6mm}\epsfig{file=colormap.eps, bb= -98 393 637 438, scale=0.71}}\vspace{-9.9cm}\\ \hspace{-1mm}\epsfig{file=avet25.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4} & \hspace{-3mm}\epsfig{file=oslst25.eps, bb= -95 4 708 805,clip=,width=8.8cm,scale=0.4}\vspace{-6mm}\\ \hspace{3.8cm}\mbox{\color{white}\large{\bf (a)}}\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\mbox{\color{white}\large{\bf 7.63 min}}&\hspace{3.5cm}\mbox{\color{white}\large{\bf (b)}}\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\mbox{\color{white}\large{\bf 19.43 min}}\vspace*{1cm}\\ \end{array}$ Fig. \[FigureB2\] shows the residual images obtained by the LS and the OS-LS calibration, utilizing $m=2$ number of OS iterations and $n=9$ number of LS iterations. The processing time in min is shown at the bottom right corner of every image. In image (a) of Fig. \[FigureB2\], LS calibration is applied on averaged data obtained from $\tau=25$ time samples. In this image, the bright source is highly underestimated (almost not subtracted at all) and there exist elongated radial features. This is due to the de-correlation by averaging the visibilities. However, in image (b) of Fig. \[FigureB2\], for which OS-LS calibration is applied on individual integrations, the bright source is perfectly subtracted and the other fainter sources are completely visible. This proves that we can not simply apply calibration on averaged visibilities to cut down the computations and reveals the need of using the OS calibration. We have also executed LS calibration on non-averaged data sets of $\tau=25$ time samples. The resulted residual image has been exactly the same as image (b) of Fig. \[FigureB2\], which is generated by OS-LS calibration. The reason is that the Jones matrices on every twenty five seconds calibrated data are invariant. Therefore, the solution which is obtained by OS calibrations, using few integrations (sub-observations) within twenty five seconds, is the same as the one obtained by non-OS calibrations using all the data. However, in reality, Jones matrices vary with time. In such a case, the result of the non-OS calibrations is always better than, or equivalent to, the one of OS calibrations. It is because finding a global solution which fits all data is generally more efficient than solving only for a part of dataset. The execution time of the LS calibration was $78.15$ min, which is indeed longer than the one of OS-LS calibration ($19.43$ min). Conclusions {#Conclusions} =========== This paper introduces OS-LS and OS-SAGE radio interferometric calibration, as combinations of the OS method with LS and SAGE calibration techniques. We show that the OS calibration provide a significant improvement in the execution speed compared to the conventional (non-OS) calibration algorithms. The key idea is to partition the observed data into groups of sub-observations for which the gain errors are considered to be fixed. OS type calibrations solve for every group by iteratively updating the solutions for that group’s sub-observations in an ordered sequence. The calibrations benefit from very fast computations and preserve almost the same quality as the one obtained by the non-OS calibrations. But, we must take in to account that their accuracy never becomes higher than the one of the non-OS calibration. Simulations show that OS calibration methods have considerable computational improvements compared to the conventional non-OS calibration methods. They also indicate that the OS-SAGE calibration provides a better quality results in a shorter time compared to the OS-LS calibration, as it is the case for the conventional SAGE and LS calibrations. In Future work, we address a novel accuracy of calibration obtained via a hybrid of non-OS and OS calibration techniques which has a computational cost almost as cheap as the one of OS calibrations. \[lastpage\] [^1]: E-mail: kazemi@astro.rug.nl [^2]: http://www.skatelescope.org [^3]: http://www.mwatelescope.org [^4]: http://astro.berkeley.edu/\~dbacker/eor [^5]: http://21cma.bao.ac.cn [^6]: http://www.reionization.org [^7]: http://lwa.unm.edu [^8]: http://www.lofar.org [^9]: A complete graph of order N has N nodes and every pair of nodes are connected to each other by a unique edge.
--- abstract: 'We show that the electron in the Riemann-Cartan spacetime with extra dimensions has a finite size that is much larger than the experimental upper limit on its radius. Thus the Arkani-Hamed-Dimopoulos-Dvali and Randall-Sundrum models of the weak/Planck hierarchy in particle physics are not viable if spin produces torsion according to the Einstein-Cartan theory of gravity.' author: - 'Nikodem J. Pop[ł]{}awski' title: 'Einstein-Cartan gravity excludes extra dimensions' --- Newton’s gravitational constant in the natural system of units ($\hbar=c=1$) is given by $G=M_{\text{Pl}}^{-2}$, where $M_{\text{Pl}}\sim10^{19}\text{GeV}$ is the Planck energy. The Planck energy is much larger than the energy scale of the electroweak unification $M_{\text{EW}}\sim10^3\text{GeV}$. Thus gravity is very weak compared to the other interactions. The Arkani-Hamed-Dimopoulos-Dvali (ADD) model explains this relative weakness, which is called the hierarchy problem in particle physics, by introducing large extra dimensions [@ADD]. If spacetime has $n$ extra compact spatial dimensions of radius $R$ then the gravitational potential $V(r)$ from a point mass $m$ at small distances $r\ll R$ is $$V(r)\sim\frac{m}{M_{\text{Pl}(4+n)}^{n+2}r^{n+1}},$$ while at large distances $r\gg R$ it must be equal to the usual Newtonian potential, $$V(r)\sim\frac{m}{M_{\text{Pl}(4+n)}^{n+2}R^nr}.$$ Thus the Planck energy in this $(4+n)$-dimensional spacetime, $M_{\text{Pl}(4+n)}$, is related to the Planck energy of the 4-dimensional spacetime by $$M_{\text{Pl}(4+n)}^{n+2}R^n=M_{\text{Pl}}^2.$$ In the ADD model there is only one fundamental energy scale, $$M_{\text{EW}}\sim M_{\text{Pl}(4+n)}.$$ Thus the radius $R$, which reproduces the observed $M_{\text{Pl}}$, is given by $$R\sim10^{30/n-19}\text{m}.$$ The value $n=1$ gives the size of the compactification radius $R\sim10^{11}\text{m}$, which would lead to deviations from Newtonian gravity at distances on the order of the size of the Solar System, and therefore excluded. The value $n=2$ gives $R\sim10^{-4}\text{m}$, which is on the order of current experimental upper limits on the distance at which new macroscopic forces may exist [@Long]. Since the electroweak and strong forces have been tested at electroweak scale distances, which are much smaller than $10^{-4}\text{m}$, particles are localized in the 4-dimensional spacetime and cannot propagate in the extra dimensions. As $n\rightarrow\infty$, $R\rightarrow10^{-19}\text{m}$. The Schwarzschild radius $r_S$ for a mass $m$ in Einstein’s general relativity (GR) is on the order of $Gm$. For the electron, $r_S\sim10^{-57}$m. In the presence of $n$ large extra dimensions ($R\gg r_S$), it is given by [@MP]: $$r_S\sim(G_n m)^{\frac{1}{n+1}},$$ where the gravitational constant $G_n$ of the $(4+n)$-dimensional spacetime is related to the corresponding Planck energy by $$G_n M_{\text{Pl}(4+n)}^{n+2}=1.$$ Thus $$r_S\sim(GmR^n)^{\frac{1}{n+1}}. \label{Schw}$$ For the electron, $n=2$ gives $r_S\sim10^{-22}$m. As $n\rightarrow\infty$, $r_S\rightarrow10^{-19}$m. While the Schwarzschild radius of the electron in GR is much smaller than the upper limit on the particle’s radius $\sim10^{-22}\text{m}$ observed in a Penning trap [@rad], the Schwarzschild radius of the electron in the ADD model is on the order of this limit, imposing strong constraints on the physically possible parameters of this model (since the size of a particle is expected to be on the order of its Schwarzschild radius [@Wald]). In the nonrelativistic limit of the ADD model, the Einstein equations reduce to the Poisson equation $\triangle V=4\pi G\rho$, so for objects of sizes $r\ll R$, the mass density is $$\rho\sim\frac{mR^n}{r^{n+3}}.$$ The Planck mass $M_{\text{Pl}(4+n)}\sim10^3$GeV of the $(4+n)$-dimensional spacetime gives also the order of the theoretical minimum mass of a black hole in the ADD model. Therefore if this model is true, the LHC, which operates at energies on the order of $10^3$GeV, will be able to produce micro black holes [@DL]. The Randall-Sundrum (RS) models provide another scenario that explains why gravity is weak relative to the other interactions [@RS1; @RS2]. They are based on the metric $$ds^2=e^{-2kR\phi}g_{\mu\nu}dx^\mu dx^\nu-R^2 d\phi^2,$$ where $k$ is a scale on the order of the Planck scale, $g_{\mu\nu}$ is the metric tensor of the 4-dimensional subspace of this 5-dimensional warped spacetime, and $\phi\in[0,\pi]$ is the coordinate for an extra dimension of size $R$. The Planck energy in the 5-dimensional spacetime, $M_{\text{Pl}(5)}$, is related to the Planck energy of the 4-dimensional spacetime by $$M_{\text{Pl}}^2=\frac{M_{\text{Pl}(5)}^3}{k}(1-e^{-2\pi kR}).$$ In the RS model with a small extra dimension [@RS1], the Standard Model fields are localized on the brane at $\phi=\pi$, so the warp factor $w=e^{\pi kR}=10^{16}$. The effective 4-dimensional metric tensor $\tilde{g}_{\mu\nu}$ is conformally related to $g_{\mu\nu}$ by $$g_{\mu\nu}=e^{-2\pi kR}\tilde{g}_{\mu\nu},$$ from which a field with the fundamental mass parameter $m_0$ appears to have the physical mass $$m=e^{-\pi kR}m_0.$$ If $kR\sim12$ then $m_0\sim10^{19}\mbox{GeV}$ gives $m\sim10^3\mbox{GeV}$, reproducing the observed hierarchy between the gravitational and electroweak energy scales. General relativity, which is the current theory of gravitation, has been confirmed by many experimental and observational tests [@Will]. However, this theory has one problematic feature - the appearance of curvature singularities, which are points in spacetime where the density of matter and curvature are infinite and thus the laws of physics break down. The Einstein-Cartan (EC) theory of gravity naturally extends GR to include matter with intrinsic spin, which produces torsion, providing a more complete account of local gauge invariance with respect to the Poincaré group. It is a viable theory of gravity, which differs significantly from GR only at densities of matter much larger than the density of nuclear matter, and thus it passes all the experimental and observational tests of GR. In this theory, the curvature of the Riemann-Cartan spacetime, represented by the Einstein tensor $G_{ik}$, is related to the matter distribution, represented by the energy-momentum tensor $T_{ik}$ via the first Einstein-Cartan equation [@Hehl]: $$\begin{aligned} & & G_{ik}=8\pi G\,T_{ik}-(S^l_{\phantom{l}ij}+2S_{(ij)}^{\phantom{(ij)}l})(S^j_{\phantom{j}kl}+2S_{(kl)}^{\phantom{(kl)}j}) \nonumber \\ & & +4S_i S_k+\frac{1}{2}g_{ik}(S^{mjl}+2S^{(jl)m})(S_{ljm}+2S_{(jm)l}) \nonumber \\ & & -2g_{ik}S^j S_j. \label{EC}\end{aligned}$$ The torsion tensor $S^j_{\phantom{j}ik}$ is related to the spin tensor $s^{\phantom{ik}j}_{ik}$ via the second Einstein-Cartan equation [@Hehl]: $$S^j_{\phantom{j}ik}-S_i \delta^j_k+S_k \delta^j_i=-4\pi G\,s^{\phantom{ik}j}_{ik},$$ where $S_i$ is the torsion vector. In GR, the torsion tensor vanishes, reducing (\[EC\]) to the usual Einstein equations. We recently showed, using the Papapetrou-Nomura-Shirafuji-Hayashi method of deriving the equations of motion for a test body from the conservation laws for the energy-momentum and spin tensors [@Pap], that the EC theory prevents the formation of singularities if matter is composed of Dirac particles, i.e. quarks and leptons (which form all stars) [@Niko]. The presence of torsion implies that a spinor particle in the Riemann-Cartan spacetime cannot be a point or a system of points, otherwise it would contradict the gravitational field equations. Instead, such a particle is an extended object whose size is determined by the conditions at which torsion introduces significant corrections to the energy-momentum tensor, i.e. when $8\pi G\,T_{ik}$ and the terms after it in (\[EC\]) are on the same order [@Niko]. For a particle with mass $m$ and spin $s$, its size is on the order of the Cartan radius $r_C$: $$\frac{m}{r_C^3}\sim G\biggl(\frac{s}{r_C^3}\biggr)^2. \label{order}$$ For the electron, $r_{Ce}\sim10^{-27}$m, which is much larger than its Schwarzschild radius $\sim10^{-57}$m, so the electron (as well as the other fermions) is nonsingular. The Cartan density for the electron, $\rho_{Ce}\sim m_e/r_{Ce}^3\sim10^{51}\text{kg}\,\text{m}^{-3}$, gives the order of the maximum density of matter composed of quarks and leptons, averting gravitational singularities in the EC theory, even if a black hole forms. In GR, the size of a spinor particle is on the order of its Schwarzschild radius and such a particle is represented by a singular worldline. The mass density of a black hole in the EC theory cannot exceed $\rho_{Ce}$. This condition gives the order of the minimum mass of a black hole in the EC theory $\sim10^{43}\text{GeV}$ [@Niko], while in GR this mass is on the order of the Planck mass. Therefore if the EC theory is true, the LHC will not be able to produce micro black holes. The EC theory, as well as the ADD and RS models, are physically very appealing. The extra dimensions in the ADD and RS models explain why gravity is so weak relative to the other interactions and introduce only one fundamental energy scale, while the torsion of spacetime in the EC theory prevents the formation of singularities from ordinary matter and introduces an effective ultraviolet cutoff in quantum field theory at distances on the order of the Cartan radius of the electron. Thus combining either the ADD or RS model with EC theory should result in a theory with all the above advantages. Since the experimental size of the electron is at least $10^3$ times smaller than the radius $R$ of the extra dimensions in the ADD model, the mass and spin densities in (\[order\]) must be modified, leading to $$\frac{mR^n}{r_C^{n+3}}\sim G\biggl(\frac{sR^n}{r_C^{n+3}}\biggr)^2. \label{mod}$$ For the electron, $n=2$ gives $r_C\sim10^{-18}\mbox{m}>r_S$, so the electron is nonsingular. As $n\rightarrow\infty$, $r_S\rightarrow10^{-19}\mbox{m}\sim r_S$. The presence of torsion in the ADD model implies that the size of the electron must be on the order of $10^{-19}-10^{-18}$m, which is at least $\sim10^3$ times larger than the experimental upper limit on its radius [@rad]. Therefore large extra dimensions and torsion produced by spin are incompatible: the EC theory of gravity excludes large extra dimensions and the ADD model [@ADD] of the weak/Planck hierarchy excludes torsion. In the RS model with a small extra dimension, the electron corresponds to a Dirac field with the fundamental mass $m_{0e}=e^{\pi kR}m_e\sim10^{13}\mbox{GeV}$. Substituting this mass into (\[mod\]) with $n=1$ and $R\sim10^{-33}\mbox{m}$ gives $r_{C0e}\sim10^{-33}\mbox{m}$, which is the Cartan radius in the fundamental 5-dimensional theory. The correspoding Cartan radius observed in the 4-dimensional spacetime is obtained by the conformal scaling, $$l=e^{\pi kR}l_0,$$ which gives $r_{Ce}=e^{\pi kR}r_{C0e}\sim10^{-17}\mbox{m}$. This result agrees with the Cartan radius obtained from (\[mod\]) with $m=m_e$ for the ADD model with $n=1$. This agreement is consistent with the relation between large extra dimensions in [@ADD] and exponential determination of the weak/Planck hierarchy [@equ]. The Schwarzschild radius for the electron in the RS model observed in the 4-dimensional spacetime is given by (\[Schw\]) with the conformal scaling [@BH]: $r_{Se}\sim w(GwmR)^{1/2}\sim10^{-22}\mbox{m}$, as in the ADD model with $n=1$. Thus the electron is nonsingular in the EC+RS model, $r_{Ce}>r_{Se}$. The presence of torsion in the RS model implies that the size of the electron must be on the order of $10^{-17}\mbox{m}$, which is $\sim10^5$ times larger than the experimental upper limit on its radius [@rad]. Therefore small extra dimensions and torsion produced by spin are also incompatible: the EC theory of gravity excludes small extra dimensions and the RS model [@RS1] of the weak/Planck hierarchy excludes torsion. If $kR=12$ [@BH] then the Cartan radius of the electron in the RS model depends on the size of an extra dimension $R$ according to $$r_{Ce}\sim\xi^{1/4}e^{9\pi\xi}10^{-29}\mbox{m},$$ where $$\xi=\frac{R}{10^{-33}\mbox{m}}.$$ If $R$ is larger than $10^{-33}\mbox{m}$ then $r_{Ce}$ is larger than $10^{-17}\mbox{m}$. For very large $R$, as in [@RS2], $r_{Ce}$ significantly exceeds the experimental upper limit on the radius of the electron [@rad]. Thus the RS model with an infinite fifth dimension [@RS2] cannot be valid if spin produces torsion according to the EC theory of gravity. To conclude, we found that the ADD and RS models of the weak/Planck hierarchy cannot be combined with the EC theory gravity in which spin of matter produces torsion of spacetime. Any signal showing that extra dimensions exist would indicate that torsion vanishes or at least that spin cannot produce torsion. Any signal showing that spin produces torsion would indicate that extra dimensions do not exist. 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--- author: - \| Jean-Louis Krivine & Yves Legrandgérard\ \ title: 'Valid formulas, games and network protocols' --- Introduction {#introduction .unnumbered} ------------ We describe a remarkable relation between a fundamental notion of mathematical logic – that is [valid formula of predicate calculus]{} – and the specification of network protocols. We explain here in detail several simple examples : the acknowledgement of one or two packets, and then of an arbitrary number. We show that, using this method, it is possible to specify the composition of protocols. We tried to write a self-contained paper, as far as possible, in what concerns the basic notions of the calculus of predicates. In particular, the notion of [valid formula]{} is defined with the help of the tools introduced in the present paper (specifically, the game associated with the formula). The equivalence with the usual definition of this notion in logic is explained in the appendix, but is never used in the paper. Logical framework {#logical-framework .unnumbered} ----------------- The language we use is described below. It is the well known [predicate calculus]{}, fundamental in mathematical logic ; important restriction : the only allowed logical symbols are $\to,\bot,\pt$, respectively read as “ implies ”, “ false ”, “ for all ”. In fact, every other logical symbol can be defined with them (see below). This restriction is therefore only syntactic, but not semantic. We suppose given an infinite set of [variables]{} : $\{x,y,\ldots\}$, an infinite set of [constants]{} : ${\cal C}=\{a,b,\ldots\}$ and some [predicate symbols]{} $P\,,Q,R\,,\ldots$ ; each of them has an arity which is an integer $\ge0$. [Atomic formulas]{} are of the form $\bot$ (read [false]{}) or $Pt_1\ldots t_k$ (denoted also as $P(t_1,\ldots,t_k$)) where $P$ is a predicate symbol of arity $k$ and $t_1,\ldots,t_k$ are variables or constants.\ Formulas of the predicate calculus are built with the following rules : $\bullet$ An atomic formula is a formula.\ $\bullet$ If $F$ and $G$ are formulas, then $F\to G$ is a formula (read $F$ implies $G$).\ $\bullet$ If $F$ is a formula and $x$ is a variable, then $\pt x\,F$ is a formula (read “ for all $x$, $F$ ”). is contained in predicate calculus : it has the only logical symbols $\to$ and $\bot$, and only predicate symbols of arity $0$, usually called [propositional variables]{}. We shall systematically use the notation $A,B\to C$ for $A\to(B\to C)$ and, more generally $A_1,A_2\ldots,A_n\to B$ for $A_1\to(A_2\to(\cdots(A_n\to B)\cdots))$.\ Usual connectives $\neg,\land,\lor,\dbfl$ of propositional calculus are considered as abbreviations, and defined as follows :\ $\neg F$ is $F\to\bot$ ; $F\land G$ is $(F\,,G\to\bot)\to\bot$ ; $F\lor G$ is $\neg F\,,\neg G\to\bot$ ;\ $F\dbfl G$ is $(F\to G)\land(G\to F)$ that is $((F\to G),(G\to F)\to\bot)\to\bot$. The connective [XOR]{}, usual in computer science and often denoted as $F\verb?^?\!G$, can be defined as $\neg F\dbfl G$. This abbreviation is not used in the formulas of predicate calculus. The existential quantifier $\ex$ (read “ there exists ”) is also considered as an abbreviation : $\ex x\,F$ is defined as $\neg\pt x\neg\,F$ that is $\pt x(F\to\bot)\to\bot$. The notation $\vec{x}$ will denote a finite sequence of variables $x_1,\ldots,x_n$.\ Therefore, we shall write $\pt\vec{x}$ for $\pt x_1\ldots\pt x_n$ and the same with $\ex$. In a formula such as $\pt x\,A$, the subformula $A$ is called the [scope]{} of the quantifier $\pt x$. An [occurrence]{} of a variable $x$ in a formula $F$ is called [bounded]{} if it is in the scope of a quantifier $\pt x$ ; otherwise, this occurrence is called [free]{}. Given a bounded occurrence of $x$, the quantifier which bounds it, is by definition, the nearest quantifier $\pt x$ which has this occurrence in its scope.\ For instance, in the formula $\pt x[\pt x(Rx\to Ry)\to\pt y(Ry\to Rx)]$ there are a bound and a free occurrence of the variable $y$ and two bounded occurrences of the variable $x$. These two occurrences of $x$ are not bounded by the same quantifier.\ A variable $x$ is called [free]{} in the formula $F$ if there is at least one free occurrence of $x$. The formula $F$ is called [closed]{} if it contains no free variable.\ The notation $F[x_1,\ldots,x_n]$ (or $F[\vec{x}]$) will mean that the free variables of the formula $F$ are [amongst]{} $x_1,\ldots,x_n$. Then, the formula $\pt x_1\ldots\pt x_n\,F[x_1,\ldots,x_n]$ (or $\pt\vec{x}\,F[\vec{x}]$) is closed. In any formula $F$, we can rename the [bounded]{} variables in an arbitrary way, provided that no [capture of variable]{} occurs. This means that no free occurrence becomes bound ; and that any bound occurrence must remain bounded [by the same quantifier]{}. Any formula $G$, obtained from $F$ in this way is considered as identical with $F$.\ For instance, $\pt z[\pt y(Rx\to Ry)\to Rz]$ is identified with $\pt y[\pt y(Rx\to Ry)\to Ry]$. For any formula $F[x_1,\ldots,x_k]\equiv F[\vec{x}]$ and constants $a_1,\,\ldots,a_k$, we denote by\ $F[a_1,\ldots,a_k]\equiv F[\vec{a}]$ the closed formula we obtain by replacing each [free]{} occurrence of $x_i$ with $a_i$ $(1\le i\le k)$. Any [atomic closed]{} formula $\not\equiv\bot$ has the form $Pa_1\ldots a_k$, where $P$ is a predicate symbol of arity $k$ and $a_1,\ldots,a_k$ are constants. In the interpretation in terms of network protocols which is given below, such a formula represents a [packet]{}, the predicate symbol $P$ represents the [datas]{} and $a_1,\ldots,a_k$ represent the [header fields]{} of the packet. When $k=0$, i.e. when $P$ is a propositional variable, $P$ represents a pure data packet. ### Normal form of a formula {#normal-form-of-a-formula .unnumbered} A formula is said to be [in normal form]{} or [normal]{}, if it can be obtained by means of the following rules :\ $\bullet$ an atomic formula $A$ is normal ;\ $\bullet$ if $\Phi_1,\ldots,\Phi_n$ are normal, if $A$ is atomic and if $\vec{x}=(x_1,\ldots,x_k)$ is a finite sequence of variables, then $\pt\vec{x}(\Phi_1,\ldots,\Phi_n\to A)$ is a normal formula.\ If $n=0$, by definition, this formula is $\pt\vec{x}\,A$.\ In the same way, if $k=0$ this formula is $\Phi_1,\ldots,\Phi_n\to A$. For instance, if $R$ is a unary predicate symbol, the formula $\pt x\,Rx\to\pt x\,Rx$ is not a normal form ; but $\pt y(\pt x\,Rx\to Ry)$ is a normal form. With any formula $F$, we associate its [normal form]{} $\wh{F}$, which is obtained as follows :\ $\bullet$ if $F$ is atomic, $\wh{F}\equiv F$ ;\ $\bullet$ if $F$ is $\pt x\,G$, then $\wh{F}$ is $\pt x\,\wh{G}$ ;\ $\bullet$ if $F$ is $G\to H$, we write $\wh{H}\equiv\pt\vec{x}(\Phi_1,\ldots,\Phi_n\to A)$. We first rename the (bounded) variables $\vec{x}$ so that they become not free in $G$ (a good method is to use variables that do not appear in $G$) ; then, $\wh{F}$ is $\pt\vec{x}[\wh{G},\Phi_1,\ldots,\Phi_n\to A]$. Obviously, $F$ and $\wh{F}$ have the same free variables. In particular, if $F$ is closed, then $\wh{F}$ is also closed.\ Note that any formula of the propositional calculus is in normal form. For instance, the normal form of the formula $(Rx\to\pt x\,Rx)\to\pt x\,Rx$ is :\ $\pt z[\pt y(Rx\to Ry)\to Rz]$ or $\pt y[\pt y(Rx\to Ry)\to Ry]$. The game associated with a closed formula {#the-game-associated-with-a-closed-formula .unnumbered} ----------------------------------------- Given a closed formula $F$, we define a two players’ game ; the players will be called [$\ex$loise]{} and [$\pt$belard]{} or, more briefly, $\ex$ and $\pt$ (the same notation as the quantifiers, but no confusion is possible). $\ex$loise is also called the “ player ” or the “ defender ” and $\pt$belard is called the “ opponent ”.\ Intuitively, the player $\ex$ defends the formula $F$, i.e. pretends this formula is “ true ” and the opponent $\pt$ attacks it, i.e. pretends it is “ false ”.\ Be careful, there is no symmetry between the players, as it will be seen by the rule of the game. To make the intuitive idea more precise, we can say that $\ex$ pretends the formula $F$ is “ always true ” and that $\pt$ pretends it is “ sometimes false ”. Now, we assume that the closed formula $F$ has been put in normal form.\ Here is the rule of the game associated with this formula \[jlk\] : We have three finite sets of normal closed formulas, denoted by ${\cal U},{\cal V},{\cal A}$, which change during the play. The elements of the set ${\cal A}$ are closed [atomic]{} formulas. The sets ${\cal U}$ and ${\cal A}$ [increase]{} during the play. At the beginning of the play, we have ${\cal U}=\{F\to\bot\}$, ${\cal V}=\{F\}$ and ${\cal A}=\{\bot\}$ (one-element sets). The first move is done by the opponent $\pt$.\ Consider now, during the play, a moment when the opponent $\pt$ must play.\ If, at this moment, the set ${\cal V}$ is empty, the game stops and $\pt$ has lost.\ Otherwise, he chooses a formula $\Phi\equiv\pt\vec{x}(\Psi_1[\vec{x}],\ldots,\Psi_m[\vec{x}]\to A[\vec{x}])$ which is in ${\cal V}$ and a sequence $\vec{a}$ of constants, of the same length as $\vec{x}$.\ Then [he adds]{} the formulas $\Psi_1[\vec{a}],\ldots,\Psi_m[\vec{a}]$ to the set ${\cal U}$ and also the atomic formula $A[\vec{a}]$ to the set ${\cal A}$. Then the defender $\ex$ must play.\ She chooses, in the set ${\cal U}$, a formula $\Psi\equiv\pt\vec{y}(\Phi_1[\vec{y}],\ldots,\Phi_n[\vec{y}]\to B[\vec{y}])$ ; she chooses also a sequence $\vec{b}$ of constants, of the same length as $\vec{y}$, [in such a way that $B[\vec{b}]\in{\cal A}$]{} ; this is always possible, since she can, at least, choose $F\to\bot$ which is in ${\cal U}$.\ Then, [she replaces]{} the content of the set ${\cal V}$ with $\{\Phi_1[\vec{b}],\ldots,\Phi_n[\vec{b}]\}$.\ Then $\pt$ must play, and so on. We observe that the opponent $\pt$ wins if, and only if, the play is infinite. The play ends after a finite time if, and only if, ${\cal V}$ becomes empty (and then, the player $\ex$ wins). Just before, the player $\ex$ has chosen an atomic formula which is in ${\cal U}\cap{\cal A}$.\ The intuitive meaning of the rule of this game is as follows : at each moment, the defender $\ex$ pretends that one of the formulas of ${\cal U}$ is false and that every formula of ${\cal V}$ is true. On the other hand, the opponent $\pt$ pretends that every formula of ${\cal U}$ is true and that one of the formulas of ${\cal V}$ is false. Now, both agree on the fact that every formula of ${\cal A}$ is false. In the examples below, we shall interpret a play of this game as a session of communication following a certain protocol. In this interpretation, the opponent $\pt$ is the [sender]{} and the defender $\ex$ is the [receiver]{}. The disymmetry of the game is well expressed by a celebrated sentence of Jon Postel (known as “ Postel’s law ” \[jp\]) : “ Be conservative in what you send, be liberal in what you receive ”. Examples {#examples .unnumbered} -------- 1\) $F\equiv P\to P$. We can describe an arbitrary play by the following table : $\dsp \begin{array}{ccccl} {\cal U} & {\cal V} & {\cal A} & & \\ (P\to P)\to\bot & P\to P & \bot & &\pt\mbox{ has no choice }\\ (P\to P)\to\bot,P &\hspace{2em}\mbox{unchanged}\hspace{2em} &\bot,P & &\ex\mbox{ chooses }(P\to P)\to\bot\\ \mbox{unchanged} &\mbox{unchanged}&\mbox{unchanged} & &\pt\mbox{ has no choice }\\ \mbox{unchanged}&\mbox{unchanged} &\mbox{unchanged}& &\ex\mbox{ chooses }(P\to P)\to\bot\\ \vdots &\vdots &\vdots & &\hspace{2em}\vdots\\ \mbox{unchanged}&\mbox{unchanged} &\mbox{unchanged}& &\ex\mbox{ chooses }P\\ \mbox{unchanged}&\vide &\mbox{unchanged}& & \ex\mbox{ wins} \end{array}$ The different possible plays depend only on the number $n$ of times when $\ex$ chooses the formula $(P\to P)\to\bot$. If $n$ is infinite, $\ex$ loses. 2\) $F\equiv P\to Q$ $\dsp \begin{array}{ccccl} {\cal U} & {\cal V} & {\cal A} & & \\ (P\to Q)\to\bot & P\to Q & \bot & &\pt\mbox{ has no choice }\\ (P\to Q)\to\bot,P &\hspace{2em}\mbox{unchanged}\hspace{2em} &\bot,Q & &\ex\mbox{ cannot choose } P\\ \mbox{unchanged} &\mbox{unchanged}&\mbox{unchanged} & &\pt\mbox{ has no choice }\\ \mbox{unchanged}&\mbox{unchanged} &\mbox{unchanged}& &\ex\mbox{ cannot choose} P\\ \vdots &\vdots &\vdots & &\hspace{2em}\vdots \end{array}$ There is only one possible play and $\pt$ wins, since this play is infinite. 3\) The reader is invited to study by himself the following two examples :\ $((Q\to Q)\to P)\to P$ ; $((P\to Q)\to P)\to P$ (Peirce’s law). 4\) $F\equiv\pt x\,Px\to\pt x\,Px$. The normal form of $F$ est $G\equiv\pt y(\pt x\,Px\to Py)$. $\dsp \begin{array}{ccccl} {\cal U} & {\cal V} & {\cal A} & & \\ \neg G & G & \bot & &\pt\mbox{ chooses } b_0\\ \neg G,\pt x\,Px &\hspace{2em}\mbox{unchanged}\hspace{2em} &\bot,Pb_0 & &\ex\mbox{ chooses }\neg G\\ \mbox{unchanged} &\mbox{unchanged}&\mbox{unchanged} & &\pt\mbox{ chooses }b_1\\ \mbox{unchanged}&\mbox{unchanged} &\bot,Pb_0,Pb_1& &\ex\mbox{ chooses }\neg F\\ \vdots &\vdots &\vdots & &\hspace{2em}\vdots\\ \mbox{unchanged}&\mbox{unchanged} &\bot,Pb_0,\ldots,Pb_n& &\ex\mbox{ chooses }\pt x\,Px\mbox{ and }b_i\\ \mbox{unchanged}&\vide &\mbox{unchanged}& & \ex\mbox{ wins} \end{array}$ Like in example 1, the play only depends on the moment when the player $\ex$ chooses the formula $\pt x\,Px$ and one of the $b_i$’s already chosen by the opponent $\pt$. We can give the following interpretation, in terms of network : the player $\pt$ sends the data packet $P$ with the headers $b_0$, then $b_1$, … The acknowledgement by the receiver $\ex$ only happens at the $n$-th step, and it is the packet $Pb_i$ that is acknowledged. Then the play, that is to say the session, stops immediately. From the network point of view, this means that the acknowledgement cannot be lost ; in other words, that the channel from the receiver $\ex$ to the sender $\pt$ is reliable.\ In the following section, we treat a particularly important example : the acknowledgement of a packet in a channel which is not reliable. The formula $\ex x(Px\to\pt y\,Py)$ {#the-formula-ex-xpxtopt-ypy .unnumbered} ----------------------------------- Let us call $F$ the normal form of this formula, i.e. $\pt x(\pt y(Px\to Py)\to\bot)\to\bot$. For the sake of clarity, let us put $G[x]\equiv\pt y(Px\to Py)$ ; thus, we have $F\equiv\neg\pt x\neg\,G[x]$. The tables I and II below represent what happens during a play, in the (very particular) case when $\ex$ plays in such a way as to win as quickly as possible. There are two possibilities, following what the opponent $\pt$ plays at line 3 : Table I $\dsp \begin{array}{cccccl} & {\cal U} & {\cal V} & {\cal A} & & \\ 1 &\neg F & F & \bot & &\pt\mbox{ chooses }F\\ 2 &\neg F,\pt x\,\neg G[x] &\hspace{2em}\mbox{unchanged}\hspace{2em} &\mbox{unchanged}& &\ex\mbox{ chooses }\pt x\,\neg G[x]\mbox{ and }a\\ 3 &\mbox{unchanged} &G[a]\equiv\pt y(Pa\to Py)&\mbox{unchanged} & &\pt\mbox{ chooses }b\mbox{, with }b\ne a\\ 4 &\neg F,\pt x\,\neg G[x],Pa & \mbox{unchanged} &\bot,Pb& &\ex\mbox{ chooses }\pt x\neg G[x]\mbox{ and }b\\ 5 &\mbox{unchanged} & G[b]\equiv\pt y(Pb\to Py) &\mbox{unchanged} & &\pt\mbox{ chooses }c\\ 6 &\neg F,\pt x\,\neg G[x],Pa,Pb &\mbox{unchanged} &\bot,Pb,Pc & &\ex\mbox{ chooses }Pb\\ 7 &\mbox{unchanged}&\vide &\mbox{unchanged}& & \ex\mbox{ wins} \end{array}$ Table II $\dsp \begin{array}{cccccl} & {\cal U} & {\cal V} & {\cal A} & & \\ 1 &\neg F & F & \bot & &\pt\mbox{ chooses }F\\ 2 &\neg F,\pt x\,\neg G[x] &\hspace{2em}\mbox{unchanged}\hspace{2em} &\mbox{unchanged}& &\ex\mbox{ chooses }\pt x\,\neg G[x]\mbox{ and }a\\ 3 &\mbox{unchanged} &G[a]\equiv\pt y(Pa\to Py)&\mbox{unchanged} & &\pt\mbox{ chooses }a\\ 4 &\neg F,\pt x\,\neg G[x],Pa & \mbox{unchanged} &\bot,Pa& &\ex\mbox{ chooses }Pa\\ 5 &\mbox{unchanged}&\vide &\mbox{unchanged}& & \ex\mbox{ wins} \end{array}$ But this is only a particular case. The game we are considering presents, in fact, a great variety of possible plays. We shall see that these various plays correspond exactly to the various possibilities which may happen during the acknowledgement of a packet.\ The play which is described in table I represents the case when the communication occurred in the best possible way. We can interpret it as follows : the receiver $\ex$ begins the session by sending the header $a$ (line 3) ; then the sender $\pt$ sends the packet $Pb$ (line 4) ; $\ex$ receives the packet $Pb$ and sends the acknowledgement (line 5) ; then $\pt$ correctly receives this acknowledgement and sends a signal $Pc$ to terminate the session (line 6). Several variants are possible :\ i) The player $\ex$ can, at each moment, choose the formula $\neg F$. This corresponds to a reinitialisation of the session.\ ii) She can also choose the formula $\pt x\neg\,G[x]$ with an arbitrary header $a'$, which corresponds to no acknowledgement. Then, the opponent $\pt$ must send the packet again. This situation corresponds to the loss of the acknowledgement.\ iii) In this case, the sender $\pt$ has the possibility of sending $Pa'$ again, which gives to the receiver $\ex$ the possibility of finishing the session immediately by choosing precisely the formula $Pa'$ (since it is now both in ${\cal U}$ and ${\cal A}$). This corresponds to the case when the sender asks to finish the session. This may happen at the very beginning : it is the case in the play which is described in table II (line 3 : $\pt$ chooses $a$) ; this corresponds to a refusal of opening the session ; then $\ex$ can only close the session, by choosing $Pa$ (again, it is what happens in table II) or to re-initialise it (by choosing $\neg F$ or $\pt x\neg\,G[x]$).\ iv) The player $\ex$ can terminate the play by choosing the formula $Pb$, where $b$ is any of the headers sent by $\pt$. This corresponds to a successfull communication session, perhaps after some loss of acknowledgements. Any session is a combination of an arbitrary number of such variants. Sending several packets {#sending-several-packets .unnumbered} ----------------------- We consider now the case of the acknowledgement of a fixed number $n$ of packets, $n$ being a previously given integer ; the order of the packets must be preserved. The associated formula $F_n$ is defined by recurrence :\ $F_1\equiv\ex x\pt y(P_1x\to P_1y)$ ; $F_{n+1}\equiv\ex x\pt y((F_n\to P_{n+1}x)\to P_{n+1}y)$ ; $F_n$ is in normal form. For the sake of simplicity, we consider only the case $n=2$. We have the formula :\ $F'\equiv\ex x\pt y((F\to Px)\to Py)$ with $F\equiv\ex x\pt y(Qx\to Qy)$.\ We put $G[x]\equiv\pt y((F\to Px)\to Py)$, $H[x]\equiv\pt y(Qx\to Qy)$ ;\ thus, we have $F'\equiv\pt x\neg G[x]\to\bot$ and $F\equiv\pt x\neg H[x]\to\bot$.\ The table below describes once more what happens during a play where $\ex$ finishes in the quickest possible way. For the sake of clarity, in the columns ${\cal U}$ and ${\cal A}$, we shall put, at each line, only the [new]{} formulas. $\dsp \begin{array}{rccccl} & {\cal U} & {\cal V} & {\cal A} & & \\ 1 &\neg F' & F' & \bot & &\pt\mbox{ has no choice}\\ 2 &\pt x\,\neg G[x] &\hspace{2em}\mbox{unchanged}\hspace{2em} &\mbox{unchanged}& &\ex\mbox{ chooses }\pt x\,\neg G[x]\mbox{ and }a\\ 3 &\mbox{unchanged} &G[a]\equiv\pt y((F\to Pa)\to Py)&\mbox{unchanged} & &\pt\mbox{ chooses }b\mbox{, with }b\ne a\\ 4 &F\to Pa & \mbox{unchanged} & Pb& &\ex\mbox{ chooses }\pt x\neg G[x]\mbox{ and }b\\ 5 &\mbox{unchanged} & G[b]\equiv\pt y((F\to Pb)\to Py) &\mbox{unchanged} & &\pt\mbox{ chooses }c\\ 6 &F\to Pb &\mbox{unchanged} & Pc & &\ex\mbox{ chooses }Pb\\ 7 &\mbox{unchanged}& F &\mbox{unchanged}& &\pt\mbox{ has no choice}\\ \multicolumn{6}{c}{\dotfill\mbox{\small\it \ acknowledgement of the first packet\ }\dotfill}\\ 8 &\pt x\,\neg H[x] &\hspace{2em}\mbox{unchanged}\hspace{2em} &\mbox{unchanged}& &\ex\mbox{ chooses }\pt x\,\neg H[x]\mbox{ and }d\\ 9 &\mbox{unchanged} &H[d]\equiv\pt y(Qd\to Qy)&\mbox{unchanged} & &\pt\mbox{ chooses }e\mbox{, with }e\ne d\\ 10 &Qd & \mbox{unchanged} & Qe& &\ex\mbox{ chooses }\pt x\neg H[x]\mbox{ and }e\\ 11 &\mbox{unchanged} & H[e]\equiv\pt y(Qe\to Qy) &\mbox{unchanged} & &\pt\mbox{ chooses }f\\ 12 &Qe &\mbox{unchanged} & Qf & &\ex\mbox{ chooses }Qe\\ 13 &\mbox{unchanged}& \vide &\mbox{unchanged}& &\ex\mbox{ wins}\\ \multicolumn{6}{c}{\dotfill\mbox{\small\it \ acknowledgement of the second packet\ }\dotfill} \end{array}$ In this particular case, we essentially get twice the table I of the previous example. Of course, we may get all the variants already described. But new variants may appear : indeed, after the acknowledgement of the first packet (lines 8, 10 and 12), the player $\ex$ can, for instance, come back to line 4, that is to say ask again for the first packet. Thus the receiver may lose a packet, even after having correctly acknowledged it. It is interesting to notice that she has not to acknowledge it again. Strategies and valid formulas {#strategies-and-valid-formulas .unnumbered} ----------------------------- Let us consider the game associated with a normal closed formula $F$. A [strategy]{} for $\ex$ in this game is, by definition, a function ${\cal S}$, which takes as an argument a finite sequence of triples $({\cal U}_i,{\cal V}_i,{\cal A}_i)_{0\le i\le n}$ (${\cal U}_i,{\cal V}_i$ are finite sets of normal closed formulas and ${\cal A}_i$ is a finite set of atomic closed formulas) and gives as a result an ordered pair $(\Psi,\vec{b})$ with $\Psi\in{\cal U}_n$, $\Psi\equiv\pt\vec{y}(\Phi_1[\vec{y}],\ldots,\Phi_k[\vec{y}]\to B[\vec{y}])$, $\vec{b}$ has the same length as $\vec{y}$ and $B[\vec{b}]\in{\cal A}_n$.\ Intuitively, a strategy ${\cal S}$ for $\ex$loise is a general method which, each time she must play, chooses for her a possible play, given all the moves already played.\ The strategy ${\cal S}$ is called a [winning strategy]{} if $\ex$ wins every play following this strategy, whatever be the choices of $\pt$.\ We could define in the same way the winning strategies for $\pt$. A normal closed formula $F$ is called [valid]{} if there exists a winning strategy for $\ex$, in the game associated with $F$. Valid formulas are exactly those which correspond to network protocols. [Games associated with a conjunction or a disjunction.]{}\ Given two formulas $F,G$, the game which is associated with the formula $F\land G$, i.e. $(F,G{\to}\bot){\to}\bot$ consists essentially in the following (this is easily checked) :\ The opponent $\pt$ chooses one of these two formulas and the game goes on, following the chosen formula ; however, the player $\ex$ can, at every moment, decide to start again the play from the beginning.\ With the formula $F\lor G$, it is the player $\ex$ who chooses the formula. ### Composition of protocols {#composition-of-protocols .unnumbered} Let us consider two valid formulas $F$ and $G$, which correspond respectively to the “protocols” (i.e. games) ${\cal P}_F$ and ${\cal P}_G$ ; we propose now to build a valid formula $H$ such that the associated protocol ${\cal P}_H$ is : ${\cal P}_F$ then ${\cal P}_G$. Let $A=P(x_1,\ldots,x_n)$ (or $\bot$) be an atomic formula and $F$ a normal formula. An “occurrence” of $A$ in $F$ is simply one of the places, in $F$, where $A$ appears.\ Each occurrence of an atomic formula $A$ in $F$ appears at the end of a subformula of $F$, of the form $\pt\vec{x}(\Psi_1,\ldots,\Psi_k\to A)$ ; $k$ will be called the number of hypothesis of this occurrence of $A$. Each occurrence of an atomic formula $A$ in $F$ is either positive or negative. This property is defined in the following way, by recurrence on the length of $F$ : $\bullet$ If $F$ is atomic, then $F\equiv A$ and the occurrence of $A$ in $F$ is positive.\ $\bullet$ If $F\equiv G\to H$, the occurrence of $A$ in $F$ that we consider, is either in $G$, or in $H$. If it is in $H$, its sign is the same in $F$ as in $H$. If it is in $G$, it has opposite signs in $F$ and in $G$.\ $\bullet$ If $F\equiv\pt x\,G$, the occurrence of $A$ we consider, has the same sign in $F$ and in $G$. An atomic occurrence $A$ in $F$, which is negative and without hypothesis, will be called a final atomic occurrence. Indeed, it corresponds to the end of a play. Now, we can build the formula $H$ we are looking for : it is obtained by replacing, in $F$, each final atomic occurrence $A$ with $G\to A$. It is easy to show that, if $F$ and $G$ are valid, then the formula $H$ defined in this way is also valid (see the appendix). Take the formula $F\equiv\pt x[\pt y(Px\to Py)\to\bot]\to\bot$ which corresponds to the sending and the acknowledgement of a packet.\ Then, we get : $H\equiv\pt x[\pt y((G\to Px)\to Py)\to\bot]\to\bot$.\ Indeed, there are, in $F$, two atomic negative occurrences, which are $Px$ and the first occurrence of $\bot$. The only atomic occurrence without hypothesis is $Px$.\ In particular, if we take $G\equiv\pt x[\pt y(Qx\to Qy)\to\bot]\to\bot$, we get the protocol which corresponds to the sending of two packets (see above). Formulas and protocols using integer variables {#formulas-and-protocols-using-integer-variables .unnumbered} ---------------------------------------------- We now consider formulas written with a new type of variables : the “ integer type ” ; to denote variables of this type, we shall use the letters $i,j,k,l,m,n$. Thus, there are now two types of variables : the type “ integer ” and the type already defined, which we shall call the type “ acknowledgement ” ; for the variables of this type, we use as before the letters $x,y,z$.\ Moreover, we have function symbols [on the integer type]{} (they denote functions from integers to integers), in particular the constant $0$ and the successor $s$ (which represents the function $n\mapsto n+1$). Each function symbol $f$ has an arity $k\in\ennl$ and represents a well determined function from $\ennl^k$ to $\ennl$ which is also denoted by $f$. We define the [terms]{} of integer type by the following rules : $\bullet$ an integer variable or a function symbol of arity $0$ (integer constant) is a term of integer type.\ $\bullet$ if $f$ is a function symbol of arity $k$ and $t_1,\ldots,t_k$ are terms of integer type, then $f(t_1,\ldots,t_k)$ is a term of integer type. We note that a term of integer type without variable (closed term) represents an integer. Predicate symbols are also typed. For example, $Pnx$ or $P(n,x)$ (the first argument of $P$ is of integer type, the second is of type acknowledgement). [Definition of formulas.]{}\ $\bullet$ Atomic formulas : $Pt_1\ldots t_k$ ; $t_i$ is a constant or a variable of type acknowledgement if the $i$-th place of $P$ is of this type ; a term of type integer, if the $i$-th place of $P$ is of integer type.\ $\bullet$ If $F\,,G$ are formulas, $F\to G$ is also a formula.\ $\bullet$ If $F$ is a formula, $\pt x\,F$ and $\pt n\,F$ are also formulas.\ $\bullet$ If $F$ is a formula and $t,u$ are terms of integer type, then $t=u\to F$ is also a formula. Be careful, the expression $t=u$ alone is not a formula. [Normal forms.]{}\ They are defined as follows :\ $\bullet$ An atomic formula is normal.\ $\bullet$ If $F$ is normal, $\pt x\,F$ and $\pt n\,F$ are normal.\ $\bullet$ If $A$ is atomic and $\Phi_1,\ldots,\Phi_k$ are normal formulas or expressions of the form $t=u$, then $\Phi_1,\ldots,\Phi_k\to A$ is a normal formula. We put the formulas under normal form exactly as in the previous case. [Game associated with a closed formula under normal form.]{}\ We indicate here only the additions to the game rule which has been already given : i\) when one of the players has chosen a formula $\pt\vec{\xi}(\Phi_1,\ldots,\Phi_n\to A)$, ($\vec{\xi}=(\xi_1,\ldots,\xi_n)$ where $\xi_i$ is a variable of type integer or acknowledgement) : - first, he or she chooses some values $\vec{a}$ for $\vec{\xi}$. - then, he or she computes the (boolean) expressions $\Phi_j$ of the form $t_j=u_j$. - if all of them are true, we get rid of them and the play goes on as before with the (simpler) formula obtained in this way. - if any of them is false : if $\ex$ is playing, she must choose other values $\vec{a}$ or another formula (which is always possible, as we already saw). if the opponent $\pt$ is playing, then he has lost and the play stops. ii\) as in the previous game, when the player $\ex$ chooses, in the set ${\cal U}$, a formula :\ $\Psi\equiv\pt\vec{y}(\Phi_1[\vec{y}],\ldots,\Phi_n[\vec{y}]\to B[\vec{y}])$ and a sequence $\vec{b}$ of constants, of the same length as $\vec{y}$, she have to check that $B[\vec{b}]\in{\cal A}$, i.e. to check that two atomic closed formulas are identical. These formulas may contain closed terms of type integer and these terms must be computed before comparing them. [$\omega$-valid formulas.]{}\ A normal closed formula $F$ will be called [$\omega$-valid]{} if there exists a winning strategy for $\ex$, in the game associated with $F$. The $\omega$-valid formulas are exactly those which correspond to network protocols (as before, with valid formulas). [Example.]{}\ First, $\pt$ send an integer $n$ ; after that, acknowledgement of $n$ packets. The formula is :\ $F\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Ui\to Pix)\to Piy)]\to Uj\}\to\pt n\,Un$.\ $U$ is a predicate symbol with one argument of integer type ; $P$ has two arguments, the first is of integer type, the second of type acknowledgement. Put $G\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Ui\to Pix)\to Piy)]\to Uj\}$ and\ $H[i,x]\equiv\pt y((Ui\to Pix)\to Piy)$.\ We put the formula $F$ in normal form, so it is written as $F\equiv\pt n(G\to Un)$. The following table shows the particular session in which every packet is acknowledged in the quickest possible way. In order to avoid a supplementary complication, we did not ask that the integer $n$ (the number of packets to transmit) which is sent by $\pt$, be acknowledged. But if we want this integer to be acknowledged, we must add a field “ acknowledgement ” to the predicate symbol $U$, which therefore becomes binary. In this case, we write the following formula :\ $F\equiv\pt n\ex x'\pt y'((\neg\,G[x']\to Unx')\to Uny')$ with\ $G[x']\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Uix'\to Pix)\to Piy)]\to Ujx'\}$ The following table shows the beginning of a communication session and the acknowledgement of the integer $n$.\ We put $H[n,x']\equiv\pt y'((\neg\,G[x']\to Unx')\to Uny')$. $\dsp \begin{array}{cccccl} & {\cal U} & {\cal V} & {\cal A} & & \\ 1 &\neg F & F & \bot & &\pt\mbox{ chooses }n_0\\ 2 &\pt x'\,\neg H[n_0,x'] &\hspace{2em}\mbox{unchanged}\hspace{2em} &\mbox{unchanged}& &\ex\mbox{ chooses }\pt x'\,\neg H[n_0,x']\mbox{ and }x'_0\\ 3 &\mbox{unchanged} & H[n_0,x'_0] &\mbox{unchanged} & &\pt\mbox{ chooses }y'_0\\ 4 &\neg G[x'_0]\to Un_0x'_0 & \mbox{unchanged} & Un_0y'_0& &\ex\mbox{ chooses }\pt x'\neg H[n_0,x']\mbox{ and }y'_0\\ 5 &\mbox{unchanged} & H[n_0,y'_0] &\mbox{unchanged} & &\pt\mbox{ chooses }y'_1\\ 6 &\neg G[y'_0]\to Un_0y'_0 & \mbox{unchanged} & Un_0y'_1& &\ex\mbox{ chooses }\neg G[y'_0]\to Un_0y'_0\\ 7 &\mbox{unchanged}&\neg\,G[y'_0] &\mbox{unchanged}& & \pt\mbox{ has no choice}\\ 8 & G[y'_0]&\mbox{unchanged}&\mbox{unchanged}& &\ex\mbox{ chooses }G[y'_0]\mbox{ and }n_0\\ &\vdots&\vdots&\vdots & &\hspace{2em}\vdots\\ \end{array}$ From now on, the play continues as in the previous example (with the formula $G[y'_0]$ instead of the formula $G$). A somewhat simpler formula for the same protocol can be written as :\ $\pt n\ex x'\pt y'(G[x']\to Uny')$ with, as before :\ $G[x']\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Uix'\to Pix)\to Piy)]\to Ujx'\}$.\ The reader will check this easily. Appendix {#appendix .unnumbered} -------- [Valid formulas.]{}\ The usual definition of a valid formula of the predicate calculus uses the notion of model. The interested reader will find it, for example in \[rcdl\] or \[jrs\]. A formula is called valid if it is satisfied in every model. A fundamental theorem of logic, known as the [completeness theorem]{}, says that a formula is valid if, and only if, it is provable by means of the deduction rules of “ pure logic ”.\ This notion of validity is equivalent to that introduced in the present paper, which is given in terms of strategies (see a proof in \[jlk\]).\ It is often much easier to check the validity of a formula with the help of models. For example, it is immediately seen in this way that the formula $F\equiv\ex x(Px\to\pt y\,Py)$ is valid : indeed, either the model we consider satisfies $\pt y\,Py$ and therefore also $F$, either it satisfies $\ex x\neg\,Px$ and thus again $F$. Let us consider two valid formulas $F$ and $G$, and let $H$ be the formula defined above, such that the protocol ${\cal P}_H$ associated with $H$ is obtained by composing the protocols associated with $F$ and $G$. Then, it is easy to show that $H$ is valid. Indeed, we obtained the formula $H$ by replacing, in $F$, some subformulas $A$ with $G\to A$. But $A$ and $G\to A$ are obviously equivalent, since $G$ is valid. Thus, we get finally a formula $H$ wich is equivalent to $F$, and therefore a valid formula. For the formulas with two types (integer and acknowledgement), the situation is a bit more complex. The $\omega$-valid formulas are the formulas which are satisfied in every [$\omega$-model]{}, that is to say the models in which the integer type has its standard interpretation. Again, in this case, it is often much easier to use this definition in order to check that a formula is $\omega$-valid.\ For instance, it is not difficult to show that the formula (that we have already used before) $F\equiv G\to\pt n\,Un$, with $G\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Ui\to Pix)\to Piy)]\to Uj\}$ is $\omega$-valid.\ Indeed, we assume $G$ and we show $Un$ by recurrence on the integer $n$.\ Proof of $U0$. We put $j=0$ in $G$ ; since $0=si$ is false, we get $U0$.\ Proof of $Un\to U(n+1)$. We put $j=n+1$ in $G$. Then, it suffices to show :\ $\pt i[n+1=si\to\ex x\pt y((Ui\to Pix)\to Piy)]$ with $Un$ as an hypothesis. Since $n+1=si$ is equivalent to $i=n$, we now need to show $\ex x\pt y((Un\to Pnx)\to Pny)$, that is to say $\ex x\pt y(Pnx\to Pny)$ (because we assume $Un$). But this last formula is already shown. With some simple changes, the same proof works for the formulas :\ $\pt n\ex x'\pt y'((\neg\,G[x']\to Unx')\to Uny')$ and $\pt n\ex x'\pt y'(G[x']\to Uny')$\ with $G[x']\equiv\pt j\{\pt i[j=si\to\ex x\pt y((Uix'\to Pix)\to Piy)]\to Ujx'\}$.\ Indeed, you only need to show the first one, since the second is trivially weaker. [Determination of games.]{}\ A game is called [determined]{} if one of the players has a winning strategy. It is always the case for the games considered in this paper (Gale-Stewart theorem).\ Sketch of proof. Suppose that $\ex$ has no winning strategy. Then, the following strategy is winning for the opponent $\pt$ : to play, at each step, in such a way that the player $\ex$ has no winning strategy from this step. Then the play lasts infinitely long, and $\pt$ wins. References {#references .unnumbered} ---------- [\[]{}rcdl\] René Cori & Daniel Lascar. [Mathematical logic.]{} Oxford University Press (2001).\ [\[]{}jlk\] Jean-Louis Krivine. [Realizability : a machine for analysis and set theory.]{} Geocal06, Marseille (2006).\ [http://cel.archives-ouvertes.fr/cel-00154509]{}. More recent version at :\ [http://www.pps.jussieu.fr/ krivine/articles/Mathlog07.pdf]{}\ [\[]{}jp\] Jon Postel. [RFC 793 - Transmission Control Protocol specification.]{} (1981).\ [\[]{}jrs\] Joseph R. Schoenfield. [Mathematical logic.]{} Addison Wesley (1967).
--- abstract: 'It is known that non-holomorphic corrections are necessary in order to get duality invariant free energy and entropy function. However the present methods of incorporating non-holomorphic corrections are in conflict with special geometry properties of moduli space. The moduli fields do not transform in the duality covariant way and their duality transformation also involves the graviphoton field strength. In the present note we construct duality covariant moduli fields for STU-model perturbatively in powers of the graviphoton field strength and demonstrated their existence upto second order.' --- [Duality covariant variables for STU-model in presence of non-holomorphic corrections]{} Shamik Banerjee, Rajesh Kumar Gupta Harish-Chandra Research Institute Chhatnag Road, Jhusi, Allahabad 211019, INDIA E-mail: bshamik@mri.ernet.in, rajesh@mri.ernet.in Introduction ============ The four dimensional $N=2$ supergravity lagrangian [@0007195] coupled non-minimally to $(n+1)$-vector multiplets is described by a holomorphic prepotential $F(Y^I,\Upsilon)$, which is a homogeneous function of degree 2 in $Y^I(I=0,....,n)$ and $\Upsilon$. Here the $Y^I$ denotes the (rescaled) complex scalar field which sits in the abelian vector multiplet and $\Upsilon$ denotes the (rescaled) square of the auxiliary, antiselfdual, antisymmetric Lorentz tensor $T_{ab}^{ij}$ which sits in the weyl multiplet. The $\Upsilon$ dependent term in the prepotential gives rise to higher derivative curvature terms in the effective action. Under electric/magnetic duality transformation $Sp(2n+2,\mathbb{Z})$, $(Y^I,F_I(Y,\Upsilon))$ transform as a symplectic vector.\ Vector multiplet scalars assume fixed values at the black hole horizon determined by the attractor equation. The attractor equation relates two symplectic vectors $(p^I,q_I)$ and $(Y^I,F_I(Y,\Upsilon))$ and has the following form $$\label{attreqn} Y^I-\bar {Y}^I=ip^I ,\quad F_I(Y,\Upsilon)-\bar{F}_I(\bar Y,\bar \Upsilon)=iq_I, \quad \Upsilon=-64$$ where $(p^I,q_I)$ denotes the magnetic and electric charges of the black hole. This equation is manifestly symplectic invariant. The macroscopic entropy of a static, supersymmetric black hole computed [@0007195] from the effective lagrangian is given by $$\label{entropy} S_{macro}(p,q)=\pi\left[{\vert Z\vert}^2 -256 Im(F_{\Upsilon}(Y,\Upsilon))\right]_{\Upsilon=-64}$$ where $$\vert Z \vert^2=p^IF_I(Y,\Upsilon)-q_IY^I$$ There exist a proposal [@0601108] for the BPS entropy function whose stationary points are attractor equations and the value of the entropy function at the attractor points equals the macroscopic entropy (\[entropy\]). In terms of the entropy function, the macroscopic degeneracy of black hole is given by $$\label{degeneracy} d(p,q)_{macro}=\int d(Y^I+\bar Y^I)d(F_I+\bar F_I)e^{\pi \Sigma(Y,\bar Y,p,q)}$$ where the entropy function is given by $$\label{entropyfn} \Sigma(Y,\bar Y, \Upsilon, \bar \Upsilon)=\mathscr{F}(Y,\bar Y,\Upsilon, \bar \Upsilon)-q_I(Y^I+\bar Y^I)+p^I(F_I+\bar F_I)$$ Here $\mathscr{F}(Y,\bar Y,\Upsilon, \bar \Upsilon)$ denotes the free energy and is given by $$\label{freeenrg} \mathscr{F}(Y,\bar Y,\Upsilon, \bar \Upsilon)=-i\left(\bar Y^IF_I-Y^I\bar F_I\right)-2i\left(\Upsilon F_{\Upsilon}-\bar \Upsilon\bar F_{\Upsilon}\right)$$ Since $(Y^I,F_I(Y,\Upsilon))$ transforms as a symplectic vector under symplectic transformation, the free energy (\[freeenrg\]), the entropy function (\[entropyfn\]) and hence the $d(p,q)_{macro}$ are symplectic invariant.\ A subgroup of the symplectic transformation is called “duality symmetry” when the function $F(Y,\Upsilon)$ remains unchanged i.e. $\tilde F(\tilde Y,\Upsilon)=F(\tilde Y,\Upsilon)$. This implies that the substitution of $Y^I\rightarrow\tilde Y^I$ into the derivatives $F_I(Y,\Upsilon)$ will induce the correct transformations on the $F_I(Y,\Upsilon)$.\ The attractor equations (\[attreqn\]) follow from the extremization of (\[entropyfn\]) with respect to $Y^I$ keeping $\Upsilon$ fixed and the macroscopic entropy (\[entropy\]) is given by $$S_{macro}(p,q)=\pi\Sigma\vert_{attractor}$$ The entropy formula (\[entropy\]) was based on the effective action approach. In this approach, coupling functions multiplying the higher derivative terms, like square of Riemann tensor, is proportional to Im$F_{\Upsilon}(Y,\Upsilon)$. The physical effective couplings in one P.I. effective action which one obtain by integrating out the fields at one loop are different from wilsonian coupling [@9906094; @0601108] because of presence of infrared divergence due to massless fields. Although one does not know the complete effective action with non-local terms it is clear that the holomorphic function $F(Y,\Upsilon)$ which give rise to local higher derivative terms will not give duality invariant entropy (\[entropy\]) and entropy function(\[entropyfn\]). However one can still make the entropy function (\[entropyfn\]), that appears in the expression (\[degeneracy\]) for $d(p,q)$, duality invariant by modifying the function $F(Y,\Upsilon)$. One does this by relaxing the holomorphicity properties of the function $F(Y,\Upsilon)$ and taking into account a non-holomorphic corrections to $F(Y,\Upsilon)$.\ The nonholomorphic coorections are added to the entropy function by replacing $F(Y,\Upsilon)$ by[@0601108; @0808.2627] $$F(Y,\bar Y,\Upsilon,\bar \Upsilon)= F^{(0)}(Y)+2i\Omega(Y,\bar Y,\Upsilon,\bar\Upsilon)$$ Here $\Omega(Y,\bar Y,\Upsilon,\bar\Upsilon)$ is a real, homogeneous function of degree 2 and $F^{(0)}(Y)$ is $\Upsilon$ independent part of the prepotential. The homogeneity condition implies that $$\label{homog} Y^I\Omega_I+\bar Y^I\Omega_{\bar I}+2\Upsilon\Omega_\Upsilon+2\bar\Upsilon\Omega_{\bar\Upsilon}=2\Omega$$ The form of $\Omega$ should be such that under duality transformation the substitution of $Y^I\rightarrow \tilde Y^I$ into the derivatives $F_I(Y,\bar Y,\Upsilon,\bar \Upsilon)$ will induce the correct duality transformation on $F_I(Y,\bar Y,\Upsilon,\bar \Upsilon)$.\ Thus one obtains the duality invariant expressions by replacing $F(Y,\Upsilon)$ by $F(Y,\bar Y,\Upsilon,\bar \Upsilon)$ into (\[degeneracy\]-\[freeenrg\]). For example the duality invariant free energy (\[freeenrg\]) becomes $$\label{modfrenrg} \mathscr{F}(Y,\bar Y,\Upsilon, \bar \Upsilon)=-i(\bar Y^IF^{(0)}_I-Y^I\bar F^{(0)}_I)-2(Y^I-\bar Y^I)(\Omega_I-\Omega_{\bar I})+4\Omega$$ Here we have used the eqn (\[homog\]).\ Recently the form of $\Omega$ has been determined for specific $N=2$-models [@0808.2627]. In [@0808.2627], the form of $\Omega$ was guessed from the transformation rule for the derivatives of $\Omega$. These transformation rule was obtained by assuming that the duality transformation constitute an invariance of the model and duality transformation of the $Y^I$ induce the expected transformation of $F_I=F^{(0)}_I+2i\Omega_I$.\ In presence of non-holomorphic corrections determined by $\Omega$, the new duality transformation mixes $Y^I$ and $\bar Y^I$. However this is in contrast with string theory where duality transformations act holomorphically on the physical moduli. Furthermore, string theory duality transformations of moduli fields does not involve $\Upsilon$ dependent terms. Hence the variables used in [@0808.2627] are not string theory variables. Furthemore, since the duality transformation mixes various powers of $\Upsilon$, the terms in the expression of free energy for a given power of $\Upsilon$ is not duality invariant.\ In this note, we shall try to obtain new variables ${Y^{\prime}}^I$ for STU-model which ,even in presence of nonholomorphic corrections, transform as string theory variables. The old variables $Y^I$ are function of the new variables ${Y^{\prime}}^I$. These functions are detetrmined by requiring that under the standard duality transformation of ${Y^{\prime}}^I$, the $Y^I$ transform as in [@0808.2627]. These functions are obtained as a power series in $\Upsilon$. In the limit $\Upsilon\rightarrow 0$, ${Y^{\prime}}^I$ coincides with $Y^I$. Furthermore the terms in the expression of free energy for a given power in $\Upsilon$ written in the new variables ${Y^{\prime}}^I$ are duality invariant. Although we will explicitly work in STU-model, the method can be applied for other models like FHSV-model also.\ This paper is organised as follows. In section 2 we briefly introduce STU-model and determine the non-holomorphic corrections upto $\Upsilon^2$. We will closely follow [@0808.2627]. In section 3 we will introduce new variables which transform in the standard way under duality transformation. In section 4 we re-expressed the free energy in terms of these new variables. In section 5 we present our conclusion. STU-model and Nonholomorphic Corrections ======================================== The STU-model [@0711.1971; @9508064] is based on four fields $Y^0,Y^1,Y^2$ and $Y^3$ of which later three appear symmetrically. The special coordinates S, T and U are defined as $$\label{stu} S=\frac{-iY^1}{Y^0},\quad T=\frac{-iY^2}{Y^0},\quad U=\frac{-iY^3}{Y^0}.$$ The tree level prepotential for the STU-model is given by $$\label{treeprepot} F^{(0)}(Y)=-\frac{Y^1Y^2Y^3}{Y^0}=i(Y^0)^2STU.$$ The complete duality group of the STU-model is $$\label{group} (\Gamma(2)_S\otimes\Gamma(2)_T\otimes\Gamma(2)_U)\times \mathbb{Z}_2^{T-U}\times\mathbb{Z}_2^{S-T}\times\mathbb{Z}_2^{S-U}$$ where $\Gamma(2)\subset SL(2;\mathbb Z)$ with $a,d\in2\mathbb Z+1$ and $b,c\in2\mathbb Z$, with $ad-bc=1$.\ The action of S-duality group $\Gamma(2)_S$ is defined as $$\label{dualtransf} \begin{split} & Y^0\rightarrow dY^0+cY^1 , \qquad F^{(0)}_0\rightarrow aF^{(0)}_0-bF^{(0)}_1\\ & Y^1\rightarrow aY^1+bY^0 ,\qquad F^{(0)}_1\rightarrow dF^{(0)}_1-cF^{(0)}_0\\ & Y^2\rightarrow dY^2-cF^{(0)}_3 ,\qquad F^{(0)}_2\rightarrow aF^{(0)}_2-bY^3\\ & Y^3\rightarrow dY^3-cF^{(0)}_2 ,\qquad F^{(0)}_3\rightarrow aF^{(0)}_3-bY^2. \end{split}$$ One has similar expression for the T(U)-duality transformation by interchanging the labels $1\leftrightarrow 2 (1\leftrightarrow 3)$.\ Next we want to include nonholomorphic corrections to STU-model. As mentioned above, this is done by considering the new prepotential $$\label{prepot} F(Y,\bar Y,\Upsilon,\bar\Upsilon)=i(Y^{0})^2STU+2i\Omega(Y^{0},Y^1,Y^2,Y^3,\bar{Y^0},\bar{Y^1},\bar{Y^2},\bar{Y^3},\Upsilon,\bar\Upsilon).$$ Here $\Omega$ is a real homogeneous function of degree 2 in $Y$ and $\Upsilon$ and contains the non-classical contribution. Throughout our analysis we will assume that $\Upsilon$ is real - this can be achieved by working in an appropriate gauge condition.\ Assuming that $\Omega$ is analytic at $\Upsilon=0$, one can expand it as $$\label{omegaexp} \Omega(Y^{0},S,T,U,\bar{Y^0},\bar{S},\bar{T},\bar{U},\Upsilon)=\Upsilon \Omega^{(1)}+\displaystyle\sum_{g=2}^{\infty} \Upsilon^g \Omega^{(g)}$$ Since $\Omega$ is a homogeneous functon of degree 2 , $\Omega^{(1)}$ will depend on $S,T,U$ and their complex conjugate but will not depend on $Y^0$ and $\Omega^{(g)}$ will be a homogeneous function of degree $2-2g$ in the $Y^{I}$’s.\ The holomorphic derivatives of function $F$ with respect to $Y^I$ are $$\begin{split} &F_0=-iY^0STU-\frac{2i}{Y^0}\left[-Y^0\frac{\partial}{\partial Y^0}+S\frac{\partial}{\partial S}+T\frac{\partial}{\partial T}+U\frac{\partial}{\partial U}\right]\Omega\\ &F_1=Y^0TU+\frac{2}{Y^0}\frac{\partial\Omega}{\partial S}\\ &F_2=Y^0SU+\frac{2}{Y^0}\frac{\partial\Omega}{\partial T}\\ &F_3=Y^0ST+\frac{2}{Y^0}\frac{\partial\Omega}{\partial U}.\\ \end{split}$$ Under $\Gamma(2)_S$ , the duality transformations (\[dualtransf\]) on the fields become $$\begin{split} & Y^0\rightarrow \Delta_S Y^0, \quad Y^1\rightarrow aY^1+bY^0\\ &Y^2 \rightarrow \Delta_S Y^2-\frac{2c}{Y^0}\frac{\partial \Omega}{\partial U},\\& Y^3 \rightarrow \Delta_S Y^3-\frac{2c}{Y^0}\frac{\partial \Omega}{\partial T}. \end{split}$$ In terms of special coordinates, these duality transformations become $$\begin{aligned} \label{duality} Y^0\rightarrow \Delta_S Y^0 , \qquad S\rightarrow \frac{aS-ib}{icS+d},\\\nonumber T\rightarrow T +\frac{2ic}{\Delta_S(Y^0)^2}\frac{\partial\Omega}{\partial U} , \quad U\rightarrow U +\frac{2ic}{\Delta_S(Y^0)^2}\frac{\partial\Omega}{\partial T}.\end{aligned}$$ Here $$\Delta_S=d+icS$$ One can similarly obtain the T(U)-duality transformation by interchanging the labels $1\leftrightarrow 2 (1\leftrightarrow 3)$.\ Now requiring that the duality transformation is an invariance of the model and the transformation of $Y^I$ induce the transformation of $F_I$, the derivatives of $\Omega$ transform under $\Gamma(2)_S$ as [@0808.2627] $$\left(\frac{\partial \Omega}{\partial T}\right)^{\prime}_S=\frac{\partial \Omega}{\partial T} \qquad \left(\frac{\partial \Omega}{\partial U}\right)^{\prime}_S=\frac{\partial \Omega}{\partial U}\\\nonumber$$ $$\left(\frac{\partial \Omega}{\partial S}\right)^{\prime}_S -\Delta^2_S\frac{\partial \Omega}{\partial S}=\frac{\partial(\Delta^2_S)}{\partial S}\left[-\frac{1}{2}Y^0\frac{\partial \Omega}{\partial Y^0}-\frac{ic}{\Delta_S(Y^0)^2}\frac{\partial \Omega}{\partial T}\frac{\partial \Omega}{\partial U}\right]\\\nonumber$$ $$\label{omegatransf} \left(Y^0\frac{\partial \Omega}{\partial Y^0}\right)^{\prime}_S=Y^0\frac{\partial \Omega}{\partial Y^0}+\frac{4ic}{\Delta_S(Y^0)^2}\frac{\partial \Omega}{\partial T}\frac{\partial \Omega}{\partial U}.$$ Similarly one can obtain the T(U)-duality transformation by replacing $S\leftrightarrow T (S\leftrightarrow U)$ in (\[omegatransf\]).\ Although for our analysis, we do not need explicit expression for $\Omega$, we will just mention the results for $\Omega^{(1)}$ and $\Omega^{(2)}$ for STU-model. The result for $\Omega^{(1)}$ was derived in [@0808.2627]. One can follow the procedure of [@0808.2627] to get $\Omega^{(2)}$. $$\begin{split} &\Omega^{(1)}=\frac{1}{256\pi}\left[4ln[\vartheta_2(iS)\vartheta_2(iT)\vartheta_2(iU)]+c.c\right]+\frac{1}{126\pi}\left[ln[(S+\bar S)(T+\bar T)(U+\bar U)]\right]\\ &\Omega^{(2)}=\frac{2}{(Y^0)^2}\left[\frac{1}{S+\bar S}\frac{\partial \Omega^{(1)}}{\partial T}\frac{\partial \Omega^{(1)}}{\partial U}+\frac{1}{T+\bar T}\frac{\partial \Omega^{(1)}}{\partial U}\frac{\partial \Omega^{(1)}}{\partial S}+\frac{1}{U+\bar U}\frac{\partial \Omega^{(1)}}{\partial S}\frac{\partial \Omega^{(1)}}{\partial T}\right]+c.c \end{split}$$ where $$\vartheta_2(iS)=\frac{2\eta^2(2iS)}{\eta(iS)}$$ Here $\eta(iS)$ is the Dedekind function and $\eta^{24}(iS)$ is a modular form of weight 12 under SL(2;$\mathbb{Z}$). The expressions for $\Omega^{(1)}$ and $\Omega^{(2)}$ are determined upto $\Upsilon$ dependent S-, T-, U-duality invariant expressions.\ Now we define the following $new$ variables $$\label{news} S^{\prime}=S+\sum_{n=1}^{\infty}\Upsilon^n s_n(Y^{0},S,T,U,\bar{Y^0},\bar{S},\bar{T},\bar{U},\Omega)\\\nonumber$$ $$T^{\prime}=T+\sum_{n=1}^{\infty}\Upsilon^n t_n(Y^{0},S,T,U,\bar{Y^0},\bar{S},\bar{T},\bar{U},\Omega)\\\nonumber$$ $$U^{\prime}=U+\sum_{n=1}^{\infty}\Upsilon^n u_n(Y^{0},S,T,U,\bar{Y^0},\bar{S},\bar{T},\bar{U},\Omega)$$ $${Y^0}^{\prime}=Y^0+\sum_{n=1}^{\infty}\Upsilon^n k_n(Y^{0},S,T,U,\bar{Y^0},\bar{S},\bar{T},\bar{U},\Omega)\\\nonumber$$ The above expansions are in positive powers of $\Upsilon$. The functions $\{s_n,t_n,u_n\}$ and $\{k_n\}$ are chosen such that under duality transformation (\[duality\]) and (\[omegatransf\]) the primed variables transform as a tree level variables i.e under $\Gamma(2)_S$ the primed variables should transform as $$\label{dualityprime} \begin{split} &S^{\prime}\rightarrow \frac{aS^{\prime}-ib}{icS^{\prime}+d}, \qquad T^{\prime}\rightarrow T^{\prime}\\ &U^{\prime}\rightarrow U^{\prime}, \qquad {Y^0}^{\prime}\rightarrow (d+icS^{\prime}){Y^0}^{\prime}. \end{split}$$ Requiring this will give transformations of $s_n,t_n,u_n$ and $k_n$ from which one can determine $s_n$’s perturbatively in powers of $\Upsilon$.\ Determination of Field Redefinition =================================== In this section we will determine the form of $\{s_n,t_n,u_n\}$ and $\{k_n\}$ perturbatively. One of the advantages of considering STU-model is that if one obtains the expression of $s_n$, the expressions for $t_n$ and $u_n$ follow from triality. Furthermore, the expression for $k_n$ should be triality invariant. We will first determine the transformation of these functions and from there we will guess their form. However we found that from these transformations, one can not determine the form of these functions $uniquely$. There exist more than one such functions which give the same duality transformation (\[dualityprime\]). This suggests that there are more than one such primed variables which under duality transformation do not mix with their complex conjugates. Here we mention the results for functions $\{s_n\}$ and $\{k_n\}$ only. The results for the other functions follow from triality.\ Determination of First Order Function ------------------------------------- Till first order in $\Upsilon$, the field redefinition are $$\begin{split} &S^{\prime}=S+\Upsilon s_1,\\ &{Y^0}^{\prime}=Y^0+\Upsilon k_1. \end{split}$$ Requiring that under old S-duality transformation (\[duality\]), S$^{\prime}$ and ${Y^0}^{\prime}$ transform as (\[dualityprime\]) gives the following duality transformations for $s_1$ and $k_1$ $$s_1\rightarrow \frac{s_1}{\Delta^2_S},\\\nonumber$$ $$\label{transf1} k_1\rightarrow \Delta_S k_1 +ics_1 Y^0.$$ Similar analysis under $T$ and $U$-duality gives the following results.\ Under $T$-duality $$s_1\rightarrow s_1-\frac{2ic}{(Y^0)^2\Delta_T}\left(\frac{\partial \Omega^{(1)}}{\partial U}\right).\\\nonumber$$ Under $U$-duality $$s_1\rightarrow s_1-\frac{2ic}{(Y^0)^2\Delta_U}\left(\frac{\partial \Omega^{(1)}}{\partial T}\right).$$ The transformation rule for $t_1$ $(u_1)$ will follow from the eqn.(\[transf1\]) by interchanging $S \leftrightarrow T$ $(S\leftrightarrow U)$ and $\Gamma(2)_S \leftrightarrow \Gamma(2)_T$ $(\Gamma(2)_S \leftrightarrow \Gamma(2)_U)$.\ Following are the transformation rule for $k_1$ under $T$ and $U$-duality.\ Under $T$-duality $$k_1\rightarrow \Delta_T k_1 +ict_1 Y^0.\\\nonumber$$ Under $U$-duality $$\label{transk1} k_1\rightarrow \Delta_U k_1 +icu_1 Y^0.$$ From the above equation (\[transf1\])- (\[transk1\]) and (\[omegatransf\]) one can guess the follwing expressions for $s_1$ and $k_1$\ $$\label{s_1} s_1=\frac{2}{(Y^0)^2}\left[\frac{1}{T+\bar T}\left(\frac{\partial \Omega^{(1)}}{\partial U}\right)+\frac{1}{U+\bar U}\left(\frac{\partial \Omega^{(1)}}{\partial T}\right)\right],$$ $$k_1=-\frac{Y^0}{2}\left[\frac{s_1}{S+\bar S}+\frac{t_1}{T+\bar T}+\frac{u_1}{U+\bar U}\right].$$ The expression of $k_1$ is invariant under triality as required.\ Similar expressions for $t_1$ and $u_1$ follows from triality. The expressions are $$t_1=\frac{2}{(Y^0)^2}\left[\frac{1}{S+\bar S}\left(\frac{\partial \Omega^{(1)}}{\partial U}\right)+\frac{1}{U+\bar U}\left(\frac{\partial \Omega^{(1)}}{\partial S}\right)\right],$$ $$u_1=\frac{2}{(Y^0)^2}\left[\frac{1}{T+\bar T}\left(\frac{\partial \Omega^{(1)}}{\partial S}\right)+\frac{1}{S+\bar S}\left(\frac{\partial \Omega^{(1)}}{\partial T}\right)\right].$$ As we have mentioned earlier, there is an ambiguity in the obtained expression for $s_1$ (similarly one can find ambiguity in $t_1$, $u_1$ and $k_1$). One finds that one can add terms in $s_1$ keeping it’s transformation rule intact. One such term is $$f=\frac{1}{(Y^0)^2}\frac{\partial \Omega^{(1)}}{\partial U}\frac{\partial \Omega^{(1)}}{\partial T}.$$ Under S-duality, $$\frac{\partial \Omega^{(1)}}{\partial T}\rightarrow \frac{\partial \Omega^{(1)}}{\partial T}, \quad \frac{\partial \Omega^{(1)}}{\partial U}\rightarrow \frac{\partial \Omega^{(1)}}{\partial U} ,\quad Y^0\rightarrow \Delta_SY^0.$$ Hence $$f\rightarrow \frac{f}{\Delta_S^2}.$$ Under T-duality, $$\frac{\partial \Omega^{(1)}}{\partial T}\rightarrow\Delta_T^2 \frac{\partial \Omega^{(1)}}{\partial T} ,\quad \frac{\partial \Omega^{(1)}}{\partial U}\rightarrow \frac{\partial \Omega^{(1)}}{\partial U}, \quad Y^0\rightarrow \Delta_TY^0.$$ Hence $$f\rightarrow f.$$ Similarly $f$ is invariant under U-duality also.\ Hence one can add $f$ in $s_1$ and still we will have correct transformation for $s_1$. One can find the same ambiguity in other functions also. This ambiguity just reflects the fact that one can have more than one duality covariant complex variables which under duality transformation do not mix with their complex conjugates. Determination of Second Order Function -------------------------------------- In this section we repeat our calculations for second order function. We will do the analysis only for $s_2$ and $k_2$. One can find $t_2$ and $u_2$ by interchanging the fields.\ We first need to find the duality transformation for $s_2$. Repeating the same procedure as we did for $s_1$, we get the following result\ Under $S$-duality $$\begin{split} s_2\rightarrow &\frac{s_2}{\Delta^2_S}-\frac{ic}{\Delta^3_S}s_1^2-\frac{4ic}{\Delta^3_S(Y^0)^4}\frac{\partial \Omega^{(1)}}{\partial U}\left[-\frac{1}{(T+\bar T)^2}\frac{\partial \Omega^{(1)}}{\partial U}+\frac{1}{U+\bar U}\frac{\partial^2 \Omega^{(1)}}{\partial T^2}\right]\\& -\frac{4ic}{\Delta^3_S(Y^0)^4}\frac{\partial \Omega^{(1)}}{\partial T}\left[-\frac{1}{(U+\bar U)^2}\frac{\partial \Omega^{(1)}}{\partial T}+\frac{1}{T+\bar T}\frac{\partial^2 \Omega^{(1)}}{\partial U^2}\right]\\&+\frac{4ic}{\bar \Delta_S\Delta^2_S(Y^0\bar Y^0)^2} \frac{\partial \Omega^{(1)}}{\partial \bar U}\left[-\frac{1}{(T+\bar T)^2}\frac{\partial \Omega^{(1)}}{\partial U}+\frac{1}{U+\bar U}\frac{\partial^2 \Omega^{(1)}}{\partial T\partial {\bar T}}\right]\\&+\frac{4ic}{\bar \Delta_S\Delta^2_S(Y^0\bar Y^0)^2} \frac{\partial \Omega^{(1)}}{\partial \bar T}\left[-\frac{1}{(U+\bar U)^2}\frac{\partial \Omega^{(1)}}{\partial T}+\frac{1}{T+\bar T}\frac{\partial^2 \Omega^{(1)}}{\partial U\partial {\bar U}}\right]. \end{split}$$ Under $T$-duality $$\begin{split} s_2\rightarrow & s_2 -\frac{2ic}{\Delta_T(Y^0)^2}\frac{\partial \Omega^{(2)}}{\partial U}-\frac{4ic}{\Delta_T(Y^0)^4}\frac{\partial \Omega^{(1)}}{\partial S}\left[\frac{1}{T+\bar T}\frac{\partial^2\Omega^{(1)}}{\partial U^2}-\frac{1}{(U+\bar U)^2}\frac{\partial \Omega^{(1)}}{\partial T}\right]\\&+\frac{4ic}{\bar \Delta_T(Y^0\bar Y^0)^2}\frac{\partial \Omega^{(1)}}{\partial \bar S}\left[\frac{1}{T+\bar T}\frac{\partial^2\Omega^{(1)}}{\partial U \partial \bar U}-\frac{1}{(U+\bar U)^2}\frac{\partial \Omega^{(1)}}{\partial T}\right]\\&-\frac{4c^2}{\Delta_T^2(Y^0)^4}\frac{\partial \Omega^{(1)}}{\partial S}\frac{\partial^2 \Omega^{(1)}}{\partial U^2}+\frac{4c^2}{\Delta_T \bar \Delta_T(Y^0\bar Y^0)^2}\frac{\partial \Omega^{(1)}}{\partial \bar S}\frac{\partial^2 \Omega^{(1)}}{\partial U \partial \bar U}. \end{split}$$ We have similar expression under $U$-duality with $T\leftrightarrow U$ in the above expression for T-duality.\ The expression for $s_2$-consistent with above transformation is $$\begin{split} s_2=&\frac{2}{(Y^0)^2}\left[\frac{1}{U+\bar U}\frac{\partial \Omega^{(2)}}{\partial T}+\frac{1}{T+\bar T}\frac{\partial \Omega^{(2)}}{\partial U}\right]-\frac{2}{Y^0}\frac{1}{(T+\bar T)(U+\bar U)}\frac{\partial \Omega^{(2)}}{\partial Y^0}\\&-\frac{2s_1}{(T+\bar T)(U+\bar U)}\left[\frac{1}{(Y^0)^2}\frac{\partial \Omega^{(1)}}{\partial S}+\frac{1}{(\bar Y^0)^2}\frac{\partial \Omega^{(1)}}{\partial \bar S}\right]\\& -\frac{4}{(S+\bar S)(Y^0\bar Y^0)^2}\left[\frac{1}{(T+\bar T)^2}\frac{\partial \Omega^{(1)}}{\partial \bar U}\frac{\partial \Omega^{(1)}}{\partial U}+\frac{1}{(U+\bar U)^2}\frac{\partial \Omega^{(1)}}{\partial \bar T}\frac{\partial \Omega^{(1)}}{\partial T}\right]. \end{split}$$ The expressions for $t_2$ and $u_2$ will follow from triality symmetry.\ Similarly one can obtain the expression for $k_2$. We will not write here the transformation of $k_2$, but just mention the result for $k_2$ $$\begin{split} k_2=&-\frac{Y^0}{2}\left[\frac{s_2}{S+\bar S}+\frac{t_2}{T+\bar T}+\frac{u_2}{U+\bar U}\right]+\frac{1}{(S+\bar S)(T+\bar T)(U+\bar U)}\frac{\partial \Omega^{(2)}}{\partial Y^0}\\&+\frac{Y^0}{2}\left[\frac{s_1^2}{(S+\bar S)^2}+\frac{t_1^2}{(T+\bar T)^2}+\frac{u_1^2}{(U+\bar U)^2}\right]\\&+\frac{1}{Y^0(S+\bar S)(T+\bar T)(U+\bar U)}\left[(s_1+\bar s_1)\frac{\partial \Omega^{(1)}}{\partial S}+(t_1+\bar t_1)\frac{\partial \Omega^{(1)}}{\partial T}+(u_1+\bar u_1)\frac{\partial \Omega^{(1)}}{\partial U}\right]. \end{split}$$ The expression for $k_2$ is also triality invariant. Free Energy =========== In this section we express the free energy as a function of the new variables. The expression for free energy (\[modfrenrg\]) in terms of old variables upto first order in $\Upsilon$ is $$\begin{split} \mathscr F=&-\vert Y^0\vert^2(S+\bar S)(T+\bar T)(U+\bar U)+4\Upsilon\Omega^{(1)}\\&-2\Upsilon\left\{\frac{\bar Y^0}{Y^0}\left[(S+\bar S)\frac{\partial \Omega^{(1)}}{\partial S}+(T+\bar T)\frac{\partial \Omega^{(1)}}{\partial T}+(U+\bar U)\frac{\partial \Omega^{(1)}}{\partial U}\right]+h.c\right\}. \end{split}$$ The above expression written in terms of new variables is $$\mathscr F=-{\vert {Y^0}^{\prime}\vert}^2(S^{\prime}+\bar S^{\prime})(T^{\prime}+\bar T^{\prime})(U^{\prime}+\bar U^{\prime})+4\Upsilon{\Omega^{(1)}}^{\prime}+O(\Upsilon^2).$$ where $$\Omega^{(1)\prime}=\frac{1}{256\pi}\left[4ln[\vartheta_2(iS^{\prime})\vartheta_2(iT^{\prime})\vartheta_2(iU^{\prime})]+c.c\right]+\frac{1}{126\pi}\left[ln[(S^{\prime}+\bar S^{\prime})(T^{\prime}+\bar T^{\prime})(U^{\prime}+\bar U^{\prime})]\right].$$ One can check that the leading and the $\Upsilon$ dependent terms are separately invariant under the duality transformation(\[dualityprime\]). Discussion ========== It has been demonstrated in [@0808.2627; @9906094; @0601108] that the nonholomorphic corrections are necessary in order to obtain duality invariant expressions for the free energy and the entropy function. But at the same time, our present method [@0808.2627] of incorporating such corrections is manifestly in conflict with the special geometry properties of the moduli space. The moduli fields mix with their complex conjugates under duality transformation and moreover these transformations involve powers of $\Upsilon$. In this note we have shown that it is possible to find a set of new variables perturbatively in powers of $\Upsilon$ which have the usual tree-level duality transformation (\[dualityprime\]). We have demonstrated their existence upto $\Upsilon^2$ order in the STU-model, but it seems likely that one can find such variables in any model and to any order in $\Upsilon$. Furthermore, these primed varibles are more natural from the point of view of string theory and topological string. The reason is that the partition function for the STU- model propsed in [@0711.1971] involves the moduli which transform in the usual manner under the S-T-U duality transformation (\[dualityprime\]). The primed variables are also more natural in the context of topological string because the integrability of the holomorphic anamoly equations depends on the special geometry properties of the moduli space[@9309140]. Furthermore in these new varibles the expression of the free energy ,at each order in $\Upsilon$, is invariant under tree level duality transformation (\[dualityprime\]) and hence is the natural candidate to compare with the topological string result for the free energy [@0702187].\ [Acknowledgement:]{} We would like to thank Arjun Bagchi, Justin David, Dileep Jatkar and Ashoke Sen for various useful discussion. We would specially like to thank Ashoke Sen for taking intrest in our work from beginning and also useful comments on the first draft. [99]{} T. Mohaupt, “Black hole entropy, special geometry and strings,” Fortsch. Phys. [49]{}, 3 (2001) \[arXiv:hep-th/0007195\]. G. L. Cardoso, B. de Wit and S. Mahapatra, “Subleading and non-holomorphic corrections to N=2 BPS black hole entropy,” JHEP [0902]{}, 006 (2009) \[arXiv:0808.2627 \[hep-th\]\]. G. L. Cardoso, J. R. David, B. de Wit and S. Mahapatra, “The mixed black hole partition function for the STU model,” JHEP [0812]{}, 086 (2008) \[arXiv:0810.1233 \[hep-th\]\]. G. Lopes Cardoso, B. de Wit and T. Mohaupt, “Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes,” Nucl. Phys. B [567]{}, 87 (2000) \[arXiv:hep-th/9906094\]. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, “Black hole partition functions and duality,” JHEP [0603]{}, 074 (2006) \[arXiv:hep-th/0601108\]. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, “Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy,” JHEP [0412]{}, 075 (2004) \[arXiv:hep-th/0412287\]. J. R. David, “On the dyon partition function in N=2 theories,” JHEP [0802]{}, 025 (2008) \[arXiv:0711.1971 \[hep-th\]\] T. W. Grimm, A. Klemm, M. Marino and M. Weiss, “Direct integration of the topological string,” JHEP [0708]{}, 058 (2007) \[arXiv:hep-th/0702187\]. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes,” Commun. Math. Phys. [165]{}, 311 (1994) \[arXiv:hep-th/9309140\]. B. de Wit, G. Lopes Cardoso, D. Lust, T. Mohaupt and S. J. Rey, “Higher-order gravitational couplings and modular forms in N = 2, D = 4 heterotic string compactifications,” Nucl. Phys. B [481]{}, 353 (1996) \[arXiv:hep-th/9607184\]. A. Sen, “How does a fundamental string stretch its horizon?,” JHEP [0505]{}, 059 (2005) \[arXiv:hep-th/0411255\] A. Sen and C. Vafa, “Dual pairs of type II string compactification,” Nucl. Phys. B [455]{}, 165 (1995) \[arXiv:hep-th/9508064\].
--- abstract: 'In this paper we prove the existence of infinitely many small energy solution of a semilinear $Schr\ddot{o}dinger$ equation via the dual form of the generalized fountain theorem. This equation is with periodic potential and concave-convex nonlinearities.' author: - \| Long-Jiang Gu, Hong-Rui Sun$\thanks{Corresponding author}$ [^1]\ \ [School of Mathematics and Statistics, Lanzhou University,]{}\ [Lanzhou, Gansu 730000, P.R. China ]{}\ title: 'Infinitely many small energy solutions of a semilinear $\mathbf{Schr\ddot{o}dinger}$ equation ' --- Introduction ============ In recent years, strongly indefinite problems have attracted many authors’ attention. Early in 1998, Kryszewski and Szulkin built the generalized linking theorem which is a powerful tool to study the strongly indefinite problems, see chapter 6 of [@MW] and [@SK]. Using the similar method Batkam and Colin built the generalized fountain theorem and its dual form in[@BC1; @BC2; @BC4; @BC5] which may be used to find infinitely many large and small energy solutions of strongly indefinite problems. In [@He] (see[@G]as well)£¬ Barstch and Willem firstly studied the elliptic equation with concave and convex nonlinearities and proved the exintence of infinitely many small energy solutions via the dual fountain theorem. In [@BC4] the authors discussed the strongly indefinite elliptic systems with concave and convex nonlinearities defined on a bounded domain. A natural question is if we can get similar results to $Schr\ddot{o}dinger$ equation with periodic potential. At this time the problem may be strongly indefinite and there does not exist the embedding from $H^{1}(\mathbb{R}^{N})$ to $L^{q}(\mathbb{R}^{N})$ when $1<q<2$. So we discuss the following equation $$\label{P} \left\{ \begin{array}{ll} -\Delta u + V(x)u=g(x)\| u\|^{q-2}u-h(x)\| u\|^{p-2}u,\\ u\in H^{1}(\mathbb{R}^{N}) , N\geq3 , \end{array} \right.$$ whose nonlinearity is with a weight and the weight is of appropriate attenuation such that it may bring us the embeddings we need. In quantum mechanics, the $Schr\ddot{o}dinger$ equation is used to depict the motion law of microscopic particles. The nonlinearity of the $Schr\ddot{o}dinger$ equation means the interaction of two particles. And $g(x)>0$ means the interaction of two particles mainly acts as attraction when the energy of the particles is small. The weight $g(x)$ looks like a permittivity which means the medium in the space is not so well-distributed and thus leads to that the attraction between two particles becomes weak when they are far away from the origin. The existence of nontrivial solutions shows that these two particles will concentrate at where they can attract each other and form a stable state. In both [@SK] and [@BC2] the authors demanded the nonlinearity to be 1-periodic as well, this condition make the energy functional to be invariant under the action of group $\mathcal{Z}_{N}$ thus is of benefit to the proof of the nontriviality of solutions. However, in our work we can not expect the nonlinearity with such a weight to be periodic and we get the nontriviality of solutions through the $(PS)_{c}$ condition if the weight $h(x)$ attenuates quickly enough so that it provides us the compact embedding. If $h(x)$ attenuates slowly we use a variant concentrating-compactness lemma noticing that the solution can not concentrate at $\infty$ when the weight vanishes at $\infty$. Preliminary =========== In this section we first introduce the abstract critical point theorems which we will need. Let Y be a closed subspace of a separable Hilbert space X endowed with the usual inner product $(\cdot)$ and the associated norm $\\| \cdot \\|$. We denote by P :$ X$ $\longrightarrow$ $Y$ and $Q : X \longrightarrow$ $Z = Y^{\perp}$ the orthogonal projections. We fix an orthonormal basis $\{e_{j} \}_{j\geq0}$ of $Y$ and an orthonormal basis $\{f_{j} \}_{j\geq0}$ of $Z$, and we consider on $X$ = $Y \oplus Z$ the $\tau$- topology introduced by Kryszewski and Szulkin in [@SK] that is, the topology associated to the following norm $\\| u \\|_{\tau}:= max\{\sum_{j=0}^{\infty}\frac{1}{2^{j+1}}\|(Pu,e_{j})\|, \\| Qu \\|\}$ , $u\in X$ . It is easy to see that if $\{u_{n}\}\subset X$ is a bounded sequence, then $u_{n}\rightarrow u$ in $\tau$- topology $\Leftrightarrow$ $Pu_{n}\rightharpoonup Pu$ and $Qu_{n}\rightarrow Qu$. For readers’ convenience, we recall the following well-known definitions. Let $\varphi \in C^{1}(X,\mathbb{R})$ and $c \in \mathbb{R}$. 1. $\varphi$ is $\tau$-upper$($resp.$\tau$-lower$)$semicontinuous if for every $C\in\mathbb{R}$ the set {$u\in X$;$\varphi(u)\geq C$}$($resp.{$u\in X$;$\varphi(u)\leq C$}$)$ is $\tau$-closed. 2. $\varphi'$ is weakly sequentially continuous if the sequence $\{\varphi'(u_{n})\}$ converges weakly to $\varphi'(u)$ whenever $\{u_{n}\}$ converges weakly to u in X. 3. $\varphi$ satisfies the Palais-Smale condition((PS) condition for short) if any sequence $\{u_{n}\}\subset X$ such that $\{\varphi(u_{n})\}$ is bounded and $\varphi'(u_{n})\rightarrow 0$, has a convergent subsequence. 4. $\varphi$ is said to satisfy the Palais-Smale condition at level c ($(PS)_{c}$ condition for short) if any sequence $\{u_{n}\}\subset X$ such that $\varphi(u_{n})\rightarrow c$ and $ \varphi'(u_{n})\rightarrow 0$ has a convergent subsequence. Now we introduce the dual form of the generalized fountain theorem built by Batkam and Colin in [@BC4]. This theorem is the key to find small energy solutions. We adopt the following notations: $Y^{k}:=\overline{\bigoplus_{j=k}^{\infty}\mathbb{R}e_{j}}$ and $Z^{k}:=(\bigoplus_{j=0}^{k}\mathbb{R}e_{j})\bigoplus Z$ . Let $\varphi \in C^{1}(X,\mathbb{R})$ be an even functional which is $\tau$-lower semicontinuous and such that $\varphi'$ is weakly sequentially continuous. If there exist a $k_{0}>0$ such that for every $k\geq k_{0}$ there exists $\sigma_{k}>s_{k}>0$ such that:\ $(B_{1})$ $a^{k}:=inf_{u\in Y^{k},\\|u\\|=\sigma_{k}}\varphi(u)\geq 0$,\ $(B_{2})$ $b^{k}:=sup_{u\in Z^{k},\\|u\\|=s_{k}}\varphi(u)<0$,\ $(B_{3})$ $d^{k}:=inf_{u\in Y^{k},\\|u\\|\leq \sigma_{k}}\varphi(u)\rightarrow 0,k\rightarrow\infty$.\ Then there exists a sequence $\{u_{k}^{n}\}$ such that $\varphi'(u_{k}^{n})\rightarrow 0$ and $\varphi(u_{k}^{n})\rightarrow c_{k}$ as $n\rightarrow\infty$, where $c_{k}\rightarrow0$. In order to apply these abstract theory to elliptic systems restrict to a bounded domain $\Omega\subset\mathbb{R}^{n}$ when there is the compact embeddings, Batkam and Colin introduced the theorems with $(PS)$ condition [@BC4] Thm 6 . Under the assumptions of Theorem 2.1, $\varphi$ satisfies in addition:\ $(B_{4})$ $\varphi$ satisfies the $(PS)_{c}$condition, for all $ c \in [d^{k_{0}},0]$. Then $\varphi$ has a sequence of critical points $\{u_{k}\}$ such that $\varphi(u_{k})<0$ and $\varphi(u_{k})\rightarrow 0$ as $k\rightarrow\infty$. Throughout this paper, $H^{1}(\mathbb{R}^{N})$ is the standard Sobolev space with the norm $\\| u \\|=\int_{\mathbb{R}^{N}} (\|u \|^{2} + \| \nabla u\|^{2}) dx$. $L^{p}(a(x),\mathbb{R}^{N})$ is the Lebesgue space with positive weight $a(x)$ endowed with the norm $\\| u \\|_{L^{p}(a(x),\mathbb{R}^{N})}:= (\int_{\mathbb{R}^{N}}\| u \|^{p} a(x) dx)^{\frac{1}{p}}$. By $B(x,r)$ we denote the ball centered at $x$ with radius $r$ . And the positive constants whose exact value are not important will be denoted by $C$ only. Main results ============ In this section, we discuss the existence of infinitely many small energy solutions of problem (1.1) with the dual form of the generalized fountain theorem and our basic assumptions are: $(H_{1})$ : $1<q<2<p<2^{\ast}$, where $2^{\ast}=\frac{2N}{N-2}$. $(H_{2})$ : The function $V(x):\mathbb{R}^{N}\rightarrow \mathbb{R}$ is continuous and 1-periodic in $x_{1},...,x_{N}$ and 0 lies in a gap of the spectrum of $-\Delta+V$. $(H_{3})$ : $g\in L^{q_{0}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ with $g(x)>0$, a.e. in $\mathbb{R}^{N}$, where $q_{0}=\frac{2N}{2N-qN+2q}$. $(H_{4})$ : $h\in L^{p_{0}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ with $h(x)\geq0$, a.e. in $\mathbb{R}^{N}$, where $p_{0}=\frac{2N}{2N-pN+2p}$. $(H_{4})'$ : $h(x)\in L^{\infty}(\mathbb{R}^{N})$ with $h(x)\geq0$, and $h(x)\rightarrow0$ as $\|x\|\rightarrow\infty$. $(H_{3})$ and $(H_{4})$ mean that the weight of the nonlinearity is of appropriate attenuation so that we can get continuous and compact embeddings from $H^{1}(\mathbb{R}^{N})$ to the Lebesgue space with weight. The condition $(H_{4})'$ is weaker than $(H_{4})$. In fact we may construct an example as $h(x)=\frac{1}{log\|x\|}$, $\|x\|>R$, for some $R>0$. The nonlinearity of the $Schr\ddot{o}dinger$ equation means the interaction of two particles. And $g(x)>0$ means the interaction of two particles mainly acts as attraction when the energy of the particles is small. The weight $g(x)$ looks like a permittivity which means the medium in the space is not so well-distributed and thus leads to that the attraction between two particles becomes weak when they are far away from the origin. So our conclusion shows that these two particles will concentrate at where they can attract each other. Let us introduce the variational setting first and for more information we refer the readers to [@SK; @BC1; @BC2]. We define a functional $\varphi$ on $H^{1}(\mathbb{R}^{N})$ as $$\varphi(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}( \| \nabla u\|^{2}+V(x)\| u \|^{2}) dx - \frac{1}{q}\int_{\mathbb{R}^{N}} g(x)\| u \| ^{q}dx +\frac{1}{p}\int_{\mathbb{R}^{N}} h(x)\| u \| ^{p}dx.$$ It is easy to see from condition $(H_{1})$, $(H_{2})$, $(H_{3})$ and $(H_{4})$ or $(H_{4}')$ that $\varphi (u)$ is well defined and is of class $\mathcal{C}^{1}$, then its critical points are weak solutions of $(3.1)$. Moreover, any $u\in H^{1}(\mathbb{R}^{N})$ satisfies that if $\phi\in H^{1}(\mathbb{R}^{N})$ then there holds $$\langle \varphi' (u),\phi\rangle=\int_{\mathbb{R}^{N}}\nabla u\nabla\phi dx +\int_{\mathbb{R}^{N}}V(x)u \phi dx-\int_{\mathbb{R}^{N}}g(x)\| u \|^{q-2}u\phi dx +\int_{\mathbb{R}^{N}}h(x)\| u \|^{p-2}u\phi dx.$$ By $(H_{2})$, the $Schr\ddot{o}dinger$ operator $-\Delta+V(x)$ in $L^{2}(\mathbb{R}^{N})$ has purely continuous spectrum, and the space $H^{1}(\mathbb{R}^{N})$ can be decomposed into $H^{1}(\mathbb{R}^{N})=Y\bigoplus Z$ such that the quadratic form: $$u\in H^{1}(\mathbb{R}^{N}) \rightarrow \int_{\mathbb{R}^{N}} ( \| \nabla u\|^{2}+V(x)\| u \|^{2}) dx$$ is positive and negative definite on $Y $and $ Z$ respectively and both $Y$ and $Z$ are infinite-dimensional. Now let $L: H^{1}(\mathbb{R}^{N})\rightarrow H^{1}(\mathbb{R}^{N})$ be the self-adjoint operator defined by $$(Lu,v)_{1}:=\int_{\mathbb{R}^{N}}(\nabla u \nabla v +V(x)uv)dx ,$$ where $( \cdot )_{1}$ is the usual inner product in $H^{1}(\mathbb{R}^{N})$. We denote $P :X\rightarrow Y$ and $Q :X \rightarrow Z $ the orthogonal projections, and thus we can introduce a new inner product which is equivalent to $(\cdot)_{1}$ by the formula $(u,v):=(L(Qu-Pu),v)_{1}$ , $u,v \in X$ and in this section $\\| \cdot\\|$ denotes the corresponding norm $\\| u \\|:=(u,u)^{\frac{1}{2}}$ . Since the inner products $ (\cdot)$ and $(\cdot)_{1}$ are equivalent, $Y$ and $Z$ are also orthogonal with respect to $ (\cdot)$. One can verify easily that $(3.2)$ reads $$\varphi(u):=\frac{1}{2}(\\| Pu \\|^{2}-\\| Qu \\|^{2}) - \frac{1}{q}\int_{\mathbb{R}^{N}} g(x)\| u \| ^{q}dx +\frac{1}{p}\int_{\mathbb{R}^{N}} h(x)\| u \| ^{p}dx.$$ In this section, we set $Y_{k}:=\overline{\bigoplus^{\infty}_{j=k}e_{j}}$ and $ Z_{k}:=[\bigoplus_{j=0}^{k}e_{j}]\bigoplus Z$, where $\{e_{j}\}_{j\geq0}$ is an orthonormal basis of $(Y,\parallel\cdot\parallel)$.\ Our main results in this section are Assume that the conditions $(H_{1})$,$(H_{2})$,$(H_{3})$ and $(H_{4})$ hold. Then problem $(1.1)$ has a sequence of nontrival solutions {$u_{k}$} with $\varphi(u_{k})< 0$, and $\varphi(u_{k})\rightarrow 0$ as $k\rightarrow \infty$ . Assume that the conditions $(H_{1})$,$(H_{2})$,$(H_{3})$ and $(H_{4}')$ hold. Then problem $(1.1)$ has a sequence of nontrival solutions {$u_{k}$} with $\varphi(u_{k})< 0$, and $\varphi(u_{k})\rightarrow 0$ as $k\rightarrow \infty$ . First let us prove the following lemmas. Assume that $1<q<2^{\ast}$ and $g\in L^{q_{0}}(\mathbb{R}^{N}) \bigcap L^{\infty}(\mathbb{R}^{N})$ with $g(x)\geq 0$ a.e. in $\mathbb{R}^{N}$, where $q_{0}=\frac{2N}{2N-qN+2q}$. Then $H^{1}(\mathbb{R}^{N})\hookrightarrow L^{q}(g(x),\mathbb{R}^{N}) $ and the embedding is compact. For $u\in H^{1}(\mathbb{R}^{n})$, from the $H\ddot{o}lder$ inequality and Sobolev inequality we have $$\begin{aligned} \int_{\mathbb{R}^{N}}g(x)\| u \|^{q} dx &\leq& \| g(x)\|_{L^{q_{0}}}\cdot \left(\int_{\mathbb{R}^{N}}\| u \|^{\frac{2N}{N-2}} dx\right)^{\frac{qN-2q}{2N}}\\ &=& \| g(x)\|_{L^{q_{0}}}\cdot \| u \| ^{q}_{L^{2^{\ast}}(\mathbb{R}^{N})} \\ &\leq& C \\| u \\|^{q}.\end{aligned}$$ So we have $$\| u \|_{L^{q}(g(x),\mathbb{R}^{N})}\leq C \\| u \\| ,$$\ which means that $$H^{1}(\mathbb{R}^{N})\hookrightarrow L^{q}(g(x),\mathbb{R}^{N}).$$ Assume that $\{u_{n}\}$ is a bounded sequence in $H^{1}(\mathbb{R}^{n})$, so it is bounded in $L^{2^{\ast}}(\mathbb{R}^{n})$ and there exists a weak convergent subsequence denoted by $\{u_{n}\}$ also, according to the Rellich imbedding theorem, when restrict to a bounded domain $\Omega$, $\{u_{n}\}$ is strongly convergent in $L^{q}(\Omega)$ . We choose $R>0$ sufficiently large such that for $\varepsilon >0$, there exists $ M>0 $ such that if $m,n>M$, we have $$\int_{B(0,R)} \| u_{n}-u_{m}\|^{q} dx <\frac{\varepsilon}{2\| g\|_{L^{\infty}}}$$ and $$\left(\int_{\mathbb{R}^{N} \backslash B(0,R)}\| g(x) \|^{q_{0}} dx\right)^{\frac{1}{q_{0}}} <\frac{\varepsilon}{4 sup_{n>0} \{\| u_{n}\|_{L^{2^{\ast}}(\mathbb{R}^{N})} \} }.$$ So, $$\begin{aligned} \int_{\mathbb{R}^{N}} g(x)\| u_{n}-u_{m}\|^{q}dx &=& \int_{\mathbb{R}^{N}\backslash B(0,R)} g(x)\| u_{n}-u_{m}\|^{q}dx +\int_{B(0,R)} g(x)\| u_{n}-u_{m}\|^{q}dx \\ & <& \left(\int_{\mathbb{R}^{N} \backslash B(0,R)}\| g(x) \|^{q_{0}} dx\right)^{\frac{1}{q_{0}}} \| u_{n}-u_{m}\|_{L^{2^{\ast}}(\mathbb{R}^{N})}\\ & & + \| g\|_{L^{\infty}(\mathbb{R}^{N})} \int_{B(0,R)} \| u_{n}-u_{m}\|^{q} dx \\ & <& \frac{\varepsilon}{2} + \frac{\varepsilon}{2} \\ & = & \varepsilon .\end{aligned}$$ This means that $\{u_{n}\}$ is a Cauchy sequence in $L^{q}(g(x),\mathbb{R}^{N})$, thus complete the proof. Under the assumptions of Lemma 3.1, we define:\ $$\beta_{k}:=sup_{u\in Y_{k},\\| u \\|=1} \| u \|_{L^{q}(g(x),\mathbb{R}^{N})} ,$$\ then $$\beta_{k}\rightarrow 0 , k\rightarrow\infty.$$ It is clear that $0<\beta_{k+1}\leq\beta_{k}$, so that $\beta_{k} \rightarrow\beta \geq 0$,$k \rightarrow \infty$. For every $k\geq0$, there exists $u_{k}\in Z_{k}$ such that $\\| u_{k} \\|=1 $ and $\| u_{k}\|_{L^{q}(g(x),\mathbb{R}^{N})}>\frac{\beta_{k}}{2}$. By the definition of $Y_{k}$, $u_{k}\rightharpoonup 0$ in $H^{1}(\mathbb{R}^{N})$. Thus Lemma 3.1 implies that $u_{k}\rightarrow 0$ in $L^{q}(g(x),\mathbb{R}^{N})$. So we proved that $\beta_{k}\rightarrow0$ as $k\rightarrow\infty$. Under the assumptions $(H_{1})$ $(H_{2})$ $(H_{3})$ and $(H_{4})$ or $(H_{4})'$, the functional $\varphi$ defined in $(3.1)$ (or $(3.5)$) is $\tau-lower$ semicontinuous, and $\varphi'$ is weakly sequentially continuous. Let $\{{u_{n}}\} \subset X$ and $c\in \mathbb{R}$ such that : $ u_{n}\rightarrow u$ in $\tau-topology$ and $\varphi(u_{n})\leq c$. We write $u_{n}=y_{n}+z_{n}$ , where $y_{n} \in Y ,z_{n} \in Z $. From the definition of $\tau-topology$, we can see that $z_{n}\rightarrow z$. $$\begin{aligned} c\geq \varphi(u_{n}) &=& \frac{1}{2}\\| y_{n}\\|^{2}-\frac{1}{2}\\| z_{n}\\|^{2} -\frac{1}{q}\| u_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})} + \frac{1}{p}\| u_{n}\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}\\ &\geq & \frac{1}{2}\\| y_{n}\\|^{2}-\frac{1}{2}\\| z_{n}\\|^{2} -\frac{1}{q}\| u_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}.\end{aligned}$$\ Now we use the Jensen inequality $$\begin{aligned} \| u_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})} &=& \| y_{n} + z_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})} \\ &\leq& ( \| y_{n}\|_{L^{q}(g(x),\mathbb{R}^{N})}+\| z_{n}\|_{L^{q}(g(x),\mathbb{R}^{N})})^{q} \\ &\leq& 2^{q-1} (\| y_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}+\| z_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}),\end{aligned}$$ with the embedding $ H^{1}(\mathbb{R}^{N}) \hookrightarrow L^{q}(g(x),\mathbb{R}^{N})$, we have $$\begin{aligned} c &\geq& \frac{1}{2}\\| y_{n}\\|^{2}-\frac{1}{2}\\| z_{n}\\|^{2} -\frac{2^{q-1}}{q} (\| y_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}+\| z_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})})\\ &\geq& \frac{1}{2}\\| y_{n}\\|^{2}-\frac{1}{2}\\| z_{n}\\|^{2} - \frac{2^{q-1}}{q} (\\| y_{n}\\|^{q}+\\| z_{n}\\|^{q}). \end{aligned}$$ We can see that $\{{y_{n}}\}$ is also bounded, thus $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ . So there exists a subsequence and we also denote it by $\{u_{n}\}$ such that $u_{n}\rightharpoonup u $. From Lemma 3.1 we have $$\| u_{n}\|_{L^{q}(g(x),\mathbb{R}^{n})} \rightarrow \| u\|_{L^{q}(g(x),\mathbb{R}^{n})},$$ and with the weakly lower semicontinuity of the norm, we have $$c \geq \varphi (u),$$ thus $\varphi$ is $\tau-lower$ semicontinuous . Now let us prove that $ \varphi'$ is weakly sequentially continuous. Assume that $u_{n}\rightharpoonup u $ in $H^{1}(\mathbb{R}^{N})$. Then $u_{n}\rightarrow u$ in $L^{2}_{loc}(\mathbb{R}^{N})$ By $(H_{3})$, $(H_{4})$ and $H\ddot{o}lder$ inequality, we can get that $\{\varphi'(u_{n})\}$ is bounded, so that $\varphi'(u_{n}) \rightharpoonup \varphi'(u) $. When the condition $(H_{4})'$ holds we can not get the (PS) condition. In order to prove that the solution with nonzero energy must be nontrivial, we need the following concentration-compactness lemma, and let us show the interpolation inequality for $L^{p}(g(x),\mathbb{R}^{N})$ first. $($interpolation inequality for $L^{p}(g(x),\mathbb{R}^{N})$ $)$Assume that $1\leq s \leq r\leq t\leq\infty$, $g \in L^{\infty}(\mathbb{R}^{N})$ with $g(x)\geq 0$ a.e. in $\mathbb{R}^{N}$ and $\frac{1}{r}=\frac{\theta}{s}+\frac{1-\theta}{t}$. If $ u \in L^{s}(g(x),\mathbb{R}^{N})\bigcap L^{t}(g(x),\mathbb{R}^{N})$, then $ u \in L^{r}(g(x),\mathbb{R}^{N})$, and $$\| u\|_{L^{r}(g(x),\mathbb{R}^{N})}\leq\| u\|^{\theta}_{L^{s}(g(x),\mathbb{R}^{N})} \| u\|^{1-\theta}_{L^{t}(g(x),\mathbb{R}^{N})}.$$ Using $H\ddot{o}lder$ inequality, we have $$\begin{aligned} \| u\|^{r}_{L^{r}(g(x),\mathbb{R}^{N})}&=& \int_{\mathbb{R}^{N}}g(x)\| u\|^{r}dx\\ &=& \int_{\mathbb{R}^{N}}(g(x)^{\frac{\theta r}{s}}\| u\|^{\theta r})(g(x)^{\frac{(1-\theta) r}{t}}\| u\|^{(1-\theta) r})dx\\ &\leq& (\int_{\mathbb{R}^{N}}g(x)\| u\|^{s}dx)^{\frac{\theta r}{s}} (\int_{\mathbb{R}^{N}}g(x)\| u\|^{t}dx)^{\frac{(1-\theta)r}{t}}\\ &=& \| u\|^{\theta r}_{L^{s}(g(x),\mathbb{R}^{N})} \| u\|^{(1-\theta)r}_{L^{t}(g(x),\mathbb{R}^{N})}.\end{aligned}$$ The idea of the following lemmas come from P.L.Lions (see chapter1 of[@MW])and we will prove it completely for readers’ convenience. (concentration-compactness) Let $r>0$ and $2\leq q<2^{\ast}$.If $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, and if $sup_{y\in\mathbb{R}^{N}}\int_{B(y,r)}\| u_{n}\|^{q} dx \rightarrow 0 $ as $n \rightarrow \infty$, $g\in L^{q_{0}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ with $g(x)\geq0$, where $q_{0}=\frac{2N}{2N-qN+2q}$. Then $u_{n}\rightarrow 0$ in $L^{p}(g(x),\mathbb{R}^{N})$, for $max\{1,\frac{(N-2)q}{N}\}<p<2^{\ast}$. Let $q<s<2^{\ast}$, using the interpolation lemma we have $$\begin{aligned} \| u\|_{L^{s}(B(y,r))} &\leq& \| u\|^{1-\theta}_{L^{q}(B(y,r))} \| u\|^{\theta}_{L^{2^{\ast}}(B(y,r))} \\ &\leq& C \| u\|^{1-\theta}_{L^{q}(B(y,r))} [\int_{B(y,r)}(\| u\|^{2}+\| \nabla u \|^{2}) dx]^{\frac{\theta}{2}},\end{aligned}$$ where $\theta=\frac{s-q}{2^{\ast}-q}\frac{2^{\ast}}{s}$. Choosing $s=\frac{2}{\theta}$ and it is easy to see that this $s$ is valid. So we get $$\int_{B(y,r)}\| u \|^{s} dx \leq C \| u\|^{(1-\theta)s}_{L^{q}(B(y,r))} \int_{B(y,r)}(\| u\|^{2}+\| \nabla u \|^{2}) dx.$$ Now covering $\mathbb{R}^{N}$ by balls of radius $r$, in such a way that each point of $\mathbb{R}^{N}$ is contained in at most $N+1$ balls, we find that $$\int_{\mathbb{R}^{n}}\| u \|^{s} dx \leq C(N+1)\int_{\mathbb{R}^{n}}(\| u\|^{2}+\| \nabla u \|^{2})dx\cdot sup_{y\in\mathbb{R}^{n}}\left(\int_{B(y,r)}\| u \|^{q} \right)^{\frac{(1-\theta)s}{q}} .$$ Thus $u_{n}\rightarrow 0$ in $L^{s}(\mathbb{R}^{N})$. And for $g\in L^{\infty}(\mathbb{R}^{N})$, we have $u_{n}\rightarrow 0$ in $L^{s}(g(x),\mathbb{R}^{N})$. For $s<p<2^{\ast}$,by the preceding lemma we have $$\| u\mid_{L^{p}(g(x),\mathbb{R}^{n})} \leq \| u\mid^{\alpha}_{L^{s}(g(x),\mathbb{R}^{n})} \| u\|^{1-\alpha}_{L^{2^{\ast}}(g(x),\mathbb{R}^{N})} < C \| u\|^{\alpha}_{L^{s}(g(x),\mathbb{R}^{N})} \| u\|^{1-\alpha}_{L^{2^{\ast}}(\mathbb{R}^{N})},$$ where $\alpha=\frac{(2^{\ast}-p)s}{(2^{\ast}-s)p}$. So we have $u_{n}\rightarrow 0$ in $L^{p}(g(x),\mathbb{R}^{N})$ , when $s<p<2^{\ast}$. For $max\{1,\frac{(N-2)q}{N}\}<p<s$, we choose $t\in(max\{1,\frac{(N-2)q}{N}\},s)$. Similarly we have $$\begin{aligned} \| u\|_{L^{p}(g(x),\mathbb{R}^{N})} &\leq& \| u\|^{\beta}_{L^{t}(g(x),\mathbb{R}^{N})} \| u\|^{1-\beta}_{L^{s}(g(x),\mathbb{R}^{N})} \\ &=& \left(\int_{\mathbb{R}^{N}} g(x)\| u \|^{t} dx\right)^{\frac{\beta}{t}} \| u\|^{1-\beta}_{L^{s}(g(x),\mathbb{R}^{n})}\\ & \leq & \left(\| g(x) \|_{L^{q_{0}}} \int_{\mathbb{R}^{N}}\| u \|^{t\frac{2N}{qN-2q}}dx\right)^{\frac{\beta}{t}} \| u\|^{1-\beta}_{L^{s}(g(x),\mathbb{R}^{N})}\\ & \leq & \left(\| g(x) \|_{L^{q_{0}}} \\| u \\|^{\frac{qN-2q}{2Nt}}\right)^{\frac{\beta}{t}} \| u\|^{1-\beta}_{L^{s}(g(x),\mathbb{R}^{N})},\end{aligned}$$ where $\beta=\frac{(s-p)t}{(s-t)p}.$ So we have $u_{n}\rightarrow 0$ in $L^{p}(g(x),\mathbb{R}^{n})$ , when $max\{1,\frac{(N-2)q}{N}\}<p<s$. Thus the proof is complete.\ Let $r>0$, $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, $g\in L^{q_{0}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ with $g(x)>0$, where $q_{0}=\frac{2N}{2N-qN+2q}$. For any $\varepsilon>0$ there exists a positive $R_{\varepsilon}<\infty$ such that if $sup_{\|y\|<R_{\varepsilon}}\int_{B(y,r)}\| u_{n}\|^{2} dx \rightarrow 0 $ as $n \rightarrow \infty$. Then $lim_{n\rightarrow\infty}\|u_{n}\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}<\varepsilon$, for $1<q<2$. For any $\varepsilon>0$, we can get from $H\ddot{o}lder$ inequality and the boundedness of $\{u_{n}\}$ that there exists a positive $R_{\varepsilon}<\infty$ such that $$\int_{\mathbb{R}^{N}\backslash B(0,R_{\varepsilon})}g(x)\|u_{n}\|^{q}dx<\frac{\varepsilon}{2}.$$ From Lemma 3.5 we can see easily that if $sup_{\|y\|<R_{\varepsilon}}\int_{B(y,r)}\| u_{n}\|^{2} dx \rightarrow 0$, as $n \rightarrow \infty$, $$lim_{n\rightarrow\infty} \int_{B(0,R_{\varepsilon})}g(x)\|u_{n}\|^{q}dx<\frac{\varepsilon}{2}.$$ So $$lim_{n\rightarrow\infty} \int_{\mathbb{R}^{N}}g(x)\|u_{n}\|^{q}dx<\varepsilon.$$ Let $r>0$, $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, $h\in L^{\infty}(\mathbb{R}^{N})$ with $h(x)\geq0$, and $h(x)\rightarrow0$ as $\|x\|\rightarrow\infty$. For any $\varepsilon>0$ there exists a positive $R_{\varepsilon}<\infty$ such that if $sup_{\|y\|<R_{\varepsilon}}\int_{B(y,r)}\| u_{n}\|^{2} dx \rightarrow 0 $ as $n \rightarrow \infty$. Then $lim_{n\rightarrow\infty}\|u_{n}\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}<\varepsilon$, for $2<p<2^{\ast}$. For any $\varepsilon>0$, we can get from the boundedness of $\{u_{n}\}$ that there exists a positive $R_{\varepsilon}<\infty$ such that $$\int_{\mathbb{R}^{N}\backslash B(0,R_{\varepsilon})}h(x)\|u_{n}\|^{p}dx<\frac{\varepsilon}{2}.$$ From the proof of Lemma 3.5 we can see that if $sup_{\|y\|<R_{\varepsilon}}\int_{B(y,r)}\| u_{n}\|^{2} dx \rightarrow 0$, as $n \rightarrow \infty$, $$lim_{n\rightarrow\infty} \int_{B(0,R_{\varepsilon})}h(x)\|u_{n}\|^{p}dx<\frac{\varepsilon}{2}.$$ So $$lim_{n\rightarrow\infty} \int_{\mathbb{R}^{N}}h(x)\|u_{n}\|^{p}dx<\varepsilon.$$ Now we are able to prove Theorem 3.1 . \ First let us verify the conditions :\ $(B_{1})$ $a^{k}:=inf_{u\in Y^{k},\\|u\\|=\sigma_{k}}\varphi(u)\geq 0$,\ $(B_{2})$ $b^{k}:=sup_{u\in Z^{k},\\|u\\|=s_{k}}\varphi(u)<0$,\ $(B_{3})$ $d^{k}:=inf_{u\in Y^{k},\\|u\\|\leq \sigma_{k}}\varphi(u)\rightarrow 0,k\rightarrow\infty$.\ \ We write $u=y+z$, where $y \in Y ,z \in Z $. For every $u\in Y_{k}$, $y=u, z=0$.\ From Lemma 3.1 and Lemma 3.2 we can see that $$\begin{aligned} \varphi(u) &=&\frac{1}{2}(\\| y \\|^{2}-\\| z \\|^{2}) - \frac{1}{q}\int_{\mathbb{R}^{N}} g(x)\|u \| ^{q}dx +\frac{1}{p}\int_{\mathbb{R}^{N}} h(x)\| u \| ^{p}dx \\ &=& \frac{1}{2}\parallel u \parallel^{2} - \frac{1}{q}\| u\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})} +\frac{1}{p} \| u\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}\\ &\geq& \frac{1}{2}\\| u \\|^{2} - \frac{1}{q}\| u\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}\\ &\geq& \frac{1}{2}\\| u \\|^{2}-\frac{1}{q}\beta^{q}_{k}\\| u \\|^{q}.\end{aligned}$$ Let $\sigma_{k}= (\frac{4\beta^{q}_{k}}{q})^{\frac{1}{2-q}}$,we get that $$a^{k}:=inf_{u\in Y^{k},\\|u\\|=\sigma_{k}}\varphi(u)\geq 0,$$ and it is easy to see from Lemma 3.2 that $\sigma_{k}\rightarrow 0$ as $k\rightarrow\infty$. Now for $u\in Z^{k}$, we use the Jensen inequality, $$\begin{aligned} \varphi(u) &=&\frac{1}{2}(\\| y \\|^{2}-\\| z \\|^{2}) - \frac{1}{q}\int_{\mathbb{R}^{N}} g(x)\| u \| ^{q}dx +\frac{1}{p}\int_{\mathbb{R}^{N}} h(x)\|u \| ^{p}dx \\ &=& \frac{1}{2}\\| y \\|^{2}-\frac{1}{2}\\| z \\|^{2} - \frac{1}{q}\| u\|^{q}_{L^{q}(h(x),\mathbb{R}^{N})} +\frac{1}{p} \| u\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}\\ &\leq& \frac{1}{2}\\| y \\|^{2}-\frac{1}{2}\\| z \\|^{2} - \frac{1}{q}\| u\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})} +\frac{2^{p-1}}{p}(\| y\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}+\| z\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}).\end{aligned}$$ Since the Sobolev space $H^{1}(\mathbb{R}^{N})$ embeds continuously in $L^{q}(g(x), \mathbb{R}^{N})$, we denote $E_{k}$ the closure of $Z^{k}$ in $L^{q}(g(x),\mathbb{R}^{N})$, then there exists a continuous projection of $E_{k}$ on $\bigoplus_{j=0}^{k}e_{j}$, thus there exists a constant $C>0$ such that $$\| y\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})}<C\| u\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})},$$ and in a finite-dimensional vector space all norms are equivalent, we have for some $C>0$ $$\\| y \\|^{q}<C \| y\|^{q}_{L^{q}(g(x),\mathbb{R}^{N})},$$ thus $$\begin{aligned} \varphi(u) &<& \frac{1}{2}\\| y \\|^{2}-\frac{1}{2}\\| z \\|^{2} - C\\| y\\|^{q} +\frac{2^{p-1}}{p}(\| y\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}+\| z\|^{p}_{L^{p}(h(x),\mathbb{R}^{N})}) \\ &<& (\frac{1}{2}\\|y\\|^{2}-C\\| y\\|^{q} +C\\| y\\|^{p}) -\frac{1}{2}\\| z\\|^{2} + C \\| z\\|^{p}.\end{aligned}$$ So we choose $s_{k}$ sufficiently small, it is easy to see $$b^{k}:=sup_{u\in Z^{k},\\|u\\|=s_{k}}\varphi(u)<0.$$\ We know that for every $u\in Y_{k}$ $$\varphi(u)\geq-\frac{1}{q}\beta_{k}\\| u \\|^{q}$$ and $$\beta_{k} , \sigma_{k}\rightarrow 0 , k\rightarrow\infty$$ we get $$d^{k}:=inf_{u\in Y^{k},\\|u\\|\leq \sigma_{k}}\varphi(u)\rightarrow 0,k\rightarrow\infty$$ thus condition$(B_{1})$,$(B_{2})$ and $(B_{3})$ of Theorem 2.1 is proved.\ Now let us show that any sequence $\{u_{n}\}$ such that $\varphi(u_{n})\rightarrow c $ and $\varphi'(u_{n})\rightarrow 0$ is bounded in $H^{1}(\mathbb{R}^{N})$. For $n$ big enough, we have $$\begin{aligned} \\| u_{n}\\| -c+1 &>& \frac{1}{2}\langle \varphi'(u_{n}),u_{n}\rangle -\varphi(u_{n})\\ &=& (\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{N}}h(x)\| u\|^{p}dx +(\frac{1}{q}-\frac{1}{2}) \int_{\mathbb{R}^{N}}g(x)\| u\|^{q}dx,\end{aligned}$$ thus $$\label{5} \int_{\mathbb{R}^{N}}h(x)\| u\|^{p}dx<C+\\| u_{n}\\|.$$ $$\\| y_{n}\\| \geq \langle \varphi'(u_{n}),y_{n}\rangle = \\| y_{n}\\|^{2}-\int_{\mathbb{R}^{N}}g(x)\| u\|^{q-2}u y_{n} dx +\int_{\mathbb{R}^{N}}h(x)\| u\|^{p-2}u y_{n} dx,$$ thus $$\\| y_{n}\\|^{2}\leq \\| y_{n}\\|+\int_{\mathbb{R}^{N}}g(x)\| u\|^{q-1} y_{n} dx + \int_{\mathbb{R}^{N}}h(x)\| u\|^{p-1} y_{n} dx.$$ Using $H\ddot{o}lder$ inequality and $(3.5)$ we have, for some $C>0$ $$\begin{aligned} \\| y_{n}\\|^{2} & \leq& \\| y_{n}\\|+ \|g(x)^{\frac{q-1}{q}}u_{n}^{q-1}\|_{L^{\frac{q}{q-1}}} \|g(x)^{\frac{1}{q}}y_{n}^{q-1}\|_{L^{\frac{q}{q-1}}} +\|h(x)^{\frac{p-1}{p}}u_{n}^{p-1}\|_{L^{\frac{p}{p-1}}} \|h(x)^{\frac{1}{p}}y_{n}^{p-1}\|_{L^{\frac{p}{p-1}}}\\ &=& \\| y_{n}\\|+ \| u_{n} \|^{q-1}_{L^{q}(g(x),\mathbb{R}^{N})} \| y_{n} \|_{L^{q}(g(x),\mathbb{R}^{N})} +[\int_{\mathbb{R}^{N}}h(x)\|u\|^{p} dx]^{\frac{p-1}{p}} \|y_{n}\|_{L^{p}(g(x),\mathbb{R}^{N})}\\ &\leq& \\|y_{n}\\| + C\\|u_{n}\\|^{q-1}\\| z_{n}\\| + C(1+\\|u_{n}\\|)^{\frac{p-1}{p}} \\| z_{n}\\|\\ &\leq&\\| u_{n}\\| + C\\| u_{n}\\|^{q} + C(1+\\|u_{n}\\|)^{\frac{p-1}{p}} \\| u_{n}\\|.\end{aligned}$$ Similarly we can get from $\\| z _{n}\\| \geq -\langle \varphi'(u_{n}),z_{n}\rangle$ that $$\begin{aligned} \\| z_{n}\\|^{2} & \leq& \\| z_{n}\\|+ \|g(x)^{\frac{q-1}{q}}u_{n}^{q-1}\|_{L^{\frac{q}{q-1}}}\|g(x)^{\frac{1}{q}}z_{n}^{q-1}\|_{L^{\frac{q}{q-1}}} +\|h(x)^{\frac{p-1}{p}}u_{n}^{p-1}\|_{L^{\frac{p}{p-1}}}\|h(x)^{\frac{1}{p}}z_{n}^{p-1}\|_{L^{\frac{p}{p-1}}}\\ &\leq& \\| u_{n}\\| + C\\| u_{n}\\|^{q} + C (1+\\| u_{n}\\|)^{\frac{p-1}{p}} \\| u_{n}\\|.\end{aligned}$$ For $\\|u_{n}\\|^{2}=\\| y_{n}\\|^{2}+\\| z_{n}\\|^{2}$ , we have $$\\| u_{n}\\|^{2} \leq \\| u_{n}\\| + C\\| u_{n}\\|^{q} + C (1+\\| u_{n}\\|)^{\frac{p-1}{p}} \\| u_{n}\\|,$$ thus $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$.\ We write $u=y+z$ and $u_{n}=y_{n}+z_{n}$, where $y$, $y_{n}\in Y$ and $z$, $z_{n}\in Z$, so that $\langle \varphi'(u_{n})-\varphi'(u),y_{n}-y\rangle\rightarrow 0$ as $n\rightarrow\infty$. By the boundedness of $\{u_{n}\}$, we may assume, up to a subsequence, that $y_{n}\rightharpoonup y$ in $H^{1}(\mathbb{R}^{N}),$ $z_{n} \rightharpoonup z$ in $H^{1}(\mathbb{R}^{N}).$ $$\begin{aligned} \langle \varphi'(u_{n})-\varphi'(u),y_{n}-y\rangle=\\|y_{n}-y\\|^{2} &+& \int_{\mathbb{R}^{N}}g(x)(\|u\|^{q-2}u-\|u_{n}\|^{q-2}u_{n})(y_{n}-y)dx \\ &-& \int_{\mathbb{R}^{N}}h(x)(\|u\|^{p-2}u-\|u_{n}\|^{p-2}u_{n})(y_{n}-y)dx.\end{aligned}$$ Using the $H\ddot{o}lder$ inequality we can get that $y_{n} \rightarrow y$ in $H^{1}(\mathbb{R}^{N}),$ similarly we have $z_{n}\rightarrow z$ in $H^{1}(\mathbb{R}^{N}),$ so $u_{n}\rightarrow u$ in $H^{1}(\mathbb{R}^{N}).$ Thus the $(PS)_{c}$ condition holds for all $c\neq 0$. and we get the conclusion we need from Corollary2.2. \ In order to prove Theorem 3.2, we need only to show that the weak limit of $\{u_{n}\}$ is nontrivial. Now let $\varepsilon=min \{\frac{2\|c\|q}{3(2-q)},\frac{2\|c\|p}{3(p-2)}\} $ and $\delta:=\overline{lim}_{n\rightarrow\infty}sup_{\|y\|<R_{\varepsilon}} \int_{B(y,1)}\| u_{n} \|^{2}dx =0$ we get from Lemma 3.6 and Lemma 3.7 that $$lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}g(x)\| u_{n}\|^{q}dx<\frac{\|c\|}{3},$$ and $$lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}h(x)\| u_{n}\|^{p}dx<\frac{\|c\|}{3}.$$ So $$\begin{aligned} \|c\| &=& lim_{n\rightarrow\infty}\|\varphi(u_{n})-\frac{1}{2}\langle\varphi'(u_{n}), u_{n}\rangle\| \\ &\leq&lim_{n\rightarrow\infty} \|(\frac{1}{2}-\frac{1}{q})\int_{\mathbb{R}^{N}}g(x)\| u_{n}\|^{q}dx\| + lim_{n\rightarrow\infty}\|(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{N}}h(x)\| u_{n}\|^{q}dx\|\\ &\leq& \frac{2\|c\|}{3}.\end{aligned}$$ This is a contradiction. Thus $\delta>0$ and the weak limit of $\{u_{n}\}$ is nontrivial. We can get the conclusion easily from the weakly sequentially continuity of $\varphi'$. [99]{} M. 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--- abstract: 'We consider thermal machines powered by locally equilibrium reservoirs that share classical or quantum correlations. The reservoirs are modelled by the so called collisional model or repeated interactions model. In our framework, two reservoir particles, initially prepared in a thermal state, are correlated through a unitary transformation and afterwards interact locally with the two quantum subsystems which form the working fluid. For a particular class of unitaries, we show how the transformation applied to the reservoir particles affect the amount of heat transferred and the work produced. We then compute the distribution of heat and work when the unitary is chosen randomly proving that the total swap transformation is the optimal one. Finally, we analyse the performance of the machines in terms of classical and quantum correlations established among the microscopic constituents of the machine.' author: - Gabriele De Chiara - Mauro Antezza bibliography: - 'Refs.bib' title: Quantum machines powered by correlated baths --- Introduction ============ The interest in thermal machines powered by quantum working media has recently surged thanks to the technological advancement in the realisation and control of individual quanta [@ThermoBook; @XuerebReview; @GooldReview; @AndersReview; @Mitchison2019; @deffner2019quantum]. This tremendous progress has led to the first realisations of quantum engines and thermal devices [@RossnagelScience2016; @ronzani2018tunable; @Maslennikov19; @LindenfelsPRL2019; @KlatzowPRL2019; @PetersonPRL2019; @gluza2020]. The theoretical modelling of such devices usually involves the system in contact with equilibrium uncorrelated baths at different temperatures. However, some papers have generalised this picture to non equilibrium reservoirs [@LeggioPRA2015; @LeggioPRE2016; @ReidEPL2017; @AssisPRL2019; @cherubim2019; @PezzuttoQST2019; @CarolloPRL2020], including the case of the Otto engine in contact with squeezed reservoirs [@RossnagelPRL2014; @Manzano2016; @AgarwallaPRB2017; @singh2020performance], which can lead to efficiencies and performances beyond the Otto and Carnot limit. This conclusion, obviously, does not take into account the cost of maintaining a non equilibrium reservoir which is then considered as a free resource but shows how to best employ these resources (see also [@niedenzu2018quantum]). Other works have considered thermal devices coupled to spatially separated reservoirs which share correlations, classical or quantum [@DoyeuxPRE2016; @TurkpenceEPL2017; @KarimiPRB2017; @HewgillPRA2018; @LatuneQST2019; @ManzanoNJP2019; @pusuluk2020]. Here we propose a general framework based on collisional models [@ScaraniPRL2002; @ZimanPRA2002; @Karevski2009; @PalmaGiovannetti; @CiccarelloPRA2013; @LorenzoPRA2015; @Landi2014b; @LorenzoPRL2015; @Barra2015; @Strasberg2016; @Manzano2016; @Ciccarello2017; @PezzuttoNJP2016; @cusumano2018entropy; @CampbellPRA2018; @GrossQST2018; @MohammadyPRA2018; @PezzuttoQST2019; @RodriguesPRL2019; @ManatulyPRE2019; @StrasbergPRL2019; @SeahPRE2019; @CakmakPRA2019; @GuarnieriPLA2020; @garcia2020ibm; @Li_2020; @cilluffo2020collisional] which allows us to analyse, in a consistent thermodynamic sense as recently proven in Ref. [@DeChiaraNJP2018], the effect of classical and quantum correlations between reservoirs in the functioning of quantum thermal machines. Our setup, depicted in Fig. \[fig:setup\], consists of a working medium made of two quantum systems $S_1$ and $S_2$. Each of these is in contact with a reservoir modelled by the repeated interaction of flying auxiliary particles. These particles are first prepared in a thermal state at $T_1$ and $T_2$, respectively, and then undergo a unitary operation $U$ which correlates them before their collision with the systems $S_1$ and $S_2$. We study the steady state of the system after many collisions with the flying particles. ![Setup of our scheme: thermal particles emerging from the cold and hot baths and prepared in the states $\tilde\rho_{\rm th}(n_1)$ and $\tilde\rho_{\rm th}(n_1)$, respectively, are made to collide to each other under a unitary operation $U$. The emerging correlated particles in the global state $\rho_B'$ collide with the system’s particles $S_1$ and $S_2$.[]{data-label="fig:setup"}](setup.pdf){width="0.7\columnwidth"} Such a microscopic model has the advantage that all thermodynamic contributions, e.g. energy, heat, work and entropy, are accountable and that it is consistent with the laws of thermodynamics [@Barra2015; @Strasberg2016; @DeChiaraNJP2018]. From the point of view of open quantum systems, our model leads to a non trivial evolution of a quantum system in contact with correlated reservoirs. As we will show, under the assumption of continuous evolution in the limit of collisions lasting an infinitesimal amount of time, the system’s evolution can be cast in the form of a Markovian Lindblad master equation with collective jump operators acting non trivially on both systems (see also Ref. [@DaryanooshPRA2018]). In this paper we showcase the functioning of our model assuming the system and environment’s particles to be qubits. However our framework is general and could be equally applied to higher-dimensional systems including infinite-dimensional ones, for instance quantum harmonic oscillators. After discussing the setup in more detail in Sec. \[sec:setup\] and discussing its thermodynamics in Sec. \[sec:thermodynamics\], we assume in Sec. \[sec:swap\] the unitary operation $U$ to be a partial swap. We will show how the amount of swapping between the flying qubits controls the amount and direction of the heat flow among the system’s qubits. We then consider in Sec. \[sec:random\] the most general two-qubit unitary operations and study the distribution of work produced and heat exchanged by the system when the unitaries are chosen randomly. We find that the extremal points of the distribution correspond to eight non-correlating unitaries with the optimal one corresponding to the complete two-qubit swap. In this context we analyse the quantum and classical correlations established among the quantum constituents of our setup. We find that, while correlations among the flying qubits are not necessary for the machine to work, quantum and classical correlations among the system’s particle must be nonzero for the optimal performance. Finally in Sec. \[sec:conclusions\] we summarise and conclude. Setup and preliminaries on repeated interactions {#sec:setup} ================================================ We assume the system to be composed of two coupled qubits described by the XXZ Hamiltonian: $$H_S= J(\sigma_{x1}\sigma_{x2}+\sigma_{y1}\sigma_{y2}+\Delta \sigma_{z1}\sigma_{z2})+B_1\sigma_{z1}+B_2\sigma_{z2}$$ where $J,\Delta$ and $B_i$ are the interaction strength, anisotropy and local magnetic field, respectively. Here and throughout the paper we assume $\hbar=k_B=J=1$ expressing all physical quantities in these units. The operators $\sigma_{xi},\sigma_{yi},\sigma_{zi}$ are the Pauli matrices for the qubit $i$. Notice that the total magnetisation $S_z=\sigma_{z1}+\sigma_{z2}$ is a conserved quantity as it commutes with the system Hamiltonian. We assume the system to be affected by a reservoir modelled by the so-called repeated interactions [@ScaraniPRL2002; @ZimanPRA2002; @Karevski2009; @PalmaGiovannetti; @CiccarelloPRA2013; @LorenzoPRA2015; @Landi2014b; @LorenzoPRL2015; @Barra2015; @Strasberg2016; @Manzano2016; @Ciccarello2017; @PezzuttoNJP2016; @cusumano2018entropy; @CampbellPRA2018; @GrossQST2018; @MohammadyPRA2018; @PezzuttoQST2019; @RodriguesPRL2019; @ManatulyPRE2019; @StrasbergPRL2019; @SeahPRE2019; @CakmakPRA2019; @GuarnieriPLA2020; @garcia2020ibm; @Li_2020; @cilluffo2020collisional]. In this model, represented in Fig. \[fig:setup\], each qubit of the system interacts with a stream of uncoupled environment qubits (or flying qubits). The interaction between the system qubit and a flying qubit only lasts for a short time $\tau$ during which the interaction Hamiltonian is constant and given by: $$H_{SB} =\sum_{i=1,2} \sqrt{\frac{\gamma(2n_i+1)}{2\tau}}(\sigma_{xi}\tilde\sigma_{xi}+\sigma_{yi}\tilde\sigma_{yi})$$ where the operators $\tilde\sigma_{xi},\tilde\sigma_{yi},\tilde\sigma_{zi}$ are the Pauli matrices for a flying qubit interacting with the system qubit $i=1,2$. The coefficient $\gamma$ determines the strength of the interaction while $n_i$ models the thermal occupation of the flying qubit. Additionally each flying qubit is subject to the local Hamiltonian: $$H_{Bi}=B_i\tilde\sigma_{zi},\quad i=1,2,$$ and we define $H_B=H_{B1}+H_{B2}$. Usually, in the literature, the state of the environment qubits for different reservoirs has been assumed to be uncorrelated. In this paper we challenge this assumption and introduce some correlations, not necessarily quantum ones, between the flying qubits. These are initially uncorrelated and prepared in a thermal state corresponding to a thermal occupation $n_i$: $$\rho_B=\tilde\rho_{\rm th}(n_1)\otimes\tilde\rho_{\rm th}(n_2)$$ where $$\tilde \rho_{\rm th}(n_1)=\frac 12\left[{\mathds{1}}+(1+2n_i)^{-1}\tilde\sigma_{zi} \right]$$ and ${\mathds{1}}$ is the identity operator. We then correlate the flying qubits with a unitary transformation $U$ so that their state becomes: $$\rho_B'=U\rho_BU^\dagger.$$ After this initial preparation, each flying qubit collides with a system’s qubit (see Fig. \[fig:setup\]). The collision, lasting for a time $\tau$, is described by the unitary operator: $$U_{\rm collision}=e^{-i H_{\rm tot}\tau}$$ where $H_{\rm tot}=H_S+H_B+H_{SB}$ is the total Hamiltonian. If we call $\rho_S(t)$ the state of the system at time $t$, then its state at time $t+\tau$ after the collision becomes $$\rho_S(t+\tau)={\rm Tr}_B \left[U_{\rm collision}\,\rho_S(t)\otimes\rho_B'U_{\rm collision}^\dagger\right ].$$ In the following we will always consider the steady state that the system approaches after many collisions which is defined by the relation: $$\rho_S^{\rm steady}(t+\tau)=\rho_S^{\rm steady}(t)$$ and for simplicity we will drop its time dependence and write simply $\rho_S^{\rm steady}$. Thermodynamics {#sec:thermodynamics} ============== We now discuss the different thermodynamics contributions arising in our setup after the system has reached its steady state. As the state of the system does not change anymore, the internal energy variation is zero: $$\Delta E ={\rm Tr}\left[H_S\left(\rho_{SB}'-\rho_{SB}\right) \right]= 0$$ where $\rho_{SB}=\rho_S^{\rm steady}\otimes\rho_B'$ and $\rho'_{SB}=U_{\rm collision}\;\rho_{SB}\,U^\dagger_{\rm collision}$. To define heat and work we distinguish two scenarios, called [partial ]{} and [complete]{} scenarios, which we explain in more details in the following. Partial scenario ---------------- In the partial scenario, we assume that we are provided with the flying qubits in the state $\rho_B'$ and that we are not paying for the work of the unitary $U$. In this scenario, we are provided with a non-equilibrium reservoir and thus the second law of thermodynamics may not apply. This is still an interesting scenario to study for two reasons: first, it continues the investigation of the functioning of thermodynamic machines in the presence of non thermal environments [@RossnagelPRL2014; @LeggioPRA2015; @LeggioPRE2016; @Manzano2016; @ReidEPL2017; @AgarwallaPRB2017; @AssisPRL2019; @cherubim2019; @PezzuttoQST2019; @CarolloPRL2020]; second, it gives us an opportunity to study the open quantum system dynamics in the presence of correlated reservoirs as developed in Sec. \[sec:swap\]. Under these assumptions the work is potentially produced or injected during the system-environment collision [@DeChiaraNJP2018]. This is given by: $$W_{\rm partial}={\rm Tr}\left[(H_S+H_B) \left(\rho_{SB}'-\rho_{SB}\right) \right]$$ Similarly the heat exchanged by the system with the flying qubits is equal to the energy balance of the latter ones: $$Q^{(i)}_{\rm partial}=-{\rm Tr}\left[H_{Bi}\left(\rho_{SB}'-\rho_{SB}\right) \right], \quad i=1,2.$$ At steady state, the first law for the two-qubit system reads: $$\Delta E = Q_{\rm partial}+W_{\rm partial}=0,$$ where $Q_{\rm partial}=Q^{(1)}_{\rm partial}+Q^{(2)}_{\rm partial}$. Notice that throughout this paper we employ the convention that positive work or heat corresponds to energy injected into the system contributing to the [increase]{} of the system energy. Complete scenario ----------------- In the complete scenario we account for the extra work needed to implement the correlating unitary $U$: $$W_U = {\rm Tr}\left[H_B\left(\rho_{B}'-\rho_{B}\right) \right]$$ so that the total work in the complete scenario is the sum of the two contrinutions: $$W_{\rm complete} = W_{\rm partial}+W_U.$$ Equally, the heat exchanged is the energy balance of the environment during the whole process: from their preparation into the product thermal state $\rho_B$ to their final state after the two unitaries $U$ and $U_{\rm collision}$: $$Q^{(i)}_{\rm complete} = -{\rm Tr}\left[H_{Bi}\left(\rho_{SB}'-\rho^{\rm steady}_{S}\otimes \rho_B\right) \right], \quad i=1,2.$$ and $Q_{\rm complete}=Q^{(1)}_{\rm complete}+Q^{(2)}_{\rm complete}$ which in general differs from $Q_{\rm partial}$. A modified first law holds also in this scenario: $$\Delta E = Q_{\rm complete}+W_{\rm complete}=0.$$ Reservoirs in a partially swapped locally thermal state {#sec:swap} ======================================================= Here, we consider the special case in which the unitary $U$ correlating the flying qubits is a partial swap operation: $$\label{eq:USWAP} S_\phi = \exp\left\{-i \frac \phi 2 (\tilde\sigma_{x1}\tilde\sigma_{y2}-\tilde\sigma_{y1}\tilde\sigma_{x2}) \right\}$$ which is a total swap for $\phi=\pi/2$. This leads to the following expression for the bath density matrix after the action of the partial swap $S_\phi$: $$\begin{aligned} \label{eq:UrhoB} \rho'_B&=&\frac{1}{(1+2n_1)(1+2n_2)}\times \\ &\times& \left(\begin{array}{cccc} n_1 n_2 & 0 & 0 & 0 \\ 0 &\frac 12( n_1+n_2+2n_1n_2+(n_1-n_2)\cos2\phi)& (n_2-n_1)\sin\phi\cos\phi & 0 \\ 0 & (n_2-n_1)\sin\phi\cos\phi &\frac 12( n_1+n_2+2n_1n_2-(n_1-n_2)\cos2\phi )& 0 \\ 0 & 0 & 0 & (1+n_1)(1+n_2) \end{array}\right)\end{aligned}$$ where the basis of eigenstates of $\sigma_{z1}$ and $\sigma_{z2}$ has been used to write the matrix representation of $\rho'_B$. The action of the partial swap is to partially exchange populations between the auxiliary qubits. Indeed, the two flying qubits are still locally in a thermal state although at a different temperature compared to their state before the partial swap: $$\tilde\rho'_{i}={\rm Tr}_{\bar i}\; \rho'_B= \tilde\rho_{th}(N_i)$$ where $i=1,2$ and ${\rm Tr}_{\bar i}$ represents the partial trace with respect to the qubit other than $i$. The effective population after the application of $S_\phi$ is: $$N_i=\frac 12 \frac{n_1+n_2+4n_1n_2+(-1)^i (n_2-n_1)\cos2\phi} {1+n_1+n_2-(-1)^i (n_2-n_1)\cos2\phi}$$ The effective populations $N_i$ of the two flying qubits are shown in Fig. \[fig:effectivepopulationsconcurrence\] where it is evident that for $\phi=\pi/2$ the populations are completely swapped. ![Top: Effective populations of the two flying qubits as a function of $\phi$ for $n_1=1$ and $n_2=3$. Bottom: Concurrence of the two flying qubits as a function of $n_1$ and $n_2$ for $\phi=0.2\pi$.[]{data-label="fig:effectivepopulationsconcurrence"}](effectivepopulations.pdf "fig:"){width="0.7\columnwidth"} ![Top: Effective populations of the two flying qubits as a function of $\phi$ for $n_1=1$ and $n_2=3$. Bottom: Concurrence of the two flying qubits as a function of $n_1$ and $n_2$ for $\phi=0.2\pi$.[]{data-label="fig:effectivepopulationsconcurrence"}](rhobconcurrence.pdf "fig:"){width="0.7\columnwidth"} The partial swap creates quantum correlations, measured for instance by quantum discord [@PhysRevLett.88.017901; @henderson2001classical], among the two auxiliary qubits, except for the special values of $\phi=m\pi/2, \; m\in \mathbb{Z}$, regardless of the values $n_1$ and $n_2$. This happens because, as it is evident from Eq. , $\rho'_B$ is not a product state. Depending on the values of $n_1,n_2$, but with the condition that $n_1\neq n_2$, the state $\rho_B$ can furthermore be entangled as evidenced by the analysis of the concurrence in Fig. \[fig:effectivepopulationsconcurrence\], whose complicated analytical expression is not amenable of a simple interpretation. Notice how unbalancing the two populations $n_1$ and $n_2$ leads to larger entanglement. The work for implementing the partial swap is given by: $$\begin{aligned} \label{eq:WUswap} W_U &=& -\frac{2(B_1-B_2)(n_1-n_2)\sin^2\phi}{(1+2n_1)(1+2n_2)}.\end{aligned}$$ Notice that similarly to what is found in Ref. [@DeChiaraNJP2018; @CampisiNJP2015], the work vanishes for $B_1=B_2$ or $n_1=n_2$ and is maximum for the total swap $\phi=\pi/2$. We set the parameters in such a way that for $\phi=0$ the two-qubit system acts as an engine ($W<0, Q^{(1)}<0,Q^{(2)}>0$ for both scenarios). It is possible to find the analytical expression of the steady state, the work and heats exchanged per cycle. While their expressions are quite long and hard to read, it is possible to show that all their ratios are proportional to ratios of the local magnetic fields, $B_1$ and $B_2$. As a consequence, and similarly to other models [@CampisiNJP2015; @DeChiaraNJP2018; @HewgillPRA2018], the efficiency $\eta=\|W\|/Q^{(2)}$, if the setup works as an engine, or the coefficient of performance (COP) $\eta_{\rm COP}=Q^{(1)}/W$, if the setup works as a refrigerator, correspond to the Otto value. For example in the former case, the efficiency is given by ($B_2<B_1$): $$\label{eq:etaotto} \eta=1-\frac{B_2}{B_1}$$ and in the latter case the COP is given by ($B_1<B_2$): $$\eta_{\rm COP}=\frac{B_1}{B_2-B_1}$$ In the partial scenario, $\phi$ can be used to control the functioning of the thermodynamic machine making it switch from an engine to a refrigerator. In the interval $0\le \phi \le \pi/2$, the machine behaves an engine for $0\le \phi < \pi/4$ and as a refrigerator for $\pi/4< \phi \le \pi/2$. For values of $\phi$ outside the interval $[0,\pi/2]$, the situation described repeats periodically. The switching points are found for $N_1=N_2$ which occurs for $\cos 2\phi=0$, i.e. $\phi=(2m+1) \pi/4, m\in \mathbb{Z}$. These switching points correspond to effective Carnot points where work and heat exchanged are zero (see also [@DeChiaraNJP2018] where the condition $n_1=n_2$ corresponds to the Carnot point). Numerical results for the partial scenario are plotted in Fig. \[fig:workSWAP\] which shows the periodic change of the machine operating as an engine and as a refrigerator. Interestingly, the largest absolute values of the thermodynamic quantities are obtained for the values $\phi=m \pi/2, \; m\in \mathbb{Z}$ at which the partial swap operation $S_\phi$ corresponds to the identity (even $m$) and the total swap (odd $m$). Thus the partial swap operation can be employed as a valve to control the direction of the heat flowing between the system’s qubits without altering the equilibrium reservoirs that prepare the flying qubits. These can be particularly useful in physical implementations in which one does not have full control of the environment. ![Thermodynamic quantities: work and heat exchanged per collision for the steady state of the system in contact with the flying qubits prepared with the partial swap Eq. . Top: partial scenario. Bottom: complete scenario. Parameters: $\Delta=1$, $\gamma=1$, $B_1=0.1$, $B_2=0.3$, $n_1=0.1, n_2=2,\tau=0.1$.[]{data-label="fig:workSWAP"}](thermopartial.pdf "fig:"){width="0.8\columnwidth"} ![Thermodynamic quantities: work and heat exchanged per collision for the steady state of the system in contact with the flying qubits prepared with the partial swap Eq. . Top: partial scenario. Bottom: complete scenario. Parameters: $\Delta=1$, $\gamma=1$, $B_1=0.1$, $B_2=0.3$, $n_1=0.1, n_2=2,\tau=0.1$.[]{data-label="fig:workSWAP"}](thermocomplete.pdf "fig:"){width="0.8\columnwidth"} Let us now move to the complete scenario. In this case, since we are taking into account all work contributions including those coming from the action of the swap operation, both first and second law are fulfilled. The corresponding results plotted in Fig. \[fig:workSWAP\] show that the work and heats exchanged do not change sign and the machine always behaves as an engine. The action of the partial swap is to amplify the work production and the heats exchanged, while keeping the same performance, reaching a maximum for $\phi=m \pi/2, m=\pm 1,3,5, \dots$ at which the swap is total. This shows that given the two flying qubits, initially in equilibrium states with fixed temperatures, the maximum work that can be extracted is achieved by completely swapping their states before making them to collide with the system’s qubit. As we will see in Sec. \[sec:random\] this is the maximum value that can be obtained for any unitary operation between the flying qubits. It is possible to prove, using the methods developed in Refs [@DeffnerPRL2011; @Strasberg2016; @DeChiaraNJP2018], that in both the complete and partial scenarios the second law is fulfilled by showing that the entropy production is always non negative, while being zero at the Carnot point $n_1=n_2$. We end up this section by considering the continuous limit of $\tau\to 0$. Up to first order in $\phi$ one obtains the following Lindblad master equation: $$\begin{aligned} \label{eq:me} \dot\rho_S &=& -i[H_S,\rho_S]+\gamma\sum_{i=1,2} (1+n_i)\mathcal L_{\sigma_{-i}}(\rho_S)+n_i\mathcal L_{\sigma_{+i}}(\rho_S) \\ &+&\frac{\gamma \phi(n_2-n_1)\left[\mathcal M(\sigma_{+1},\sigma_{-2},\rho_S)+\mathcal M(\sigma_{-2},\sigma_{+1},\rho_S)+h.c.\right]}{\sqrt{(1+2n_1)(1+2n_2)}} \nonumber\end{aligned}$$ where $\mathcal L_{a}(\rho) = 2a\rho a^\dagger-a^\dagger a \rho-\rho a^\dagger a$ is the usual Lindblad operator and $\mathcal M(a,b,\rho) = 2a\rho b-b a \rho-\rho b a$ is a modified Lindblad-like operator. In Eq. we have also defined the jump operators $\sigma_{\pm i} =\frac 12 (\sigma_{xi}\pm i \sigma_{yi})$. The presence of quantum correlations in the initial state of the bath is the reason for the appearance in the master equation of the collective term $\mathcal M$ proportional to $\phi$ which corresponds to environment-induced processes of emission of an excitation from one qubit and absorption from the other. In this continuous limit, it is possible to define work power and heat currents both in the partial and complete scenario. However, in the complete scenario, one should assume that the work necessary for the swap operation given in Eq. scales to zero as $\tau\to 0$. This is indeed the case up to first order in $\phi$ consistently with our master equation. Reservoirs prepared by random unitaries {#sec:random} ======================================= Here we generalise the approach developed in the previous section by considering thermal flying qubits which are subject, before the collision with the system qubits, to a unitary transformation $U_R$. The unitary transformation is always the same for all collisions for a given setup and allows the system to reach a steady state which we then analyse. We repeat the same procedure for an ensemble of $6\times 10^6$ random unitaries $U_R$ chosen with uniform distribution according to the Haar measure [@Ozols]. For a generic $U_R$, the creation of finite quantum coherences and correlations between the environment qubit may lead to difficulties in the derivation of a consistent master equation in the continuous limit $\tau\to 0$, see Ref. [@RodriguesPRL2019]. In this section, we will therefore restrict our analysis to a small but finite $\tau$. Under this assumption, the repeated interactions remain a discrete map for the system which, after many applications reaches a steady state, which we analyse. ![Histogram of the partial work $W_{\rm partial}$ with the vertical axis measuring the corresponding probability density function (PDF) for $6\times 10^6$ random unitaries. The solid line is the best fit normal distribution with average $9.6 \times 10^{-7}$ and standard deviation $1.1\times 10^{-3}$. The vertical dashed lines signpost the minimum and maximum possible values of $W_{\rm partial}$ obtained with non-correlating unitaries (see text). Parameters as in Fig. \[fig:workSWAP\] except for the local magnetic fields which are $B_1=0.1$ and $B_2=0.15$.[]{data-label="fig:randomworkpartial"}](Random_W_vs_Q1_partial.pdf){width="0.8\columnwidth"} We consider a setup which, for $U_R= \mathds 1$, corresponding to the situation in which no unitary is applied to the flying qubits before their collision with the system, operates as an engine. Analogous results can be obtained for the refrigerator regime. In the partial case, as in Sec. \[sec:swap\], work and heat fluxes are proportional to each other with the prefactor being a ratio of the local magnetic fields, $B_1$ and $B_2$, related to the Otto efficiency. For this reason we only show the analysis of the work probability distribution shown in Fig. \[fig:randomworkpartial\]. Its probability density function (PDF) is approximately Gaussian with a very small average statistically compatible with zero. The maximum and minimum values of the partial work $W_{\rm partial}$ are obtained with unitaries that do not create correlations, classical or quantum, among the auxiliary qubits. As we will see in more detail in the complete scenario, this type of unitary operations do not create any mutual information between the flying qubits but simply rearrange the populations of the four basis states. In the partial scenario we find that the minimum negative value of the work (largest value of produced work) is obtained with the unitary that inverts the populations of the qubit prepared in the hottest temperature and leaves unchanged the qubit prepared in the coldest temperature: $U_R=\sigma_{x2}$ (corresponding to operation III defined later). In contrast, the maximum positive value of the work (largest value of the work injected), is obtained with the unitary that inverts the populations of the qubit prepared in the coldest temperature and leaves unchanged the qubit prepared in the hottest temperature: $U_R=\sigma_{x1}$ (corresponding to operation VI defined later). ![[Top]{}: Joint histogram of the complete work $W_{\rm complete}$ versus heat $Q_{\rm complete}^{(2)}$. The dashed gray line represents the Otto efficiency. The blue dots, connected by thin solid lines, are obtained for non correlating unitaries marked by the corresponding number. These special points form an extremal octagon within which all other machines obtained for random unitaries must be located. The colour code identifies the number of values in each bin of the histogram. Parameters: $\Delta=1$, $\gamma=1$, $B_1=0.1$, $B_2=0.15$, $n_1=0.1, n_2=2,\tau=0.1$. [Bottom]{}: Extremal octagons obtained for the same parameters of the top panel but different values of $B_2=0.15,0.3,0.6,0.9$, corresponding to vertexes denoted by circles, squares, triangles and diamonds, respectively.[]{data-label="fig:randomheatwork"}](Random_W_vs_Q2_complete.pdf "fig:"){width="0.99\columnwidth"} ![[Top]{}: Joint histogram of the complete work $W_{\rm complete}$ versus heat $Q_{\rm complete}^{(2)}$. The dashed gray line represents the Otto efficiency. The blue dots, connected by thin solid lines, are obtained for non correlating unitaries marked by the corresponding number. These special points form an extremal octagon within which all other machines obtained for random unitaries must be located. The colour code identifies the number of values in each bin of the histogram. Parameters: $\Delta=1$, $\gamma=1$, $B_1=0.1$, $B_2=0.15$, $n_1=0.1, n_2=2,\tau=0.1$. [Bottom]{}: Extremal octagons obtained for the same parameters of the top panel but different values of $B_2=0.15,0.3,0.6,0.9$, corresponding to vertexes denoted by circles, squares, triangles and diamonds, respectively.[]{data-label="fig:randomheatwork"}](Octagons.pdf "fig:"){width="0.9\columnwidth"} Let us now pass to the complete scenario in which the work produced or extracted is not necessarily proportional to the two heat fluxes. We plot in Fig. \[fig:randomheatwork\] the joint histogram of the complete work and the heat input from the hot environment. The distribution is confined by a non-regular octagon whose eight vertexes correspond to unitary operations that do not create quantum or classical correlations between the environment qubits. Let $\{p_1,p_2,p_3,p_4\}$ be the vector of the populations of the density matrix $\rho_B$ of the flying qubits after their preparation in equilibrium states but before undergoing the unitary $U$. These eight unitary operations only affect the populations of the flying qubits according to:\ Label Populations Unitary ---------- ------------------------ --------------------------------------- [I]{} $\{p_1,p_2,p_3,p_4\}$ ${\mathds{1}}_1{\mathds{1}}_2$ [II]{} $\{p_1,p_3,p_2,p_4\}$ $ S_{\pi/2}$ [III]{} $\{p_2,p_1,p_4,p_3\}$ $ {\mathds{1}}_1 \sigma_{x2}$ [IV]{} $ \{p_2,p_4,p_1,p_3\}$ $S_{\pi/2}{\mathds{1}}_1 \sigma_{x2}$ [V]{} $\{p_3,p_1,p_4,p_2\}$ $S_{\pi/2} \sigma_{x1}{\mathds{1}}_2$ [VI]{} $ \{p_3,p_4,p_1,p_2\}$ $ \sigma_{x1}{\mathds{1}}_2$ [VII]{} $\{p_4,p_2,p_3,p_1\}$ $S_{\pi/2}\sigma_{x1}\sigma_{x2}$ [VIII]{} $ \{p_4,p_3,p_2,p_1\}$ $ \sigma_{x1}\sigma_{x2}$ where $S_{\pi/2}$ is the transformation that completely swap the state of the two qubits, see Eq. for $\phi=\pi/2$. Notice the symmetry between the unitary operations and the position of the corresponding vertex of the octagon. In Fig. \[fig:randomheatwork\], two vertexes are opposite on the octagon if their transformations can be obtained one from the other by applying the total swap operation. Moreover, one of the sides of the octagon, delimited by the vertexes I and II, corresponds to the Otto efficiency given in Eq. , so that points above the line corresponds to engines operating at a lower efficiency or, for $W_{\rm complete}\ge 0$ operating as an accelerator, if $Q^{(2)}_{\rm complete}>0$ or a heater if $Q^{(2)}_{\rm complete}<0$. In the complete scenario, there are no unitaries that lead to refrigeration, similarly to what we discussed in Sec. \[sec:swap\] for the partial swap. The optimal point II yielding the larger amount of work produced (minimum negative value) corresponds to the complete swap operation. This can be related to an engine proposed by Campisi, Pekola and Fazio also based on a complete swap transformation but applied to the qubits of the working fluid [@CampisiNJP2015]. In Fig. \[fig:randomheatwork\] we also show how the shape and size of the extremal octagon change when $B_2$ is varied while all other parameters are kept fixed. Notice that, as we increase $B_2$, the region corresponding to the engine increases, leading to a higher probability of achieving this operation mode. This can be understood as follows. First, $Q^{(2)}_{\rm complete}\propto B_2$, thus if we rescaled the horizontal axis of the lower diagram in Fig. \[fig:randomheatwork\], all the vertexes would have the same horizontal coordinates. Second, for $W_{\rm complete}=W_{\rm partial}+W_U$ the situation is more involved. While the partial work always fulfils $W_{\rm partial}\propto B_2-B_1$, the work $W_U$ needed to implement the non-correlating unitaries is a linear function of $B_2$ and $B_1$ which depends on the actual transformation. Therefore there exists no rescaling of the vertical axis that would bring the vertexes of different octagons to the same vertical coordinate. Furthermore, the vertex IV, corresponding to an accelerator for $B_2=0.15$, turns into an engine for the larger values of $B_2$ we analysed. To get more insight into the relationship between the functioning of the whole system as an engine and the quantum features of the working medium quantum steady state, we looked at two measures of correlation between two parts of the engine. The mutual information between two quantum objects $O_1$ and $O_2$ is defined as: $$I_{O1O2}=S(\rho_{O1})+S(\rho_{O2})-S(\rho_{O1O2})$$ where $S(\rho)=-{\rm Tr}\rho\ln\rho$ is the von Neumann entropy and $\rho_i, \, i=O_1,O_2,O_1O_2$ are the density matrices of the corresponding objects. The mutual information quantifies both classical and quantum correlations. If one subtracts the maximum amount of classical correlations that can be obtained by local measurements, the quantum discord, a genuine measure of quantum correlations, is obtained [@PhysRevLett.88.017901; @henderson2001classical] . This can be defined as: $$D_{O1O2}=I_{O1O2}-J_{O1O2},$$ where the classical information $J_{O1O}$ is the maximum information that can be extracted on $O_2$ if we perform local measurements on $O_1$: $$\label{eq:JO1O2} J_{O1O2}=S(\rho_{O2})-\min_{\{\Pi_i\}}\sum_{i=1}^{N}q_i S(\tilde\rho_i).$$ In Eq. , we have defined the probabilities $q_i ={\rm Tr}[\Pi_i\rho_{O1O2}\Pi_i]$ of the outcome $i$ and the post-measurement states $\tilde\rho_i= {\rm Tr}_{O1}[\Pi_i\rho_{O1O2}\Pi_i]$ of the object $O_2$. The minimisation is done over all possible sets of measurements $\{\Pi_i\}$ on $O_1$, not necessarily orthogonal projectors. The results are shown in Fig. \[fig:randomquantumwork\]. We start with the distribution of the mutual information $I_{S_1S_2}$ between the system qubits in the steady state. This shows that, although the optimal point II corresponds to uncorrelated flying qubits, the system qubits are nevertheless correlated as a result of their direct interaction and of reaching a non-equilibrium steady state due to the multiple collisions with the environment. A similar distribution is also obtained for the quantum discord $D_{S_1S_2}$, which shows that the system qubit are also genuinely quantum correlated. We have also considered the mutual information $I_{A_1A_2}$ between the flying qubits after the unitary but before the collision with the system. This shows that the preparation unitaries applied before the correlations do indeed create a lot of correlations, quantum or classical, but these do not necessarily lead to large values of work produced or injected. The extremal operation II in fact corresponds to zero mutual information between the flying qubits, which are thus in a product state. Similar conclusions, although quantitatively different, are reached when analysing concurrence and discord between the two flying qubits. Finally, we have analysed the mutual information $I_{AS}$ between the flying qubits and the system after the collision. As before, the state of the system is steady. Thus, although it does not change during the collision, it sustains correlations to be created between system and each pair of environmental qubits. These correlations are necessary for allowing exchange of heat between the two reservoirs through the system. The distribution of $I_{AS}$ plotted in Fig. \[fig:randomquantumwork\], shows that to achieve large amounts of work, and consequently heat exchange, it is sufficient a small value of $I_{AS}$. In particular the non-correlating operations I-VIII, that leave the flying qubits in a product state, correspond to the smallest values of $I_{AS}$. Finally, in all these figures of merit, the non-correlating operations I-VIII do not necessarily correspond to extremal points since these functions, mutual information and discord, are not linear functions of the state in contrast to the average work and heat. ![image](QInfovsW.pdf){width="1.99\columnwidth"} Conclusions {#sec:conclusions} =========== In this paper we have proposed a general framework to model quantum thermal machines in contact with correlated reservoirs using repeated interactions. Different conclusions are found depending on whether we assume the partial or the complete scenario, the latter one being always consistent with the laws of thermodynamics. We have shown how, in the partial scenario, the amount of partial swapping among the flying qubits, allows one to control the operating mode of the thermal machine switching it from an engine to a refrigerator. In the case of random unitaries, we found a complex geometrical structure in the distribution of heat and work bounded by a non regular octagon whose vertexes correspond to non correlating unitaries. We analysed the discord and mutual information between the flying qubits, the system qubits and between system and environment. We found that correlations in the system steady state and between system and environment are necessary to achieve the optimal performance. Our work leads the way to future studies of open quantum systems with correlated environments consistently with thermodynamics. We thank Marco Cattaneo, Francesco Ciccarello, Adam Hewgill and Gabriel Landi for useful discussions. GDC thanks the CNRS and the group Theory of Light-Matter and Quantum Phenomena of the Laboratoire Charles Coulomb for hospitality during his stay in Montpellier. We acknowledge support from the UK EPSRC EP/S02994X/1.
--- abstract: 'We present numerical simulations of jets modelled with relativistic radiation hydrodynamics (RRH), which evolve across two environments: i) a stratified surrounding medium and ii) a 16TI progenitor model. We consider opacities consistent with various processes of interaction between the fluid and radiation, specifically free-free, bound-free, bound-bound, and electron scattering. We explore various initial conditions, with different radiation energy densities of the beam in hydrodynamical and radiation-pressure-dominated scenarios, considering only highly relativistic jets. In order to investigate the impact of the radiation field on the evolution of the jets, we compare our results with purely hydrodynamical jets. Comparing among jets driven by an RRH, we find that radiation-pressure-dominated jets propagate slightly faster than gas pressure dominated ones. Finally, we construct the luminosity light curves (LCs) associated with the two cases. The construction of LCs uses the fluxes of the radiation field that is fully coupled to the hydrodynamics equations during the evolution. The main properties of the jets propagating on the stratified surrounding medium are that the LCs show the same order of magnitude as the gamma-ray luminosity of typical Long gamma-ray Bursts $10^{50}-10^{54}$ erg/s, and the difference between the radiation and gas temperatures is of nearly one order of magnitude. The properties of jets breaking out from the progenitor star model are that the LCs are of the order of magnitude of low-luminosity GRBs $10^{46}-10^{49}$ erg/s, and in this scenario, the difference between the gas and radiation temperature is of four orders of magnitude, which is a case far from thermal equilibrium.' author: - \| F. J. Rivera-Paleo [^1] and F. S. Guzmán [^2]\ Laboratorio de Inteligencia Artificial y Supercómputo, Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo.\ Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoacán, México.\ title: 'Evolution of jets driven by relativistic radiation hydrodynamics as long and low-luminosity GRBs' --- \[firstpage\] opacity; radiative transfer; methods: numerical; gamma-rays: general Introduction {#sec:introduction} ============ There is observational evidence that long gamma-ray bursts (LGRBs) are produced after the death of massive stars [@Woosley; @Galama; @Stanek; @Hjorth; @WoosleyBloom], whose spectrum agrees with those of Type Ic supernovae (SNe). Although the LGRBs have been identified spectroscopically with SNe, many of these events show smaller luminosity than those of standard LGRBs. These events are called low-luminosity GRBs (LLGRBs). The emission mechanisms and the surrounding medium density profiles that make the difference between LGRBs and LLGRBs are still a matter of debate. As a consequence of this, several authors have developed numerical models for the jet propagation in a stratified surrounding medium applied to LGRBs and massive star models applied to LLGRBs. In this context, several numerical studies have been done. For instance, the evolution of jets within a surrounding medium using different approximations has been studied in [@Aloy; @DeColle; @MacFadyen; @Meliani_2007; @Meliani_2010; @Mizuta; @MizutaAloy; @Morsony; @MizutaII; @Nagakura; @Lazzati; @Lopez; @ZhangI; @ZhangII] using pure relativistic hydrodynamics. More elaborate models include the use of ideal magnetohydrodynamics [@Bromberg; @Obergaulinger] or radiation hydrodynamics simulations [@Van_2010; @Van_2011; @Vlasis; @Cuesta; @DeColleII]. However, in these later works the radiation is not coupled with hydrodynamics during the evolution of the system, but the radiation is analysed by post-processing the hydrodynamical variables. From a theoretical point of view, in the RRH simulations there are two key variables at modelling the system, these are: the rest-mass density of the fluid and the radiation energy density of the radiation field, which in turn has an impact on the optical depth of the fluid and the radiation pressure [@Mihalas]. In accordance with the later, the radiative transport does not significantly affect the fluid dynamics as long as the optical depth is in the optically thin regime or the fluid pressure dominates. In this case, the radiation attached to the system in a post-processing of the numerical simulations is a good approach because emitted photons can be assumed to escape with no further interactions. Otherwise, in the optically thick regime or in the radiation-pressure-domination scenario, the effect of radiative transport cannot be negligible because the photons carry significant momentum and energy that affect the dynamics of the fluid around. In this case, the radiation and fluid feedback each other’s dynamics, and we need to solve the equations of radiative transport during the evolution at the same time as Euler’s equations. In view of the above, and under the assumption that the radiation carried by jets goes from an optically thick region to an optically thin region, coupling the radiation field and the fluid during the evolution, is expected to be a better approximation than post-processing or pure hydrodynamics. Therefore, in this paper, we assume a model of the jet, where the fluid is coupled to the radiation field, and the evolution is dictated by the RRH equations. In our numerical simulations, we assume, initially, that the radiation field and matter are in local thermal equilibrium (LTE) and evolve according to the RRH equations. However, after initial time, the system loses the LTE. We explore various initial conditions for the jet, with different radiation energy density and a fixed but high Lorentz factor. In order to describe GRB events in a more realistic astrophysical scenario, we study the dynamics of the jet when it propagates in two different environments. In the first one, the jets propagate in a stratified external medium with rest mass toy density profile $\rho\sim r^{-2}$, which is associated with LGBRs [@Mignone]. In the second one, the jet propagates within its progenitor star. For the later, we consider a pre-supernova 16TI model as the progenitor star that is considered to be a progenitor of LLGBRs. In each of the models, we construct the LCs associated with particular processes of interaction between the fluid and radiation, specifically, free-free, bound-free, bound-bound, and electron scattering opacities. The paper is organized as follows. In section \[sec:RRH\], we describe the system of the RRH equations. In section \[sec:Opacities\], we describe the opacities used in our numerical simulations. In section \[sec:IntSetup\], we describe the initial conditions for the parameters of our simulations. In section \[sec:GRBJet\], we study the evolution of jets on a stratified medium and illustrate how the LCs can be associated to LGRBs. Also, we discuss the implications of the radiation field in the evolution of the system. In section \[sec:LLGRBs\], we present the evolution of jets starting from inside a progenitor and their relation to LLGRBs. Finally, in section \[sec:conclusions\], we discuss our results. Equations of evolution and numerical methods {#sec:RRH} ============================================ As mentioned before, the model assumed for the jet corresponds to a fluid interacting with a radiation field. This implies the need to solve the equations coupling such a system in order to capture the back-reaction of one of the components on to the other. One important advantage of considering the radiation field is that the construction of LCs is natural because one is constantly calculating the variables of the radiation field. The RRH equations governing the evolution of this system are form [@Farris; @Fragile]: $$\begin{aligned} \nabla_\alpha(\rho u^\alpha) &=&0,\\ \nabla_\alpha T^{\alpha\beta}_\text{m} &=&G^\beta_\text{r},\\ \nabla_\alpha T^{\alpha\beta}_\text{r} &=&-G^\beta_\text{r}, \label{eq:rad-hyd}\end{aligned}$$ where $T^{\alpha\beta}_m$ is the stress-energy tensor of a perfect fluid $$T^{\alpha\beta}_m = \rho hu^\alpha u^\beta + Pg^{\alpha\beta},$$ where $g^{\alpha\beta}$ is the metric of the space-time, $u^\alpha$ is the four-velocity of fluid elements, $\rho$, $h=1+\epsilon+P/\rho$, $\epsilon$ and $P$ are the rest-mass density, specific enthalpy, specific internal energy, and the thermal pressure, respectively. The thermal pressure is related to $\rho$ and $\epsilon$ through a gamma-law equation of state $P=\rho\epsilon(\Gamma-1)$, where $\Gamma$ is the adiabatic index of the fluid. Here, $T^{\alpha\beta}_\text{r}$ is the stress-energy tensor that describes the radiation field and is given by $$T^{\alpha\beta}_\text{r} = (E_\text{r}+P_\text{r})u^\alpha u^\beta +F^\alpha_\text{r} u^\beta +u^\alpha F^\beta_\text{r} + P_\text{r}g^{\alpha\beta},$$ where $E_\text{r}$, $F^\alpha_\text{r}$, and $P_\text{r}$ are the radiated energy density, radiated flux, and radiation pressure, respectively, measured in the comoving reference frame. The source term $G^{\alpha}_\text{r}$ is the radiation four-force that describes the interaction between the fluid and the radiation field. Among the various regimes of coupling between radiation and fluid, we choose the ‘grey-body’ approximation, which technically means that the radiation field variables do not depend on its frequency [@Mihalas]. In this case the radiative four-force is given by [@Farris]: $$G^\alpha_\text{r} = \chi^\text{t}(E_\text{r} - 4\pi B)u^\alpha + (\chi^\text{t} + \chi^\text{s})F^\alpha_\text{r}, \label{eq:source}$$ with $\chi^\text{t}$ and $\chi^\text{s}$ the coefficients of thermal and scattering opacities, respectively. Finally $B=\frac{1}{4\pi}a_\text{r}T_\text{fluid}^4$, is the Planck function, $T_\text{fluid}$ the temperature of the fluid and $a_\text{r}$ the radiation constant. The above set of equations is completed with a closure relation that identifies the second moment of radiation with one of the lower order moments. The simplest approach is the Eddington approximation, which assumes a nearly isotropic radiation field and in the fluid frame shows a pressure tensor with the form $P^{ij}_\text{r} = \frac{1}{3}E_\text{r}\delta^{ij}$ [@Mihalas]. This assumption is valid only in the optically thick regime within the diffusion limit. The radiation field in the optically thin regime requires a more general assumption. A scheme that allows a description of the radiation field in both optically thick and thin regimes is the M1 [@Levermore; @Dubroca; @Gonzalez]. The M1 closure provides a better approximation than Eddington’s to the radiation field because it describes the diffusion limit as well as the free-streaming limit, where the radiative energy is transported at the speed of light. Explicitly, this closure relation is given by $$P^{ij}_\text{r} = \left(\frac{1-\zeta}{2}\delta^{ij} + \frac{3\zeta-1}{2}\frac{f^if^j}{\|f\|^2}\right) E_\text{r},$$ where $f^i=F_\text{r}^i/cE_\text{r}$ is the reduced radiative flux and $\zeta=\frac{3+4f^if_i}{5+2\sqrt{4-3f^if_i}}$ is the Eddington factor [@Levermore]. This closure relation recovers the two regimes of radiative transfer. In the optically thick regime $F_\text{r}^i\approx 0$, $f^i=0$, and $\zeta=1/3$ that correspond to Eddington’s approximation. On the other hand, in the optically thin regime $F_\text{r}^i= cE_\text{r}$, $f^i=1$, and $\zeta=1$ that correspond to the free-streaming limit. The fluid temperature is estimated taking into account contributions of both, baryons and radiation pressure. An approximate expression for the total pressure is written as [@Cuesta] $$P_\text{t} = \frac{k_B}{\mu m_\text{p}}\rho T_\text{fluid} + (1 - e^{-\tau})\zeta(T) a_\text{r}T_\text{rad}^4, \label{eq:tem}$$ where $k_B$ is the Boltzmann constant, $\mu$ the mean molecular weight, and $m_\text{p}$ the mass of the proton, $\tau=\int(\rho\chi^\text{t}+\rho\chi^\text{s})d\text{s}$ is the total optical depth. Finally, $T_\text{rad} = (E_\text{r}/a_\text{r} )^{1/4}$, is the temperature of radiation. Here, $\tau$ depends on the temperature only, if any of the opacity coefficients does. The path to integrate $\tau$ in our simulations is straight lines parallel to the $z$-axis. In general, the Eddington factor depends on the temperature $\zeta=\zeta(T_\text{fluid})$. When the fluid and radiation are in LTE, that is $T_\text{fluid}=T_\text{rad}$, the temperature approximately obeys a fourth degree equation similar to (\[eq:tem\]) (see [@Cuesta]). We programmed a code that solves the 3D RRH equations above, together with the M1-closure relation, using the following numerical methods. First, the RRH equations are written in flux balance form $\partial_t\textbf{U}+\partial_{i}\textbf{F}^{i}=\textbf{S}$, where $\textbf{U}$ is the vector of conserved variables, $\textbf{F}^i$ are the fluxes, and $\textbf{S}$ the sources [@ZanottiI; @Roeding]. Based on this structure of the equations, we apply high-resolution shock capturing methods that use a finite volume discretization, the HLLE flux formula, and the minmod-slope limiter. For the evolution, we use the Method of Lines, with an explicit-implicit Runge-Kutta (IMEX RK) time integrator with second-order accuracy, as done in [@Panchos]. In order to benefit from efficient parallelization and standard I/O, we mounted our code on the Cactus frame [@Cactus], using the Carpet driver [@Carpet], and in all the cases, we use an unigrid discretization. In the appendices, we present canonical tests showing that our implementation works properly. Opacities {#sec:Opacities} ========= An essential ingredient in our analysis is the use of appropriate opacities, because their values determine the radiative processes associated with the GRB emission. When temperature is of the order of $\sim 10^9 K$, the energy of photons becomes an appreciable fraction of the electron rest mass, photons may be scattered only on some electrons, and the electron-scattering opacity is given by [@Buchler]: $$\chi^\text{s} = 0.2(1 + X)\left[1 + 2.7\times 10^{11}\frac{\rho}{T^2}\right]^{-1}\left[1 + \left(\frac{T}{4.5\times 10^8}\right)^{0.86}\right]^{-1}. \label{eq:ScOp}$$ Moreover, at these high temperatures $(\sim 10^9 K)$ and low densities, a primary source of opacity comes from the creation of pairs. On the other hand, at intermediate temperatures $(<10^6 K)$ and low densities $(\sim 1 \text{gr}/\text{cm}^3)$, bound-free opacity may be dominant. Finally, at sufficiently low temperatures and densities, bound-bound absorption in the ultraviolet (UV) and far UV dominate the opacity. This effect is relevant in the low-density regions, both in the stratified model and the progenitor model used later, specially in the wind-like structure surrounding the progenitor star. The opacity due to free-free, bound-free and bound-bound emission can be roughly approximated with the so-called Kramers formula [@George; @Hayashi; @Schwarzschild]: $$\begin{aligned} \kappa^\text{ff} &\simeq& 3.8\times10^{22}(1+X)(X+Y+Z)\rho T^{-7/2} \text{cm}^2/\text{gr}, \\ \kappa^\text{bf} &\simeq& 4.3\times10^{25}Z(1+X)\rho T^{-7/2} \text{cm}^2/\text{gr}, \\ \kappa^\text{bb} &\simeq& 10^{25}Z\rho T^{-7/2} \text{cm}^2/\text{gr}. \label{eq:opacities}\end{aligned}$$ The total coefficients of thermal opacity may be approximated over a wide range of temperatures, $10^4 \leq T \leq 10^9 K$ with the sum of free-free, bound-free, and bound-bound coefficients $(\chi^\text{t} = \kappa^\text{ff} +\kappa^\text{bf} +\kappa^\text{bb})$. In all opacities, $X$, $Y$, and $Z$ represent the mass fractions of hydrogen, helium, and elements heavier than helium, respectively. A more accurate treatment of opacities consists in using the mean thermal Gaunt factor in the free-free opacity, and the mean Gaunt and guillotine factors in the bound-free opacity [@Sutherland; @Van_2015; @Hayashi; @Schwarzschild], or directly calculating the Rosseland mean opacities by using the metallicity and the microphysics involved in the processes [@opal]. Simulations setup {#sec:IntSetup} ================= The GRB jet model that we study here, is produced by the injection of a relativistic beam evolving through a fluid at rest, starting from a nozzle with radius $r_\text{b}$ and velocity $v_\text{b}$. The process is characterized by the ratio between the density of the beam (subindex b) and that of the medium (subindex m) $\eta=\rho_\text{b}/\rho_\text{m}$, as well as by the ratio between their pressures $K = P_\text{b}/P_\text{m} $. The relativistic Mach number of the beam is $M_\text{b}={\cal M}_\text{b}W_\text{b}\sqrt{1-c_\text{s}^2}$, where ${\cal M}_\text{b}$, $W_\text{b}$, and $c_\text{s}^2$ are the Newtonian Mach number, Lorentz factor, and speed of sound, respectively. Outflow boundary conditions are used at the boundaries, except inside the nozzle radius, where the values of the variables are kept constant in time during the time window in which we inject the beam. In order to study the jet interaction with the surrounding medium, we assume the radiation field of the beam starts in an optically thick region and propagates towards an optically thin region [@Pe'er; @Giannios]. This fact allows one to assume LTE between the fluid and the radiation field initially. This assumption is satisfied by the external medium density $\rho_\text{m}$ and pressure profiles $P_\text{m}$ that decrease with distance. We use density and pressure profiles described by the following power law [@Mignone; @DeColle]: $$\rho_\text{m}=\rho_0\left(\frac{r_\text{b}}{r}\right)^{2}, \ \ P_\text{m}= P_0\left(\frac{r_\text{b}}{r}\right)^{2},\label{eq:stratified}$$ where the parameters of the surrounding medium are $\rho_0=10^5 \text{g} \text{cm}^{-3}$ and $P_0 = 10^{22} \text{g} \text{cm}^{-1} \text{s}^{-2}$. This is a simplified model for the propagation of a relativistic jet through a collapsing, non-rotating massive star with $0.1$ times solar metallicity [@Mignone]. Other interesting properties of GRBs are the luminosity and total injected energy $L_\text{j}$ and $E_\text{j}$, respectively. The injected jet luminosity is given by the flux of the momentum density equation times the surface of injection, $A_\text{b}$, which in an optically thick regime is $$L_\text{j} \simeq \left[ \left(\rho_\text{b} h_\text{b} + 4/3 E_\text{r,b} \right) W_\text{b}^2 v_{z,\text{b}} + F_\text{r}^z W_\text{b} \left(v_{z,\text{b}} +1\right) \right] A_\text{b},$$ where $h_\text{b}$, $E_\text{r,b}$, $W_\text{b}$, and $v_{z,\text{b}}$ are the specific enthalpy, radiated energy density, Lorentz factor, and velocity of the beam. Finally, the total injected energy is approximately $$E_\text{j} \simeq L_\text{j} t_\text{inj},$$ where $t_\text{inj}$ is the injection time. We explore various initial conditions for a highly relativistic jet, with different radiation energy densities, in both gas and radiation-pressure-dominated scenarios. In order to determine whether a jet is radiation or matter dominated initially, we measure the ratio between the effective inertia of the radiation field $(4/3 E_\text{r,b}W^2_\text{b}\text{b})$ and the effective inertia in the purely hydrodynamical case $(\rho_\text{b} h_\text{b} W^2_\text{b})$, as well as the ratio between radiation and gas pressures. These are, respectively, $$\begin{aligned} g_\text{1,b} = \frac{4/3 E_\text{r,b}}{\rho_\text{b} h_\text{b}}, \\ g_\text{2,b} = \frac{1/3 E_\text{r,b}}{P_\text{b}}.\end{aligned}$$ A very important part of our simulations’ diagnostics is the LC curve that we calculate by directly integrating the radiation fluxes in the laboratory frame $\vec{F^\prime}_\text{r}$, on a given surface $$L=\int \vec{F^\prime}_\text{r}\cdot\hat{n} dA. \label{eq:luminosity}$$ We use the second-order trapezoidal rule to calculate the integral. The relation between laboratory and comoving frames of the radiation moments is given by a Lorentz transformation [@Mihalas; @Park]: $$\begin{aligned} \nonumber E^\prime_\text{r} &=& W^2\left( E_\text{r} + 2v_iF^i_\text{r} +v_iv_jP^{ij}_\text{r}\right), \\ \nonumber F^{\prime i}_\text{r} &=& W^2v^iE_\text{r} + \left[\delta^i_j +\left(\frac{W-1}{v^2} + W\right)v^iv_j\right]F^j_\text{r} \\ \nonumber & & +Wv_j\left(\delta^i_k + \frac{W-1}{v^2}v^iv_k\right)P^{jk}_\text{r},\\ \nonumber P^{\prime ij}_\text{r} &=& W^2v^iv^jE_\text{r} + W\left(v^i\delta^j_k + v^j\delta^i_k -2\frac{W-1}{v^2}v^iv^jv_k\right)F^k_\text{r} \\ \nonumber & & +\left(\delta^i_k +\frac{W-1}{v^2}v^iv_k\right)\left(\delta^j_k +\frac{W-1}{v^2}v^jv_l\right)P^{kl}_\text{r}, \nonumber\end{aligned}$$ where the primed variables are those measured by an observer in the laboratory frame. As a detecting surface in the laboratory frame, we choose $A$ to be a plane included in the numerical domain. The surface is located in a region, where the optically thin regime holds. We calculate this luminosity in two different planes, the first one perpendicular to $\hat{z}$, whereas the second one is a plane perpendicular to $(\hat{x}+\hat{z})/\sqrt{2}$. In order to evaluate the radiation LC seen by a distant observer, we need to compute quantities in an observer frame that take into account the cosmological effects induced by the redshift at which the source is located. We consider a distant observer whose line of sight makes an angle $\theta$ with respect to the jet axis. We define the time at which the observer sees the radiation coming from a fluid element located at a distance $z_i$ at time $t$ (both measured in a laboratory frame) as $t_\text{det}=t-z_i\cos\theta / c$. Assuming that the emitting source is located at redshift $z$, the time measured in the observer’s frame is $t_\text{obs}= t_\text{det}(1 + z)$ [@Cuesta; @Chucho]. On the other hand, the total luminosity in the observer’s frame is given by $L_\text{obs}(t_\text{obs}) = (1 + z)D^2L$, where $D = [W(1-v_z\cos\theta/c)]^{-1}$ is the Doppler factor [@dermer]. In our calculations we assume a generic $z=1$ and $\theta = 0$ for the perpendicular detector $\hat{n}=\hat{z}$ and $\theta=\pi/4$ for the inclined detector $\hat{n}=(\hat{x}+\hat{z})/\sqrt{2}$. Different jet models {#sec:GRBJet} ==================== The aim of this paper is to study the evolution of jets and their LCs across interesting surrounding media. The first case assumes the medium density has the toy density profile (\[eq:stratified\]), and we analyse various scenarios. The parameters of the cases studied are summarized in Table \[t1\]. In these models, we choose the density and pressure ratios to be initially $\eta=0.01$ and $K = 0.01$, a nozzle radius of $r_\text{b}= 8\times10^8 \text{cm}$ and a beam Lorentz factor $W_\text{b} = 10$. We inject the jet during a finite time $t_\text{inj}=12 \text{s}$, which is a lapse consistent with the amount of total energy of a generic GRB. In all these simulations, the jet propagates along the $z$-direction and the numerical parameters have been standardized such that we use a resolution of $\Delta x=\Delta y=\Delta z=1\times10^{8}$ cm, in a numerical domain of $[-1,1]\times[-1,1]\times[0,5]\times 10^{10}\text{cm}$. This resolution is enough to contain eight zones per beam radius, which is a recommended resolution to properly resolve the internal structure and the jet/external medium interaction [@Aloy]. Model 1 corresponds to a purely hydrodynamical jet and will serve as reference to learn how much the radiation field affects the dynamics of the evolution. This is important, because in the ideal case one would like to avoid the use of post-processing to include the radiation effects that do not back react into the hydrodynamics, and these hydrodynamical jets show how much of the dynamics one loses when the radiation is not considered during the evolution. Even though the jets propagate through a stratified surrounding medium, they are collimated along their entire length and present a morphology similar to that of jets propagating on a constant surrounding medium [@Marti; @MartiII; @AloyII; @Hughes]. The properties are pretty much same among models 1, 2, and 3, namely the beam, bow shock, contact discontinuity, cocoon, back-flow, and internal shocks. However, quantitatively there are differences among these models. In Fig. \[fig:ultra\], we show the rest-mass density profile along the $z$-axis of the purely hydrodynamical jet (model 1), the gas-pressure-dominated jet (model 2), and the radiation-pressure-dominated jet (model 3) at time $t=11$ s in the laboratory frame. From Fig. \[fig:ultra\], we can distinguish a shocked region, which is in front of the jet’s head, the profile is nearly steady, and the radiation field does not contribute significantly because in the three cases the profile is similar. Behind the jet’s head, the radiation field plays a more important role because there the rest-mass density is higher for the radiation dominated case by a 10%. In the following subsections, we will analyse in more detail the implications of the interaction of the radiation field with the fluid. We explore two cases in which the radiation photon field is coupled with the fluid. Model $L_\text{j}(\text{erg}/\text{s})$ $E_\text{j}(\text{erg})$ $E_\text{r,b}\left(\frac{\text{erg}}{\text{cm}^3}\right)$ $g_\text{1,b}$ $(g_\text{2,b})$ ------- ----------------------------------- -------------------------- ----------------------------------------------------------- --------------------------------- -- 1 $2.4\times 10^{51}$ $2.88\times 10^{52}$ $0$ $0$ $(0)$ 2 $3.18\times 10^{51}$ $3.81\times 10^{52}$ $1\times10^{20}$ $0.33$ $(0.33)$ 3 $8.22\times 10^{52}$ $9.8\times 10^{53}$ $1\times10^{22}$ $33.3$ $(33.3)$ : \[t1\] Parameters of the jet models. In all cases, we use the opacities that emulate the free-free, bound-free, bound-bound, and electron-scattering precesses, adiabatic index $\Gamma=4/3$, and Lorentz factor $W_\text{b} = 10$. ![Rest-mass density along $z$-axis of models 1, 2, and 3 in Table \[t1\] at $t=11 \text{s}$. We remind that model 1 is purely hydrodynamical, model 2 is a case where hydrodynamical pressure dominates, and model 3 is radiation pressure dominated. In all the cases, the Lorentz factor is $W_\text{b}=10$.[]{data-label="fig:ultra"}](rho_ultra.eps){width="8cm"} ![Model 2. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted line are obtained measuring the flux through the perpendicular and inclined planes, respectively. Bottom: Maximum of radiation and fluid temperatures, which is located behind the working surface of the jet. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:ultra-gaspress:a"}](lum_ultragaspress.eps "fig:"){width="7cm"} ![Model 2. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted line are obtained measuring the flux through the perpendicular and inclined planes, respectively. Bottom: Maximum of radiation and fluid temperatures, which is located behind the working surface of the jet. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:ultra-gaspress:a"}](temp_ultragaspress.eps "fig:"){width="7cm"} ![Model 3. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted line corresponds to the perpendicular and inclined planes. Bottom: Maximum of both radiation and fluid temperatures, which is located behind the working surface of the jet. The solid and dotted line corresponds to the radiation and fluid temperature, respectively, measured in the laboratory frame.[]{data-label="fig:ultra-radpress:a"}](lum_ultraradpress.eps "fig:"){width="40.00000%"} ![Model 3. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted line corresponds to the perpendicular and inclined planes. Bottom: Maximum of both radiation and fluid temperatures, which is located behind the working surface of the jet. The solid and dotted line corresponds to the radiation and fluid temperature, respectively, measured in the laboratory frame.[]{data-label="fig:ultra-radpress:a"}](temp_ultraradpress.eps "fig:"){width="40.00000%"} ![Lorentz factor for jet models 1, 2 and 3 at $t=11 \ {\rm s}$. The front shock moves faster in the radiation-pressure-dominated case. The inset shows a zoom in at the front shock region.[]{data-label="fig:ultra-LF"}](LF_ultra.eps){width="40.00000%"} ![image](ultra-radrho0.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radrho3.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radrho5.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radrho7.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radW0.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radW3.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radW5.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radW7.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radFr0.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radFr3.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radFr5.eps){width="23.50000%" height="0.35\textheight"} ![image](ultra-radFr7.eps){width="23.50000%" height="0.35\textheight"} Gas-pressure-dominated case {#subsec:gpu} --------------------------- This is model 2 of Table \[t1\] and in Fig. \[fig:ultra-gaspress:a\], we show the LCs measured in the two different planes mentioned above for comparison. In the top right-hand panel of Fig. \[fig:pl-10radrho\], we indicate the surface of the two detectors with white lines, used to calculate the LC using (\[eq:luminosity\]). From now on we will refer to these planes as perpendicular and inclined planes where the LC is measured. Also, in Fig. \[fig:ultra-gaspress:a\], we show the maximum of the temperature of the fluid and radiation for this model 2, which is found to be right behind the bow shock. This figure also illustrates that the initial thermal equilibrium is lost, and the matter temperature is bigger than radiation temperature. The difference is of one order of magnitude, which indicates how close to equilibrium the gas and radiation are in this case. Morphologically, the three highly relativistic jets in Table \[t1\] have a similar structure. The high pressure at the cocoon compacts the jet and some material is convected backwards. It is known that the shock reflected backwards from the contact discontinuity modifies the structure of the jet head and influences the further propagation into the surrounding medium [@Massaglia]. In this context, the radiation pressure helps to push material towards the contact discontinuity by making the internal shock behind the jet head a little bit different. From Fig. \[fig:ultra\], the rest-mass density of the purely hydrodynamical case (model 1) and the gas-pressure-dominated case (model 2) are very similar. Quantitatively though, the maximum of the density of model 2 is slightly higher than that of model 1 by a 2% only in the region behind the front shock. Radiation-pressure-dominated case {#subsec:rpu} --------------------------------- This is model 3 in Table \[t1\]. In Fig. \[fig:ultra-radpress:a\], we show the LCs for this model in the two detectors. The difference with respect to its gas-pressure-dominated counterpart is the amplitude of the LC, which is of the order of $\sim 10^{52} \ \text{erg}/\text{s}$, two orders of magnitude bigger than that of model 2. Also, in Fig. \[fig:ultra-radpress:a\], we show the maximum of the radiation and fluid temperatures. The fluid temperature is bigger than the radiation temperature throughout evolution by approximately one order of magnitude. With respect to the morphology, in Fig. \[fig:pl-10radrho\], we show the evolution of the rest-mass density and Lorentz factor corresponding to model $3$. At $t \sim 5 \text{s}$, we can see the basic properties of the jet, namely a collimation shock in the beam, a bow shock, and the reverse shock produced by the interaction between the jet/external medium and the formation of a cocoon. At this time, the Lorentz factor of the beam remains similar to its initial value, whereas the Lorentz factor of the head and cocoon is smaller. Later on, the jet starts to propagate in a very dilute medium, and consequently the pressure of the cocoon drops and the jet begins to expand laterally into the circumstellar matter. We can see a snapshot of this behaviour at $t \sim 9 \text{s}$. Also, at this time, the beam’s Lorentz factor grows. When the head of the jet reaches the boundary along the $z-\text{axis}$ at $t \sim 13.12 \text{s}$, the Lorentz factor of the beam starts to decrease because the jet is not being injected anymore. We also can see an indication of a vortical flow formed behind the head of the jet similar to a Kelvin-Helmholtz instability that could be better resolved using higher resolution. Notice that a higher resolution would help to capture finer structures such as in Fig. \[fig:Jets\], where snapshot of the rest-mass density of model 3 using three different resolutions at $t \sim 9.37$ s is shown, and the vortical flow behind the jet head is better resolved with the high resolution. Another example is the jets evolved with high resolution in [@Meliani_2010] can capture the fine-scale instabilities at the head of the jet. Comparing this case with its counterpart models, gas pressure dominated and purely hydrodynamics, in Fig. \[fig:ultra\], we can see a difference along the polar axis of the jet. In this model, we see that the rest-mass density of the jet is bigger than that of models 1 and 2. Quantitatively, at time $t = 11 \text{s}$, the difference between the maximum rest mass density of model 3 with respect to its hydrodynamical version is of the order of $\sim 10\%$. In order to learn the effect of radiation pressure in the case when the radiation pressure is dominant, we compare the Lorentz factor profile of model 3 with the gas-pressure dominated (model 2) and the purely hydrodynamical (model 1) cases, at the same time $t=11 \ {\rm s}$. This is shown in Fig. \[fig:ultra-LF\], where we can see that the radiation-pressure-dominated jet propagates slightly faster than the gas-pressure dominated and the purely hydrodynamic models. This illustrates the influence of radiation pressure in the dynamics of the jet. Another important parameter that helps to appreciate the role of the radiative effects incorporated in models 2 and 3 is the optical depth $\tau$. In Fig. \[fig:tao\], we see the evolution of $\tau$, integrated all along the $z-$axis. In this figure, we can see that the optically thick ($\tau >1$) and optically thin ($\tau <1$) regimes are very close to each other in both models. This suggests that the thermal radiation field does not significantly affect the morphology of the system, which explains why the outcomes are so similar despite the injection of radiation energy $E_\text{r}$ in model 3. ![Time evolution of optical depth $\tau$ for models 2 and 3, integrated in the laboratory frame along the $z$-axis. The solid and dotted lines correspond to the models 2 and 3 respectively. We also plot the line $\tau=1$ as a threshold of optically thick and thin.[]{data-label="fig:tao"}](tao.eps){width="40.00000%"} To finalize this section, in Fig. \[fig:pl-10radfr\], we show the $z$ component of the radiative flux during the whole evolution. Because we are assuming that the jet starts in an optically thick regime, the radiative flux in comoving frame at $t = 0$ is equal to zero. At $t\sim 5 \text{s}$, we can see that the highest radiative flux takes place in the head of the jet, collimation, oblique, and blow shock. At $t \sim 9 \text{s}$, the radiative flux decreases in the blow shock. Finally, at $t\sim 13 \text{s}$, we can see how the radiative flux in the jet is dissipating because the matter and radiation was switched off at time $t = 12 \text{s}$. Application to models of Low-Luminosity GRBs {#sec:LLGRBs} ============================================ Now, we study the scenario of jet propagation within a progenitor star, which has been applied to model Low-luminosity GRBs [@BrombergI; @Mizuta; @DeColleII; @Senno; @Geng]. In order to study the LLGRBs, we evolve a jet propagating through its progenitor star assuming the 16TI progenitor density profile [@WoosleyHeger]. This model consists of a pre-supernova star with radius of $R=4\times10^{10}$ cm, $13.95$ solar masses, and $1\%$ of solar metallicity. From $10^9$ cm to $6\times10^9$ cm the density falls quickly as a power law $\sim r^{-1.5}$, from this point to a radius of $4\times10^{10}$ cm, it decays exponentially. Finally, the surrounding medium from the surface to $1.8\times10^{11}$ cm, the density falls off like $\sim r^{-2}$. In Fig. \[fig:16TI\], we show the initial rest-mass density profile of this progenitor model. The rest-mass density and pressure of the beam are $10^3\text{g}~\text{cm}^{-3}$ and $10^{19}\text{g}~ \text{cm}^{-1} \text{s}^{-2}$, respectively. The radius of the jet is $r_\text{b}=2\times10^9 \text{cm}$, and the ratio between their pressures is $K = 0.01$. In our evolution, we do not consider the gravitational field of the progenitor because the jet is moving with high enough velocity such that the dominant effects are due to the interaction of the jet with the matter of the star. [Set up of jets.]{} Initially, we launched the jet at a distance of $10^9$ cm from the centre of star in the $z$ direction, as done in [@Mizuta]. Unlike in the previous scenario, where the jet was injected during $12$ s, in this case the jet is injected during $t_\text{inj}=20$ s. ![Rest-mass density profile of the 16TI progenitor model used in the numerical simulations. This model describes a pre-supernova star with radius of $R=4\times10^{10}$ cm and mass $13.95M_{\odot}$.[]{data-label="fig:16TI"}](16TI.eps){width="40.00000%"} We carried out simulations with a Lorentz factor $W_\text{b}=10$ and consider the case, where the gas and radiation pressure are dominant. The specific values for these parameters are in Table \[t2\]. These models have been standardized with a resolution of $1.25\times10^8 \text{cm}$ for the numerical domain. Model $L_j(\text{erg}/\text{s})$ $E_j(\text{erg})$ $E_{r,b}\left(\text{erg}\text{cm}^{-3}\right)$ $g_\text{1,b}$ $(g_\text{2,b})$ ------- ---------------------------- ---------------------- ------------------------------------------------ --------------------------------- -- 4 $4.97\times 10^{50}$ $9.94\times 10^{51}$ $1\times10^{19}$ $0.33$ $(0.33)$ 5 $1.28\times 10^{52}$ $2.58\times 10^{53}$ $1\times10^{21}$ $33.3$ $(33.3)$ : \[t2\] Parameters of the jets that we evolve on the progenitor model. We use the opacities that emulate the free-free, bound-free, bound-bound and electron-scattering opacities with adiabatic index $\Gamma=4/3$. In the dynamical evolution of model $4$, there are two important phases shown in Fig. \[fig:16ti-10gasrho\]. In the first phase, when the jet is propagating through the star, the Lorentz factor of the beam around the nozzle begins to grow up as expected because the nozzle is continuously injecting energy to this region, whereas the head of the jet propagates with smaller velocity due to the interaction with the stellar envelope. Also, as a consequence of the interaction between the jet and stellar envelope a reverse shock is formed, which interacts with the jet and when the beam crosses the reverse shock a cocoon is created, and the beam is deflected sideways. We show a snapshot of the rest-mass density and Lorentz factor in Fig. \[fig:16ti-10gasrho\] at $t=3 \ {\rm s}$ in the laboratory frame. At this time, the jet is confined by the pressure in the cocoon. The second phase of the evolution starts when the jet breaks out the progenitor star. In Fig. \[fig:16ti-10gasrho\], we show the rest mass density and Lorentz factor at various times. In particular at time $t\sim 7 \ {\rm s}$ after the breakout, the jet expands into a rarefied medium, the gas pressure is smaller than inside of the star, as a result of this, the cocoon starts to expand laterally. Later on, at $t=13 \ {\rm s}$, we can see a stratified cocoon close to the head, which allows the advance of the head. At this time, the jet propagates with high velocity whereas the cocoon expands with a slower velocity. Likewise, we can see that after crossing the collimation shock, which is located around $5\times 10^{10}~\text{cm}$, the jet is not confined enough as to keep the cylindrical radius fixed and it starts a lateral expansion. Finally, at $t\sim 18 \text{s}$ the jet continues to expand laterally whereas the collimation shock is larger than at previous times. ![image](16ti-gas0.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gas1.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gas2.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gas3.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gasW0.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gasW1.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gasW2.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-gasW3.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-rad0.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-rad1.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-rad2.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-rad3.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-radW0.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-radW1.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-radW2.eps){width="23.50000%" height="0.35\textheight"} ![image](16ti-radW3.eps){width="23.50000%" height="0.35\textheight"} The LC and the difference of the gas and radiation temperatures for this model 4 are shown in Fig. \[fig:16ti-gas:lum\]. The first peak in the luminosity measured by a perpendicular plane is produced by the breakout of the jet from the stellar surface. The main peak is due to the one produced by the material near the working surface when it crosses the detector location. The main characteristic of this luminosity curve is that the amplitude of the LC lies within the LLGRBs range ($10^{46} - 10^{49} \ {\rm erg ~s}^{-1}$). The radiation and fluid temperatures have a behaviour similar to those measured in the previous section for LGRBs, that is, the fluid temperature is higher than the radiation temperature. Nevertheless, the fluid temperature is of the order of $10^8 \ {\rm K}$, whereas the radiation temperature is of the order of $10^4 \ {\rm K}$. Unlike in the jets evolving on the stratified medium from the previous section, in this case, the gas and radiation are not as close to thermal equilibrium, showing a difference in temperature, including four orders of magnitude. It is worth noticing that the opacities used in our simulations are appropriate for this range of temperature. ![Model 4. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted lines correspond to the perpendicular and inclined detector planes. Bottom: Maximum of both radiation and fluid temperatures. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:16ti-gas:lum"}](lum_16tigas.eps "fig:"){width="40.00000%"} ![Model 4. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted lines correspond to the perpendicular and inclined detector planes. Bottom: Maximum of both radiation and fluid temperatures. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:16ti-gas:lum"}](temp_16tigas.eps "fig:"){width="40.00000%"} ![Model 5. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted lines correspond to the perpendicular and inclined detector planes. Bottom: Maximum of both radiation and fluid temperatures. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:16ti-rad:lum"}](lum_16tirad.eps "fig:"){width="40.00000%"} ![Model 5. Top: The LCs measured in two different planes transformed to the observer’s frame. The solid and dotted lines correspond to the perpendicular and inclined detector planes. Bottom: Maximum of both radiation and fluid temperatures. The solid and dotted lines correspond to the radiation and fluid temperatures, respectively, measured in the laboratory frame.[]{data-label="fig:16ti-rad:lum"}](temp_16tirad.eps "fig:"){width="40.00000%"} ![Opening angle as a function of time after the jet breakout. The solid and dotted lines correspond to models 5 and 4, respectively, measured in the laboratory frame.[]{data-label="fig:angles"}](angles.eps){width="40.00000%"} The dynamical evolution of model $5$ is similar to that of model $4$. The results are shown in Fig. \[fig:16ti-10radrho\] for model $5$ at the same instances of times of model $4$. At $t \sim 3 \text{s}$, before the jet reaches the surface of the progenitor star, the reverse shock produced by the jet/stellar-envelope interaction forms a cocoon and the pressure in the cocoon confines the jet. At this time, the only difference with respect to the jet in the gas-pressure dominated scenario is that in this case, the jet is propagating faster. At $t = 7.5 \ {\rm s}$, we can see a collimated beam after the jet breaks out the progenitor star, and at this time there are two important differences with respect to the model $4$. The first one is that the collimation shock appears at $\sim 2\times 10^{10} \text{cm}$ further ahead on the beam jet. The second is the jet opening angle $\Theta_\text{b}$ that we show for models $4$ and $5$ in Fig. \[fig:angles\]. This shows that the jet opening angle not only depends on the initial Lorentz factor [@MizutaIII] but on other variables as well, such as the radiative energy density. At $t = 13.12 \ {\rm s}$, a collimated jet continues propagating through a stratified external medium. In this case, the pressure of the cocoon is enough to keep the structure collimated. As time goes on, $t= 18.7 \text{s}$, the matter continues flowing along the forward direction, and the jet remains collimated. In the bottom of Fig. \[fig:16ti-10radrho\], we show the evolution of the Lorentz factor along the progenitor and external medium. By $t = 3.75 \ {\rm s}$ the jet is still within the star, and the jet Lorentz factor near the zone of injection grows, whereas the Lorentz factor in the head of the jet decreases. When the jet breaks out the surface of the star, the jet not only continues propagating faster compared to the jet of model $4$ but with a Lorentz factor slightly bigger. In Fig. \[fig:16ti-rad:lum\], we show the LCs for model $5$. The initial injected energy arrives to the detector with sufficient energy as to obtain the LCs of the order of $10^{48} \ {\rm ergs} \ \text{s}^{-1}$, two orders of magnitude bigger than the LCs of model $4$. The LC measured by the inclined plane shows a plateau following the first main peak. Our results suggest that this plateau regime is due to two effects (see Fig. \[fig:16ti-rad\]): (1) The radiation flux released during the evolution that moves ahead of the GRB jet and interact with the surrounding medium before the outflow drives a shock wave into the external medium and (2) the radiation flux emitted by the GRB jet is detected by an observer in the inclined plane (${\cal O}_1$) before an observer in the perpendicular plane (${\cal O}_2$). Thus, the ${\cal O}_1$ measures a radiation flux, not only before ${\cal O}_2$ but also before the external shock of the jet crosses the inclined plane. This allows the rarefaction signal to travel inwards towards the jet axis and carry radiation flux that contributes to the amplitude of the LC and thereby slow down the decay of the luminosity. Consequently, the rarefaction signal seen by ${\cal O}_2$ arrives with a delay that is enough for the radiation flux to decrease, making ${\cal O}_2$ aware of a deficit of the radiation flux with respect to ${\cal O}_1$, and therefore its LC starts to fall more quickly. The bottom of Fig. \[fig:16ti-rad:lum\] shows the maximum temperatures, in this case the fluid temperature is about $\sim 10^ 8 \ {\rm K}$, whereas the radiation temperature is around $\sim 10^4 \ {\rm K}$, which shows the difference of temperatures during the evolution. Finally, in Fig. \[fig:tao16ti\], we show the evolution of the optical depth $\tau$ for models 4 and 5, where we can see the transition from optically thick ($\tau >1$) to optically thin ($\tau <1$) regimes. Notice that the optical depth associated with the radiation-pressure-dominated scenario, becomes optically thin before the gas-pressure-dominated case. This result is in accordance with the fact that the radiation boosts the jet in the radiation dominated model 5. ![Model 5. The pseudo-colour slice represents the rest-mass density of the outflow, and vectors represent the vectorial field of the radiation flux, measured in the laboratory frame. We can see how the radiation flux moves ahead of the GRB jet, and also how it arrives first at the inclined plane. The white lines indicate the position of the perpendicular and inclined detecting planes.[]{data-label="fig:16ti-rad"}](16ti-rad00.eps "fig:"){width="20.00000%"} ![Model 5. The pseudo-colour slice represents the rest-mass density of the outflow, and vectors represent the vectorial field of the radiation flux, measured in the laboratory frame. We can see how the radiation flux moves ahead of the GRB jet, and also how it arrives first at the inclined plane. The white lines indicate the position of the perpendicular and inclined detecting planes.[]{data-label="fig:16ti-rad"}](16ti-rad.eps "fig:"){width="20.00000%"} ![Optical depth $\tau$ for models 4 and 5, integrated all along the $z$-axis and measured in the laboratory frame. The solid and dotted lines correspond to the models 5 and 4, respectively. We also plot $\tau=1$ as a threshold for optically thick and thin cases.[]{data-label="fig:tao16ti"}](tao_16ti.eps){width="40.00000%"} Discussion and conclusions {#sec:conclusions} ========================== We implemented an RRH code in 3D, with the main objective of constructing the LCs produced by jets. The LC is calculated by the integration of the radiation flux, which is fully coupled to the hydrodynamics during the simulations. We present essential tests that validate our code and applications to the jet propagation in stratified toy media and on to a progenitor star model. As a first application of our code, we considered a model of LGRBs jet evolving through a stratified surrounding medium. For the surrounding medium, we use density and pressure profiles that decrease as power law $\sim r^{-2}$. We also study the dynamics of jets propagating through a progenitor star using the 16TI progenitor model. The simulations assume LTE between the fluid and the radiation field initially. For the definition of the jet, one needs nine parameters: three components for the velocity, the rest-mass density, pressure of matter for hydrodynamics, three components for the radiative flux, and the radiated energy density for the radiation. In particular, we explored the regime in which the jet goes along $z-\text{axis}$ with a highly relativistic velocity. We have combined each one of the jets with values of radiated energy density that created scenarios, where the radiation pressure or gas pressure are dominant, one at a time. Our model is also restricted to a single frequency and opacities associated with free-free, bound-free, bound-bound, and electron-scattering precesses but can be extended to other scenarios. In order to close the RRH system of equations, we use a constant adiabatic law EoS for the fluid, while for the radiation field, we use the M1-closure relation. The use of a more realistic EoS for the fluid in stellar models could provide interesting outcomes because the fluid temperature that could be obtained would be different. While in purely hydrodynamical models, the temperature is an auxiliary quantity, for an RRH model, it is essential because it may substantially change the opacities and subsequently the dynamics of the GRB jets. On the other hand, we try -as a first approach- to describe the dominant physical processes that contribute to the opacity in stellar interiors, using a classical approximation given by Kramers opacities. This simple approximation provides a qualitative and quantitative description of how important fully coupling the radiation with the hydrodynamics is. Under these conditions, we have compared the evolution of jet models with and without the coupling of hydrodynamics to the radiation field and have also shown the effects of gas and radiation-pressure domination. We have found that when gas pressure dominates, the dynamics of the jet is pretty similar to its purely hydrodynamical counterpart. On the other hand, when the radiation pressure is dominant, the effect of radiation is noticeable in comparison to gas pressure and purely hydrodynamics versions because the radiation field acts as a boost accelerating the material density around. This effect is more important in the case of jets propagating across the progenitor density model. Regarding the luminosity LCs, depending on the combination of radiated energy density, the maximum amplitude of the luminosity lies within the range $\sim [10^{50}-10^{52}] \ {\rm erg} ~ {s}^{-1}$ for the jets propagating along the stratified medium. The scenario with the smallest amplitude is the gas pressure dominated, whereas the biggest amplitude is achieved when the jet is dominated by radiation pressure. This is physically consistent with the fact that the energy injected in the radiation-pressure-dominated scenario is bigger by two orders of magnitude than in the scenario, where the gas pressure dominates. Additionally, we compute the maximum of the radiation and fluid temperatures for each of the jets, found to be right behind the working surface. For the gas-pressure-dominated scenario the fluid temperature is of $\sim 10^8 \ {\rm K}$, and the radiation temperature is around one order of magnitude smaller. In the case where the radiation pressure dominates, the gas temperature is of order of $10^{9} \ {\rm K}$ and the radiation temperature is $\sim 10^{8} \ {\rm K}$. An important point here to highlight is that during a large part of the evolution, the fluid temperature is bigger than the radiation temperature, which is consistent with the notion that radiation carries energy and acts as a fluid-cooling mechanism. We also applied our code to evolve jets triggered inside a progenitor. We verified that the jets propagate inside the progenitor and successfully breakout the surface. For our initial conditions, the LCs peak are of order of $\sim 10^{48} \ {\rm erg} ~ {s}^{-1}$ and $\sim 10^{46} \ {\rm erg} ~ {s}^{-1}$ for the radiation-pressure- and gas-pressure-domination scenarios, respectively. This is comparable to the luminosity of LLGRBs. Similarly to the previous scenarios, the fluid temperature is bigger than radiation temperature, however, in this case, the difference is of four orders of magnitude. Even though initially the gas and radiation are assumed in thermal equilibrium, during the evolution the temperature difference between the two reaches four orders of magnitude, which indicates how far from thermal equilibrium these components are. Acknowledgments {#acknowledgments .unnumbered} =============== We appreciate the comments and recommendations from the members of the Valencia Group and from the anonymous Referee. This research is supported by grants CIC-UMSNH-4.9 and CONACyT 258726 (Fondo Sectorial de Investigación para la Educación). Most of the simulations were carried out in the computer farm funded by CONACyT 106466 and the Big Mamma machine of the Laboratory of Artificial Intelligence at the IFM. The authors also acknowledge the computer resources, technical expertise and support provided by the Laboratorio Nacional de Supercómputo del Sureste de México, CONACyT network of national laboratories. We also thank ABACUS Laboratorio de Matemáticas Aplicadas y Cómputo de Alto Rendimiento del CINVESTAV-IPN, grant CONACT-EDOMEX-2011-C01-165873, for providing computer resources. Basic tests {#sec:NT} =========== We only present the standard tests in two dimensions, because the standard 1D tests of our code were presented in [@Panchos]. These tests show the performance of our code in both, optically thick and optically thin regimes. The initial conditions for the tests are the following: 1. Single beam test: This test is intended to verify that our code can work properly in optically thin media, where gas and radiation are decoupled ($\kappa_a=\kappa_{total}=0$). This test consists in injecting a simple beam of radiation and checking that the beam does not present any rupture during its evolution. The test was solved in the plane $z=0$ on a $31\times31$ grid. The boundary conditions for all borders are outflow, except in the given region delimited by $y\in[0.4,0.6]$, where the beam is injected with energy density $100$ times larger than that of the environment. The value of the $a_\text{r}$ and adiabatic index of the gas are $1.118\times10^{17}$, code units, and $4/3$, respectively. In Fig. \[fig:t1\], we can see, at $t=10$, that radiation beam through the whole domain without presenting any rupture. The standard snapshot can be compared with [@Sadowski]. ![\[fig:t1\] Radiation energy density at $t=10$. The radiation beam is located at the left boundary.](singleB.eps){width="8.9cm"} 2. Shadow: In order to verify that the M1 approximation works properly, and to illustrate the difference between the M1 and Eddington approximations we solve this problem. The test consists of an optically thick gas lego circle, immersed in an optically thin environment. We solve the test on the plane $z=0$ on a $100\times50$ grid, with a fixed mass density within a lego circle given by $$\rho_0 = \rho_a + (\rho_\text{b} - \rho_a)e^{(-\sqrt{x^2+y^2+z^2}/\omega^2)},$$ where $\rho_a=10^{-4}$, $\rho_\text{b}=10^3$ and $\omega=0.22$. Initially the system is in thermal equilibrium, and has velocities and radiative fluxes equal to zero. The boundary conditions are as follows: inflow at the left border and outflow at the right border. On all other borders, we use periodic boundary conditions. At the border where we imposed the inflow. The values for radiated energy density, radiated flux and gas temperature are $E_L=a_\text{r}T^4_{g,L}$, $F^x=0.99999E_L$, y $T_{g,L}=100T_a$, respectively. The value of $a_\text{r}=351.37$ and $\kappa_a=\kappa_{total}=\rho_0$. The gas temperature is given such that the pressure is constant throughout the domain, $$T_g = T_a\frac{\rho_a}{\rho_0},$$ with an adiabatic index $\Gamma=1.4$. In Fig. \[fig:t2\], we show the results when the incoming radiation beam passes through the entire domain and reaches a steady state ($t\sim10$). In the upper panel, we show the solution obtained with the Eddington approximation. This approximation treats the radiation field isotropically, as consequence the radiation diffuses rapidly behind the sphere and a shadow cannot be formed. In the lower panel, we show the solution with the M1 approximation, contrary to the Eddington approximation, here we can see that a shadow is formed behind the circle because it is designed to keep moving the flow parallel to itself in optically thin regions for $F_\text{r}\approx E_\text{r}$. The standard snapshot can be compared with [@Sadowski]. ![\[fig:t2\] Radiation energy density at $t=10$. The source of radiation is located at the left boundary. Top: result corresponding to the Eddington approximation. Bottom: result corresponding to the M1 approximation.](shadow.eps){width="8.9cm"} 3. Double shadow: To test the performance and efficiency of our code with multiple sources of light, whose radiative flux is not parallel to the direction of propagation, we implement the double-shadow problem described in [@Sadowski]. In this test, a beam of light is injected into a static environment, where the photons move in different directions than the direction of propagation of the beam. The initial conditions for gas and radiation are exactly same as for the simple shadow test, but unlike that test, the beam is injected into the left border for $y>0.3$ with a radiative flux given by $F^x_\text{r}=0.93E_\text{r}$, $F^y_\text{r}=-0.37E_\text{r}$. We also establish a symmetry of reflection in $y=0$. As a consequence, the domain is illuminated by two radiation beams that intersect. In the region near the upper and lower left corners, where the beams do not overlap, the flow direction follows the direction imposed by the boundary conditions. In the region of the overlap, the density of radiative energy increases twice, whereas the flow is purely horizontal because the vertical component is cancelled in this region. The fact that the incident beam is inclined has an effect on the shadow produced behind the lego sphere. On the one hand, we have regions of partial shadow (penumbra) that result from perpendicular photons, while the region of total shadow (umbra) is limited by the edges of the penumbra. This test shows the limits of the M1 approximation that, in principle, does not limit the specific intensity of radiation to a particular direction. But in the case of multiple light sources, it should be used with caution, as seen in Fig. \[fig:t3\], the M1 approximation produces an extra horizontal shadow along the x-axis, where the penumbra overlaps. In this region, it is expected to be uniform and without shadow [@Jiang]. The standard snapshot, Fig. \[fig:t3\], can be compared with [@Sadowski]. ![\[fig:t3\] Radiation energy density at $t=10$. The source of radiation is located at the left boundary. This result corresponds to the M1-closure approximation.](dshadow.eps){width="8.9cm"} Convergence test {#subsec:rs} ================ In order to test the convergence of the evolution of the jets and select an adequate mesh resolution, we performed a convergence test. For this we use three different resolutions: low resolution which uses $6$ zones per beam radius ($\Delta x_{1}=1.333\times 10^8 \text{cm}$), medium resolution that uses $8$ zones per beam radius ($\Delta x_{2}=3\Delta x_1/4$), employed in all the simulations listed in Tables \[t1\] and \[t2\] and high resolution with $10$ zones per beam radius ($\Delta x_{3}=3\Delta x_1/5$), the high resolution. These resolutions were chosen so that the simulations could be done during the time it takes the jet to travel through the domain. Fig. \[fig:Jets\] shows the morphology of the rest mass density for model $3$ at $t\sim 9.37 \text{s}$. With the three resolutions the morphology is consistent in the sense that the jet head has reached the same position in the z-axis in all the cases. Also, the transverse expansion of the jet is consistent. However, as expected, higher resolution reveals smaller structures and the exact morphology of the turbulent internal part of the jet is not exactly the same for the different resolutions. The exact details of that region are not expected to contribute to the thermal emission, which is dominated by the jet/external interaction, nevertheless they have an effect on the LCs which are different for different resolutions, however their time series should converge. This is the reason why we practice a self-convergence test on the final result of the simulations, namely the LC. Using the three resolutions mentioned $\Delta x_{3} < \Delta x_{2} < \Delta x_{1}$ we calculate the respective LCs $L_1,~L_2,~L_3$ and perform a self convergence test by comparing the differences among them. The convergence factor is given by $$CF(L_1,L_2,L_3)=\frac{L_1 - L_2}{L_2 - L_3} \simeq \frac{(\Delta x_{1})^Q - (\Delta x_{2})^Q}{(\Delta x_{2})^Q - (\Delta x_{3})^Q}. \label{eq:conver}$$ where $Q$ is the accuracy order of the methods. In our case, the numerical methods used, that is, the piecewise linear reconstructor of variables, the HLLE flux formula and the IMEX integrator, all combined in our simulations, in the presence of shocks are expected to converge within first and second order. The convergence factor for for $Q=1$ is $CF_{Q=1}=5/3$, whereas for $Q=2$ is $CF_{Q=2}=175/81\simeq 2.16$. We show in Fig. \[fig:fc\] that the Convergence Factor has a value between these two, which shows our results self-converge with the expected accuracy and thus also shows that our simulations use a resolution within a convergence regime, except for the spikes that usually appear when the numerator or denominator in (\[eq:conver\]) decrease with respect to the other. ![Morphology of the rest-mass density for Model $3$ using three different resolutions. From left to right: lower, standard and high resolution.[]{data-label="fig:Jets"}](Jet_L.eps "fig:"){width="15.60000%"} ![Morphology of the rest-mass density for Model $3$ using three different resolutions. From left to right: lower, standard and high resolution.[]{data-label="fig:Jets"}](Jet_M.eps "fig:"){width="15.60000%"} ![Morphology of the rest-mass density for Model $3$ using three different resolutions. 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--- abstract: \| In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $SU(2)$, are not approximately amenable. [[2010 Mathematics subject classification:]{} primary 46H20; secondary 43A20, 43A62, 46H10, 46J10.\ [Keywords and phrases:]{} approximately amenable Banach algebra, Segal algebra, abstract Segal algebra, locally compact abelian group, compact group, hypergroup, Leptin condition.]{}\ author: - date: 10 February 2012 --- [Introduction]{}\[s:introduction\] The notion of approximate amenability of a Banach algebra was introduced by Ghahramani and Loy in [@aa]. A Banach algebra $\cal A$ is said to be [approximately amenable]{} if for every $\cal A$-bimodule $X$ and every bounded derivation $D:{\cal A}\rightarrow X$, there exists a net $(D_\alpha)$ of inner derivations such that $$\lim_{\alpha} D_\alpha(a) = D(a)\ \ \text{ for all\ } a \in {\cal A}.$$ This is not the original definition but it is equivalent. In [@aa], it is shown that approximately amenable algebras have approximate identities (possibly unbounded) and that approximate amenability is preserved where passing to quotient algebras and to closed ideals that have a bounded approximate identity. In this paper, we study the approximate amenability of proper abstract Segal algebras of Fourier algebras and subsequently, Segal algebras on abelian groups and compact groups. The definition of Segal algebras will be given in Section \[s:SA(G)-abelian\]. 1.5em Approximate amenability of Segal algebras has been studied in several papers. Dales and Loy, in [@da], studied approximate amenability of Segal algebras on $\Bbb{T}$ and $\Bbb{R}$. They showed that certain Segal algebras on $\Bbb{T}$ and $\Bbb{R}$ are not approximately amenable. It was further conjectured that no proper Segal algebra on $\Bbb{T}$ is approximately amenable. Choi and Ghahramani, in [@ch], have shown the stronger fact that no proper Segal algebra on $\Bbb{T}^{d}$ or $\Bbb{R}^{d}$ is approximately amenable. We extend the result of Choi and Ghahramani to apply to all locally compact abelian groups, not just $\Bbb{T}^d$ and $\Bbb{R}^d$. Our approach, like that of Choi-Ghahramani and Dales-Loy, is to apply the Fourier transform and work with abstract Segal subalgebras of the Fourier algebra of a locally compact abelian group. In fact we prove a more general result in Section \[s:SA(G)-abelian\]: when $G$ is an amenable locally compact group, no proper Segal subalgebra of Fourier algebra is approximately amenable. The proof makes use of equivalence between amenability of $G$ and the so-called Leptin condition on $G$. In the rest of the paper, we try to apply similar tools, this time for compact groups. In Section \[s:hypergroup-approach\] our idea is to view the dual of compact groups as a discrete hypergroup. The Fourier space of hypergroups studied as well. In Section \[ss:Folner-condition-on-\^G\], we introduce an analogue for hypergroups of the classical Leptin condition, and show that this holds for certain examples. Eventually, in Section \[ss:on-S\^1(\^SU(2))\], We apply these tools to show that for a class of compact groups including $SU(2)$, every proper Segal algebra is not approximately amenable. [Abstract Segal algebras of Fourier algebra on amenable groups]{}\[s:SA(G)-abelian\] Let $\cal A$ be a commutative Banach algebra. We denote by $\sigma(\cal A)$ the [spectrum]{} of ${\cal A}$ which is also called [maximal ideal space]{} or [character space]{} of ${\cal A}$. A commutative Banach algebra $\cal A$ is called [regular]{}, if for every $\phi$ in $\sigma({\cal A})$ and every $U$, open neighborhood of $\phi$ in the Gelfand topology, there exists an element $a\in {\cal A}$ such that $\phi(a)=1$ and $\psi(a)=0$ for each $\psi\in \sigma({\cal A})\setminus U$. Using Gelfand representation theory, for a commutative semisimple regular Banach algebra $\cal A$, it can be viewed as an algebra of continuous functions on its spectrum, $\sigma(\cal A)$. So for each $a\in \cal A$, ${\operatorname{supp}}(\hat{a})$ is defined the support of function $\hat{a}$ as a subset of $\sigma(\cal A)$. \[p:regularity-of-asa\] Let ${\cal A}$ be a commutative semisimple regular Banach algebra and let $\cal B$ be an abstract Segal algebra of $\cal A$. Then $\cal B$ is also semisimple and regular. Moreover, $\cal B$ contains all elements $a\in\cal A$ such that ${\operatorname{supp}}(\hat{a})$ is compact. By [@bu Theorem 2.1], $\sigma({\cal B})$ is homeomorphic to $\sigma(\cal A)$ and ${\cal B}$ is semisimple. Theorem 3.6.15 and Theorem 3.7.1 of [@ric] imply that for a commutative regular Banach algebra $\cal A$ and a closed subset $E$ of $\sigma({\cal A})$ in the Gelfand spectrum topology, if $a\in {\cal A}$ such that $\|\phi(a)\|\geq\delta>0$ for every $\phi \in E$, then there exists some $a'\in {\cal A}$ such that $\phi(aa')=1$ for every $\phi\in E$. We call this property [local invertibility]{}. On the other hand, [@re Proposition 2.1.14] says that if $\cal A$ is a commutative semisimple algebra with local invertibility and $\cal I$ an ideal of $\cal A$ such that $\bigcap_{a\in \cal I}{\operatorname{Ker}}\hat{a}=\emptyset$. Then $\cal I$ contains all elements $a\in {\cal A}$ such that $\hat{a}$ has a compact support in $\sigma(\cal A)$. Now since ${\cal A}$ is regular, for a closed set $E\subseteq \sigma(\cal A)$ and $\phi\in\sigma({\cal A})\setminus E$, we have some $a\in \cal A$ such that $\hat{a}\|_E\equiv 0$, $\hat{a}(\phi)=1$, and ${\operatorname{supp}}(\hat{a})$ is a compact subset of $\sigma(\cal A)$. Therefore $a\in\cal B$ implying that ${\cal B}$ is regular. Let $G$ be a locally compact group, equipped with a fixed left Haar measure $\lambda$. The Fourier algebra of $G$ was defined and studied by Eymard in [@ey]. In the following lemma, we summarize the main features of the Fourier algebra, denoted by $A(G)$, which we need here. We denote by $C_c(G)$ the space of continuous, compactly supported, complex-valued functions on $G$. \[l:A(G)-general-construction\] Let $G$ be a locally compact group and $K$ be a compact subset of $G$, and let $U$ be an open subset of $G$ such that $K \subset U$. For each measurable set $V$ such that $0<\lambda(V)<\infty$ and $KVV^{-1} \subseteq U$, we can find $u_{V}\in A(G) \cap C_c(G)$ such that 1. [$u_{V}\big(G\big) \subseteq [0,1]$.]{} 2. [$u_{V}\|_{K} \equiv 1$.]{} 3. [${ {\operatorname{supp}}}u_{V} \subseteq U$.]{} 4. [$\\|u_{V}\\|_{A(G)} \leq (\lambda(KV)/\lambda(V))^{1 \over 2}$.]{} The existence of $f_{V} \in A(G) \cap C_{c}(G)$ satisfying $(1)$, $(2)$, and $(3)$ is proved in [@ey Lemma 3.2]. Also based on the proof of [@ey Lemma 3.2] and the definition of the norm of $A(G)$, one can show $(4)$. Since $\sigma(A(G))=G$, Lemma \[l:A(G)-general-construction\] shows that $A(G)$ is a commutative semisimple regular algebra. We say that the Banach algebra $({\cal B},\\|\cdot\\|_{{\cal B}})$ is an [abstract Segal algebra]{} of a Banach algebra $({\cal A},\\|\cdot\\|_{\cal A})$ if\ 1. [ ${\cal B}$ is a dense left ideal in ${\cal A}$.]{} 2. [There exists $M>0$ such that $\\|b\\|_{\cal A} \leq M \\|b\\|_{\cal B}$ for each $b\in {\cal B}$.]{} 3. [There exists $C>0$ such that $\\|ab\\|_{\cal B}\leq C\\|a\\|_{\cal A}\\|b\\|_{\cal B}$ for all $a,b \in {\cal B}$.]{} If ${\cal B}$ is a proper subalgebra of ${\cal A}$, we call it a [proper ]{} abstract Segal algebra of ${\cal A}$. 1.5em Let $G$ be a locally compact group. A linear subspace $S^1(G)$ of $L^1(G)$, the group algebra of $G$, is said to be a [Segal algebra]{} on $G$, if it satisfies the following conditions:\ 1. [$S^1(G)$ is dense in $L^1(G)$.]{} 2. [$S^1(G)$ is a Banach space under some norm ${\Vert \cdot \Vert}_{S^1}$ and ${\Vert f \Vert}_{S^1}\geq {\Vert f \Vert}_1$ for all $f \in S^1(G)$.]{} 3. [$S^1(G)$ is left translation invariant and the map $x\mapsto L_xf$ of $G$ into $S^1(G)$ is continuous where $L_xf(y)=f(x^{-1}y)$.]{} 4. [${\Vert L_xf \Vert}_{S^1} ={\Vert f \Vert}_{S^1}$ for all $f \in S^1(G)$ and $x \in G$.]{} Note that every Segal algebra on $G$ is an abstract Segal algebra of $L^1(G)$ with convolution product. Similarly, we call a Segal algebra on $G$ [proper]{} if it is a proper subalgebra of $L^1(G)$. For the sake of completeness we will give some examples of Segal algebras. - Let ${\cal L}A(G):=L^1(G)\cap A(G)$ and $\|\|\|h\|\|\|:=\\|h\\|_1+\\|h\\|_{A(G)}$ for $h\in{\cal L}A(G)$. Then ${\cal L}A(G)$ with norm $\|\|\|.\|\|\|$ is a Banach space; this space was studied extensively by Ghahramani and Lau in [@gl]. They have shown that ${\cal L}A(G)$ with the convolution product is a Banach algebra called [Lebesgue-Fourier algebra]{} of $G$; moreover, it is a Segal algebra on locally compact group $G$. ${\cal L}A(G)$ is a proper Segal algebra on $G$ if and only if $G$ is not discrete. Also, ${\cal L}A(G)$ with pointwise multiplication is a Banach algebra and even an abstract Segal algebra of $A(G)$. Similarly, ${\cal L}A(G)$ is a proper subset of $A(G)$ if and only if $G$ is not compact. - [The convolution algebra $L^1(G)\cap L^p(G)$ for $1\leq p <\infty$ equipped with the norm ${\Vert f \Vert}_{1}+{\Vert f \Vert}_{p}$ is a Segal algebra.]{} - [Similarly, $L^1(G) \cap C_0(G)$ with respect to the norm ${\Vert f \Vert}_{1}+{\Vert f \Vert}_{\infty}$ is a Segal algebra where $C_0(G)$ is the $C^$-algebra of continuous functions vanishing at infinity.]{} - [ Let $G$ be a compact group, $\cal F$ denote the Fourier transform, and ${\cal L}^p(\widehat{G})$ be the space which will be defined in (\[eq:cal L\^p(\^G)\]). We can see that ${\cal F}^{-1}\big({\cal L}^p(\widehat{G})\big)$, we denote by ${{{\textswab C}^p(G)}}$, equipped with convolution is a subalgebra of $L^1(G)$. For ${\Vert f \Vert}_{{{\textswab C}^p(G)}}:={\Vert {\cal F}f \Vert}_{{{\cal L}^p(\widehat{G})}}$, one can show that for each $1\leq p \leq 2$, $\big({{{\textswab C}^p(G)}},{\Vert \cdot \Vert}_{{{{\textswab C}^p(G)}}}\big)$ is a Segal algebra of $G$.]{} In [@ch], a nice criterion is developed to prove the non-approximate amenability of Banach algebras. At several points below we will rely crucially on that criterion. For this reason, we present a version of that below. Recall that for a Banach algebra $\cal A$, a sequence $(a_n)_{n\in \Bbb{N}}\subseteq \cal A$ is called [multiplier-bounded]{} if $\sup_{n\in\Bbb{N}}{\Vert a_n b \Vert}\leq M {\Vert b \Vert}$ for all $b\in \cal A$. \[t:SUM\] Let $\cal A$ be a Banach algebra. Suppose that there exists an unbounded but multiplier-bounded sequence $(a_n)_{n\geq 1}\subseteq {\cal A}$ such that $$a_na_{n+1}=a_n=a_{n+1}a_n$$ for all $n$. Then $\cal A$ is not approximately amenable. The following theorem is the main theorem of this section. \[t:SA(G)-non-a-a-abelian\] Let $G$ be a locally compact amenable group and $SA(G)$ a proper abstract Segal algebra of $A(G)$. Then $SA(G)$ is not approximately amenable. Since $SA(G)$ is a proper abstract Segal algebra of $A(G)$, the norms ${\Vert \cdot \Vert}_{SA(G)}$ and ${\Vert \cdot \Vert}_{A(G)}$ can not be equivalent. On the other hand, $M {\Vert f \Vert}_{A(G)}\leq {\Vert f \Vert}_{SA(G)}$ for $f\in SA(G)$ and some $M>0$. Therefore, we can find a sequence $(f_n)_{n\in \Bbb{N}}$ in $C_c(G)\cap A(G)$, and hence by Proposition \[p:regularity-of-asa\] in $SA(G)$, such that $$\label{eq:condition-of-f_n} n {\Vert f_n \Vert}_{A(G)} \leq {\Vert f_n \Vert}_{SA(G)}\ \ \text{for all $n\in \Bbb{N}$}.$$ If $G$ is a locally compact group, then $G$ is amenable if and only if it satisfies the [Leptin condition]{} i.e. for every $\epsilon>0$ and compact set $K\subseteq G$, there exists a relatively compact neighborhood $V$ of $e$ such that $\lambda(KV)/\lambda(V) <1+\epsilon$, [@pi Section 2.7]. Fix $D>1$. Using the Leptin condition and Lemma \[l:A(G)-general-construction\], we can generate a sequence $(u_n)_{n\in \Bbb{N}}$ inductively in $A(G)\cap C_c(G)\subseteq SA(G)$ such that $u_n\|_{{\operatorname{supp}}f_n} \equiv 1$, ${ {\operatorname{supp}}}u_{n} \subseteq \{x\in G: u_{n+1}(x)=1\}$, and $\\|u_n\\|_{A(G)} \leq D$. Hence $u_{n}f_n=f_n$ and $u_n u_{n+1}=u_n$ for every $n\in \Bbb{N}$. So here we only need to prove unboundedness of $(u_n)_{n\geq 1}$ in ${\Vert \cdot \Vert}_{SA(G)}$. Suppose otherwise then $\sup_{n\in\Bbb{N}}{\Vert u_n \Vert}_{SA(G)}=C'$ for some $0<C' <\infty$. Then for each $n\in \Bbb{N}$, one can write $${\Vert f_n \Vert}_{SA(G)}={\Vert u_{n} f_n \Vert}_{SA(G)}\leq C {\Vert f_n \Vert}_{A(G)} {\Vert u_{n} \Vert}_{SA(G)} \leq C'C {\Vert f_n \Vert}_{A(G)}$$ for some fixed $C>0$. But this violates the condition (\[eq:condition-of-f\_n\]); therefore, $(u_n)_{n\in\Bbb{N}}$ is unbounded in ${\Vert \cdot \Vert}_{SA(G)}$. Consequently, Theorem \[t:SUM\] shows that $SA(G)$ is not approximately amenable. We should recall that Leptin condition played a crucial role in the proof. Indeed we have used Leptin condition to impose ${\Vert \cdot \Vert}_{A(G)}$-boundedness to the sequence $(u_n)$. As we mentioned before, the approximate amenability of all proper Segal algebras on $\Bbb{R}^d$ has been studied by Choi and Ghahramani in [@ch]. We are therefore motivated to conclude a generalization of their result in the following corollary. \[c:non-a-a-of-proper-segal-algebras-of-abelian-groups\] Let $G$ be a locally compact abelian group. Then every proper Segal subalgebra of $L^1(G)$ is not approximately amenable. Let $S^1(G)$ be a proper Segal algebra on $G$. Applying the Fourier transform on $L^1(G)$, we may transform $S^1(G)$ to a proper abstract Segal algebra $SA(\widehat{G})$ of $A(\widehat{G})$, the Fourier algebra on the dual of group $G$. By Theorem \[t:SA(G)-non-a-a-abelian\], $SA(\widehat{G})$ is not approximately amenable. \[eg:lebesgue-Fourier-algebra\] In particular, by Theorem \[t:SA(G)-non-a-a-abelian\], for amenable locally compact group $G$ the Lebesgue Fourier algebra with pointwise multiplication is approximately amenable if and only if $G$ is a compact group, in which case it equals the Fourier algebra of $G$. Moreover, by Corollary \[c:non-a-a-of-proper-segal-algebras-of-abelian-groups\], for a locally compact abelian group $G$, the Lebesgue Fourier algebra with convolution product is approximately amenable if and only if $G$ is discrete, in which case it equals $\ell^1(G)$. 2.0em [Hypergroups and their Fourier algebra]{}\[s:hypergroup-approach\] Although the dual of a compact group is not a group, in general, it is a (commutative discrete) hypergroup. We give the background needed for this result in Subsection \[ss:dual-of-compact\]. Muruganandam, [@mu1], gave a definition of the [Fourier space]{}, $A(H)$, of a hypergroup $H$ and showed that $A(H)$ is a Banach algebra with pointwise product for certain commutative hypergroups. In Subsection \[ss:Fourier-on-\^G\], we study the Fourier space on the dual of a compact group $G$, denoted by $A(\widehat{G})$. We show that indeed for each compact group $G$, $A(\widehat{G})$ is a Banach algebra. [Preliminaries and notations]{}\[ss:dual-of-compact\] For studying hypergroups, we mainly rely on [@bl]. As a short summary for hypergroups we give the following definitions and facts. Let $(H,,\tilde{ }\;)$ be a (locally compact) [hypergroup]{} possessing a Haar measure $h$. The notation $AB$ stands for $$\bigcup\{{\operatorname{supp}}(\delta_x\delta_y):\; \text{for all }\;x\in A, y\in B\}$$ for $A,B$ subsets of the hypergroup $H$. With abuse of notation, we use $xA$ to imply $\{x\}A$. Let $C_c(H)$ be the space of all complex valued compact supported continuous functions over $H$. We define $$L_xf(y)=\int_H f(t) d\delta_x\delta_y(t) \ \ f\in C_c(H),\ x,y\in H.$$ Defining $$f_hg(x):=\int_{H} f(t) L_{\tilde{t}}g(x) dh(t), \ \ \tilde{f}(x):=f(\tilde{x}),\ \text{ and}\ \ f^:=\overline{(\tilde{f})},$$ we can see that all of the functions $f:H\rightarrow \Bbb{C}$ such that $${\Vert f \Vert}_{L^1(H,h)}:=\int_{H} \|f(t)\| dh(t)<\infty$$ form a Banach $$-algebra, denoted by $(L^1(H,h),_h,{\Vert \cdot \Vert}_h)$; it is called the [hypergroup algebra]{} of $H$. 1.5em If $H$ is discrete and $h(e)=1$, we have $$h(x)=(\delta_{\tilde{x}}\delta_x(e))^{-1}.$$ \[l:Haar-convolution-of-dirac-functions\] Let $H$ be a discrete hypergroup. For each pair $x,y\in H$, $$\delta_x _h \delta_y(z) = \delta_x \delta_y(z)\frac{h(x)h(y)}{h(z)}$$ for each $z\in H$. 1.5em In this section, let $G$ be a compact group and $\widehat{G}$ the set of all irreducible unitary representations of $G$. In this paper we follow the notation of [@du] for the dual of compact groups. Where ${\cal H}_\pi$ is the finite dimensional Hilbert space related to the representation $\pi\in\widehat{G}$, we define $\chi_\pi:=Tr\pi$, the group character generated by $\pi$ and $d_\pi$ denotes the dimension of ${\cal H}_\pi$. Let $\phi=\{\phi_\pi:\; \pi\in\widehat{G}\}$ if $\phi_\pi\in{\cal B}({\cal H}_\pi)$ for each $\pi$ and define $${\Vert \phi \Vert}_{{\cal L}^\infty(\widehat{G})}:=\sup_\pi{\Vert \phi_\pi \Vert}_\infty$$ for ${\Vert \cdot \Vert}_\infty$, the operator norm. The set of all those $\phi$’s with ${\Vert \phi \Vert}_{{\cal L}^\infty(\widehat{G})}<\infty$ forms a $C^$-algebra; we denote it by ${{\cal L}^\infty(\widehat{G})}$. It is well known that ${\cal L}^\infty(\widehat{G})$ is isomorphic to the von Neumann algebra of $G$ i.e. the dual of $A(G)$, see [@du 8.4.17]. We define $$\label{eq:cal L^p(^G)} {\cal L}^p(\widehat{G})=\{\phi\in{\cal L}^\infty(\widehat{G}): {\Vert \phi \Vert}_{{\cal L}^p(\widehat{G})}^p:=\sum_{\pi \in\widehat{G}} d_\pi {\Vert \phi_\pi \Vert}_p^p<\infty\},$$ for ${\Vert \cdot \Vert}_p$, the $p$-Schatten norm. For each $p$, ${\cal L}^p(\widehat{G})$ is an ideal of ${\cal L}^\infty(\widehat{G})$, see [@du 8.3]. Moreover, we define $${\cal C}_0(\widehat{G})=\{\phi\in{\cal L}^\infty(\widehat{G}):\;\lim_{\pi\rightarrow\infty} {\Vert \phi_\pi \Vert}_\infty=0\}.$$ For each $f\in L^1(G)$, ${\cal F}(f)=(\hat{f}(\pi))_{\pi\in\widehat{G}}$ belongs to ${\cal C}_0(\widehat{G})$, where $\cal F$ denotes Fourier transform and $$\hat{f}(\pi)=\int_G f(x) \pi(x^{-1}) dx.$$ Indeed, ${\cal F}(L^1(G))$ is a dense subset of ${\cal C}_0(\widehat{G})$ and ${\cal F}$ is an isomorphism from Banach algebra $L^1(G)$ onto its image. For each two irreducible representations $\pi_{1},\pi_{2}\in\widehat{G}$, we know that $\pi_1 \otimes \pi_2$ can be written as a product of $\pi'_1,\cdots,\pi'_n$ elements of $\widehat{G}$ with respective multiplicities $m_1,\cdots,m_n$, i.e. $$\pi_1\otimes \pi_2 \cong \bigoplus_{i=1}^n m_i \pi'_i.$$ We define a convolution on $\ell^1(\widehat{G})$ by $$\label{eq:hypergroup-convolution-on-^G} \delta_{\pi_1} \delta_{\pi_2}:=\sum_{i=1}^n \frac{m_i d_{\pi'_i}}{d_{\pi_1}d_{\pi_2}}\delta_{\pi'_i}$$ and define an involution by $ \tilde{\pi}=\overline{\pi}$ for all $\pi,\pi_1,\pi_2\in \widehat{G}$. It is straightforward to verify that $(\widehat{G}, * ,\tilde{ }\;)$ forms a discrete commutative hypergroup such that $\pi_0$, the trivial representation of $G$, is the identity element of $\widehat{G}$ and $h(\pi)=d_\pi^2$ is the Haar measure of $\widehat{G}$. 2.0em \[eg:SU(2)\] Let $\widehat{SU(2)}$ be the hypergroup of all irreducible representations of the compact group $SU(2)$. We know that $$\widehat{SU(2)}=(\pi_\ell)_{\ell \in 0,\frac{1}{2},1,\frac{3}{2}, \cdots}$$ where the dimension of $\pi_\ell$ is $2\ell+1$, see [@he2 29.13]. Moreover, $$\pi_\ell \oplus \pi_{\ell'}= \bigoplus_{r=\|\ell-\ell'\|}^{\ell+\ell'} \pi_r = \pi_{\|\ell-\ell'\|} \oplus \pi_{\|\ell-\ell'\|+1 } \oplus \cdots \oplus \pi_{\ell+\ell'} \ \ \ \text{(by \cite[Theorem 29.26]{he2})}.$$ So using Definition \[eq:hypergroup-convolution-on-\^G\], we have that $$\delta_{\pi_\ell}\delta_{\pi_{\ell'}} = \sum_{r=\|\ell-\ell'\|}^{\ell+\ell'} \frac{(2r+1)}{(2\ell+1)(2\ell'+1)}\delta_{\pi_r}.$$ Also $ \tilde{\pi_\ell}=\pi_\ell $ and $h(\pi_\ell)=(2\ell+1)^2$ for all $\ell$. 2.0em \[eg:product-of-finite-groups\] Suppose that $\{G_i\}_{i\in{{\bf I}}}$ is a non-empty family of compact groups for arbitrary indexing set ${{\bf I}}$. Let $G:=\prod_{i\in{{\bf I}}}G_i$ be the product of $\{G_i\}_{i\in {{\bf I}}}$ i.e. $ G = \{(x_i)_{i\in {{\bf I}}}:\ x_i\in G_i\}$ equipped with product topology. Then $G$ is a compact group and by [@he2 Theorem 27.43], $$\widehat{G}=\{\pi=\bigotimes_{i\in{{\bf I}}}\pi_i:\ \text{such that}\ \pi_i\in\widehat{G}_i \ \text{and}\ \pi_i=\pi_0\ \text{except for finitely many $i\in{{\bf I}}$}\}$$ equipped with the discrete topology. Moreover, for each $\pi=\bigotimes_{i\in{{\bf I}}}\pi_i \in \widehat{G}$, $d_\pi=\prod_{i\in{{\bf I}}} d_{\pi_i}$. If $\pi_k=\bigotimes_{i\in{{\bf I}}} \pi_i^{(k)} \in \widehat{G}$ for $k=1,2$, one can show that $$\delta_{\pi_1} \delta_{\pi_2}(\pi)= \prod_{i\in {{\bf I}}} \delta_{\pi_i^{(1)}} _{\widehat{G}_i} \delta_{\pi_i^{(2)}}(\pi_i)\ \ \ \ \text{for}\ \pi=\bigotimes_{i\in{{\bf I}}}\pi_i \in \widehat{G},$$ where $_{\widehat{G}_i}$ is the hypergroup product in $\widehat{G}_i$ for each $i\in{{\bf I}}$. Also, each character $\chi$ of $G$ corresponds to a family of characters $(\chi_i)_{i\in{{\bf I}}}$ such that $\chi_i$ is a character of $G_i$ and $\chi(x)=\prod_{i\in{{\bf I}}}\chi_i(x_i)$ for each $x=(x_i)_{i\in{{\bf I}}}\in G$. Note that $\chi_1\equiv 1$ for all of $i\in{{\bf I}}$ except finitely many. [The Fourier algebra of the dual of a compact group]{}\[ss:Fourier-on-\^G\] For a compact hypergroup $H$, first Vrem in [@vr] defined the [Fourier space]{} similar to the Fourier algebra of a compact group. Subsequently, Muruganandam, [@mu1], defined the [Fourier-Stieltjes space]{} on an arbitrary (not necessary compact) hypergroup $H$ using irreducible representations of $H$ analogous to the Fourier-Stieltjes algebra on locally compact groups. Subsequently, he defines the [ Fourier space]{} of hypergroup $H$, as a closed subspace of the Fourier-Stieltjes algebra, generated by $\{f_h\tilde{f}:\; f\in L^2(H,h)\}$. Also Muruganandam shows that where $H$ is commutative $A(H)$ is $ \{f_h\tilde{g}:\; f,g\in L^2(H,h)\}$ and ${\Vert u \Vert}_{A(H)}=\inf {\Vert f \Vert}_2 {\Vert g \Vert}_2$ for all $f,g\in L^2(H,h)$ such that $u=f\tilde{g}$. He calls hypergroup $H$ a [regular Fourier hypergroup]{}, if the Banach space $({A}(H),{\Vert \cdot \Vert}_{{A}(H)})$ equipped with pointwise product is a Banach algebra. We prove a hypergroup version of Lemma \[l:A(G)-general-construction\] which shows some important properties of the Banach space $A(H)$ for an arbitrary hypergroup $H$ (not necessarily a regular Fourier hypergroup) and $\widehat{G}$ is a regular Fourier hypergroup. Some parts of the following Lemma have already been shown in [@vr] for compact hypergroups and that proof is applicable to general hypergroups. Here we present a complete proof for the lemma. \[l:A(H)-properties\] Let $H$ be a hypergroup, $K$ a compact subset of $H$ and $U$ an open subset of $H$ such that $K\subset U$. Then for each measurable set $V$ such that $0<h_H(V)<\infty$ and $\overline{K V\tilde{V}} \subseteq U$, there exists some $u_V\in A(H) \cap C_c(H)$ such that:\ 1. [$u_V(H)\geq 0$.]{} 2. [$u_V\|_K=1$.]{} 3. [${\operatorname{supp}}(u_V) \subseteq U$.]{} 4. [${\Vert u_V \Vert}_{A(H)} \leq \big( h_H(KV)/{h_H(V)}\big)^{\frac{1}{2}}$.]{} Let us define $$u_V:=\frac{1}{h_H(V)} 1_{KV} _h \tilde{1}_{V}.$$ Clearly $u_V\geq 0$. Moreover, for each $x\in K$ , $$\begin{aligned} h_H(V) u_V(x) &=& 1_{KV} _h \tilde{1}_{V}(x)\\ &=& \int_{ H} 1_{KV}(t) L_{\tilde{t}}\tilde{1}_V(x) dh_H(t) \\ &=& \int_{ H} 1_{KV}(t) L_{\tilde{x}} {1}_V(t) dh_H(t)\\ &=& \int_{t\in H} L_{x}1_{KV}(t) {1}_V(t) dh_H(t)\ \ \ \text{(by \cite[Theorem~1.3.21]{bl})}\\ &=& \int_{ V} \langle 1_{KV} , \delta_{{x}}\delta_{t} \rangle dh_H(t)\\ &=& {h_H(V)}.\end{aligned}$$ Also [@bl Proposition 1.2.12] implies that $${\operatorname{supp}}( 1_{KV} _h \tilde{1}_{V})\subseteq \overline{\left(KV \tilde{V}\right)}\subseteq U.$$ Finally, by [@mu1 Proposition 2.8], we know that $$\begin{aligned} {\Vert u_V \Vert}_{A(H)}&\leq & \frac{{\Vert 1_{KV} \Vert}_2 {\Vert 1_V \Vert}_2}{h_H(V)}= \frac{h_H(KV)^{\frac{1}{2}} h_H(1_V)^{\frac{1}{2}}}{h_H(V)}= \frac{ h_H(KV)^{\frac{1}{2}}}{h_H(V){\frac{1}{2}}}.\end{aligned}$$ \[r:existence-of-the-V\] For each pair $K,U$ such that $K \subset U$, we can always find a relatively compact neighborhood $V$ of $e_H$ that satisfies the conditions in Lemma \[l:A(H)-properties\]. But the proof is quite long and in our application the existence of such $V$ will be clear. Given a commutative hypergroup, it is not immediate that it is a regular Fourier hypergroup or not. We will show that when $G$ is a compact group, the hypergroup $\widehat{G}$ is a regular Fourier hypergroup. For $A(G)$, Fourier algebra on $G$, we define $$ZA(G):=\{f\in A(G): f(yxy^{-1})=f(x)\ \text{for all $x\in G$}\}.$$ which is a Banach algebra with pointwise product and ${\Vert \cdot \Vert}_{A(G)}$. \[t:Fourier-of-\^G\] Let $G$ be a compact group. Then $\widehat{G}$ is a regular Fourier hypergroup and $A(\widehat{G})$ is isometrically isomorphic with the center of the group algebra $G$, i.e. $A(\widehat{G})\cong ZL^1(G)$. Moreover, the hypergroup algebra of $\widehat{G}$, $L^1(\widehat{G},h)$, is isometrically isomorphic with $ZA(G)$. Let $\cal F$ be the Fourier transform on $L^1(G)$. We know that ${\cal F}\|_{L^2(G)}$ is an isometric isomorphism from $L^2(G)$ onto $ {\cal L}^2(\widehat{G})$. By the properties of the Fourier transform, [@du Proposition 4.2], for each $f\in ZL^2(G)$ and $g\in L^1(G)$ we have $$\label{eq:ZL^2-commutes-with-everything} {\cal F}(f)\circ {\cal F}(g)={\cal F}(fg)={\cal F}(gf)={\cal F}(g)\circ {\cal F}(f).$$ So ${\cal F}(f)$ commutes with all elements of ${\cal C}_0(\widehat{G})$; therefore, ${\cal F}(f)=(\alpha_\pi I_{d_\pi \times d_\pi})_{\pi\in\widehat{G}}$ for a family of scalars $(\alpha_\pi)_{\pi \in \widehat{G}}$ in $\Bbb{C}$. Hence, $$\begin{aligned} {\Vert {\cal F}f \Vert}_2^2 = \sum_{\pi\in\widehat{G}} d_\pi {\Vert \widehat{f}(\pi) \Vert}_2^2 = \sum_{\pi\in\widehat{G}} d_\pi \alpha_\pi^2 {\Vert I_{d_\pi\times d_\pi} \Vert}_2^2 = \sum_{\pi\in\widehat{G}} \alpha_\pi^2 {d_\pi}^2 = \sum_{\pi\in\widehat{G}} \alpha_\pi^2 h(\pi).\end{aligned}$$ Using the preceding identity, we define $ {\cal T}: ZL^2(G) \rightarrow L^2(\widehat{G},h)$ by ${\cal T}(f)=(\alpha_\pi)_{\pi\in\widehat{G}}$. Note that $\{\chi_\pi\}_{\pi\in\widehat{G}}$ forms an orthonormal basis for $ZL^2(G)$. Since ${\cal F}(\chi_{\tilde{\pi}})=d_{\tilde{\pi}}^{-1} I_{d_{\tilde{\pi}\times {\tilde{\pi}}}}$, ${\cal T}(\tilde{f})={\cal T}(f\tilde{)}$ for each $f\in L^2(G)$ where $\tilde{f}(x)=f(x^{-1})$. So ${\cal T}$ is an isometric isomorphism from $ZL^2(G)$ onto $L^2(\widehat{G},h)$. We claim that ${\cal T}(f \tilde{g})={\cal T}(f) _h {\cal T}(g\tilde{)}$ for all $f,g\in ZL^2(G)$. To prove our claim it is enough to show that ${\cal T}(\chi_{\pi_1} \chi_{\pi_2})={\cal T}(\chi_{\pi_1}) _h {\cal T}(\chi_{\pi_2})$ for $\pi_1,\pi_2\in \widehat{G}$. Therefore, using Lemma \[l:Haar-convolution-of-dirac-functions\], for each two representations $\pi_1,\pi_2\in\widehat{G}$, we have $$\begin{aligned} {\cal T}(\chi_{\pi_1}\chi_{\pi_2}) &=& {\cal T}\left( \sum_{i=1}^n m_i \chi_{\pi'_i}\right)\\ &=& \sum_{i=1}^n m_i {\cal T}(\chi_{\pi'_i})\\ &=& \sum_{i=1}^n m_i d_{\pi'_i}^{-1} \delta_{\pi'_i}\\ &=& d_{\pi_1}^{-1} \delta_{\pi_1} _h d_{\pi_2}^{-1}\delta_{\pi_2}\\ &=& {\cal T}(\chi_{\pi_1})_h {\cal T}(\chi_{\pi_2}).\end{aligned}$$ 1.0em Now we can define a surjective extension $ {\cal T}:ZL^1(G) \rightarrow {A}(\widehat{G})$, using the fact that $lin\{\chi_\pi\}_{\pi\in\widehat{G}}$ is dense in $ZL^1(G)$ as well and ${\Vert f \Vert}_1=\inf {\Vert g_1 \Vert}_2{\Vert g_2 \Vert}_2$ for all $g_1,g_2\in L^2(G)$ such that $f=g_1\tilde{g_2}$. Using the definition of the norm of $A(\widehat{G})$, ${\Vert {\cal T}(f) \Vert}_{{A}(\widehat{G})}={\Vert f \Vert}_1$ for each $f\in ZL^1(G)$. To show that the extension of ${\cal T}$ is onto, for each pair $g_1,g_2\in ZL^2(G)$, we note that $g_1\tilde{g_2}\in ZL^1(G)$. So, ${\cal T}$ is an isometric isomorphism. This implies that ${A}(\widehat{G})$ is a Banach algebra with pointwise product and hence $\big({A}(\widehat{G}),\cdot,{\Vert \cdot \Vert}_{{A}(\widehat{G})}\big)\cong \big(ZL^1({G}),,{\Vert \cdot \Vert}_{1}\big)$. 1.5em The second part is similar to the first part of the proof. This time we consider the restriction of the Fourier transform from $ZA(G)$ onto $ {\cal L}^1(\widehat{G})$. Again by an argument similar to (\[eq:ZL\^2-commutes-with-everything\]), we define an isometric mapping $\cal T'$ from $ZA(G)$ onto $L^1(\widehat{G},h)$. Since $lin\{\chi_\pi\}_{\pi\in \widehat{G}}$ is dense in $ZA(G)$, we observe that ${\cal T}'$ is an isometric isomorphism from $ZA(G)$ as an $$-algebra with complex conjugate and pointwise product onto $L^1(\widehat{G},h)$ as an $$-algebra with $_h$ convolution. 2em [The Leptin condition on Hypergroups]{}\[ss:Folner-condition-on-\^G\] In the proof of Theorem \[t:SA(G)-non-a-a-abelian\], we used the Leptin condition for amenable groups. In this subsection we study the Leptin condition for the dual of hypergroups. In [@sk], the Reiter condition was introduced for amenable hypergroups. Although the Reiter condition on hypergroups is defined similar to amenable groups, the Leptin condition is a problem for hypergroups. There are some attempts to answer to this question for some special hypergroups in [@la4]. Recall that for each two subsets $A$ and $B$ of $X$, we denote the set $(A\setminus B) \cup (B\setminus A)$ by $A\bigtriangleup B$. \[d:Folner-Leptin-condition\] Let $H$ be a hypergroup. [ We say that $H$ satisfies the [Leptin condition]{} if for every compact subset $K$ of $H$ and $\epsilon>0$, there exists a measurable set $V$ in $H$ such that $0<h(V)<\infty$ and $h(KV)/h(V) < 1+\epsilon$.]{} We will use the Leptin condition, in the case where $H$ is the dual of a compact group $G$, to study approximate amenability for Segal algebras on $G$. \[r:relatively-compact\] In the definitions of the conditions mentioned above, we can suppose that $V$ is a compact measurable set. To show this fact suppose that $H$ satisfies the Leptin condition. For compact subset $K$ of $H$ and $\epsilon>0$, there exists a measurable set $V$ such that $h(KV)/h(V) < 1 + \epsilon$. Using regularity of $h$, we can find compact set $V_1\subseteq V$ such that $h(V\setminus V_1) < h(V)/n$ for some positive integer $n>0$. It implies that $0< h(V_1)$ and $ h(V)/h(V_1) < {n}/{(n-1)}$. Therefore, $$\frac{h(KV_1)}{h(V_1)} \leq \frac{h(V)}{h(V_1)} \left( \frac{h(KV_1)}{h(V)}\right)< \frac{n}{n-1}(1+\epsilon).$$ So we can add compactness of $V$ to the definition of the Leptin condition. Note that since the duals of compact groups are commutative, they are all amenable hypergroups, [@sk], but as we mentioned it does not say anything about the Leptin condition on those hypergroups. So the next question is for which compact groups $G$ do the hypergroups $\widehat{G}$ satisfy those conditions. We will now show that Examples \[eg:SU(2)\] and \[eg:product-of-finite-groups\] sometimes satisfy these conditions. \[p:Leptin-SU(2)\^\] The hypergroup $\widehat{SU(2)}$ satisfies the Leptin condition. Given compact subset $K$ of $\widehat{SU(2)}$ and $\epsilon>0$. Let $K$ and $\epsilon>0$ are given. $k:=\sup\{\ell:\; \pi_\ell\in K\}$. We select $m \geq k$ such that for $V=\{\pi_\ell\}_{\ell=0}^m$, $$\begin{aligned} \label{eq:proof-of-Leptin} \frac{h(\pi_k V)}{h(V)} &=& \frac{ \sum_{\ell=1}^{2m+2k+1}\ell^2 }{\sum_{\ell=1}^{2m+1} \ell^2}\\ &=& \frac{\frac{1}{3}(2m+2k+1)^3 + \frac{1}{2}(2m+2k+1)^2 + \frac{1}{6}(2m+2k+1)}{\frac{1}{3}(2m+1)^3 + \frac{1}{2}(2m+1)^2 + \frac{1}{6}(2m+1)}< 1+ \epsilon.\nonumber\end{aligned}$$ But also for each $x\in K$, $xV\subseteq \pi_kV$. So using (\[eq:proof-of-Leptin\]), $$\frac{h(K V)}{h(V)} = \frac{h(\pi_k V)}{h(V)} < 1+ \epsilon.$$ \[p:Leptin-for-Prod-of-finite-groups\] Let $\{G_i\}_{i\in {{\bf I}}}$ be a family of compact groups whose duals have the Leptin condition and $G=\prod_{i\in{{\bf I}}}G_i$ is their product equipped with product topology. Then $\widehat{G}$ satisfies the Leptin condition. Let $K \subseteq \widehat{G}$ be an arbitrary compact subset. Then, there exists some $F\subseteq {{\bf I}}$ finite such that $K\subseteq \bigotimes_{i\in F} K_i \otimes E_F^c$ where $K_i$ is a compact subset of $\widehat{G}_i$ and $E_F^c=\bigotimes_{i\in{{\bf I}}\setminus F} \pi_0$ where $\pi_{0}$’s are the trivial representations of the corresponding $\widehat{G}_i$. Using the Leptin condition for each $G_i$, there exists some compact set $V_i$ which satisfies the Leptin condition for $K_i$ and $\epsilon>0$ i.e. $h_{G_i}(K_iV_i)<(1+ \epsilon) h_{G_i}(V_i)$. Therefore, for the compact set $V=(\bigotimes_{i\in F} V_i )\otimes E_F^c$, $$\frac{h(KV)}{h(V)} \leq \prod_{i\in F} \frac{ h_{G_i}(K_iV_i)}{h_{G_i}(V_i)} < (1+\epsilon)^{\|F\|}.$$ If $G$ is finite then $\widehat{G}$ satisfies the Leptin condition; hence, for a family of finite groups say $\{G_i\}_{i\in{{\bf I}}}$, $G:=\prod_{i\in{{\bf I}}}G_i$, $\widehat{G}$ satisfies the Leptin condition. [Segal algebras on compact groups whose duals satisfy Leptin condition]{}\[ss:on-S\^1(\^SU(2))\] In this section, we apply hypergroup approach to the original questions for Segal algebras. We show that every proper Segal algebra on ${G}$ is not approximately amenable if $G$ is a compact group if $\widehat{G}$ satisfies the Leptin condition. First we need a general lemma for Banach algebras. \[l:characters-in-ZS\^1(G)\] Let ${\cal A}$ be a Banach algebra and ${\cal J}$ be a dense left ideal of ${\cal A}$. Then for each idempotent element $p$ in the center of algebra ${\cal A}$ i.e. $p^2=p \in Z({\cal A})$, $p$ belongs to ${\cal J}$. Since $\cal J$ is dense in $\cal A$, there exists an element $a \in \cal J$ such that $ {\Vert p - a \Vert}_{\cal A} <1$. Let us define $$b:= p + \sum_{n=1}^\infty (p - a)^{n}.$$ One can check that $p b - p b ( p - a) = p b a$, which is an element in $\cal J$. On the other hand, $$\begin{aligned} p b - p b ( p - a) &=& p \left( p + \sum_{n=1}^\infty(p -a)^{ n}\right) - p \left( p + \sum_{n=1}^\infty (p -a)^{ n} \right) (p -a)\\ &=& p + p \sum_{n=1}^\infty(p -a)^{ n} - p \sum_{n=2}^\infty (p -a)^{ n}- p (p -a)\\ &=& p + p (p -a) - p (p -a) =p.\end{aligned}$$ 2.0em The main theorem of this section is as follows. \[t:Segal-of-S\^1(G)-G-Leptin\] Let $G$ be a compact group such that $\widehat{G}$ satisfies the Leptin condition. Then every proper Segal algebra on $G$ is not approximately amenable. Let $S^1(G)$ be a proper Segal algebra on $G$. By Lemma \[l:characters-in-ZS\^1(G)\], $S^1(G)$ contains all central idempotents $d_\pi\chi_\pi$ for each $\pi\in \widehat{G}$. If $\cal T$ is the map defined in the proof of Theorem \[t:Fourier-of-\^G\], then ${\cal T}^{-1}(\delta_\pi)=d_\pi\chi_\pi\in S^1(G)$. So in order to use Theorem \[t:SUM\], we look for a suitable sequence in $A(\widehat{G})$ with compact supports. 1.0em Fix $D>1$. Using the Leptin condition on $\widehat{G}$, for every arbitrary non-void compact set $K$ in $\widehat{G}$, we can find a finite subset $V_K$ of $\widehat{G}$ such that $h(KV_K)/h(V_K)<D^{2}$. Using Lemma \[l:A(H)-properties\] for $$v_K:=\frac{1}{h(V_K)} 1_{K V_K} _h \tilde{1}_{V_K}$$ we have ${\Vert v_K \Vert}_{A(\widehat{G})} < D$ and $v_{K}\|_{K}\equiv 1$. We consider the net $\{v_K: K\subseteq \widehat{G}\ \text{compact}\}$ in $A(\widehat{G})$ where $v_{K_1} \preccurlyeq v_{K_2}$ whenever $v_{K_1}v_{K_2} = v_{K_1}$. So $(v_K)_{ K\subseteq \widehat{G}}$ forms a ${\Vert \cdot \Vert}_{A(\widehat{G})}$-bounded net in $A(\widehat{G}) \cap c_c(\widehat{G})$. Let $f\in A(\widehat{G})\cap c_c(\widehat{G})$ with $K={\operatorname{supp}}f$. Then $v_K f= f$. Therefore, $(v_K)_{K\subseteq \widehat{G}}$ is a bounded approximate identity of $A(\widehat{G})$. Using ${\cal T}$ defined in the proof of Theorem \[t:Fourier-of-\^G\], we can define the net $(u_K)_{K\subseteq \widehat{G}}$ in $S^1(G)$ by $ u_K:={{\cal T} }^{-1}(v_K)$. Now, we show that $(u_K)_{K\subseteq \widehat{G}}$ satisfies all the conditions of the Theorem \[t:SUM\]. First of all, since ${\cal T}$ is an isometry from $ZL^1(G)$ onto $A(\widehat{G})$, $(u_K)_{K\subseteq \widehat{G}}$ is a ${\Vert \cdot \Vert}_1$-bounded sequence in $S^1(G)$. Therefore, it forms a multiplier-bounded sequence in the Segal algebra. Moreover, since ${\cal T}$ is an isomorphism, $$u_{K_1} u_{K_2} = {\cal T}^{-1}(v_{K_1}) * {\cal T}^{-1}(v_{K_2})={\cal T}^{-1} (v_{K_1} v_{K_2}) = {\cal T}^{-1}(v_{K_1}) = u_{K_1}$$ for $u_{K_1}\preccurlyeq u_{K_2}$. Toward a contradiction, suppose that $ \sup_{K\subseteq \widehat{G}} {\Vert u_K \Vert}_{S^1(G) }\leq C$ for some $C>0$. We know that for a Segal algebra $S^1(G)$, the group $G$ is a SIN group if and only if $S^1(G)$ has a central approximate identity which is bounded in $L^1$-norm, [@ko]. So let $(e_\alpha)_{\alpha}$ be a central approximate identity of $S^1(G)$ which is ${\Vert \cdot \Vert}_1$-bounded. Since $ZS^1(G)=\overline{lin\{\chi_n\}_{n\in\Bbb{N}}}^{{\Vert \cdot \Vert}_{S^1(G)}}$ and $(u_K)_{K\subseteq \widehat{G}}$ is a ${\Vert \cdot \Vert}_1$-bounded approximate identity for $ZL^1(G)=\overline{lin\{\chi_n\}_{n\in\Bbb{N}}}^{{\Vert \cdot \Vert}_{1}}$, we can show that for each $\alpha$, ${\Vert e_\alpha \Vert}_{S^1(G)} = \lim_{K \rightarrow \widehat{G}} {\Vert e_\alpha * u_K \Vert}_{S^1(G)}$. Consequently, $${\Vert e_\alpha \Vert}_{S^1(G)} = \lim_{K \rightarrow \widehat{G}} {\Vert e_\alpha * u_K \Vert}_{S^1(G)} \leq \lim_{K \rightarrow \widehat{G}} {\Vert e_\alpha \Vert}_1 {\Vert u_K \Vert}_{S^1(G)} \leq C {\Vert e_\alpha \Vert}_1.$$ Hence, $(e_\alpha)_\alpha$ is ${\Vert \cdot \Vert}_{S^1(G)}$-bounded. But, a Segal algebra cannot have a bounded approximate identity unless it coincides with the group algebra, see [@bu], which contradicts the properness of $S^1(G)$. Hence, $(u_K)_{K\subseteq \widehat{G}}$ is not ${\Vert \cdot \Vert}_{S^1(G)}$-bounded. So by Theorem \[t:SUM\], $S^1(G)$ is not approximately amenable. \[c:Segal-of-S\^1(SU(2))\] Every proper Segal algebra on $SU(2)$ is not approximately amenable. \[c:Segal-of-Product-G\_i\] Let $\{G_i\}_{i\in{{\bf I}}}$ be a non-empty family of compact groups whose duals satisfy the Leptin condition, and $G=\prod_{i\in{{\bf I}}}G_i$ equipped with product topology. Then every proper Segal algebra on $G$ is not approximately amenable. [Further questions]{} - [For which other compact groups do their duals satisfy the Leptin condition? The best suggestions to study the Leptin condition for them are Lie groups which are the natural next step after $SU(2)$.]{} - [For locally compact groups, the existence of a bounded approximate identity of the Fourier algebra implies the Leptin condition. In the hypergroup case, it seems that we cannot prove this implication. Can we find a hypergroup $H$ whose Fourier algebra has a bounded approximate identity while $H$ does not satisfies the Leptin condition.]{} 2.0em [Acknowledgements]{}\ The author was supported by Dean’s Ph.D. scholarship at the University of Saskatchewan. The author would like to express his deep gratitude to Yemon Choi and Ebrahim Samei, his supervisors, for their kind helps and constant encouragement. 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Anal.]{} [208]{} (2004), no. 1, 229-–260. , Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Die Grundlehren der mathematischen Wissenschaften, Band 152 [Springer-Verlag, New York-Berlin]{} 1970. , Means and Følner condition on polynomial hypergroups. [Mediterr. J. Math.]{} [7 ]{}(2010), no. 1, 75-–88. , Segal algebras on non-abelian groups, [Trans. Amer. Math. Soc.]{} [237]{} (1978), 271-–281. , Fourier algebra of a hypergroup. I. [J. Aust. Math. Soc.]{} [82]{} (2007), no. 1, 59–-83. , Amenable locally compact groups. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. [John Wiley & Sons, Inc., New York]{}, 1984. , Classical harmonic analysis and locally compact groups. Second edition. London Mathematical Society Monographs. New Series, 22. [The Clarendon Press, Oxford University Press, New York]{}, 2000. , General theory of Banach algebras. The University Series in Higher Mathematics [D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York]{} 1960. , Amenable hypergroups. [Illinois J. Math.]{} [36]{} (1992), no. 1, 15-–46. , Harmonic analysis on compact hypergroups, Pacific. J. Math. 85 (1979), 239-–251. Mahmood Alaghmandan\ Department of Mathematics and Statistics,\ University of Saskatchewan,\ Saskatoon, SK S7N 5E6, CANADA\ E-mail: `mahmood.a@usask.ca`\
[Three-tangle for high-rank mixed states]{} [Shu-Juan He$^1$, Xiao-Hong Wang$^2$, Shao-Ming Fei$^{2}$, Hong-Xiang Sun$^3$ and Qiao-Yan Wen$^1$]{} [$ ^1$ State Key Laboratory of Networking and Switching Technology, Beijing University of Post and Telecommunication, China]{}\ [$ ^2$ Department of Mathematics, Capital Normal University, Beijing, China]{}\ [$ ^3$ School of Science, Beijing University of Post and Telecommunication, China]{} Abstract A family of rank-n ($n=5,6,7,8$) three-qubit mixed states are constructed. The explicit expressions for the three-tangle and optimal decompositions for all these states are given. The CKW relations for these states are also discussed. Key words: Three-tangle, Optimal decomposition PACS number(s): 03.67.Mn, 03.65.Ud 0.4cm Introduction ============= Quantum entangled states are important physical resource and play key roles in quantum information processing such as teleportation, superdense coding, quantum cloning, quantum cryptography [@information; @teleportation; @cloning]. Characterizing and quantifying entanglement of quantum states are of great importance. To quantify entanglement some measures like concurrence [@Rungta01-AlbeverioFei01], entanglement of formation [@BDSW; @Horo-Bruss-Plenioreviews] are studied. Though entanglement of bipartite states have been understood well in many aspects [@two-qubit], there is still no generally accepted theory for characterizing and quantifying entanglement for multipartite qubit systems, especially for mixed states. For three-qubit systems, some results have been presented [@oufan; @gfw; @gf; @cw]. An important quantity for three-qubit entanglement is the so called residual entanglement or three-tangle [@ckw-relation], which is a polynomial invariant for three-qubit states, the modulus of the hyperdeterminant [@Caley; @Miyake]. For a pure three-qubit state $\|\psi\rangle=\sum_{i,j,k=0}^{1}a_{ijk}\|ijk\rangle\in {\cal C}^2 \otimes {\cal C}^2 \otimes {\cal C}^2$, its three-tangle is defined by \[three-tangle-pure\] \_[3]{}(\|)=4\|d\_[1]{}-2d\_[2]{}+4d\_[3]{}\|, where $$d_{1}=a_{000}^{2}a_{111}^{2}+a_{001}^2a_{110}^2+a_{010}^{2}a_{101}^{2}+a_{100}^{2}a_{011}^{2},$$ $$d_{2}=a_{000}a_{111}a_{011}a_{100}+a_{000}a_{111}a_{101}a_{010}+a_{000}a_{111}a_{110}a_{001}$$ $$+a_{011}a_{100}a_{101}a_{010}+ a_{011}a_{100}a_{110}a_{001}+a_{101}a_{010}a_{110}a_{001},$$ $$d_{3}=a_{000}a_{110}a_{101}a_{011}+a_{111}a_{001}a_{010}a_{100}.$$ For a mixed three-qubit state $\rho=\sum_i p_i\rho_i$, $0<p_i\leq 1$, $\rho_i=\|\psi_i\rangle\langle\psi_i\|$, $\|\psi_i\rangle\in{\cal C}^2 \otimes {\cal C}^2 \otimes {\cal C}^2$, the three-tangle is defined in terms of convex roof [@convex-roof] \[mix\] \_[3]{}()=min\_[i]{}p\_[i]{}\_[3]{}(\_[i]{}). A decomposition that realizes the above minimum is called optimal. It is a challenge to find the optimal decomposition even for the simplest case of rank-2 mixed states. Nice analytical results have been obtained for some classes of three-qubit mixed states. In [@rank2; @grank2] Lohmayer et al. have constructed the optimal decomposition for a family of rank-2 three-qubit states. Jung et al.[@rank3; @rank4] have also provided analytical formulae of three-tangle for a class of rank-3 and rank-4 three-qubit mixed states. In [@guogc] a numerical method has been also presented to compute the three tangle for general three-qubit states. In this paper we analyze the optimal decomposition for some families of high rank-$n$ ($n=5,6,7,8$) three-qubit mixed states. The analytical expressions for the three-tangles and explicit optimal decompositions for all these states are given. The CKW relations for these states are also investigated. The Three-tangle for some high-rank mixed states ================================================ Recently, Jung et al.[@rank4] have provided an analytic quantification of the three-tangle for a rank-4 three-qubit mixed state which is composed by GHZ-type states. In this paper, we extend the method of [@rank4] to the high-rank mixed states. We use the following notations [@rank4] in studying three tangle of high rank mixed states: $$\ba{l} \|GHZ,1\pm\rangle=\frac{1}{\sqrt{2}}(\|000\rangle\pm\|111\rangle),~~~ \|GHZ,2\pm\rangle=\frac{1}{\sqrt{2}}(\|110\rangle\pm\|001\rangle),\\ \|GHZ,3\pm\rangle=\frac{1}{\sqrt{2}}(\|101\rangle\pm\|010\rangle),~~~ \|GHZ,4\pm\rangle=\frac{1}{\sqrt{2}}(\|011\rangle\pm\|100\rangle). \ea$$ [The case of rank-5 states]{} We first consider the following rank-5 states: \[rank5\] (p)=p\|GHZ,1+GHZ,1+\|+(1-p)\_[GHZ]{},where $\Gamma_{GHZ}=\frac{1}{10}\|GHZ,1-\rangle\langle GHZ,1-\|+\frac{3}{10}\|GHZ,2+\rangle\langle GHZ,2+\| +\frac{3}{10}\|GHZ,3+\rangle\langle GHZ,3+\|+\frac{3}{10}\|GHZ,4+\rangle\langle GHZ,4+\|$. We first consider the state $\Gamma_{GHZ}$. In order to calculate the three-tangle of the state $\Gamma_{GHZ}$, we first investigate the following rank-4 state: $$\rho(p)=p\|GHZ,1-\rangle\langle GHZ,1-\|+(1-p)\Pi_{GHZ},$$where $\Pi_{GHZ}=\frac{1}{3}[\|GHZ,2+\rangle\langle GHZ,2+\|+\|GHZ,3+\rangle\langle GHZ,3+\|+\|GHZ,4+\rangle\langle GHZ,4+\|]$ which has vanishing three-tangle [@rank4]. By straightforward calculation the three tangle of the following pure state $$\|Z(p,\varphi_{1},\varphi_{2},\varphi_{3})\rangle =\sqrt{p}\|GHZ,1-\rangle-\sqrt{\frac{1-p}{3}}(e^{i\varphi_{1}}\|GHZ,2+\rangle +e^{i\varphi_{2}}\|GHZ,3+\rangle+e^{i\varphi_{3}}\|GHZ,4+\rangle)$$ is given by \[4\] \_[3]{}(\|Z(p,\_[1]{},\_[2]{},\_[3]{}))&=& \|p\^2+(e\^[4i\_[1]{}]{}+e\^[4i\_[2]{}]{}+e\^[4i\_[3]{}]{}) +p(1-p)(e\^[2i\_[1]{}]{}+e\^[2i\_[2]{}]{}+e\^[2i\_[3]{}]{})\ &&-(e\^[2i(\_[1]{}+\_[2]{})]{}+e\^[2i(\_[1]{}+\_[3]{})]{} +e\^[2i(\_[2]{}+\_[3]{})]{})\|. Note that $\tau_{3}(\|Z(p,\varphi_{1},\varphi_{2},\varphi_{3})\rangle)$ is zero at $\varphi_{1}=\varphi_{2}=\varphi_{3}=0$ and $p_{0}=\frac{2-\sqrt{3}}{2}\doteq0.134.$ And $\rho(p)$ can be decomposed into $\rho(p)=\frac{p}{8p_{0}}\sum\Gamma_{i}(p_{0})+\frac{p_{0}-p}{p_{0}}\prod_{GHZ}$ for $0\leq p\leq p_{0}$, where \[def1\]\_1(p\_[0]{})=\|Z(p\_[0]{},0,0,0)Z(p\_[0]{},0,0,0)\|, \_2(p\_[0]{})=\|Z(p\_[0]{},0,0,)Z(p\_[0]{},0,0,)\|,\ \_3(p\_[0]{})=\|Z(p\_[0]{},0,,0)Z(p\_[0]{},0,,0)\|, \_4(p\_[0]{})=\|Z(p\_[0]{},0,,)Z(p\_[0]{},0,,)\|,\ \_5(p\_[0]{})=\|Z(p\_[0]{},,0,0)Z(p\_[0]{},,0,0)\|, \_6(p\_[0]{})=\|Z(p\_[0]{},,0,)Z(p\_[0]{},,0,),\ \_7(p\_[0]{})=\|Z(p\_[0]{},,,0)Z(p\_[0]{},,,0)\|, \_8(p\_[0]{})=\|Z(p\_[0]{},,,)Z(p\_[0]{},,,)\|.All $\Gamma_{i}(p_{0})$ and $\prod_{GHZ}$’s three-tangle are zero, therefore the three-tangle for the mixed state $\rho(p)$ is zero for $0\leq p\leq p_{0}$. Thus $\Gamma_{GHZ}$ has vanishing three-tangle. Now consider the three-qubit pure state constituted by linear combinations of $\|GHZ,1+\rangle$, $\|GHZ,1-\rangle$, $\|GHZ,2+\rangle$, $\|GHZ,3+\rangle$ and $\|GHZ,4+\rangle$: \[5\] \|Z(p,\_[1]{},\_[2]{},\_[3]{},\_[4]{})&=&\|GHZ,1+-e\^[i\_[1]{}]{}\|GHZ,1--e\^[i\_[2]{}]{}\|GHZ,2+\ &&-e\^[i\_[3]{}]{}\|GHZ,3+- e\^[i\_[4]{}]{}\|GHZ,4+. The corresponding three-tangle is \[6\] \_[3]{}(\|Z(p,\_[1]{},\_[2]{},\_[3]{},\_[4]{})) &=&\|p\^2+e\^[4i\_[1]{}]{}+ (e\^[4i\_[2]{}]{}+e\^[4i\_[3]{}]{}+e\^[4i\_[4]{}]{})\ &&-p(1-p)e\^[2i\_[1]{}]{}-p(1-p)(e\^[2i\_[2]{}]{}+e\^[2i\_[3]{}]{}+e\^[2i\_[4]{}]{})\ &&+(e\^[2i(\_[1]{}+\_[2]{})]{}+e\^[2i(\_[1]{}+\_[3]{})]{}+e\^[2i(\_[1]{}+\_[4]{})]{})\ &&-(e\^[2i(\_[2]{}+\_[3]{})]{} +e\^[2i(\_[2]{}+\_[4]{})]{}+e\^[2i(\_[3]{}+\_[4]{})]{})\ &&-e\^[i(\_[2]{}+\_[3]{}+\_[4]{})]{}\|. Since the three-tangle $\tau_3(\|Z(p,\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4})\rangle)=0$ at $p=p_{0}=0.7377$ and $\varphi_{1}=\varphi_{2}=\varphi_{3}=\varphi_{4}=0$, the state $\sigma(p)$ can be expressed in terms of $\|Z(p,\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4})\rangle$, \[optidec1\] (p)&=&\_[i]{}(p), where \[def1\]\_1(p)=\|Z(p,0,0,0,0)Z(p,0,0,0,0)\|, \_2(p)=\|Z(p,0,0,,)Z(p,0,0,,)\|,\ \_3(p)=\|Z(p,0,,0,)Z(p,0,,0,)\|, \_4(p)=\|Z(p,0,,,0)Z(p,0,,,0)\|,\ \_5(p)=\|Z(p,,0,0,0)Z(p,,0,0,0)\|, \_6(p)=\|Z(p,,0,,)Z(p,,0,,)\|,\ \_7(p)=\|Z(p,,,0,)Z(p,,,0,)\|, \_8(p)=\|Z(p,,,,0)Z(p,,,,0)\|. When $0\leq p\leq p_{0}$, we have the optimal decomposition of $\sigma(p)$: \[optidec\] (p)&=&\_[i]{}(p\_0)+\_[GHZ]{}, where $\Pi_{i}(p)$ are defined as (\[def1\]), $i=1,2,\cdots,8$. Since all $\Pi_{i}(p_0)$ and $\Gamma_{GHZ}$ has vanishing three-tangle, we have that $\tau_{3}(\sigma(p))=0$ when $0\leq p\leq p_{0}$. For $p>p_{0}$, the decomposition in Eqs.(\[optidec1\]) also is a trial optimal decomposition for $\sigma(p)$. Its three-tangle is \[tI\] g\_[I]{}(p)=p\^2-2p(1-p)--, p>p\_[0]{}. We need to check whether the function $g_{I}(p)$ is convex or not for $p>p_{0}$. It can be verified that the function $g_{I}(p)$ is convex for $p<p_{}=0.9750$, but concave for $p>p_{}$. For large $p$ let us propose a decomposition of $\sigma(p)$ as follows: \[optidec2\] (p)&=&\_[i]{}(p\_1)+ \|GHZ,1+GHZ,1+\|, where $p_{1}\leq p\leq 1$, $p_{1}\leq p_{\ast}$, $\Pi_{i}(p)$ are defined as (\[def1\]), $i=1,2,\cdots,8$. The three tangle of (\[optidec2\]) is given by \[tII\] g\_[II]{}(p)=+g\_[I]{}(p\_[1]{}). Since $d^{2}g_{\Pi}/dp^2=0$ for all $p$, from $\partial g_{II}/\partial p_{1}=0$ we have $$3\sqrt{30}p_{1}^{\frac{1}{2}}(1-p_{1})^{-\frac{1}{2}}-3 \sqrt{30}p_{1}^{-\frac{1}{2}}(1-p_{1})^{\frac{1}{2}}=73,$$ which gives rise to $$p_{1}=\frac{1}{2}+\frac{73\sqrt{6409}}{12818}\doteq0.9559.$$ Therefore the three tangle of the rank-5 state $\sigma(p)$ is given by \[imc\] \_[3]{}((p))={ [ll]{} 0, &\ g\_[I]{}(p), &\ g\_[II]{}(p), & . where $p_{0}$=0.7377, $p_{1}$=0.9559, $g_{I}(p)$ is given by (\[tI\]) and $g_{II}(p)$ by (\[tII\]). And the corresponding optimal decomposition are (\[optidec\]), (\[optidec1\]) and (\[optidec2\]) respectively. In order to show that (\[imc\]) is genuine optimal, we plot the p-dependence of the three-tangles for various $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\varphi_{4}$. These curves have been referred as the characteristic curves [@curve]. As [@curve] indicated, the three-tangle is a convex hull of the minimum of the characterisitic curves. Fig.1 indicates that the three-tangles plotted as black solid line are the convex characteristic curves, which implies that (\[imc\]) is really optimal. [\[33\]]{} [The case of rank-6 states]{} We consider now the three-tangle for a family of rank-6 mixed states: \[rank6states\] (t)=t\|GHZ,2-GHZ,2-\|+(1-t),where \[8\] &=&\|GHZ,1+GHZ,1+\|+\|GHZ,1-GHZ,1-\|+ \|GHZ,2+GHZ,2+\|\ &&+\|GHZ,3+GHZ,3+\|+ \|GHZ,4+GHZ,4+\|. From the analysis for our rank-5 states, we know that $\sigma$ has vanishing three-tangle, and the three-tangle for $\varrho(t)$ is given by \[imp\] \_[3]{}((t))={ [ll]{} 0, &\ g\_[I]{}(t), &\ g\_[II]{}(t), & . where $$\ba{rcl} g_{I}(t)&=&t^2+\frac{6}{11}t(1-t)-\frac{27-24\sqrt{3}}{121}(1-t)^2- \frac{24\sqrt{11}}{121}\sqrt{t(1-t)^3},\\[3mm] g_{II}(t)&=&\frac{t-t_{1}}{1-t_{1}}+\frac{1-t}{1-t_{1}}g_{I}(t_{1}),~~~t_{0}=0.2143,~~~t_{1}=0.8290. \ea$$ We have the optimal decomposition \[imp\] (t)={ [ll]{} \_[i]{}(t\_0)+, &\ \_[i]{}(t), &\ \_[i]{}(t\_1)+ \|GHZ,2-GHZ,2-\|, & . where \[def2\]\_1(t)=\|Z(t,0,0,0,0,0)Z(t,0,0,0,0,0)\|, \_2(t)=\|Z(t,0,,,0,0)Z(t,0,,,0,0)\|,\ \_3(t)=\|Z(t,,0,,0,)Z(t,,0,,0,)\|, \_4(t)=\|Z(t,,,0,0,)Z(t,,,0,0,)\|,\ \_5(t)=\|Z(t,,0,,,0)Z(t,,0,,,0)\|, \_6(t)=\|Z(t,,,0,,0)Z(t,,,0,,0)\|,\ \_7(t)=\|Z(t,0,0,0,,)Z(t,0,0,0,,), \_8(t)=\|Z(t,0,,,,)Z(t,0,,,,)\|.Obvious, all $\Pi_{i}(t_0)$ have vanishing three-tangle. [The case of rank-7 states]{} The three tangle of the following rank-7 mixed states can be similarly calculated: \[rank7states\] (s)=s\|GHZ,3-GHZ,3-\|+(1-s),where \[rank7\] &=&\|GHZ,2-GHZ,2- \|+\|GHZ,1+GHZ,1+\|+\|GHZ,1-GHZ,1- \|\ &&+\|GHZ,2+GHZ,2+\|+\|GHZ,3+GHZ,3+\|+ \|GHZ,4+GHZ,4+\|. Applying the similar approach above we get \[imp\] \_[3]{}((s))={ [ll]{} 0, &\ g\_[I]{}(s), &\ g\_[II]{}(s), & . where $$\ba{rcl} g_{I}(s)&=&s^2+\frac{8}{17}s(1-s)-\frac{56-72\sqrt{3}}{289}(1-s)^2- \frac{24\sqrt{102}}{289}\sqrt{s(1-s)^3},\\[3mm] g_{II}(s)&=&\frac{s-s_{1}}{1-s_{1}}+\frac{1-s}{1-s_{1}}g_{I}(s_{1}),~~ s_{0}=0.2062,~ s_{1}=0.8375. \ea$$ We also can get the corresponding optimal decomposition for $\gamma(s)$: \[imp\] (s)={ [ll]{} \_[i]{}(s\_0)+, &\ \_[i]{}(s), &\ \_[i]{}(s\_1)+ \|GHZ,3-GHZ,3-\|, & . where \[def3\] \_1(s)=\|Z(s,0,0,0,0,0,0)Z(s,0,0,0,0,0,0)\|,\ \_2(s)=\|Z(s,0,,0,,,)Z(s,0,,0,,,)\|,\ \_3(s)=\|Z(s,,0,0,,0,)Z(s,,0,0,,0,)\|,\ \_4(s)=\|Z(s,,,0,0,,0)Z(s,,,0,0,,0)\|,\ \_5(s)=\|Z(s,0,,,0,0,)Z(s,,0,,,0,0,)\|,\ \_6(s)=\|Z(s,0,0,,,,0)Z(s,0,0,,,,0)\|,\ \_7(s)=\|Z(s,,,,,0,0)Z(s,,,,,0,0)\|,\ \_8(s)=\|Z(s,,0,,0,,)Z(s,,0,,0,,)\|. [The case of rank-8 states]{} The rank of a three-qubit mixed state could be at most $8$. We now introduce a family of rank-8 mixed states: $$\rho(r)=r\|GHZ,4-\rangle\langle GHZ,4-\|+(1-r){\eta},$$ where \[rank-8\] &=&\|GHZ,3-GHZ,3-\|+\|GHZ,2-GHZ,2-\|+\|GHZ,1+GHZ,1+\|\ &&+\|GHZ,1-GHZ,1-\|+\|GHZ,2+GHZ,2+\|\ &&+\|GHZ,3+GHZ,3+\|+\|GHZ,4+GHZ,4+\|. Obviously, $\tau_{3}({\eta})=0$. The three-tangle of $\rho(r)$ is given by: \[imp\] \_[3]{}((r))={ [ll]{} 0, &\ g\_[I]{}(r), &\ g\_[II]{}(r), & . where $$\ba{rcl} g_{I}(r)&=&r^2+\frac{2}{5}r(1-r)-\frac{207-384\sqrt{3}}{1225}(1-r)^2 -\frac{128\sqrt{105}}{1225}\sqrt{r(1-r)^{3}},\\[3mm] g_{II}(r)&=&\frac{r-r_{1}}{1-r_{1}}+\frac{1-r}{1-r_{1}}g_{I}(r_{1}),~~ r_{0}=0.2490,r_{1}=0.8649. \ea$$ The optimal decomposition for $\rho(r)$ can be similarly obtained: \[imp\] (r)={ [ll]{} \_[i]{}(r\_0)+, &\ \_[i]{}(r), &\ \_[i]{}(r\_1)+ \|GHZ,4-GHZ,4-\|, & . where \[def4\] \_1(r)=\|Z(r,0,0,0,0,0,0,0)Z(P,0,0,0,0,0,0,0)\|,\ \_2(r)=\|Z(r,0,0,0,,,,)Z(r,0,0,0,,,,)\|,\ \_3(r)=\|Z(r,0,,,0,0,,)Z(r,0,,,0,0,,)\|,\ \_4(r)=\|Z(r,0,,,,,0,0)Z(r,0,,,,,0,0)\|,\ \_5(r)=\|Z(r,,0,,0,,0,)Z(r,,0,,0,,0,)\|,\ \_6(r)=\|Z(r,,0,,,0,,0)Z(r,,0,,,0,,0)\|,\ \_7(r)=\|Z(r,,,0,0,,,0)Z(r,,,0,0,,,0),\ \_8(r)=\|Z(r,,,0,,0,0,)Z(r,,,0,,0,0,)\|. CKW inequality ============== Given a family of mixed three-qubit states with the corresponding three-tangle, one might check the CKW relations [@ckw-relation]. For a pure three-qubit state $\|\psi\rangle\in {\cal C}^2 \otimes {\cal C}^2 \otimes {\cal C}^2$, with the reduced two-qubit density matrices $\rho_{AB}=Tr_C(\|\psi\rangle\langle\psi\|)$, $\rho_{AC}=Tr_B(\|\psi\rangle\langle\psi\|)$ and $\rho_{A}=Tr_{BC}(\|\psi\rangle\langle\psi\|)$, one has the monogamy relation $4det(\rho_{A})=C(\rho_{AB})^{2}+ C(\rho_{AC})^{2}+\tau_{3}(\|\psi\rangle)$, where $C(\rho_{AB})$ (resp. $C(\rho_{AC})$) is the concurrence for the corresponding reduced state $\rho_{AB}$ (resp. $\rho_{AC}$), $\tau_{3}(\|\psi\rangle)$ is the three-tangle of $\|\psi\rangle$. For mixed states, the following CKW inequality holds, $4min[det(\rho_{A})]\geq C(\rho_{AB})^{2}+C(\rho_{AC})^{2}$. The CKW inequality has been examined for the mixture of GHZ and W states in [@rank2] and the mixture of GHZ, W and flipped-W states in [@rank3]. In the following we check if the CKW inequality holds for the states introduced in our paper. As an example, we consider the case of rank-5 states. It is direct to verify that these states satisfy $C(\rho_{AB})^{2}+C(\rho_{AC})^{2}=0$. And the minimum one-tangle is given by \[9\] 4min\[det(\_[A]{})\]=1-p(1-p)-(1-p)\^2+ . [\[33\]]{} From Fig. 2 we see that the CKW inequality is obviously satisfied. Moreover the inequality $4min[det(\rho_{A})]\geq C(\rho_{AB})^{2}+ C(\rho_{AC})^{2}+\tau_{3}(\psi)$ is also satisfied for these rank-5 states. In particular in the region $0\leq p \leq p_{0}=0.7377$, both the concurrence and three-tangle are zero, but the one-tangle is not zero. Conclusion ========== We have constructed several classes of different ranked mixed states in three-qubit system. We have provided explicit expressions for the three-tangle and optimal decompositions for all these states. We have also studied the relations between the CKW inequality and these classes of states, and shown that the CKW inequality are satisfied for these states. Concurrence of mixed two-qubit states has been applied to study quantum phase transitions. It has been shown that the pairwise entanglement of the nearest-neighbor two sites in spin-$1/2$ lattice models has special singularity at quantum critical points [@pt]. It can be expected that multipartite entanglement would reveal further relations between the quantum phase transitions and quantum entanglement. Our results could help studies on applications of quantum entanglement in all these related researches. [Acknowledgments]{} The work is supported by NSFC (Grant Nos.10871228, 10875081, 60873191, 60903152, 60821001), SRFDP (Grant No. 200800131016, 20090005110010), Beijing Nova Program (Grant No. 2008B51), Key Project of Chinese Ministry of Education (Grant No. 109014), and Beijing Municipal Education Commission No. KM200510028021 and No.KZ200810028013. [20]{} M.A. Nilsen and I.L. Chuang, [ Quantum Computation and Quantum Information]{} (2000). Ye Yeo and Wee Kang Chua, [ Phys. Rev. Lett.]{} [96]{}, 060502 (2006). N. Gisin and S.Massar, Phys. Rev. Lett. [79]{}, 2153 (1997). A.Uhlmann, Phys. Rev. A 62, 032307 (2000); P. Rungta, V. Buzek, C.M. Caves, M. Hillery, and G.J. Milburn, Phys. Rev. A 64, 042315 (2001); S. Albeverio and S. M. Fei, J. Opt. B: Quantum Semiclass. Opt. 3, 223 (2001). C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996). M. Horodecki, Quant. Inf. Comp. 1, 3 (2001); D. Bruß, J. Math. Phys. 43, 4237 (2002); K. Chen, S. Albeverio, and S.M. Fei, Phys. Rev. Lett. [95]{}, 210501 (2005). C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A. [53]{}, 2046 (1996); B. Sutherland,[ Phys. Rev. Lett.]{} [80]{}, 2445 (1998); A. Uhlmann,[ Phys. Rev. A.]{} [62]{}, 032307 (2000); Y.C. Ou, H. Fan, and S.M. Fei, [ Phys. Rev. A.]{} [78]{}, 012311 (2008). Y.C. Ou and H. Fan,[ Phys. Rev. A.]{} [75]{}, 062308 (2007). X.H. Gao, S.M. Fei, and K. Wu, [ Phys. Rev. A.]{} [74]{}, 050303(R) (2006). X.H. Gao and S.M. Fei,[ Eur. Phys. J.Special Topics]{} [159]{}, 71 (2008). Y. Cao and A.M. Wang, [J. Phys. A: Math. Theor.]{} [40]{}, 3507 (2007). V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A [61]{}, 052306 (2000). A. Caley, 1845 Cambridge Math. J. [4]{}, 193. A. Miyake, Phys. Rev. A [67]{}, 012108 (2003). F. Benatti, A. Narnhofer, and A. Uhlmann, Rep.Math.Phys. [38]{}, 123 (1996). R. Lohmayer, A. Osterloh, J. Siewert, and A. Uhlmann, Phys. Rev. Lett. [97]{}, 260502 (2006). C. Eltschka, A. Osterloh, J. 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--- abstract: 'We propose automated techniques for the verification and control of probabilistic real-time systems that are only partially observable. To formally model such systems, we define an extension of probabilistic timed automata in which local states are partially visible to an observer or controller. We give a probabilistic temporal logic that can express a range of quantitative properties of these models, relating to the probability of an event’s occurrence or the expected value of a reward measure. We then propose techniques to either verify that such a property holds or to synthesise a controller for the model which makes it true. Our approach is based on an integer discretisation of the model’s dense-time behaviour and a grid-based abstraction of the uncountable belief space induced by partial observability. The latter is necessarily approximate since the underlying problem is undecidable, however we show how both lower and upper bounds on numerical results can be generated. We illustrate the effectiveness of the approach by implementing it in the PRISM model checker and applying it to several case studies, from the domains of computer security and task scheduling.' author: - Gethin Norman - David Parker - Xueyi Zou bibliography: - 'poptas.bib' title: 'Verification and Control of Partially Observable Probabilistic Real-Time Systems' --- [Acknowledgments.]{} This work was partly supported by the EPSRC grant “Automated Game-Theoretic Verification of Security Systems” (EP/K038575/1). We also grateful acknowledge support from Google Summer of Code 2014.
--- abstract: 'Perturbative quantum gravity in the framework of the Schwinger–Keldysh formalism is applied to compute lowest-order corrections to the actual expansion of the Universe described in terms of the spatially flat Friedman–Lema[î]{}tre–Robertson–Walker solution. The classical metric is approximated by a third order polynomial perturbation around the Minkowski metric. It is shown that the quantum contribution to the classical expansion, although extremely small, has damping properties (quantum friction), i.e. it slows down the expansion.' address: 'Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Łódź, Pomorska 149/153, 90-236 Łódź, Poland' author: - Bogusław Broda bibliography: - 'Perturbative\_quantum\_damping\_of\_cosmological\_expansion.bib' title: Perturbative quantum damping of cosmological expansion --- perturbative quantum gravity ,cosmological expansion ,Schwinger–Keldysh formalism 04.60.Gw ,04.60.Bc ,04.60.Pp ,98.80.Qc Introduction ============ The aim of our work is to explicitly show the appearance of quantum generated damping or quantum friction, i.e. slowing down, in the present (accelerating) expansion of the Universe. In general, quantum corrections to classical gravitational field can be perturbatively calculated in a number of ways. First of all, it is possible to directly calculate quantum (one-loop) corrections to classical gravitational field from the graviton vacuum polarization (self-energy), in the analogy to the case of the Coulomb potential in QED (for example, see @QuantumElectrodynamics), the so-called Uehling potential. Such a type of calculations has been already performed for the Schwarzschild solution by @duff1974quantum, as well as for the spatially flat Friedman–Lema[î]{}tre–Robertson–Walker (FLRW) metric (@broda2011one). Another approach refers to the energy-momentum tensor calculations, and it has been applied to the Newton potential (for example, see @bjerrum2003quantum, and the references therein), to the Reissner-Nordström and the Kerr-Newman solutions (see @donoghue2002quantum), and to the Schwarzschild and the Kerr solution metrics (see @jannik2002quantum). Yet another approach uses the Schwinger–Keldysh (SK) formalism to the case of the Newton potential (e.g., see @park2010solving). It is argued that only the SK formalism is adequate for time-dependent potentials, hence in particular, in the context of cosmology (e.g., see @weinberg2005quantum, and references therein). As we aim to perturbatively calculate corrections to the spatially flat FLRW metric, we should use the SK formalism, because this is exactly that case (time-dependence of gravitational field) the SK approach has been devised for. The corrections we calculate are a quantum response to the spatially flat FLRW solution which is described by a small perturbation around the Minkowski metric. Since such a type of calculations is usually plagued by infinities, we confine ourselves to the classical perturbation given by a polynomial of the third degree. Moreover, to avoid infinities on intermediate stages of our calculations, time derivative of the convolution of the time propagator with the perturbation of the metric should be performed in a suitable order. The final result is given in terms of the present time quantum correction $q_{0}^{\textrm{Q}}$ to the deceleration parameter $q_{0}$. Interestingly, it appears that $q_{0}^{\textrm{Q}}$ is positive, although obviously, it is extremely small. Quantum damping =============== Our starting point is a general spatially flat FLRW metric $$ds^{2}\equiv g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+a^{2}\left(t\right)d\mathbf{\boldsymbol{x}}{}^{2},\qquad\mu,\nu=0,1,2,3,$$ with the cosmological scale factor $a\left(t\right)$. To satisfy the condition of weakness of the perturbative gravitational field $h_{\mu\nu}$ near our reference time $t=t_{0}$ (where $t_{0}$ could be the age of the Universe — the present moment) in the expansion $$g_{\mu\nu}\left(x\right)=\eta_{\mu\nu}+h_{\mu\nu}\left(x\right),$$ the metric should be normalized in such a way that it is exactly Minkowskian for $t=t_{0}$, i.e. $$a^{2}\left(t\right)=1+h\left(t\right),\qquad h\left(t_{0}\right)=0.$$ Let us note the analogy to the Newton potential ($\sim1/r$), where the “reference radius” is in spatial infinity, i.e. $r_{0}=+\infty$. Then, in the block diagonal form, $$h_{\mu\nu}\left(t,\boldsymbol{x}\right)=\left(\begin{array}{cc} 0 & 0\\ 0 & \delta_{ij}h\left(t\right) \end{array}\right),\quad i,j=1,2,3.\label{eq:hmini}$$ To obtain quantum corrections to the classical gravitational field $h_{\mu\nu}^{\textrm{C }}\left(x\right)$, we shall use the one-loop effective field equation derived by @park2010solving $$\begin{aligned} \mathcal{D}^{\mu\nu\varrho\sigma}h_{\varrho\sigma}^{\textrm{Q}}\left(t,\boldsymbol{x}\right) & = & \frac{\kappa^{2}}{10240\pi^{3}}\textrm{D}^{\mu\nu\varrho\sigma}\partial^{4}\intop_{0}^{t}dt'\nonumber \\ & \times & \int d^{3}x'\theta\left(\Delta t-\Delta r\right)\left[\ln\left(-\mu^{2}\Delta x^{2}\right)-1\right]h_{\varrho\sigma}^{\textrm{C }}\left(t',\boldsymbol{x}'\right),\label{eq:effeq}\end{aligned}$$ where $\Delta t\equiv t-t'$, $\Delta r\equiv\left\|\boldsymbol{x}-\boldsymbol{x}'\right\|$, $\Delta x^{2}\equiv-\left(\Delta t\right)^{2}+\left(\Delta r\right)^{2}$, and the mass scale $\mu$ is coming from the renormalization procedure (see @ford2005stress). Here $\kappa^{2}=16\pi G_{N}$, where $G_{N}$ is the Newton gravitational constant. The operator $\mathcal{D}$ (the Lichnerowicz operator in the flat background) is of the form $$\mathcal{D}^{\mu\nu\varrho\sigma}=\frac{1}{2}\left(\eta^{\mu\nu}\eta^{\varrho\sigma}\partial^{2}-\partial^{\mu}\partial^{\nu}\eta^{\varrho\sigma}-\eta^{\mu\nu}\partial^{\varrho}\partial^{\sigma}-\eta^{\mu(\varrho}\eta^{\sigma)\nu}\partial^{2}+2\partial^{(\mu}\eta^{\nu)(\varrho}\partial^{\sigma)}\right),$$ whereas for the minimally coupled massless scalar field $$\begin{aligned} \textrm{D}{}^{\mu\nu\varrho\sigma} & = & \Pi^{\mu\nu}\Pi^{\varrho\sigma}+\frac{1}{3}\Pi^{\mu(\varrho}\Pi^{\sigma)\nu}\label{eq:opd}\end{aligned}$$ with $$\Pi^{\mu\nu}\equiv\eta^{\mu\nu}\partial^{2}-\partial^{\mu}\partial^{\nu}.$$ For conformally coupled fields we have $\widetilde{\textrm{D}}$ instead of $\textrm{D}$, where $$\widetilde{\textrm{D}}{}^{\mu\nu\varrho\sigma}\equiv-\frac{1}{9}\Pi^{\mu\nu}\Pi^{\varrho\sigma}+\frac{1}{3}\Pi^{\mu(\varrho}\Pi^{\sigma)\nu}.$$ Since the metric depends only on time, we can explicitly perform the spatial integration in (\[eq:effeq\]) with respect to $\boldsymbol{x}'$ obtaining the integral kernel (time propagator) $$\begin{aligned} \textrm{K}\left(\Delta t\right) & \equiv & 4\pi\intop_{0}^{\Delta t}dr\, r^{2}\left\{ \ln\left[\mu^{2}\left(\left(\Delta t\right)^{2}-r^{2}\right)\right]-1\right\} \nonumber \\ & = & \frac{4\pi}{3}\left(\Delta t\right)^{3}\left[\ln\left(4\mu^{2}\Delta t^{2}\right)-\frac{11}{3}\right].\label{eq:defk}\end{aligned}$$ For the time-dependent metric of the form $$\left(\begin{array}{cc} f\left(t\right)\\ & \delta_{ij}h\left(t\right) \end{array}\right),$$ the action of the operators $\mathcal{D}$, $\textrm{D}$ and $\widetilde{\textrm{D}}$ is given by $$\mathcal{D}\left(\begin{array}{cc} f\left(t\right)\\ & \delta_{ij}h\left(t\right) \end{array}\right)=\left(\begin{array}{cc} 0\\ & -\delta_{ij}\frac{d^{2}}{dt^{2}}h\left(t\right) \end{array}\right),\label{eq:dcalonmet}$$ $$\begin{aligned} \textrm{D}\left(\begin{array}{cc} f\left(t\right) & 0\\ 0 & \delta_{ij}h\left(t\right) \end{array}\right) & = & \left(\begin{array}{cc} 0 & 0\\ 0 & \frac{10}{3}\delta_{ij}\frac{d^{4}}{dt^{4}}h\left(t\right) \end{array}\right),\label{eq:dromonmet}\end{aligned}$$ whereas $$\widetilde{\textrm{D}}\left(\begin{array}{cc} a\left(t\right) & 0\\ 0 & \delta_{ij}h\left(t\right) \end{array}\right)=0,\label{eq:dtilonmet}$$ respectively. There are no mixing of diagonal and non-diagonal terms, and the empty blocks mean expressions which can be non-zero, but they are inessential in our further analysis. Thus, (\[eq:effeq\]) assumes the simple form $$\frac{d^{2}}{dt^{2}}h^{\textrm{Q}}\left(t\right)=-\frac{\kappa^{2}}{3072\pi^{3}}\frac{d^{8}}{dt^{8}}\left(\textrm{K}\star h^{\textrm{C}}\right)\left(t\right),\label{eq:diffeq}$$ where the integral kernel $\textrm{K}$ is given in (\[eq:defk\]), and the convolution “$\star$” is standardly defined by $$\left(\textrm{K}\star\mathrm{F}\right)\left(t\right)\equiv\intop_{0}^{t}\textrm{K}\left(t-t'\right)\mathrm{F}\left(t'\right)dt'=\intop_{0}^{t}\textrm{K}\left(t'\right)\mathrm{F}\left(t-t'\right)dt'.\label{eq:defconv}$$ One should note that due to “diagonality” of (\[eq:hmini\]) and (\[eq:dcalonmet\]–\[eq:dtilonmet\]), no non-diagonal terms of the metric enter (\[eq:diffeq\]). Now, observing that also the limit of integration in (\[eq:defconv\]) depends on $t$, one can easily derive the following differentiation formula for the convolution $$\frac{d^{n}}{dt^{n}}\,\left(\textrm{K}\star\mathrm{F}\right)\left(t\right)=\left(\frac{d^{n}}{dt^{n}}\textrm{K}\star\mathrm{F}\right)\left(t\right)+{\displaystyle \sum_{k=1}^{n}\frac{d^{\left(n-k\right)}}{dt^{\left(n-k\right)}}\textrm{K}\left(0\right)\frac{d^{\left(k-1\right)}}{dt^{\left(k-1\right)}}\textrm{F}\left(t\right).}\label{eq:diffofconv}$$ Using symmetry between $\textrm{K}$ and $\textrm{F}$ (see (\[eq:defconv\])), it is possible to distribute differentiation in (\[eq:diffofconv\]) in several different ways. For practical purposes, i.e. elimination of possible singular terms on intermediate stages of our calculations, the most convenient form of the eighth derivative is the symmetric one, i.e. $$\begin{aligned} \frac{d^{8}}{dt^{8}}\left(\textrm{K}\star h^{\textrm{C}}\right)\left(t\right) & = & \left(\frac{d^{4}}{dt^{4}}\textrm{K}\star\frac{d^{4}}{dt^{4}}h^{\textrm{C}}\right)\left(t\right)\nonumber \\ & + & \sum_{k=1}^{4}\left[\frac{d^{\left(4-k\right)}}{dt^{\left(4-k\right)}}\textrm{K}\left(0\right)\frac{d^{\left(k+3\right)}}{dt^{\left(k+3\right)}}h^{\mathrm{C}}\left(t\right)\right.\label{eq:mixdiff}\\ & + & \left.\frac{d^{\left(4-k\right)}}{dt^{\left(4-k\right)}}h^{\mathrm{C}}\left(0\right)\frac{d^{\left(k+3\right)}}{dt^{\left(k+3\right)}}\textrm{K}\left(t\right)\right].\nonumber \end{aligned}$$ To finally prevent the appearance of possible infinities, i.e. primary UV infinities in the propagator, signaled by $\mu$, as well as divergences in the convolution, which could come from singularities in the kernel (time propagator) $\textrm{K}$, we assume the following third order polynomial form of the classical metric $$h^{\textrm{C}}\left(\tau\right)=h_{0}+h_{1}\tau+h_{2}\tau^{2}+h_{3}\tau^{3}.\label{eq:polh}$$ Henceforth, for simplicity, we use the dimensionless unit of time, $\tau\equiv t/t_{0}$, instead of $t$. Non-singularity of Eq.\[eq:eightdiff\] proofs that (\[eq:mixdiff\]) and (\[eq:polh\]), have been properly selected. In fact, our choice is unique. First of all, let us observe that $$\left.\frac{d^{k}}{dt^{k}}\textrm{K}\left(t\right)\right\|_{t=0},\qquad\textrm{for}\quad k>2,$$ is singular, and $$\frac{d^{k}}{dt^{k}}\textrm{K}\left(t\right),\qquad\textrm{for}\quad k<4,$$ is $\mu$-dependent. Then, the only possibility to avoid such troublesome terms admits exactly the products in the second part of the sum in (\[eq:mixdiff\]). In turn, to nullify the unwanted first part of the sum in (\[eq:mixdiff\]), $h^{\textrm{C}}\left(\tau\right)$ should be of the form (\[eq:polh\]). The term before the summation sign in (\[eq:mixdiff\]) vanishes, and thus, it is inessential. Actually, the classical metric (\[eq:polh\]) does not belong to any favorite family of cosmological solutions, perhaps except for the linear case ($h_{0}=-1,h_{1}=1,h_{2}=h_{3}=0$), corresponding to radiation. In fact, a physically realistic metric is not precisely given, for example, by the matter-dominated cosmological scale factor $a\left(\tau\right)=\tau^{2/3}$, because firstly, the character of cosmological evolution depends on the epoch (time $\tau$), and secondly, it is “contaminated” by other “matter” components, e.g. radiation, and possibly, dark energy. Therefore, we should consider (\[eq:polh\]) as a phenomenological description, approximating actual cosmological evolution on the finite time interval $\tau\in\left[\tau_{0},1\right]$, $0\leq\tau_{0}<1$. Inserting (\[eq:defk\]) and (\[eq:polh\]) to (\[eq:mixdiff\]) we derive, by virtue of (\[eq:diffeq\]), the second order differential equation $$\ddot{h}^{\textrm{Q}}\left(\tau\right)=\lambda\left(h_{0}\tau^{-2}-\frac{h_{1}}{3}\tau^{-1}+\frac{h_{2}}{3}-h_{3}\tau\right),\label{eq:eightdiff}$$ which can be easily integrated out with respect to $\tau$, yielding $$\dot{h}^{\textrm{Q}}\left(\tau\right)=\lambda\left(-h_{0}\tau^{-1}-\frac{h_{1}}{3}\log\left\|\tau\right\|+\frac{h_{2}}{3}\tau-\frac{h_{3}}{2}\tau^{2}\right)\label{eq:seventhdiff}$$ and $$h^{\textrm{Q}}=\lambda\left(-h_{0}\log\left\|\tau\right\|-\frac{h_{1}}{3}\left(\tau\log\left\|\tau\right\|-\tau\right)+\frac{h_{2}}{6}\tau^{2}-\frac{h_{3}}{6}\tau^{3}\right),\label{eq:sixthdiff}$$ where $\lambda\equiv\kappa^{2}/32\pi^{2}t_{0}^{2}\approx\frac{1}{2}\cdot10^{-46}$. As a physical observable we are interested in, we take the deceleration parameter $$q\left(\tau\right)\equiv-\frac{a\left(\tau\right)\ddot{a}\left(\tau\right)}{\dot{a}^{2}\left(\tau\right)}=1-2\left[1+h\left(\tau\right)\right]\frac{\ddot{h}\left(\tau\right)}{\dot{h}^{2}\left(\tau\right)}.\label{eq:qzero-1}$$ The quantum contribution to the deceleration parameter, namely, the lowest order contribution of (\[eq:eightdiff\]–\[eq:sixthdiff\]) to (\[eq:qzero-1\]), i.e. $q\left(\tau\right)=q^{\textrm{C}}\left(\tau\right)+q^{\textrm{Q}}\left(\tau\right)+\mathcal{O}\left(\lambda^{2}\right)$, reads $$q^{\textrm{Q}}=-\frac{2}{\left(\dot{h}^{\textrm{C}}\right)^{2}}\left[\ddot{h}^{\textrm{C}}h^{\textrm{Q}}+\left(1+h^{\textrm{C}}\right)\left(\ddot{h}^{\textrm{Q}}-\frac{2\ddot{h}^{\textrm{C}}\dot{h}^{\textrm{Q}}}{\dot{h}^{\textrm{C}}}\right)\right].\label{eq:qquant}$$ To approximate the cosmological evolution by the (four-parameter) phenomenological metric (\[eq:polh\]), we need four conditions. First of all, we impose the following two obvious boundary conditions $$h^{\textrm{C}}\left(0\right)=-1\quad\textrm{and}\quad h^{\textrm{C}}\left(1\right)=0,$$ corresponding to $$a^{2}\left(0\right)=0\quad\textrm{and}\quad a^{2}\left(1\right)=1,$$ and implying $$h_{0}=-1\quad\textrm{and}\quad h_{1}+h_{2}+h_{3}=-h_{0}=1.\label{eq:firsttwoeq}$$ In this place various different further directions of proceedings could be assumed, depending on the question we pose. Then, let us study the quantum contribution to the actual cosmological evolution. By virtue of (\[eq:defk\]) and (\[eq:diffeq\]), the “effective” time propagator determined by the sixth order derivative of the kernel $\textrm{K}$, behaves as $\left(\Delta t\right)^{-3}$, which follows from, e.g., dimensional analysis. Thus, the largest contributions to the quantum part of the metric $h^{\textrm{Q}}\left(\tau\right)$ are coming from integration (\[eq:defconv\]) in the vicinity of $\tau\approx1$, because of the large value of $\left(\tau-1\right)^{-3}$. Therefore, we impose the next two additional conditions at the dominating point $\tau=1$. Namely, $h^{\textrm{C}}$ is supposed to yield the observed value of the Hubble constant $$H_{0}\equiv\frac{\dot{a}\left(1\right)}{a\left(1\right)}=\frac{1}{2}\dot{h}^{\textrm{C}}\left(1\right),\label{eq:hubbledef}$$ and the observed deceleration parameter $q_{0}=q^{\textrm{C}}\left(1\right)$. Solving (\[eq:qzero-1\]), (\[eq:firsttwoeq\]) and (\[eq:hubbledef\]) for $h_{k}$ ($k=1,2,3$), we obtain $$\begin{aligned} h_{1} & = & 3-\left(3+q_{0}\right)H_{0},\nonumber \\ h_{2} & = & -3+\left(4+2q_{0}\right)H_{0},\label{eq:threeh}\\ h_{3} & = & 1-\left(1+q_{0}\right)H_{0}.\nonumber \end{aligned}$$ To estimate only the order and the qualitative behavior of the present time quantum contribution to the accelerating expansion of the Universe, it is sufficient to insert to (\[eq:threeh\]) the following crude approximation: $H_{0}=1$ and $q_{0}=-\frac{1}{2}$. Now $$h_{1}=\frac{1}{2},\qquad h_{2}=0,\qquad h_{3}=\frac{1}{2},$$ yielding $$\dot{h}^{\textrm{C}}\left(1\right)=2,\qquad\ddot{h}^{\textrm{C}}\left(1\right)=3.$$ Finally, by virtue of (\[eq:eightdiff\]–\[eq:sixthdiff\]) $$h^{\textrm{Q}}\left(1\right)=\frac{\lambda}{12},\qquad\dot{h}^{\textrm{Q}}\left(1\right)=\frac{3\lambda}{4},\qquad\ddot{h}^{\textrm{C}}\left(1\right)=-\frac{5\lambda}{3},$$ and hence (see (\[eq:qquant\])) $$q_{0}^{\textrm{Q}}\equiv q^{\textrm{Q}}\left(1\right)=\frac{11\lambda}{6}=\frac{11\kappa^{2}}{192\pi^{2}t_{0}^{2}}\sim10^{-46}.\label{eq:finalnumber}$$ Summary ======= In the framework of the SK (one-loop) perturbative quantum gravity, we have derived the formula (\[eq:finalnumber\]) expressing the (approximated) value of the present time quantum contribution $q_{0}^{\textrm{Q}}$ to the deceleration parameter $q_{0}$. The present time quantum contribution, $q_{0}^{\textrm{Q}}\sim+10^{-46}$, is positive but it is negligibly small in comparison to the observed (negative) value of the deceleration parameter, $q_{0}\approx-\frac{1}{2}$. Therefore, we deal with an extremely small damping (slowing down) of the expansion of the Universe, which is of quantum origin (quantum friction). One should also stress, that in the course of our analysis, because of some technical difficulties, we have been forced to confine our work to a particular case: a conforming with the FLRW form polynomial (of the third degree) perturbation around the Minkowski metric — to avoid short distance infinities; minimally coupled massless scalar field and conformally coupled fields (trivial contributions) only, without (virtual) gravitons — to avoid calculational limitations. Finally, it would be desirable to compare our present result to our earlier computation (see @broda2011one), where we have obtained an opposite result, i.e. repulsion instead of damping. First of all, one should note that non-SK approaches are acausal, in general, for finite time intervals, as they take into account contributions coming from the future state of the Universe. This follows from the fact that the Feynman propagator has an “advanced tail”, which is not dangerous for (infinite time interval) S-matrix elements. Besides, the present work concerns scalar field contributions, whereas the result of the previous paper is determined by graviton contributions. Nevertheless, in the both approaches, quantum contributions are trivial for conformal fields, which well corresponds to conformal flatness of the FLRW metric. Supported by the University of Łódź grant.
--- abstract: 'We calculate heat invariants of arbitrary Riemannian manifolds without boundary. Every heat invariant is expressed in terms of powers of the Laplacian and the distance function. Our approach is based on a multi-dimensional generalization of the Agmon-Kannai method. An application to computation of the Korteweg-de Vries hierarchy is also presented.' address: 'Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel' author: - Iosif Polterovich title: Heat invariants of Riemannian manifolds --- \[equation\][Definition]{} \[equation\][Lemma]{} \[equation\][Theorem]{} \[equation\][Corollary]{} \[equation\][Proposition]{} Introduction and main results ============================= Heat invariants --------------- Let $M$ be a $d$-dimensional Riemannian manifold without boundary with a metric $(g_{ij})$, and $\Delta$ be the [Laplace-Beltrami]{} operator (or simply the [Laplacian]{}) on $M$. In local coordinates $(x_1,\dots,x_d)$ the Laplacian is given by $$\Delta f = - \frac{1}{\sqrt{g}}\sum_{i,j=1}^d \frac{\partial(\sqrt{g} g^{ij}(\partial f/\partial x_i))}{\partial x^j},$$ where $g=$det$(g_{ij})$, and $(g^{ij})$ denotes the inverse of the matrix $(g_{ij})$. The [heat kernel]{} $K(t,x,y)$ is the fundamental solution of the heat equation $$(\frac{\partial}{\partial t}+\Delta)f=0.$$ The function $K(t,x,y)$ is analytic in $t>0$ and $C^{\infty}$ in $x$ and $y$, and has the following asymptotic expansion on the diagonal as $t\to 0+$ (see \[G1\]): $${\mbox K}(t,x,x) \sim \sum_{n=0}^{\infty} a_n(x)t^{n-\frac{d}{2}}$$ It is called the Minakshisundaram-Plejel asymptotic expansion (\[MP\]). The coefficients $a_n(x)$ are (local) [heat invariants]{} of the manifold $M$. They are homogeneous polynomials of degree $2n$ in the derivatives of the Riemannian metric $\{g^{ij}\}$ at the point $x$ (see \[G2\]). Integrating $a_n(x)$ over the manifold one gets the coefficients $a_n$ of the expansion for the trace of the heat operator $e^{-t\Delta}$: $$\label{tr} \sum_i e^{-t\lambda_i} \sim \sum_{n=0}^{\infty}\left(\,\,\int\limits_M a_n(x)\sqrt{g}dx \right) t^{n-\frac{d}{2}} \sim \sum_{n=0}^{\infty}a_n t^{n-\frac{d}{2}}.$$ Computation of heat invariants is a well-known problem in spectral geometry (see \[Be\], \[G1\], \[Ch\], \[G3\], \[Ro\]) which has various applications (\[F\], \[P2\]). The first method for derivation of heat kernel asymptotics is due to Seeley (\[Se\]). This method was deveoped later by Gilkey (see Theorem 1.3 in \[G1\]) who presented a way to get recursive formulas for the heat invariants. However, explicit formulas for $a_n(x)$ in arbitrary dimension existed only for $n\le 5$ (\[MS\], \[Sa\], \[Av\], \[vdV\]). The reason for this is the combinatorial complexity of $a_n(x)$ which is increasing very rapidly with the growth of $n$. For the higher heat invariants only partial information is known (\[BGØ\], \[OPS\]). Let us also mention interesting recursive formulas for $a_n(x)$ obtained in \[Xu\]. In this paper we calculate all heat invariants $a_n(x)$ for an arbitrary Riemannian manifold without boundary in terms of powers of the Laplacian and the distance function. Note that the heat invariants were initially given by a recursive system of differential equations involving exactly the Laplacian and the distance function (see \[MP\]). Main result ----------- Given a point $x\in M$ denote by $\rho_x:M\to {\Bbb R}$ the corresponding distance function: for every $y\in M$ the distance between the points $y$ and $x$ is $\rho_x(y)$. \[most\] Heat invariants $a_n(x)$ are equal to $$a_n(x)=(4\pi)^{-d/2}(-1)^n\sum_{j=0}^{3n} \binom{3n+\frac{d}{2}}{j+\frac{d}{2}} \frac{1}{4^j \, j! \, (j+n)!}\left. \Delta^{j+n}(\rho_x(y)^{2j})\right\|_{y=x}.$$ The binomial coefficients for $d$ odd are defined by (\[bc\]). Structure of the paper ---------------------- In \[P1\], \[P2\] we have developed a method for computation of heat invariants based on the Agmon-Kannai asymptotic expansion of resolvent kernels of elliptic operators (\[AK\]). In \[P1\] this method is used to obtain explicit formulas for the heat invariants of $2$-dimensional Riemannian mainfolds, in \[P2\] — for computation of the Korteweg-de Vries (KdV) hierarchy via heat kernel coefficients of the $1$-dimensional Schrödinger operator. In this paper we present a multi-dimensional generalization of the Agmon-Kannai method which is described in section 2.3. In section 3.1 we apply it to get formulas for the heat invariants in normal coordinates. It turns out that combinatorial coefficients in these formulas can be substantially simplified which is done in section 4.1. In section 4.2 we present $a_n(x)$ in a completely invariant form and prove Theorem \[most\]. Main result allows to simplify the formulas for the KdV hierarchy obtained in \[P2\]. This is shown in sections 5.1 and 5.2. Asymptotics of derivatives of the resolvent =========================================== A modification of Agmon-Kannai expansion ---------------------------------------- The original Agmon–Kannai theorem (\[AK\]) deals with asymptotic behaviour of resolvent kernels of elliptic operators. In \[P1\] we have obtained a concise reformulation of this theorem which is suitable for computation of heat invariants. We start with some notations. Let $H$ be a a self–adjoint elliptic differential operator of order $p$ on a Riemannian manifold $(M, g_{ij})$ of dimension $d<p$ and let $H_0$ be the operator obtained by freezing the coefficients of the principal part $H'$ of the operator $H$ at some point $x\in M$: $H_0=H'(x)$. Denote by $R_\lambda(x,y)$ the kernel of the resolvent $R_\lambda=(H-\lambda)^{-1}$, and by $F_{\lambda}(x,y)$ — the kernel of $F_{\lambda}=(H_0-\lambda)^{-1}$. $\operatorname{([P1])}.$ \[ak\] The resolvent kernel $R_{\lambda}(x,y)$ has the following asymptotic representation on the diagonal as $\lambda \to \infty$: $$\label{s} R_\lambda(x,x) \sim \frac{1}{\sqrt{g}}\sum_{m=0}^\infty X_m F_\lambda^{m+1}(x,x),$$ where the operators $X_m$ are defined by: $$\label{y} X_m=\sum_{k=0}^m (-1)^k \binom{m}{k} H^k H_0^{m-k}, \,\, m\ge 0.$$ Derivatives of the resolvent ---------------------------- The main obstruction for using Theorem \[ak\] directly for computation of heat invariants of a $d$-dimensional Riemannian manifold is the condition $d<p$, where $p=2$ is the order of the Laplacian. In \[P1\] we avoid this difficulty for 2-dimensional manifolds taking the difference of resolvents. However, in the general case one should consider derivatives of the resolvent kernel (cf. \[AvB\]). The following asymptotic expansion on the diagonal holds for the derivatives of the resolvent kernel of the Laplacian on a $d$-dimensional Riemannian manifold $M$: $$\label{res} \frac{d^s}{d\lambda^s}R_\lambda(x,x)\sim \sum_{n=0}^{\infty} \Gamma(s+n-\frac{d}{2}+1)a_n(x) (-\lambda)^{\frac{d}{2}-s-n-1}, \,\,\, s\ge d/2,$$ where $a_n(x)$ are heat invariants of the manifold $M$. [Proof.]{} Let $\operatorname{Re} \lambda <0$. We have (formally): $$\int_0^\infty e^{-t(\Delta-\lambda)}dt=\frac{1}{\Delta-\lambda}$$ Differentiating $R_{\lambda}$ $s$ times with respect to $\lambda$ we get a self–adjoint operator from $L^2(M)$ into the Sobolev space ${\mbox H}^{2s+2}(M)$. Since $2s+2> \operatorname{dim} M$ this operator has a continuous kernel (see \[AK\]). Taking into account (\[tr\]) we formally have: $$\label{ros} \frac{d^s}{d\lambda^s}\left(\frac{1}{\Delta-\lambda}\right)= \int_0^\infty t^se^{- t(\Delta-\lambda)}dt \sim \sum_{n=0}^\infty a_n \int_0^\infty t^{s+n-d/2}e^{\lambda t}dt$$ The asymptotic expansion in (\[ros\]) is obviously valid if we integrate over a finite interval $[0,T]$. In order to show that it remains true in our case as well we need an additional argument. Indeed, it is well-known (for example, see \[Da\]) that $$\|e^{-t\Delta}\|\le ct^{-d/2}.$$ Therefore we have: $$\left\|\int_0^\infty t^se^{- t(\Delta-\lambda)}dt- \int_0^T t^se^{- t(\Delta-\lambda)}dt\right\|\le \int_T^\infty ct^{s-d/2}e^{\lambda t}dt.$$ Let us estimate the second integral. We have: $$\int_T^\infty t^{s-d/2}e^{\lambda t}dt\le e^{-\epsilon T}\int_T^{\infty} t^{s-d/2}e^{(\lambda+\epsilon)t}dt\le e^{-\epsilon T}\frac{\Gamma(s-d/2+1)} {(\lambda+\epsilon)^{s-d/2+1}}$$ Take $\epsilon=\sqrt{\|\lambda\|}$. Then for $T=1$ this is $O(e^{-\sqrt{\|\lambda\|}})$ and therefore the term $$\int_T^\infty t^{s-d/2}e^{\lambda t}dt$$ is negligent. This proves the asymptotic expansion in (\[ros\]). The right-hand side of (\[ros\]) is equal to $$\sum_{n=0}^\infty \frac{a_n}{(-\lambda)^{s+n+1-d/2}} \int_0^\infty u^{s+n-d/2}e^{-u}du =\sum_{n=0}^{\infty} \frac{\Gamma(s+n+1-d/2)a_n}{(-\lambda)^{s+n+1-d/2}},$$ and this completes the proof of the lemma. Agmon-Kannai expansion for derivatives of the resolvent ------------------------------------------------------- In the notations of Theorem \[ak\] let $H=\Delta$ be the Laplacian on a $d$-dimensional Riemannian manifold $M$, and $\Delta_0$ be the operator obtained from the principal part of the Laplacian by freezing its coefficients at a certain point $x\in M$. As before, $R_\lambda=(\Delta-\lambda)^{-1}$, $F_{\lambda}=(\Delta_0-\lambda)^{-1}$. The following asymptotic expansion on the diagonal holds for the derivatives of the resolvent kernel of the Laplacian on a $d$-dimensional Riemannian manifold $M$: $$\label{der} \frac{d^s}{d\lambda^s}R_\lambda(x,x)\sim \frac{1}{\sqrt{g}} \sum_{m=0}^{\infty}\frac{(m+s)!}{m!} X_mF_{\lambda}^{m+s+1}, \quad s\ge d/2.$$ [Proof.]{} Formally we have: $$\frac{d}{d\lambda}F_\lambda=\frac{d}{d\lambda} \left(\frac{1}{\Delta_0-\lambda}\right)=\frac{1}{(\Delta_0-\lambda)^2}= F_{\lambda}^2.$$ This implies $$\frac{d^s}{d\lambda^s}F_\lambda^{m+1}= \frac{(m+s)!}{m!}F_{\lambda}^{m+s+1}.$$ Together with (\[s\]) this completes the proof of the theorem. Let us introduce the standard multi-index notations (see \[Hö\]): if $\alpha=(\alpha_1,\dots,\alpha_d)$ is a multi-index, then $\|\alpha\|=\alpha_1+\cdots+\alpha_d$, $\alpha!=\alpha_1!\cdots\alpha_d!$. For any vector $x=(x_1,\dots,x_d)$ we denote $x^{\alpha}=x_1^{\alpha_1}\cdots x_d^{\alpha_d}$ and $$\frac{\partial^{\alpha}}{\partial x^{\alpha}}= \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}}\cdots \frac{\partial^{\alpha_d}}{\partial x_d^{\alpha_d}}.$$ We note that (\[der\]) and (\[s\]) are in fact asymptotic expansions in the powers of $-\lambda$ as well as (\[res\]). This is due to the following formula (see \[AK\]): $$\begin{gathered} \label{int} \frac{\partial^{\gamma}}{\partial x^{\gamma}} F_\lambda^{m+s+1}(x,x)=\\ (-\lambda)^{\frac{d+\|\gamma\|}{2}-m-s-1} \frac{(-1)^{\frac{\|\gamma\|}{2}}}{(2\pi)^d}\int\limits_{\Bbb R^d} \frac{\xi^{\gamma}\,d\xi} {(\Delta_0(\xi)^2+1)^{m+s+1}}, \end{gathered}$$ where $\Delta_0(\xi)$ denotes the symbol of the operator $\Delta_0$, $\gamma=(\gamma_1,\dots,\gamma_d)$ is a multi-index and $\xi=(\xi_1,\dots,\xi_d)$. Heat invariants in normal coordinates ===================================== Computation of heat invariants ------------------------------ Let $(x_1,\dots,x_d)$ be local coordinates on the Riemannian manifold $M$ such that the Riemannian metric at the origin $x=(0,\dots,0)\in M$ (in the sequel we simply write $x=0$), is Euclidean: $g_{ij}\|_{x=0}=\delta_{ij}$ For convenience we may consider normal coordinates on $M$ centered at the point $x=0$ (see \[GKM\]). \[main\] Let $M$ be a $d$-dimensional Riemannian manifold without boundary and $(x_1,\dots,x_d)$ be normal coordinates on $M$ centered at the point $x=0$. Then the heat invariants $a_n(x)$ at the point $x=0$ are equal to: $$\begin{gathered} \label{q} a_n(0)= (4\pi)^{-\frac{d}{2}}(-1)^n \sum_{m=n}^{4n}\sum_{k=n}^m \frac{1}{k!\,2^{2m-2n}}\cdot \\ \cdot \sum_{\|\alpha\|=m-k}\,\,\sum_{\|\beta\|=k-n} \frac{(2\alpha+2\beta)!}{\alpha!(\alpha+\beta)!(2\beta)!} \left.\Delta^k(x^{2\beta})\right\|_{x=0},\end{gathered}$$ where $\alpha=(\alpha_1,\dots,\alpha_d)$, $\beta=(\beta_1,\dots,\beta_d)$ are multi-indices. [Proof of Theorem \[main\]]{} Since $(x_1,\dots,x_d)$ are normal coordinates centered at $x=0$, the principal part of the Laplacian at this point coincides with the Euclidean Laplacian, i.e.: $$\label{d0} \Delta_0=-\frac{\partial^2}{\partial x_1^2}-\cdots- \frac{\partial^2}{\partial x_d^2}.$$ Due to (\[res\]), in order to compute the coefficient $a_n(x)$ we have to collect all terms in the expansion (\[der\]) containing $(-\lambda)^{d/2-s-n-1}$. From (\[int\]) we have: $$\frac{d+\|\gamma\|}{2}-m-s-1=\frac{d}{2}-s-n-1,$$ which implies $\|\gamma\|=2m-2n$ and in particular $m\ge n$. As it was shown in \[P1\], estimates on the orders of operators $X_m$ (namely, Lemma 3.1 and Theorem 5.1 in \[AK\]) imply that $m\le 4n$. Note that due to (\[d0\]) all indices $\gamma_1,\dots,\gamma_2$ should be even since otherwise the integral in (\[int\]) will vanish. Setting $\gamma=2\mu=(2\mu_1,\dots,2\mu_d)$ and taking into account that $\|\mu\|=m-n$ we compute this integral (see \[GR\]): $$\int\limits_{\Bbb R^d} \frac{\xi^{2\mu}\,d\xi}{(\xi^2+1)^{m+s+1}}= \frac{\Gamma(\mu_1+\frac12)\Gamma(\mu_2+\frac12)\cdots\Gamma(\mu_d+\frac12) \Gamma(s+n+1-\frac{d}{2})}{(m+s)!}$$ Substituting this into (\[int\]) and further on into (\[der\]) we obtain due to (\[res\]): $$a_n(x)=\left.\sum_{m=n}^{4n}\sum_{k=0}^m \frac{(-1)^{k+m-n}}{m!(2\pi)^d}\binom{m}{k} \Delta^k\Delta_0^{m-k}\left(\sum_{\|\mu\|=m-n}\frac{x^{2\mu}}{(2\mu)!} \prod_{i=1}^d\Gamma\left(\mu_i+\frac12\right) \right)\right\|_{x=0}$$ Note that $s$ has cancelled out as one could expect since heat invariants do not depend on $s$! Now let us simplify this formula. First notice that $$\label{1} \Delta_0^{m-k}=(-1)^{m-k}\sum_{\|\beta\|=m-k}\frac{(m-k)!}{\beta!} \frac{\partial^{2\beta}}{\partial x^{2\beta}},$$ where $\beta=(\beta_1,\cdots,\beta_d)$. Using the well-known representation of the $\Gamma$-function $$\label{gam} \Gamma(k+1/2)=\frac{\sqrt{\pi}(2k)!}{4^k k!}$$ we also obtain: $$\label{2} \prod_{i=1}^d\Gamma\left(\mu_i+\frac12\right)= \frac{\pi^{d/2}}{2^{2m-2n}}\frac{(2\mu)!}{\mu!}.$$ Let us substitute (\[1\]) and (\[2\]) into the above formula for the $a_n(x)$ and apply $\Delta_0^{m-k}$ to $x^{2\mu}$. Introducing the new summation multi-index $\alpha=\mu-\beta$ and noticing that all terms for $k<n$ vanish we finally obtain $$a_n(0)=(4\pi)^{-\frac{d}{2}}(-1)^n \sum_{m=n}^{4n}\sum_{k=n}^m \frac{1}{k!\,2^{2m-2n}}\sum_{\|\alpha\|=m-k}\,\,\sum_{\|\beta\|=k-n} \frac{(2\alpha+2\beta)!}{\alpha!(\alpha+\beta)!(2\beta)!} \left.\Delta^k(x^{2\beta})\right\|_{x=0}.$$ This completes the proof of the theorem. Remarks ------- The proof of the Theorem \[main\] is similar to the proofs of main theorems in \[P1\] and \[P2\]. One may check that in the particular cases of the $2$-dimensional Laplacian and the $1$-dimensional Schrödinger operator Theorem \[main\] agrees with the results obtained in \[P1\] and \[P2\]. Invariance and combinatorial identities ======================================= Combinatorial identities ------------------------ Let us rewrite (\[q\]) in the following way: $$(4\pi)^{-\frac{d}{2}}(-4)^n\sum_{k=n}^{4n}\frac{1}{k!}\left. \Delta^k\left(\sum_{m=k}^{4n} \frac{1}{4^m}\sum_{\|\beta\|=k-n}\,\left( \sum_{\|\alpha\|=m-k} \frac{(2\alpha+2\beta)!\beta!}{\alpha!(\alpha+\beta)!(2\beta)!} \right) \frac{1}{\beta!}x^{2\beta}\right)\right\|_{x=0}.$$ Observe that due to the multinomial theorem $$\label{mult} \sum_{\|\beta\|=k-n}\frac{1}{\beta!}x^{2\beta}=\frac{1}{(k-n)!} (x_1^2+\cdots +x_d^2)^{k-n}.$$ Let us recall the following generalization of the binomial coefficients (see \[Er\]). For real $z\in {\Bbb R}$ and $a\in {\Bbb N}$ set $$\begin{gathered} \label{bc} \binom{z}{a}=\binom{z}{z-a}=\\ \frac{\Gamma(z+1)}{\Gamma(a+1)\Gamma(z-a+1)}= \frac{z(z-1)\cdots (z-a+1)}{a!}.\end{gathered}$$ We also set $\binom{z}{0}=\binom{z}{z}=1$. Let us proceed with the following simple combinatorial formula. \[hren\] $$\label{vspom2} \sum_{a=0}^{u}\binom{z+a}{a}\binom{w+u-a}{u-a}=\binom{z+w+u+1}{z+w+1}.$$ [Proof.]{} Using the method of generating functions (see \[Rio\]) we have: $$\sum_{a=0}\binom{z+a}{a}q^{2a}=\frac{1}{(1-q^2)^{z+1}}$$ which implies $$\sum_{u=0}^{\infty}\sum_{a_1+a_2=u}\binom{z+a_1}{a_1}\binom{w+a_2}{a_2} q^{2u}=\frac{1}{(1-q^2)^{z+w+2}}= \sum_{u=0}^{\infty}\binom{z+w+u+1}{z+w+1}q^{2u}.$$ This completes the proof of the lemma. Now we can prove our main combinatorial identity. \[comb1\] Let $\alpha$, $\beta$ be multi-indices of dimension $d$ and let $\|\beta\|=v$. Then $$\label{expr} \sum_{\|\alpha\|=u} \frac{(2\alpha+2\beta)!\beta!}{\alpha!(\alpha+\beta)!(2\beta)!}= 4^u\,\binom{u+v-1+d/2}{u}.$$ [Proof.]{} We proceed by induction over $d$. For $d=1$ we have due to (\[gam\]): $$\frac{(2u+2v)! v!}{u! (u+v)! (2v)!}=\frac{4^{u+v}\Gamma(u+v+1/2) \sqrt{\pi}}{4^v \Gamma(v+1/2) u! \sqrt{\pi}}=4^u\binom{u+v-1/2}{u}.$$ and hence (\[expr\]) is valid. Suppose we have proved the formula (\[expr\]) in all dimensions less than some $d>1$. Let us prove it in the dimension $d$. Denote $\alpha_1=a$, $\beta_1=b$. By induction we may rewrite the sum in (\[expr\]) as $$\label{vspom3} \sum_{\|\alpha\|=u} \frac{(2\alpha+2\beta)!\beta!}{\alpha!(\alpha+\beta)!(2\beta)!}= \sum_{a=0}^u \frac{(2a+2b)! b!}{a!(a+b)!(2b)!} \binom{u-a-1+l}{u-a}4^{u-a},$$ where $l=v-b+\frac{d-1}{2}$. On the other hand, $$\frac{1}{4^a}\frac{(2a+2b)!b!}{a!(a+b)!(2b)!}=\binom{a+b-1/2}{a}$$ and hence (\[vspom3\]) is equal to $$4^u \sum_{a=0}^u \binom{a+b-1/2}{a}\binom{u-a-1+l}{u-a}.$$ By Lemma (\[hren\]) this equals to $$4^u \binom{u+b+l-1/2}{u}=4^u \binom{u+v-1+d/2}{u},$$ which completes the proof of the theorem. Proof of Theorem \[most\] ------------------------- Set $u=m-k$, $v=k-n$. Combining Theorem \[main\], Theorem \[comb1\] and formula (\[mult\]) we obtain the following reformulation of (\[q\]): $$\begin{gathered} \label{qq} a_n(0)=\\ (4\pi)^{-d/2} (-4)^n \sum_{k=n}^{4n} \left(\sum_{m=k}^{4n} \binom{m+\frac{d}{2}-n-1}{m-k}\right) \frac{\Delta^k(\|x\|^{2k-2n})\|_{x=0}}{k!\, 4^k},\end{gathered}$$ where $\|x\|^2=x_1^2+\cdots+x_d^2$. Denote $i=m-k$, $j=k-n$. By Lemma \[hren\] the inner sum may be rewritten as $$\sum_{i=0}^{4n-k}\binom{i+d/2+k-n-1}{i}=\binom{3n+d/2}{j+d/2}.$$ Therefore (\[qq\]) is equal to $$\label{qqq} a_n(0)=(4\pi)^{-d/2} (-1)^n \sum_{j=0}^{3n}\binom{3n+d/2}{j+d/2} \frac{\Delta^{j+n}(\|x\|^{2j})\|_{x=0}}{4^j\, j! \,(j+n)!}.$$ Consider the function $\rho_x(y)^2$ which is the square of the distance between the points $x$ and $y$. In normal coordinates centered at the point $x=0$ it is given locally by $$\rho_x(y)^2=\sum_{i,j=1}^d g_{ij}(0)\,y_i y_j = y_1^2+\cdots+y_d^2=\|y\|^2$$ where $y=(y_1,..,y_d)$ (see \[Du\], p. 94). Therefore we may rewrite formula (\[qqq\]) in an invariant form, namely $$a_n(x)=(4\pi)^{-d/2} (-1)^n \sum_{j=0}^{3n}\binom{3n+d/2}{j+d/2} \frac{\Delta_y^{j+n}(\rho_x(y)^{2j})\|_{y=x}}{4^j\, j! \,(j+n)!},$$ where the subscript of the Laplacian means that the operator is acting on functions in $y$-variable. This completes the proof of Theorem \[most\]. Application to computation of the Korteweg-de Vries hierarchy ============================================================= Asymptotics of Schrödinger operator and KdV hierarchy ----------------------------------------------------- In \[P2\] we have applied the Agmon-Kannai method to computation of the Korteweg-de Vries hierarchy (see \[NMPZ\]). Let us briefly recall the setting of the problem. Consider the $1$-dimensional Schrödinger operator: $$L=\frac{\partial^2}{\partial x^2}+U(x).$$ Its heat kernel $H(t,x,y)$ is the fundamental solution of the heat equation $$\left(\frac{\partial}{\partial t} - L\right)f=0.$$ It has the following asymptotic representation on the diagonal as $t\to 0+$: $$H(t,x,x)\sim\frac{1}{\sqrt{4\pi t}}\sum_{n=0}^{\infty}h_n[U]t^n,$$ where $h_n[U]$ are some polynomials in $U(x)$ and its derivatives. The KdV hierarchy is defined by (see \[AvSc\]): $$\label{kdvh} \frac{\partial U}{\partial t}=\frac{\partial}{\partial x}G_n[U],$$ where $$G_n[U]= \frac{(2n)!}{2\cdot n!}h_n[U], \quad n\in {\Bbb N}.$$ Set $U_0=U$, $U_n=\partial^n U/\partial x^n$, $n\in {\Bbb N}$, where $U_n$, $n\ge 0$ are formal variables. The sequence of polynomials $G_n[U]=G_n[U_0,U_1,U_2,\dots]$ starts with (see \[AvSc\]): $$G_1[U]=U_0, \,\, G_2[U]=U_2+3U_0^2, \,\, G_3[u]=U_4+10U_0U_2+5U_1^2+10U_0^3, \, \dots$$ In particular, substituting $G_2[U]$ into (\[kdvh\]) we obtain the familiar Korteweg-de Vries equation (see \[NMPZ\]): $$\frac{\partial U}{\partial t} =\frac{\partial^3 U}{\partial x^3} +6U\frac{\partial U}{\partial x}$$ Computation of the KdV hierarchy -------------------------------- In \[P2\] we have presented explicit formulas for the KdV hierarchy (we refer to \[P2\] for the history of this question). Theorem \[most\] allows to simplify the results of \[P2\]. \[check\] The KdV hierarchy is given by: $$G_n[U]=\frac{(2n)!}{2\cdot n!}\sum_{j=0}^{n} \binom{n+\frac{1}{2}}{j+\frac{1}{2}} \frac{(-1)^j}{4^j \, j! \, (j+n)!}P_{nj}[U], \eqno{(\ref{check})}$$ where the polynomial $P_{nj}[U]$ is obtained from $\left.L^{j+n}(x^{2j})\right\|_{x=0}$ by a formal change of variables: $U_i(0)\to U_i$, $i=0,..,2n+2j-2$. This expression can be completely expanded due to a formula for the powers of the Schrödinger operator (\[Rid\]). \[KdV1\] The polynomials $G_n[U]$, $n\in {\Bbb N}$ are equal to: $$G_n[U]= \frac{(2n)!}{2\cdot n!}\sum_{j=0}^{n} \binom{n+\frac{1}{2}}{j+\frac{1}{2}} \frac{(-1)^j (2j)!}{4^j \, j! \, (j+n)!} \sum_{p=1}^{j+n}\mskip-2\thinmuskip \sum_{k_1,\dots,k_p\atop k_1+\cdots +k_p=2(n-p)}\mskip-13\thinmuskip C_{k_1,\dots,k_p}U_{k_1}\cdots U_{k_p},$$ where $$C_{k_1,\dots,k_p}=\sum\begin{Sb} 0\le\l_0\le l_1\le \cdots\le l_{p-1}=j+n-p\\ 2l_i\ge k_1+\cdots+k_{i+1},\,\,i=0,\dots,p-1.\end{Sb} \binom{2l_0}{k_1}\binom{2l_1-k_1}{k_2}\cdots \binom{2l_{p-1}-k_1-\cdots-k_{p-1}}{k_p}.$$ Remark {#remark .unnumbered} ------ Formula (\[check\]) was checked using Mathematica (\[Wo\]) and for $1\le n\le5$ the results agreed with the already known ones (cf. \[GD\]). Acknowledgments {#acknowledgments .unnumbered} --------------- This paper is a part of my Ph.D. research at the Department of Mathematics of the Weizmann Institute of Science. I am very grateful to my Ph.D. advisor Yakar Kannai for his constant help and support. The author is indebted to Leonid Polterovich, Amitai Regev and Mikhail Solomyak for helpful discussions. I would like to thank Ivan Avramidi, Isaac Chavel, Peter Gilkey and Steven Rosenberg for their remarks on the preliminary versions of this paper. I am also grateful to Sergei Novikov for a useful discussion on the KdV hierarchy and its computation. Theorem \[comb1\] was first established with the help of a Mathematica implementation of the Wilf-Zeilberger algorithm (see \[PWZ\]). I am thankful to Marko Petkovšek and Doron Zeilberger for their help with this matter. References {#references .unnumbered} ========== \[AK\] S. Agmon, Y. 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Chavel, Eigenvalues in Riemannian geometry, Academic Press, 1984. \[Da\] E. B. Davies, Heat kernels and spectral theory. Cambridge University Press, 1989. \[Du\] G.F.D. Duff, Partial differential equations, University of Toronto Press, 1956. \[Er\] A. Erdélyi et. al., Higher transcendental functions, vol. 1, McGraw-Hill, 1953. \[F\] S.A. Fulling ed., Heat Kernel Techniques and Quantum Gravity, Discourses in Math. and its Appl., No. 4, Texas A&M Univ., 1995. \[GD\] I.M. Gelfand, L.A. Dikii, Asymptotic behaviour of the resolvent of Sturm-Liouville equation and the algebra of the Korteweg-de Vries equations, Russian Math. Surveys, 30:5 (1975), 77-113. \[G1\] P. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975), 601-618. \[G2\] P. Gilkey, The index theorem and the heat equation, Math. Lect.Series, Publish or Perish, 1974. \[G3\] P. Gilkey, Heat equation asymptotics, Proc. Symp. Pure Math. 54 (1993), 317-326. \[GR\] I.S. Gradshtein, I.M. 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--- abstract: \| We describe a systematic investigation into the existence of congruences between the mod $p$ torsion modules of elliptic curves defined over ${\mathbb{Q}}$, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod $p$ congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes $p\ge7$ to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than $\numprint{500000}$. We also show that while such congruences exist for each $p\le17$, there are none for $p \geq 19$ in the database, in line with a strong form of the Frey-Mazur conjecture. address: - 'Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom' - 'Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain' author: - John Cremona - Nuno Freitas title: Global methods for the symplectic type of congruences between elliptic curves --- [^1] [^2] Introduction ============ Let $p$ be a prime, $K$ be a number field and $G_K = \operatorname{Gal}({{\overline{K}}}/K)$ the absolute Galois group of $K$. Let $E$ and $E'$ be elliptic curves defined over $K$, and write $E[p]$ and $E'[p]$ for their $p$-torsion $G_K$-modules. Let $\phi : E[p] \to E'[p]$ be an isomorphism of $G_K$-modules. There is an element $d(\phi) \in {\mathbb{F}}_p^\times$ such that the Weil pairings $e_{E,p}$ and $e_{E',p}$ satisfy $$e_{E',p}(\phi(P), \phi(Q)) = e_{E,p}(P, Q)^{d(\phi)}$$ for all $P, Q \in E[p]$. We say that $\phi$ is a [symplectic isomorphism]{} or an [anti-symplectic isomorphism]{} if $d(\phi)$ is a square or a non-square modulo $p$, respectively. When two elliptic curves have isomorphic $p$-torsion modules we say that there is a mod $p$ [congruence]{} between them, or that they are [ congruent]{} mod $p$ or $p$-[congruent]{}. For example, suppose that $\phi$ is induced by an isogeny (also denoted $\phi$) from $E$ to $E'$, of degree $\deg(\phi)$ coprime to $p$. Then, using Weil reciprocity, we have $$e_{E',p}(\phi(P), \phi(Q)) = e_{E,p}(P, \hat\phi\phi(Q)) = e_{E,p}(P, \deg(\phi)(Q)) = e_{E,p}(P, Q)^{\deg(\phi)},$$ where $\hat{\phi}$ denotes the dual isogeny; thus $d(\phi)=\deg\phi\pmod{p}$. Hence $\phi$ is symplectic or antisymplectic according as $\deg\phi$ is a quadratic residue or nonresidue mod $p$, respectively. We will refer to this condition as the isogeny criterion. Given $G_K$-isomorphic modules $E[p]$ and $E'[p]$ as above, it is possible they admit isomorphisms with both symplectic types; this occurs if and only if $E[p]$ admits an anti-symplectic automorphism. We say that * $E[p]$ satisfies [condition [(S)]{}]{}* if $E[p]$ does not admit anti-symplectic automorphisms. The following proposition follows from [@FKSym] and gives two equivalent description of [condition [(S)]{}]{}. \[P:conditionS\] Let $E$ be an elliptic curve over a number field $K$. Let $p > 2$ be a prime and ${{\overline{\rho}}}_{E,p} : G_K \to {\operatorname{GL}}_2({\mathbb{F}}_p)$ the representation arising from the action of $G_K$ on $E[p]$. Then $E[p]$ satisfies [condition [(S)]{}]{} if and only if the centralizer of ${{\overline{\rho}}}_{E,p}(G_K)$ in ${\operatorname{GL}}_2({\mathbb{F}}_p)$ contains only matrices with square determinant or if and only if one of the following is satisfied: - The image of ${{\overline{\rho}}}_{E,p}$ is a non-abelian subgroup of ${\operatorname{GL}}_2({\mathbb{F}}_p)$ or, - we have ${{\overline{\rho}}}_{E,p} \cong \left( \begin{smallmatrix} \chi & * \\ 0 & \chi \end{smallmatrix} \right)$ where $\chi : G_K \to {\mathbb{F}}_p^$ is a character and $ \neq 0$. If ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible then $E[p]$ satisfies condition (A) and hence [condition [(S)]{}]{}. The first claim is the second part of [@FKSym Lemma 6] and the second follows from [@FKSym Lemmas 7 and 8]. The last statement follows from the proof of Corollary 3 in [loc. cit.]{} In the case $K={\mathbb{Q}}$, [condition [(S)]{}]{} is satisfied by $E[p]$ for all $p\ge7$, by [@FKSym Corollary 3 and Proposition 2]. Hence, when a $G_{\mathbb{Q}}$-isomorphism $\phi : E[p] \simeq E'[p]$ exists, normally there is only one possible symplectic type for any such $\phi$. For an example where both types exist, take $p=5$ and $E$, $E'$ to be the curves[^3] ${\href{http://www.lmfdb.org/EllipticCurve/Q/11a1}{{\text{\rm11a1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/1342c2}{{\text{\rm1342c2}}}}$, respectively (see [@FKSym Example 5.2] for more details). It is then natural to consider triples $(E,E',p)$ where $E/K$ and $E'/K$ are elliptic curves with isomorphic $p$-torsion such that the $G_K$-modules isomorphisms $\phi : E[p] \rightarrow E'[p]$ are either all symplectic or all anti-symplectic. In this case, we will say that the [symplectic type]{} of $(E,E',p)$ is respectively symplectic or anti-symplectic. The problem of determining the symplectic of $(E,E',p)$ over $K={\mathbb{Q}}$ was extensively studied by the second author and Alain Kraus in [@FKSym]. The isogeny criterion gives an easy solution when $(E,E',p)$ arises from an isogeny $h \colon E \to E'$ of degree $n$ coprime to $p$, since in such cases $d(h\|_{E[p]}) = n$ and the symplectic type of $(E,E',p)$ is symplectic if $n$ is a square mod $p$ and anti-symplectic otherwise. Given a generic triple $(E, E', p)$, in principle, one could compute the $p$-torsion fields of $E$ and $E'$, write down the Galois action on $E[p]$ and $E'[p]$ and check if they are symplectically or anti-symplectically isomorphic. However, the degree of the $p$-torsion fields grows very fast with $p$ making this method not practical already over ${\mathbb{Q}}$ for $p = 5$. One way to circumvent this computational problem, at least over ${\mathbb{Q}}$, is to use the methods presented in [@FKSym]. Indeed, the main objective of [loc. cit.]{} was to establish a complete list of [local symplectic criteria]{}, allowing one to determine the symplectic type of $(E,E',p)$ using only standard information about the local curves $E/{\mathbb{Q}}_\ell$ and $E'/{\mathbb{Q}}_\ell$ at a single prime $\ell \neq p$ and congruence conditions on $p$. Further, it is also proved in [@FKSym] that if the symplectic type of $(E,E',p)$ is encoded in local information at a single prime $\ell \neq p$, then one of the local criteria will successfully determine it. There are cases where the local methods are insufficient: however, this can occur only when the representation ${{\overline{\rho}}}_{E,p} : G_{\mathbb{Q}}\to {\operatorname{GL}}_2({\mathbb{F}}_p)$ attached to $E$ has image without elements of order $p$; see [@FKSym Proposition 16] for an example. This paper has the following main objectives: - Give global methods to determine the symplectic type of $(E,E',p)$ when the local methods of [@FKSym] may not apply, specifically when the projective images are dihedral. - To systematically identify and determine the symplectic type of all congruences between curves in the LMFDB (see [@lmfdb]) for all $p\ge7$. More specifically, towards (i), we give a complete resolution for $p=7$ using modular curves and, for general $p$, we give global criteria for when $E$ and $E'$ are twists of each other: Proposition \[P:twist\] and Theorem \[T:quadratic\] cover congruences between quadratic twists and Proposition \[P:higherTwists\] and Theorem \[T:typeHigher\] study the existence and the symplectic type of congruences between higher order twists. For objective (ii), we have implemented in [[[Magma]{}]{}]{} [@magma] and [[[SageMath]{}]{}]{} [@sage] the methods from objective (i) together with those from [@FKSym]. We have used our code to classify the symplectic types of congruences between curves in the LMFDB, namely all elliptic curves defined over ${\mathbb{Q}}$ of conductor less than $\numprint{500000}$. More precisely, Theorem \[T:cong19\] shows there are no mod $p$ congruences in LMFDB for $p \geq 19$ and we found, for $p$ in the range $7 \leq p \leq 17$, all the congruences in the database; we report on the results obtained in Section \[S:statistics\] and \[S:Frey-Mazur\]. We note that the description of our method also includes a discussion on how to determine whether $E[p]$ and $E'[p]$ are isomorphic (ignoring the symplectic structure) in both the irreducible and reducible cases. Modular parametrizations ------------------------ The modular curve $X(p)$ parametrizes elliptic curves with full level $p$ structure. It has genus $0$ for $p=2,3,5$, genus $3$ for $p=7$ and genus $\ge26$ for $p\ge11$. Fixing an elliptic curve $E/K$, the curve $X_E(p)$, which is a twist of $X(p)$ (and hence has the same genus), parametrizes pairs $(E',\phi)$ such that $\phi$ is a symplectic isomorphism $E[p]\cong E'[p]$; similarly $X_E^-(p)$ parametrizes antisymplectic isomorphisms. Note that the pair $(E,\operatorname{id})$ constituted by $E$ itself and the identity map gives a base point defined over $K$ on $X_E(p)$, while $X_E^-(p)$ may have no $K$-rational points. It follows that congruences modulo $p$ for $p\le 5$ are common. There are certainly many mod $3$ and mod $5$ congruences in the database, but we have not searched for these systematically. Indeed, since $X_E(3)$ and $X_E(5)$ have genus 0, for each fixed $E$, there will always be congruences that are not part of the database independently of its range. In contrast, for primes $p \geq 7$, the curve $X_E(p)$ has genus $\geq 3$ and so for each $E$ there are only finitely many mod $p$ congruences with $E$, hence the database might contain all such congruences; however proving this fact for a fixed $E$ is a hard problem. For convenience we sometimes also write $X_E^+(p)$ to denote $X_E(p)$. By “explicit equations” for $X_E^{\pm}(p)$, we mean the following. - An explicit model for a family of curves, with equations whose coefficients are polynomials in ${\mathbb{Q}}[a,b]$, such that specializing $a,b$ gives a model for $X_E^{\pm}(p)$ where $E$ is the elliptic curve with equation $Y^2=X^3=aX+b$; each rational point $P$ is either a cusp of the modular curve or encodes a pair $(E',\phi)$ such that $(E,E', p)$ is a symplectic (respectively, antisymplectic) triple. - A rational function with coefficients in ${\mathbb{Q}}(a,b)$ defining the map $j: X_E^{\pm}(p) \to {\mathbb{P}}^1$, taking a point $P=(E',\phi)$ to $j(E')$. The degree of this map is the index $[{\operatorname{PSL}}_2({\mathbb{Z}}):\Gamma(p)] = \|{\operatorname{PSL}}_2({\mathbb{F}}_p)\|$: for example, when $p=7$ the degree is $168$. - Rational functions $c_4$ and $c_6$ such that for each non-cuspidal point $P=(E',\phi)$, a model for $E'$ is $Y^2=X^3+a'X+b'$ where $(a',b')=(-27c_4(P),-54c_6(P))$. For $p=3$ and $p=5$, the curves themselves have genus $0$ and hence we do not need equations, but the formulas for $j$, $c_4$ and $c_6$ are still useful. They may be found in [@Rubin-Silverberg]. Equations for $X_E(7)$, in this sense, were obtained by Kraus and Halberstadt in [@Halberstadt-Kraus-XE7], though the functions $c_4$ and $c_6$ in [@Halberstadt-Kraus-XE7] are only defined away from $5$ points (which may or may not be rational). Fisher gives more complete equations for this, together with $X_E^-(7)$ and $X_E^{\pm}(11)$, in [@Fisher]. Further motivation {#S:motivation} ------------------ We finish this introduction with a discussion on how the methods of this paper complement the local methods in [@FKSym]. From the discussion so far we know that triples $(E,E',p)$ as above give rise to ${\mathbb{Q}}$-points on one of the modular curves $X_E(p)$ or $X_E^-(p)$. The Frey-Mazur conjecture states there is a constant $C \geq 18$ such that, if $E/{\mathbb{Q}}$ and $E'/{\mathbb{Q}}$ satisfy $E[p] \simeq E'[p]$ as $G_{\mathbb{Q}}$-modules for some prime $p > C$, then $E$ and $E'$ are ${\mathbb{Q}}$-isogenous. Theorem \[T:cong19\] shows that for curves of conductor at most $\numprint{500000}$, this holds with $C=18$. In view of this conjecture, any $(E,E',p)$ with $p > C$ arises from an isogeny, hence its symplectic type is easily determined by the isogeny criterion. Thus we are interested in primes $p \leq C$. As mentioned above, it follows from [@FKSym] that if no local symplectic criterion applies to $(E,E',p)$ then the image of ${{\overline{\rho}}}_{E,p}$ is irreducible and contains no element of order multiple of $p$. Moreover, when ${{\overline{\rho}}}_{E,p}$ is reducible only the local criteria at primes of multiplicative or good reduction may succeed and often the bounds on a prime $\ell$ for which a local criterion at $\ell$ applies may not be useful in practice. From the strong form of Serre’s uniformity conjecture, these ‘bad’ cases for the local methods imply that either $E$ has complex multiplication (CM), or one of the following holds: - $p=3,5,7$ and ${{\overline{\rho}}}_{E,p}$ is reducible or has image the normalizer of a Cartan subgroup; - $p=11$ and ${{\overline{\rho}}}_{E,p}$ has image the normalizer of non-split Cartan subgroup; - $p=13$ and ${{\overline{\rho}}}_{E,p}$ is reducible; - $p=13$ and ${{\overline{\rho}}}_{E,p}$ has exceptional image projectively isomorphic to $S_4$; - $p=11,17$ or $37$, and $j(E)$ is listed in [@DahmenPhD Table 2.1]; in particular, ${{\overline{\rho}}}_{E,p}$ is reducible. In [@BarinderCrem Corollary 1.9] there is a list of $3$ rational $j$-invariants of elliptic curves over ${\mathbb{Q}}$ satisfying (iv); it has recently been shown (see [@BDMTV-S4]) that the associated genus $3$ modular curve $X_{S_4}(13)$ found explicitly in [@BarinderCrem] has no more rational points, and hence that this list is complete. By Proposition \[P:twist\], none of these curves is mod $13$ congruent to a twist of another of them (including itself); the same is true for the curves and values of $p$ in case (v). Thus no examples arise in cases (iv) and (v). The case (iii) includes the infinitely many curves with ${{\overline{\rho}}}_{E,13}$ reducible (recall that $X_0(13)$ has genus 0). However, we know of no reducible mod $13$ congruences between rational elliptic curves, so there are no known examples in this case either. Nevertheless, in spite of the lack of helpful bounds as mentioned above, a putative congruence between such curves will often be addressed by local methods in practice. Indeed, the bounds are very large as they depend on Tchebotarev density theorem to predict a prime $\ell$ of good reduction for $E$ where Frobenius has order multiple of $p$ but, in practice, it is usually easy to find such a prime $\ell$ after trying a few small primes. We refer to [@FKSym Example 31.2] for an example with $p=7$ analogous to the discussion in this paragraph. Our method described in Section \[S:statistics\] for $p=7$ could be adapted by replacing the modular curves $X_E(7)$ by $X_E(p)$ when explicit equations for the latter are known, which is the case for $p=3,5$ and $11$. This method works independently of the image of ${{\overline{\rho}}}_{E,p}$, so covers the remaining cases (i), (ii) entirely and also case (v) with $p=11$. Notation {#S:notation} -------- For $p$ an odd prime, define $p^=\pm p\equiv1\pmod4$, so that ${\mathbb{Q}}(\sqrt{p^})$ is the quadratic subfield of the cyclotomic field ${\mathbb{Q}}(\zeta_p)$. Let $D_{n}$ denote the dihedral group with $2n$ elements and $C_n \subset D_{n}$ for a normal cyclic subgroup of order $n$; note that $C_n$ is unique unless $n=2$, in which case $D_{n} \simeq C_2 \times C_2$ and there are three such subgroups. We will also denote by $C \subset {\operatorname{GL}}_2({\mathbb{F}}_p)$ a Cartan subgroup (either split or non-split) and by $N$ its normalizer in ${\operatorname{GL}}_2({\mathbb{F}}_p)$. For a number field $K$ we denote by $G_K$ the absolute Galois group of $K$. Let ${\varepsilon}_d$ be the quadratic character of $G_K$ associated to the extension $K(\sqrt{d})/K$. For $a, b \in K$, we write $E_{a,b}$ for the elliptic curve defined over $K$ by the short Weierstrass equation $Y^2=X^3+aX+b$; every elliptic curve over $K$ has such a model, unique up to replacing $(a,b)$ by $(au^4,bu^6)$ with $u\in K^$. We will denote by $I$ the identity matrix in ${\operatorname{GL}}_2({\mathbb{F}}_p)$. Congruences between Twists {#S:cong-twist} ========================== Congruences between elliptic curves which are twists of each other arise in a number of ways in our study; these are often between quadratic twists but they also occur between higher order twists. In this section we study all types of congruences in detail. Let $p$ be an odd prime. Let $K$ be a number field and $E/K$ an elliptic curve. If the representation ${{\overline{\rho}}}_{E,p}$ has image contained in the normaliser $N$ of a Cartan subgroup $C$ of ${\operatorname{GL}}_2({\mathbb{F}}_p)$ but not in $C$ itself, then the projective image ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_K)$ is isomorphic to $D_{n}$ for some $n \geq 2$ (see [@LangModForms Theorem XI.2.3]). The preimage of $C$ in $G_K$ then cuts out a quadratic extension $M = K(\sqrt{d})$ satisfying ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_M) \simeq C_n$. We will see in Proposition \[P:twist\] that in this setting there is a mod $p$ congruence between $E$ and its quadratic twist by $d$. This construction is the main source of congruences between twists and it appeared already in [@Halberstadt-11nonsplit §2], including a determination of the symplectic type of the congruence; our contribution for this type of congruences is to show that congruences between quadratic twists only occur via this construction, when the projective image is dihedral: see Theorem \[T:quadratic\]. The second contribution of this section is to classify congruences between higher order twists in Proposition \[P:higherTwists\] and describe their symplectic type in Theorem \[T:typeHigher\]. Twists of elliptic curves {#S:twists} ------------------------- Here we recall some standard facts about elliptic curves and their twists. Let $E$ be an elliptic curve defined over any field $K$ of characteristic $0$. The twists of $E$ over $K$ are parametrized by $H^1(G_K,\operatorname{Aut}(E))$. If $E'$ is a twist of $E$, then by definition there exists a ${{\overline{K}}}$-isomorphism $t:E\to E'$ so that, for all $P\in E({{\overline{K}}})$, we have $\sigma(t(P))=\psi(\sigma)t(\sigma(P))$ where $\psi:G_K\to\operatorname{Aut}(E')\cong\operatorname{Aut}(E)$ is a cocycle. Here, $\operatorname{Aut}(E)\cong\mu_n$ is cyclic of order $n=2$, $4$ or $6$ according as $j(E)\not\in\{0,1728\}$, $j(E)=1728$ and $j(E)=0$ respectively. By Kummer theory, $H^1(G_K,\operatorname{Aut}(E))\cong H^1(G_K,\mu_n)\cong K^/(K^)^n$; hence each $n$-twist is determined by a parameter $u\in K^$ whose image in $K^/(K^)^n$ determines the isomorphism class of the twist. In the cases where $\operatorname{Aut}(E)\not=\{\pm1\}$, we make two elementary, but important, observations. First, $G_K$ acts non-trivially on $\operatorname{Aut}(E)$, unless $-1$ or $-3$ (respectively for the cases $n=4$ and $n=6$) are squares in $K$, so the cocycle is usually not a homomorphism; secondly, there are two isomorphisms ${\mathcal{O}}={\mathbb{Z}}[\zeta_n]\cong\operatorname{End}(E)$ (differing by complex conjugation in ${\mathcal{O}}$), when $n=4$ or $6$, and hence two actions of ${\mathcal{O}}^\cong\operatorname{Aut}(E)$ on $E$. We fix isomorphisms ${\mathcal{O}}\cong\operatorname{End}(E)$ and ${\mathcal{O}}\cong\operatorname{End}(E')$, and hence isomorphisms $\mu_n={\mathcal{O}}^\cong\operatorname{Aut}(E)\cong\operatorname{Aut}(E')$, which are normalised (in the sense of [@SilvermanII Prop. I.1.1]), so that $\zeta\cdot t(P)=t(\zeta\cdot P)$ for $P\in E({{\overline{K}}})$ and $\zeta\in{\mathcal{O}}^$. Then the twist isomorphism $t$ is an isomorphism of ${\mathcal{O}}$-modules, and we may view the twisting cocycle $\psi$ as taking values in ${\mathcal{O}}^$. Explicitly, in terms of a short Weierstrass equation $E_{a,b}$ for $E$, we fix the action of $\zeta\in{\mathcal{O}}^$ to be $(x,y)\mapsto(\zeta^2x,\zeta^3y)$, and the $n$-twist by $u\in K^$ to be $t:(x,y)\mapsto(v^2x,v^3y)$ where $v \in \overline{K}$ satisfies $v^n=u$. Then the associated cocycle is $\psi(\sigma)=\sigma(v)/v$, we have $$\label{E:twist} \sigma(t(P)) = \psi(\sigma)t(\sigma(P)) = t(\psi(\sigma)\sigma(P)) \quad \text{ for all } \sigma \in G_K, \; P \in E({{\overline{K}}})$$ and $E$ and its $n$-twist by $u$ become isomorphic over $K(\root n\of u)$, which is an extension of $K$ of degree dividing $n$. We end this subsection with a brief discussion of each kind of twists. Quadratic twists, $n=2$: we usually denote the twist parameter by $d$ instead of $u$, and use the notation $E^d$ for the quadratic twist of $E$ by $d\in K^$. If $E=E_{a,b}$ then $E^d=E_{ad^2,bd^3}$. Quartic twists, $n=4$: only exist when $j(E)=1728$. The short Weierstrass model of such a curve has the form $E_{a,0}$, and its quartic twist by $u$ is $E_{au,0}$. As a special case, when $u=d^2\in(K^)^2$, with $\pm d\notin(K^)^2$, the quartic twist by $u$ is the same as the quadratic twist by $d$. These are just the quartic twists for which the cocycle takes values in $\{\pm1\}$. Note that for elliptic curves with $j$-invariant $1728$, the quadratic twists by $+d$ and by $-d$ are exactly the same, and in particular the quadratic twist by $-1$ is trivial. This is simplest to see from the explicit model $E_{ad^2,0}$ for the twisted curve. When $-1$ is a square in $K$ then it is obvious that the quadratic twist by $-1$ is trivial, but this is also true when $K(\sqrt{-1})\not=K$. A more intrinsic explanation of this fact is that the inflation map $H^1(G_K,\mu_2)\to H^1(G_K,\mu_4)$ is not injective, but has kernel of order $2$, when $-1$ is not a square in $K$. This is most easily seen using the Kummer isomorphism, under which the inflation map becomes the squaring map $K^/(K^)^2 \to K^/(K^)^4$, which has $-1$ in its kernel. Thus, when $j(E)=1728$, the quadratic twists of $E$ are parametrized not by $K^/(K^)^2$ but by $K^/\left<-1,(K^)^2\right>$. \[R:2-isog\] The quartic twist by $u=-4$ is trivial if and only if $-1$ is a square in $K$, since $-4=(1+\sqrt{-1})^4$. The curves $E_{a,0}$ and $E_{-4a,0}$ are $2$-isogenous over $K$ (see [@SilvermanI p. 336]); over $K(\sqrt{-1})$ this $2$-isogeny is the endomorphism $1+\sqrt{-1}$. Note that this twist is not* a quadratic twist, despite the fact that the curve and its twist become isomorphic over a quadratic extension. Taking $v=1+\sqrt{-1}$ so that $v^4=-4$, the cocycle $\psi$ takes value $\psi(\sigma)=\sigma(1+\sqrt{-1})/(1+\sqrt{-1})=-\sqrt{-1}$ when $\sigma$ does not fix $\sqrt{-1}$. Sextic twists, $n=6$: only exist when $j(E)=0$. The short Weierstrass model of such a curve has the form $E_{0,b}$, and its sextic twist by $u$ is $E_{0,bu}$. The sextic twist of $E$ is isomorphic to $E$ over $K(\root6\of u)$, which in general has degree $6$, but special cases arise when $u$ is either a square or a cube in $K$. When $u=d^3\in(K^)^3$, with $d\notin(K^)^2$, we again obtain the quadratic twist by $d$ as a special case of a sextic twist; these are the sextic twists for which the cocycle takes values in $\{\pm1\}$. (Unlike the case of $j(E)=1728$ we obtain each quadratic twist exactly once, since the inflation map $H^1(G_K,\mu_2)\to H^1(G_K,\mu_6)$ is now injective, as is the cubing map $K^/(K^)^2 \to K^/(K^)^6$.) When $u\in(K^)^2$, the sextic twist by $u$ may be called a cubic twist; these are sextic twists for which the cocycle takes values in $\mu_3$, and the curve and its twist become isomorphic over an extension of $K$ of degree $3$. \[R:3-isog\] The sextic twist by $u=-27$ is the quadratic twist by $-3$, and is trivial if and only if $-3$ is a square in $K$. The curves $E_{0,b}$ and $E_{0,-27b}$ are $3$-isogenous over $K$; over $K(\sqrt{-3})$ this $3$-isogeny is the endomorphism $\sqrt{-3}$. The mod $p$ Galois representations of twists {#SS:GalRepTwist} -------------------------------------------- We now consider the effect of twisting $E$ on the associated mod $p$ Galois representations ${{\overline{\rho}}}_{E,p}$. This is straightforward in the case of quadratic twists, but more involved for higher twists. Let $K$ be a number field. Let $E$ and $E'$ be elliptic curves over $K$ having the same $j$-invariant $j = j(E) = j(E')$. Assume they are not isomorphic over $K$. Then $E$ and $E'$ are (non-trivial) twists and become isomorphic over an extension $L/K$. Write $d = [L : K]$. From the discussion in section \[S:twists\], we have a twist map $t : E \to E'$ with an associated cocycle $\psi : G_K \to \mu_m \subseteq{\mathcal{O}}^$, where $m\in\{2,3,4,6\}$, the order of the twist, is the order the subgroup of ${\mathcal{O}}^$ generated by $\psi(G_K)$. We have the following cases: - for arbitrary $j$: $m=d=2$ (quadratic twists); and additionally, - for $j = 1728$ only: $m=4$ and $d \in \{2,4\}$ (quartic twists); and - for $j = 0$ only: $m=d\in\{3,6\}$ (cubic or sextic twists). When $m=4$, the case $d=2$ occurs only for the special quartic twist by $-4$ as in Remark \[R:2-isog\]. Denote by $t_p : E[p] \to E'[p]$ the restriction of $t$ to the $p$-torsion. Then becomes $$\label{E:cocycle} \sigma(t_p(P)) = \psi(\sigma) \cdot t_p(\sigma(P)) = t_p(\psi(\sigma)\cdot\sigma(P)) \quad \text{ for all } \sigma \in G_K, \; P \in E[p].$$ Let $P_1, P_2$ be a basis of $E[p]$, so that $t_p(P_1), t_p(P_2)$ is a basis of $E'[p]$. With respect to these bases, the map $t_p$ is represented by the identity matrix in ${\operatorname{GL}}_2({\mathbb{F}}_p)$ and $\psi(\sigma)$ by a matrix $\Psi(\sigma)$ (the same matrix on both $E[p]$ and $E'[p]$). Then implies, for all $\sigma \in G_K$, the matrix equation $$\label{E:PsiMatrix} {{\overline{\rho}}}_{E',p}(\sigma) = {{\overline{\rho}}}_{E,p}(\sigma) \cdot \Psi(\sigma).$$ For quadratic twists, $\Psi(\sigma)=\pm I$, but in general $\Psi(\sigma)$ is not scalar. Note that $\det\Psi(\sigma)=1$ in all cases, since the determinants of ${{\overline{\rho}}}_{E,p}$ and ${{\overline{\rho}}}_{E',p}$ are both given by the cyclotomic character. The map $\Psi:G_K\to{\operatorname{SL}}_2({\mathbb{F}}_p)$ becomes a homomorphism over an extension $K'/K$ given by $K'=K$ if $m=2$, $K'=K(\sqrt{-1})$ if $m=4$ and $K'=K(\sqrt{-3})$ if $m=3,6$. In the quadratic case, $\Psi$ matches the quadratic character $\varepsilon_d$ associated to the quadratic extension $K(\sqrt{d})$ of $K$ over which the curves become isomorphic. In general, the representation attached to the twist of $E$ is obtained from that of $E$ itself twisting by the cocycle $\Psi$, with values in ${\operatorname{GL}}_2({\mathbb{F}}_p)$. For quadratic twists this is just the tensor product by a quadratic character. We summarize this discussion in the following. \[L:twist-rep\] 1. Let $E'$ be the twist of $E$ by a cocycle $\psi$. Then there is an isomorphism $t :E\to E'$, defined over an extension of $K$ of degree at most $6$, satisfying . In the case of a quadratic twist with $E'=E^d$, the isomorphism $t$ is defined over $K(\sqrt{d})$ and satisfies $$\label{E:iso-twist} \sigma(t (P)) = {\varepsilon}_d(\sigma)t (\sigma(P)) \qquad\text{for all~$\sigma\in G_K$ and $P\in E({{\overline{K}}})$}.$$ 2. For all primes $p$ we have in general the matrix equation . In the case of quadratic twists this simplifies to $$\label{E:rho-twist} {{\overline{\rho}}}_{E^d,p} \cong {{\overline{\rho}}}_{E,p}\otimes{\varepsilon}_d.$$ Assuming that $E$ and $E'$ are $p$-congruent, there exists an isomorphism $\phi : E[p] \to E'[p]$ of $G_K$-modules. Choosing compatible bases for $E[p]$ and $E'[p]$ as above, let $A \in {\operatorname{GL}}_2({\mathbb{F}}_p)$ be the matrix representing $\phi$ with respect to them. Then, using , we have $$\label{E:PsiM} A {{\overline{\rho}}}_{E,p}(\sigma) A^{-1} = {{\overline{\rho}}}_{E',p}(\sigma) = {{\overline{\rho}}}_{E,p}(\sigma) \cdot \Psi(\sigma) \quad \text{ for all } \sigma \in G_K$$ and the symplectic type of $\phi$ is determined by the square class of $\det A$; this latter conclusion follows from the fact that $t_p$ preserves the Weil pairing and [@FKSym Lemma 6]. Moreover, if there is another $A' \in {\operatorname{GL}}_2({\mathbb{F}}_p)$ satisfying then $\det A' = \det A \cdot \lambda^2$ by Proposition \[P:conditionS\] (under the natural assumption that $E[p]$ satisfies condition [(S)]{}) and so symplectic type of $\phi$ is also determined by $\det A'$ mod squares. Projectively dihedral images {#SS:dihedral} ---------------------------- From Proposition \[P:conditionS\], [condition [(S)]{}]{} follows from absolute irreducibility. Conversely, the next result shows that, in the presence of a $p$-congruence between twists, [condition [(S)]{}]{}implies that ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible. Clearly, elliptic curves with CM can only satisfy [condition [(S)]{}]{} when the CM is not defined over the base field $K$, or when $p$ ramifies in the CM field, as otherwise the image is abelian. \[L:noCyclic\] Let $E/K$ be an elliptic curve and $p$ an odd prime such that $E[p]$ satisfies [condition [(S)]{}]{}. If $E$ is $p$-congruent to a twist $E'$ then ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible. For a contradiction, suppose ${{\overline{\rho}}}_{E,p}$ is absolutely reducible. Thus $$\label{E:absRed} {{\overline{\rho}}}_{E,p} \otimes {{\overline{{\mathbb{F}}}}}_p \cong \begin{pmatrix} \chi_1 & h \\ 0 &\chi_2 \end{pmatrix} \quad \text{ with } \; h \neq 0,$$ where $h \neq 0$ follows from the description of the two cases in Proposition \[P:conditionS\]. Suppose that $j(E)=0$ and $p=3$. Then $\chi_1$ and $\chi_2$ are quadratic, their product is the cyclotomic character which is not trivial since by [condition [(S)]{}]{}, $K$ does not contain the CM field $K(\sqrt{-3})$. But then $\chi_1\not=\chi_2$, contradicting [condition [(S)]{}]{}, by Proposition \[P:conditionS\]. Similarly, we cannot have $j(E) = 1728$ or $j=0$ and $p\ge5$, as then $E$ would have CM by ${\mathbb{Q}}(i)$ or ${\mathbb{Q}}(\sqrt{-3})$, and the hypothesis on $p$ would imply that the image would be in the normalizer of a Cartan, contradicting . Therefore $E$ only admits quadratic twists, and we have $E' = E^d$ for some non-square $d \in K^$. Let ${\varepsilon}_d$ be the quadratic character associated to the extension $K(\sqrt{d})/K$. From the hypothesis $E[p] \simeq E^d[p]$, part 2) of Lemma \[L:twist-rep\] and it follows that $\chi_1 =\chi_1{\varepsilon}_d$ since both give the Galois action on the unique fixed line. This contradicts ${\varepsilon}_d \neq 1$. Under the natural [condition [(S)]{}]{}, the previous lemma tells us we can assume that ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible for our study of congruences between twists. In fact more is true: the next proposition says that congruences between twists arise when the projective image is dihedral of order at least 4 (equivalently, when the image is contained in the normaliser of a Cartan subgroup but not in the Cartan itself), and only then. The converse part of the following result is already contained in [@Halberstadt-11nonsplit §2], with a different proof and without the uniqueness statement. \[P:twist\] Let $E/K$ be an elliptic curve and $p \geq 5$ a prime. Suppose that $E[p]$ satisfies [condition [(S)]{}]{}. 1\) If $E$ is $p$-congruent to a twist, then the image of ${{\overline{\rho}}}_{E,p}$ is contained in the normaliser of a Cartan subgroup of ${\operatorname{GL}}_2({\mathbb{F}}_p)$ but not in the Cartan itself and ${\mathbb{P}}{{\overline{\rho}}}_{E,p} \simeq D_n$ for some $n \geq 2$. 2\) Conversely, if the image of ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible and contained in the normaliser of a Cartan subgroup of ${\operatorname{GL}}_2({\mathbb{F}}_p)$ but not in the Cartan subgroup itself, then we have $E[p]\cong E'[p]$ where $E'$ is the (non-trivial) twist associated to the quadratic extension $K(\sqrt{d})$ cut out by the Cartan subgroup. This is the quadratic twist $E^d$ unless $j(E)=1728$ and $d=-1$, in which case it is the quartic twist by $-4$. Moreover, there is a unique such twist $E'$ which is $p$-congruent to $E$, except when the projective image has order $4$, in which case there are three (non-trivial) such twists. 1\) Let $E'$ be a $p$-congruent twist of $E$. If $E'$ is a quartic or sextic twist of $E$ then $j(E)=0,1728$ and $E$ has CM (not defined over $K$, since the projective image is not abelian) by ${\mathbb{Q}}(\sqrt{-1})$ or ${\mathbb{Q}}(\sqrt{-3})$; moreover, since $p > 3$ it is not ramified in the CM field and hence the projective image is absolutely irreducible and isomorphic to $D_n$ for some $n \geq 2$. Therefore we can assume $E' = E^d$ is the quadratic twist of $E$ by a non-square $d\in K^$. From Lemma \[L:noCyclic\] we know that ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible. By (\[E:rho-twist\]), for primes ${{\mathfrak q}}$ of $K$ where both curves have good reduction (so excluding only finitely many primes), we have $$a_{{\mathfrak q}}(E^d) = {\varepsilon}_d({{\mathfrak q}}) a_{{\mathfrak q}}(E),$$ where $a_{{\mathfrak q}}(E)$ and $a_{{\mathfrak q}}(E^d)$ denotes the trace of Frobenius at ${{\mathfrak q}}$ of $E$ and $E^d$, respectively, and we view ${\varepsilon}_d$ as a function on primes of $K$ unramified in $K(\sqrt{d})$ as usual via class field theory. Moreover, since $E^d[p]\cong E[p]$, we have $$a_{{\mathfrak q}}(E^d) \equiv a_{{\mathfrak q}}(E) \pmod{p}.$$ Hence for almost all primes ${{\mathfrak q}}$ satisfying ${\varepsilon}_d({{\mathfrak q}}) = -1$ (i.e. inert in $K(\sqrt{d})/K$) we have $$a_{{\mathfrak q}}(E^d) \equiv a_{{\mathfrak q}}(E) \equiv0 \pmod{p}.$$ So the image $H={{\overline{\rho}}}_{E,p}(G_K)$ has a subgroup $H^+={{\overline{\rho}}}_{E,p}(G_{K(\sqrt{d})})$ of index $2$ such that all elements of $H\setminus H^+$ have trace $0$; such elements have order $2$ in ${\operatorname{PGL}}_2({\mathbb{F}}_p)$. From the irreducibility of $E[p]$ and the classification of subgroups of ${\operatorname{PGL}}_2({\mathbb{F}}_p)$ ([@LangModForms Theorem XI.2.3]), it follows that $H$ is contained in the normaliser of a Cartan subgroup $C$ and $H^+=H\cap C$. 2\) For the converse, suppose that ${{\overline{\rho}}}_{E,p}$ is absolutely irreducible and has image contained in the normalizer $N \subset {\operatorname{GL}}_2({\mathbb{F}}_p)$ of a Cartan subgroup $C$, but is not contained in $C$ itself. Thus ${\mathbb{P}}{{\overline{\rho}}}_{E,p} \simeq D_n$ for $n \geq 2$ and $\operatorname{Tr}{{\overline{\rho}}}_{E,p}(\sigma) = 0$ for all $\sigma \in G_K$ such that ${{\overline{\rho}}}_{E,p}(\sigma) \not\in C$. Let $K(\sqrt{d})$ be the quadratic extension cut out by ${{\overline{\rho}}}_{E,p}^{-1}(C)$, with associated character ${\varepsilon}_d$ as above. First suppose that we are not in the special case where $j(E)=1728$ and $K(\sqrt{d})=K(\sqrt{-1})$. Set $E'=E^d$, the quadratic twist. By (\[E:rho-twist\]), for all $\sigma \in G_K$ we have the following equality of traces $$\operatorname{Tr}{{\overline{\rho}}}_{{E^d},p}(\sigma) = {\varepsilon}_d(\sigma) \cdot \operatorname{Tr}{{\overline{\rho}}}_{E,p}(\sigma).$$ Clearly, if ${\varepsilon}_d(\sigma)=1$ then $\operatorname{Tr}{{\overline{\rho}}}_{E,p}(\sigma) = \operatorname{Tr}{{\overline{\rho}}}_{E^d,p}(\sigma)$. If ${\varepsilon}(\sigma) = -1$ then ${{\overline{\rho}}}_{E,p}(\sigma) \in {{\overline{\rho}}}_{E,p} (G_K) \backslash C$ and $\operatorname{Tr}{{\overline{\rho}}}_{E,p}(\sigma) = 0$, so also $\operatorname{Tr}{{\overline{\rho}}}_{E^d,p}(\sigma) = 0$. Then, $\operatorname{Tr}{{\overline{\rho}}}_{E,p}(\sigma) = \operatorname{Tr}{{\overline{\rho}}}_{E^d,p}(\sigma)$ for all $\sigma \in G_K$. Since ${{\overline{\rho}}}_{E,p}$ and ${{\overline{\rho}}}_{E^d,p}$ are absolutely irreducible and have the same traces, they are isomorphic. In the special case, $E'$ is the quartic twist of $E$ by $-4$, since these become isomorphic over $K(\sqrt{-1})$; now $E$ and $E'$ are isogenous, so are $p$-congruent for all odd $p$. For the last part, we note that $D_2\cong C_2\times C_2$ has three cyclic subgroups of index $2$, while $D_n$ for $n\ge3$ has only one such subgroup. The symplectic type of congruences between quadratic twists {#SS:typeQuadratic} ----------------------------------------------------------- A special case of the situation described in Proposition \[P:twist\] is the case of elliptic curves with CM. Here, the quadratic twists are isogenous to the original curve so we may already determine the symplectic nature of the congruence. For simplicity we state such result over ${\mathbb{Q}}$. \[C:CM-twist\] Let $E/{\mathbb{Q}}$ be an elliptic curve with CM by the imaginary quadratic order of (negative) discriminant $-D$. Set $M={\mathbb{Q}}(\sqrt{-D})$. 1. For $D\not=4$: $E[p] \simeq E^{-D}[p]$ for all primes $p \geq 5$ unramified in $M$ and of good reduction for $E$. This congruence is symplectic if and only if $\legendre{D}{p}=+1$. For each such $p$, $E^{-D}$ is the unique quadratic twist of $E$ which is $p$-congruent to $E$. 2. For $D=4$: let $E'$ be the quartic twist of $E$ by $-4$, so that $E$ and $E'$ are isomorphic over $M$ but not over ${\mathbb{Q}}$. Again, $E[p] \simeq E'[p]$ for all primes $p \geq 5$ unramified in $M$ and of good reduction for $E$. This congruence is symplectic if and only if $\legendre{2}{p}=+1$. For each such $p$, there are no quadratic twists of $E$ which are $p$-congruent to $E$. Since $E$ has CM by $M$ we know that the image of ${{\overline{\rho}}}_{E,p}$ is the normalizer of a Cartan for all primes $p$ unramified in $M$ and of good reduction for $E$. Moreover, ${\mathbb{P}}{{\overline{\rho}}}_{E,p} \simeq D_n \neq C_2 \times C_2$ (since $p \geq 5$) and ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_M) \simeq C_n$. Thus, for each such $p$, in the notation of Proposition \[P:twist\], we have $K(\sqrt{d})=M$, so $E[p]\cong E^{-D}[p]$ except for $D=4$ when $E[p]\cong E'[p]$ with $E'$ the quartic twist of $E$ by $-4$. Since the projective image is not $C_2\times C_2$, in each case there are no more $p$-congruences with curves isomorphic to $E$ over quadratic extensions (cf. Prop \[P:twist\]). For the symplectic types, observe that $E$ and $E'$ are isogenous via an isogeny of degree $2$ for $D=4$ (see Remark \[R:2-isog\]), while for $D\not=4$, one may check that $E$ and $E^{-D}$ are isogenous via an isogeny of degree $D/4$ for $D=8,12,16$, and $28$ and of degree $D$ otherwise. The reason for the isogeny degrees is as follows. We obtain an isogeny from $E$ to its twist by composing the twist isomorphism $t:E\to E^{-D}$ with an endomorphism $\alpha$ of $E$. For $t\circ\alpha$ to be defined over ${\mathbb{Q}}$ we require $\alpha^{\sigma}=\psi(\sigma)^{-1}\alpha$, where $\sigma$ generates $\operatorname{Gal}(M/{\mathbb{Q}})$ and $\psi$ is the twisting cocycle. The isogeny will be cyclic provided that $\alpha$ is primitive (not divisible in $\operatorname{End}(E)$ by any integer other than $\pm1$). These conditions uniquely determine $\alpha$ up to a unit factor (by a case-by-case computation) and hence determine the degree $N(\alpha)$. For $D\not=4$ we find that $N(\alpha)=D$ or $D/4$ so that $\legendre{N(\alpha)}{p}=\legendre{D}{p}$, while for $D=4$, $\alpha=\pm1\pm\sqrt{-1}$ of norm $2$. In our computed data in Section \[S:statistics\], we do not see usually isomorphisms arising from CM curves as in this corollary. This is because (apart from the CM cases where quartic or sextic twists occur) all such mod $p$ isomorphisms occur within an isogeny class, and we have omitted these from consideration. Note that if ${\mathbb{P}}{{\overline{\rho}}}_{E,p} (G_K)$ has order $1$ or $2$, then the image ${{\overline{\rho}}}_{E,p} (G_K)$ is cyclic of order dividing $2(p-1)$ and so not absolutely irreducible. In particular, the smallest projective images occurring in our context are ${{\overline{\rho}}}_{E,p} (G_K) \simeq D_2 \simeq C_2 \times C_2$. The next result will be used below to determine the symplectic type of congruences in the presence of this kind of image. \[L:C2xC2\] Let $p$ be an odd prime. Let $N$ be a subgroup of ${\operatorname{PGL}}_2({\mathbb{F}}_p)$ isomorphic to $C_2\times C_2$, so that there are three subgroups $C< N$ of index $2$. If $N\le{\operatorname{PSL}}_2({\mathbb{F}}_p)$, then every such $C$ is contained in a Cartan subgroup and $N$ in its normalizer, and each $C$ is split when $p\equiv1\pmod4$ and non-split when $p\equiv3\pmod4$. If $N\not\le{\operatorname{PSL}}_2({\mathbb{F}}_p)$, then for $p\equiv1\pmod4$, one such subgroup $C$ is contained in a split Cartan subgroup while the other two are contained in non-split Cartan subgroups; while if $p\equiv3\pmod4$ then one $C$ is non-split and the other two are split. First note that in ${\operatorname{PGL}}_2({\mathbb{F}}_p)$, the condition of having zero trace is well-defined, and the determinant is also well-defined modulo squares. Also, ${\operatorname{PSL}}_2({\mathbb{F}}_p)$ is the subgroup of ${\operatorname{PGL}}_2({\mathbb{F}}_p)$ of elements with square determinant, which has index $2$. Hence either $N\le{\operatorname{PSL}}_2({\mathbb{F}}_p)$, and all elements of $N$ have square determinant, or $[N:N\cap{\operatorname{PSL}}_2({\mathbb{F}}_p)]=2$, in which case exactly one of the elements of order $2$ has square determinant. Examination of the characteristic polynomial shows that the elements of order $2$ in ${\operatorname{PGL}}_2({\mathbb{F}}_p)$ are precisely those with trace zero, and these elements are split (having $2$ fixed points on ${\mathbb{P}}^1({\mathbb{F}}_p)$) if the determinant is minus a square, and non-split (having no fixed points) otherwise. Hence when $p\equiv1\pmod4$, elements of order $2$ are split if and only if they lie in ${\operatorname{PSL}}_2({\mathbb{F}}_p)$, while for $p\equiv3\pmod4$ the reverse is the case. The result follows. The next theorem describes the symplectic type of congruences between general quadratic twists. Since the projective image is dihedral (by Proposition \[P:twist\]), this is a situation where the local methods from [@FKSym] may not apply, as is illustrated by Example \[Ex:LocalFail7\] below. The first part of the theorem (with $K={\mathbb{Q}}$) is again already in [@Halberstadt-11nonsplit §2], with essentially the same proof; we include it here in order to include the second part, which describes a situation that cannot occur over ${\mathbb{Q}}$ except for very small primes. See Example \[Ex:3twists\] below for an example with $p=3$. \[T:quadratic\] Let $p > 2$ be a prime. Let $E/K$ be an elliptic curve $p$-congruent to some quadratic twist $E^d$. Assume $E[p]$ is an absolutely irreducible $G_K$-module. 1. Let $C$ be the Cartan subgroup of ${\operatorname{GL}}_2({\mathbb{F}}_p)$ associated to the extension $K(\sqrt{d})/K$ in Proposition \[P:twist\]. Then the congruence is symplectic if and only if either* $C$ is split and $p \equiv 1 \pmod{4}$, or $C$ is non-split and $p \equiv 3 \pmod{4}$. 2. When ${\mathbb{P}}{{\overline{\rho}}}_{E,p}\cong C_2\times C_2$, there are three different quadratic[^4] twists of $E$ which are $p$-congruent to $E$. If $\sqrt{p^}\in K$ then all three congruences are symplectic. Otherwise, one of the quadratic twists is by $K(\sqrt{p^})$ and is symplectic while the other two are anti-symplectic. \(1) We will define a matrix $A\in{\operatorname{GL}}_2({\mathbb{F}}_p)$ satisfying and $\det(A)=-\delta$, where $\delta \in {\mathbb{F}}_p^$ is a square if $C$ is split and a non-square if $C$ is non-split. Then, the map $E[p]\to E'[p]$ corresponding to $A$ is then a $G_K$-equivariant isomorphism which, by the discussion following , is symplectic if and only if $-\delta$ is a square. From this, part (1) follows by considering the four cases: $\delta$ square/non-square and $p\equiv\pm1\pmod4$. Let $H=G_{K(\sqrt{d})}$ be the index $2$ subgroup cut out by the homomorphism $\varepsilon_d:G_K\to\{\pm 1 \}$. The projective image ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(H)$ is a subgroup of ${\mathbb{P}}(C) \subset {\operatorname{PGL}}_2({\mathbb{F}}_p)$ and the latter is cyclic of even order $p\pm1$ (depending on whether $C$ is split or non-split). Hence ${\mathbb{P}}(C)$ contains a unique element of order $2$. Let $A \in C \subset {\operatorname{GL}}_2({\mathbb{F}}_p)$ be any lift of this element. Then $A$ is not scalar, while $A^2$ is scalar, so by Cayley-Hamilton we have $\operatorname{Tr}(A)=0$ and, by the proof of Lemma \[L:C2xC2\], we have that $-\det(A) = \delta$ is square if and only if the Cartan $C$ is split. Since $A$ is central in ${{\overline{\rho}}}_{E,p}(G_K)$ modulo scalars, for all $g\in{{\overline{\rho}}}_{E,p}(G_K)$ we have $AgA^{-1}=\lambda(g)g$ with a scalar $\lambda(g)=\pm I$ (comparing determinants). Now $\lambda(g)=I$ if and only if $g$ commutes with $A$ which—since $A$ is a non-scalar element of the Cartan subgroup—is if and only if $g$ is itself in the Cartan subgroup, that is, if and only if $g={{\overline{\rho}}}_{E,p}(\sigma)$ with $\sigma\in H$, so holds. \(2) Since the determinant of ${{\overline{\rho}}}_{E,p}$ is the mod $p$ cyclotomic character, we have ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_K)\subseteq {\operatorname{PSL}}_2({\mathbb{F}}_p)$ if and only if $\sqrt{p^}\in K$. When this is the case, by Lemma \[L:C2xC2\] we see that each index $2$ subgroup of the projective image is split when $p\equiv1\pmod4$ and each is non-split when $p\equiv3\pmod4$. By part (1), it follows in both cases that the congruences between $E$ and each of the three twists are all symplectic. Now suppose that $\sqrt{p^}\notin K$. By Lemma \[L:C2xC2\] again, exactly one of the three subgroups is split when $p\equiv1\pmod4$, and exactly one is non-split when $p\equiv3\pmod4$, so in both cases exactly one congruence is symplectic. Moreover, since the subgroup $C$ inducing the symplectic congruence is the unique one contained in ${\operatorname{PSL}}_2({\mathbb{F}}_p)$ (see the end of the proof of Lemma \[L:C2xC2\]), the associated quadratic extension is $K(\sqrt{p^})$. \[Ex:LocalFail7\] Consider the elliptic curve $$E : y^2 + y = x^3 - x^2 - 74988699621831x + 238006866237979285299,$$ which has conductor $ N_E = 7^2 \cdot 2381 \cdot 134177^2>2\cdot10^{15}$, so is not in the LMFDB database. This example was found using the explicit parametrization of curves for which the image of the mod $7$ Galois representation is contained in the normalizer of a non-split Cartan subgroup; we verified by explicit computation that the mod 7 image is equal to the full normalizer and that the Cartan subgroup cuts out the field ${\mathbb{Q}}(\sqrt{d})$ where $d = -7 \cdot 134177$. Consider $E^d$ the quadratic twist of $E$ by $d$. From Proposition \[P:twist\] we have that $E[7] \simeq E^d[7]$ as $G_{\mathbb{Q}}$-modules and part (1) of Theorem \[T:quadratic\] yields that $E[7]$ and $E^d[7]$ are symplectically isomorphic (and not anti-symplectically isomorphic). We note that the same conclusion can be obtained via the general method from Section \[S:statistics\] and, more interestingly, none of the local criteria in [@FKSym] applies for this case. \[Ex:3twists\] For an example with projective image $C_2\times C_2$, let $E$ be the elliptic curve with label [[](http://www.lmfdb.org/EllipticCurve/Q/6534a1)]{}, of conductor $6534=2\cdot3^3\cdot11^2$. The image of the mod $3$ Galois representation is the normalizer of the split Cartan, which is projectively isomorphic to $D_2 = C_2\times C_2$. The three quadratic subfields of the projective $3$-division field are ${\mathbb{Q}}(\sqrt{-3})$, ${\mathbb{Q}}(\sqrt{-11})$, and ${\mathbb{Q}}(\sqrt{33})$; the corresponding quadratic twist of $E$ are $E^{-3}={\href{http://www.lmfdb.org/EllipticCurve/Q/6534v1}{{\text{\rm6534v1}}}}$, $E^{-11}={\href{http://www.lmfdb.org/EllipticCurve/Q/6534p1}{{\text{\rm6534p1}}}}$, and $E^{33}={\href{http://www.lmfdb.org/EllipticCurve/Q/6534h1}{{\text{\rm6534h1}}}}$ respectively. All four curves have isomorphic mod $3$ representations by Proposition \[P:twist\], the isomorphism being antisymplectic between $E$ and $E^{-11}$ and $E^{33}$, and symplectic between $E$ and $E^{-3}$, in accordance with part (2) of Theorem \[T:quadratic\]. Congruences between higher order twists {#SS:higherCong} --------------------------------------- In this section we will study, under condition [(S)]{}, the congruences between an elliptic curve $E/K$ and its quartic, cubic or sextic twists by $u \in K$. Note that if $u = -s^2$ then $u = -4(s/2)^2$ and the quartic twist by $u$ is obtained by composing the quartic twist by $-4$ with a quadratic twist; similarly, if $u = -3s^2$ then $u = -27(s/3)^2$ and the sextic twist by $u$ is obtained by composing the quadratic twist by $-27$ with a cubic twist. Since the quartic twist by $-4$ and the quadratic twist by $-27$ correspond to isogenies (see Remarks \[R:2-isog\] and \[R:3-isog\]) their effect on the symplectic type is known; observe also that both these cases are covered by the theory in sections \[SS:dihedral\] and \[SS:typeQuadratic\]. In view of this, we fix the following natural assumptions for this and the next section. Let $p$ be an odd prime. Let $E/K$ be an elliptic curve with $j(E) = 0, 1728$ and let $u \in K$. - If $j(E) = 1728$, assume also $K'=K(\sqrt{-1}) \neq K$, $u \neq -1$ modulo squares; - If $j(E) = 0$, assume also $K'=K(\sqrt{-3}) \neq K$, $u \neq -3$ modulo squares and $p\not=3$. \[L:diagonalC\] Let $E/K$ be as above. 1. The projective image ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_K)$ is dihedral $D_n$ for some $n\ge2$, and the projective image of $G_{K'}$ is $C_n$. 2. After extending scalars from ${\mathbb{F}}_p$ to ${\mathbb{F}}_{p^2}$ if necessary, there is a basis for $E[p]$ with respect to which for $\sigma\in G_{K'}$ we have $$\label{E:diagonalC} {{\overline{\rho}}}_{E,p}(\sigma) = c(\sigma)\cdot{{\operatorname{diag}}}(1,{\varepsilon}(\sigma))$$ with $c(\sigma)\in\overline{{\mathbb{F}}}_p^$ and ${\varepsilon}:G_{K'}\to\overline{{\mathbb{F}}}_p^$ a character of exact order $n$. Part (1) follows from standard facts on CM curves plus our running assumptions and (2) follows by diagonalising the Cartan over ${\mathbb{F}}_p$ in the split case and over ${\mathbb{F}}_{p^2}$ in the non-split case. \[L:DiagonalPsi\] Let $E/K$ and $u\in K^$ be as above. For $m\in\{3,4,6\}$, let $\psi$ be the order $m$ cocycle associated to $u$, with values in $\operatorname{Aut}(E)\cong{\mathcal{O}}^$ where ${\mathcal{O}}\cong{\mathbb{Z}}[\zeta_m]$. After extending scalars to ${{\overline{{\mathbb{F}}}}}_p^$ and changing basis so that holds, the matrices $\Psi(\sigma)$ in become diagonal. More precisely, $$\Psi(\sigma) = {{\operatorname{diag}}}(\eta(\sigma),\eta(\sigma)^{-1})$$ where $\eta:G_K\to \overline{{\mathbb{F}}}_p^$ is a map whose restriction to $G_{K'}$ is a homomorphism of order exactly $m$. Recall that we have fixed an isomorphism $\operatorname{End}(E)\cong{\mathcal{O}}$ and that ${{\overline{\rho}}}_{E,p}(G_{K'}) \subset C$ for some Cartan subgroup $C \in {\operatorname{GL}}_2({\mathbb{F}}_p)$. Hence ${\mathcal{O}}$ acts on $E[p]$ via matrices which are in $C$ since the endomorphisms are all defined over $K'$ and so their action commutes with that of $G_{K'}$, which is not scalar. Note that the basis giving diagonalizes the whole of $C$, therefore $\Psi(\sigma)$ is diagonal. Finally, an endomorphism $\alpha$ acts via a matrix of determinant $\deg(\alpha)$ so ${\mathcal{O}}^$ acts via diagonal matrices of determinant $1$ as stated. The map $\eta$ is obtained as follows. In the split case $p{\mathcal{O}}={{\mathfrak p}}\overline{{{\mathfrak p}}}$ and $\eta(\sigma)$ is the image of $\psi(\sigma)$ under the isomorphism $\operatorname{End}(E)\cong{\mathcal{O}}$ followed by the reduction ${\mathcal{O}}\to{\mathcal{O}}/{{\mathfrak p}}\cong{\mathbb{F}}_p$ which induces a reduction homomorphism $\eta:{\mathcal{O}}^\to{\mathbb{F}}_p^$. Interchanging ${{\mathfrak p}}$ and $\overline{{{\mathfrak p}}}$ has the effect of replacing $\eta$ by its inverse; we make an arbitrary but fixed choice. In the non-split case, $p{\mathcal{O}}={{\mathfrak p}}$ and $\eta(\sigma)$ is the image of $\psi(\sigma)$ under the isomorphism followed by the reduction ${\mathcal{O}}\to{\mathcal{O}}/{{\mathfrak p}}\cong{\mathbb{F}}_{p^2}$. This induces a homomorphism $\eta: {\mathcal{O}}^\to{\mathbb{F}}_{p^2}^$ for which there are two choices since we may compose with the nontrivial automorphism of ${\mathbb{F}}_{p^2}$. By definition, the map $\eta$ becomes a homomorphism of order $m$ when restricted to $G_{K'}$ because this is the case for $\psi$ by cases (ii) and (iii) of section \[SS:GalRepTwist\], given that we have excluded the special quartic twist by $u=-4$. \[L:EpsEta\] Keeping the notation of Lemmas \[L:diagonalC\] and \[L:DiagonalPsi\], let $E'$ be the order $m$ twist of $E$ by $u$ with $m \in \{3,4,6\}$, and suppose also that $E$ and $E'$ are $p$-congruent. Then ${\varepsilon}(\sigma)=\eta(\sigma)$ for all $\sigma\in G_{K'}$, and ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_K) \simeq D_n$ where $n=m$. Let $A\in{\operatorname{GL}}_2({\mathbb{F}}_p)$ be the matrix of an isomorphism $E[p]\to E'[p]$. For $\sigma\in G_{K'}$ we have, from and after cancelling the scalar factor $c(\sigma)$ from each side, $$A\ {{\operatorname{diag}}}(1,{\varepsilon}(\sigma))\ A^{-1} = {{\operatorname{diag}}}(1,{\varepsilon}(\sigma))\; {{\operatorname{diag}}}(\eta(\sigma),\eta(\sigma)^{-1}).$$ Now since $A$ conjugates a non-scalar diagonal matrix into another diagonal matrix, it is either itself diagonal or is anti-diagonal. But $A$ cannot be diagonal since that would imply $\eta(\sigma)=1$ for all $\sigma\in G_{K'}$ (which is not the case because we have excluded the special quartic twist), so $A$ is anti-diagonal. Therefore, conjugating any diagonal matrix by $A$ just interchanges the two diagonal entries, so the previous equation becomes $${{\operatorname{diag}}}({\varepsilon}(\sigma),1) = {{\operatorname{diag}}}(1,{\varepsilon}(\sigma))\; {{\operatorname{diag}}}(\eta(\sigma),\eta(\sigma)^{-1}).$$ Hence ${\varepsilon}(\sigma)=\eta(\sigma)$ for all $\sigma\in G_{K'}$, thus $n=m$ and the statement of the lemma follows. The preceding lemma shows that, for $n\in\{4,3,6\}$, a necessary condition for the existence of a $p$-congruence between $E$ with a twist $E'$ of order $n$ is that the projective mod $p$ image is isomorphic to $D_n$. We next show that this condition is also sufficient. First we have an elementary lemma. Let $n\in\{4,3,6\}$, and let $K$ be a number field not containing the $n$th roots of unity. Let $F/K$ be a Galois extension with $\operatorname{Gal}(F/K)\cong D_n$, and assume that the subfield $K'$ of $K$ fixed by the unique cyclic subgroup of order $n$ is $K'=K(\sqrt{-1})$ if $n=4$ and $K'=K(\sqrt{-3})$ if $n=3,6$. Then there exists $u\in K$ such that $F$ is the splitting field of $X^n-u$. \[L:Dn\] Write $\operatorname{Gal}(F/K)=\left<\sigma,\tau\mid\sigma^n=\tau^2=1, \tau\sigma\tau=\sigma^{-1}\right>$. The fixed field of $\sigma$ is $K'$. Since $K'$ contains the $n$th roots of unity, by Kummer Theory, $F=K'(\rnu)$ for some $u\in K'$. Now $\sigma(\rnu)=\zeta\rnu$ with $\zeta$ a primitive $n$th root of unity, and either $\tau(u)=u$ or $\tau(u)=\overline{u}$ (the $K'/K$-conjugate of $u$). In the first case, $u\in K$ and the result follows. Suppose that $\tau(u)=\overline{u}\not=u$. Since $(\tau(\rnu))^n=\tau(u)=\overline{u}$, we may set $\rnub=\tau(\rnu)$. Using $\sigma=\tau\sigma^{-1}\tau$ we find that $\sigma(\rnub)=\zeta\rnub$. Hence $v=\rnub/\rnu\in K'$, so $\rnub=\rnu v$; applying $\tau$ gives $$\rnu = \rnub\;\overline{v} = \rnu v\overline{v},$$ so $v\overline{v}=1$. Then $v(1+\overline{v})=1+v$ and it follows that $u_1=u(1+v)^n\in K$. Replacing $u$ by $u_1$ completes the argument. \[P:higherTwists\] Let $E/K$ and $p$ be as above. Suppose that ${\mathbb{P}}{{\overline{\rho}}}_{E,p}(G_K) \simeq D_n$ where $n=4$ if $j(E)=1728$ and $n\in\{3,6\}$ if $j(E)=0$. Then $E$ is $p$-congruent to an $n$-twist. Moreover, for $n=3$ there is a unique such twist, while for $n=4$ there are two which are $2$-isogenous to each other and for $n=6$ there are two which are $3$-isogenous to each other. The projective $p$-division field of $E$ is the Galois extension $F/K$ fixed by ${\mathbb{P}}{{\overline{\rho}}}_{E,p}$ and satisfies $\operatorname{Gal}(F/K) \simeq D_n$. By Lemma \[L:Dn\] we can write it as $F=K'(\root n\of u)$ for some $u\in K^$, which is unique up to replacing $u$ by $u^{-1}$ and multiplication by an element of $K^\cap(K'^)^n$. (Here we use that fact that $\operatorname{Aut}(C_n)\cong\{\pm1\}$ for $n\in\{4,3,6\}$.) Keeping the notations from the previous lemmas, to prove the first part of the theorem we will construct a matrix $A\in{\operatorname{GL}}_2({\mathbb{F}}_p)$ giving the $G_K$-isomorphism $E[p]\to E'[p]$. Indeed, the character ${\varepsilon}:G_{K'} \to \overline{{\mathbb{F}}}_p^$ has exact order $n$ and cuts out the extension $F/K'$. Now let $\psi:G_K\to \mu_n\subseteq{\mathcal{O}}^$ be the cocycle associated to $u$, namely $$\sigma\mapsto \psi(\sigma) = \sigma(\root n\of u)/\root n\of u.$$ The restriction of $\psi$ to $G_{K'}$ is a character of order $n$ which cuts out the same extension $F/K'$, so it is either ${\varepsilon}$ or ${\varepsilon}^{-1}$. Replacing $u$ by $u^{-1}$ if necessary, we may assume that $\left.\psi\right\|_{G_{K'}} = {\varepsilon}$. Now let $E'$ be the order $n$ twist of $E$ by $u$. As before, we have $\Psi(\sigma)={{\operatorname{diag}}}(\eta(\sigma),\eta^{-1}(\sigma))$ where $\left.\eta\right\|_{G_{K'}} = \left.\psi\right\|_{G_{K'}} = {\varepsilon}$ by Lemma \[L:EpsEta\]. Thus, for $\sigma\in G_{K'}$, we have $$\Psi(\sigma) = {{\operatorname{diag}}}({\varepsilon}(\sigma),{\varepsilon}(\sigma)^{-1}),$$ and so, the same computation as in the proof of Lemma \[L:EpsEta\] gives that, for any anti-diagonal matrix $A$ and all $\sigma\in G_{K'}$, we have $$A\ {{\operatorname{diag}}}(1,{\varepsilon}(\sigma))\ A^{-1} = {{\operatorname{diag}}}({\varepsilon}(\sigma),1) = {{\operatorname{diag}}}(1,{\varepsilon}(\sigma)) \Psi(\sigma),$$ that is holds for $\sigma\in G_{K'}$. To show that holds for all $\sigma\in G_{K}$, it suffices, since both sides of are homomorphisms $G_K\to {\operatorname{GL}}_2({\mathbb{F}}_p)$, to do so for a single element $\tau\in G_K\setminus G_{K'}$. Write $\Psi(\tau)={{\operatorname{diag}}}(w,w^{-1})$ with $w\in{\mathbb{F}}_{p^2}^$. Define $$A={{\overline{\rho}}}_{E,p}(\tau){{\operatorname{diag}}}(w,1)={{\operatorname{diag}}}(1,w){{\overline{\rho}}}_{E,p}(\tau),$$ the second equality following since ${{\overline{\rho}}}_{E,p}(\tau)$ is antidiagonal. Then ${{\overline{\rho}}}_{E,p}(\tau)A^{-1}={{\operatorname{diag}}}(1,w^{-1})$, so $$A {{\overline{\rho}}}_{E,p}(\tau) A^{-1} = {{\overline{\rho}}}_{E,p}(\tau)\Psi(\tau)$$ as required. In the case of the split Cartan we are done, as the diagonalization of $C$ occurs in ${\operatorname{GL}}_2({\mathbb{F}}_p)$ and so holds with $A \in {\operatorname{GL}}_2({\mathbb{F}}_{p})$. In the non-split case, we have shown that equation holds over ${\operatorname{GL}}_2({\mathbb{F}}_{p^2})$. By undoing the initial change of coordinates we obtain that holds in ${\operatorname{GL}}_2({\mathbb{F}}_p)$. We will now prove the second statement. First, recall from the first paragraph that $u \in K$ is unique up to inverse and multiplication by an element in $K^\cap(K'^)^n$. Therefore, up to $n$th powers in $K^$, we have either four or two possible choices for $u$, namely - $\{u, u^{-1}, -4u, -4u^{-1}\}$ if $n=4$; - $\{u, u^{-1}, -27u, -27u^{-1}\}$ if $n=6$; - $\{u, u^{-1}\}$ if $n=3$. This follows from the observation that the natural map $K^/(K^)^n \to (K'^)/(K'^)^n$ is injective when $n=3$ and has kernel $\{1,-4\}$ for $n=4$ and $\{1,-27\}$ for $n=6$. Secondly, observe that the construction in the first part of the proof only works for exactly one out of each inverse pair $u,u^{-1}$ mod $n$th powers, so we have two quartic or sextic twists when $n=4$ or $n=6$ respectively, and just one cubic twist when $n=3$. The two quartic twists differ by the quartic twist by $-4$, so by a $2$-isogeny, while the two sextic twists differ by the sextic twist by $-27$, so by a $3$-isogeny (see Remarks \[R:2-isog\] and \[R:3-isog\]). See Examples \[Ex:j1728p3\] and \[Ex:j0p5\] in section \[SS:typeHigher\] for an illustration of this theorem, where we also determine the symplectic type of the congruences using Theorem \[T:typeHigher\] below. The symplectic type of congruences between higher order twists {#SS:typeHigher} -------------------------------------------------------------- We have classified above, under condition [(S)]{}, exactly when congruences between twists occur. To complete this part of our study we are left to describe the symplectic type of congruences between higher order twists. This is given by the following result. \[T:typeHigher\] Let $p$ be an odd prime, $K$ a number field, $E/K$ an elliptic curve, and $u\in K^$. Assume that - either:* $j(E)=1728$, $\sqrt{-1}\notin K$, $u\not=\pm1$ modulo squares, and $n=4$; - or: $j(E)=0$, $\sqrt{-3}\notin K$, $u\not=1,-3$ modulo squares, and $n=3$ or $6$. Let $E'/K$ be the order $n$ twist of $E/K$ by $u$. Suppose that $E$ and $E'$ are $p$-congruent. 1. If $\sqrt{p^}\in K$ then $E[p]$ and $E'[p]$ are symplectically isomorphic and $p\equiv\pm1\pmod{2n}$. 2. Assume $\sqrt{p^}\notin K$. Then: 1. if $n=3$ then $E[p]$ and $E'[p]$ are anti-symplectically isomorphic; 2. if $n=4$ then $E[p]$ and $E'[p]$ are symplectically isomorphic if and only if $\sqrt{up^}\in K$, and the congruence with the quartic twist by $-4u$ has the opposite symplectic type; moreover, $p\equiv\pm3\pmod8$; 3. if $n=6$ then $E[p]$ and $E'[p]$ are symplectically isomorphic if and only if $\sqrt{up^}\in K$, and then the sextic twist by $-27u$ is antisymplectic; moreover, $p\equiv\pm5\pmod{12}$. This proof builds on the proof of Proposition \[P:higherTwists\]. Indeed, we have chosen $\tau\in G_K$ to be such that it fixes $\root n\of u$ and is non-trivial when restricted to $K'$ and we let $A = {{\overline{\rho}}}_{E,p}(\tau)$. We have shown that $A$ satisfies and so, by the discussion following , the symplectic type of the congruence is given by the square class of $\det A$. We also know that the projective $p$-division field is $F = K'(\root n \of u)$ and $\operatorname{Gal}(F/K) \simeq D_n$. We have $\det(A)=\det{{\overline{\rho}}}_{E,p}(\tau)=\chi_p(\tau)$ where $\chi_p$ denotes the mod $p$ cyclotomic character. Since the $p$-division field of $E$ contains the $p$-th roots of unity, and the projective $p$-division field contains $\sqrt{p^}$, the congruence is symplectic if and only if $\tau$ fixes $\sqrt{p^}$. Clearly, when $\sqrt{p^}\in K$ the congruence is symplectic, proving (1) except for the congruence condition. Suppose now that $\sqrt{p^}\notin K$. We divide into cases: \(a) Suppose $n=3$. We have $\operatorname{Gal}(F/K) \simeq D_3$ and so $K'$ is the unique quadratic subfield of $F$, therefore $\sqrt{p^} \in K'$. Since $\tau$ acts non-trivially on $K'$ the congruence is anti-symplectic. \(b) Suppose $n=4$. We have $\operatorname{Gal}(F/K) \simeq D_4$ and there are exactly three quadratic sub-extensions of $F/K$. Furthermore, the fields $K'=K(\sqrt{-1})$, $K(\sqrt{p^})$, $K(\sqrt{u})$ are quadratic extensions of $K$ satisfying $K' \neq K(\sqrt{p^})$, $K(\sqrt{u})$. (b1) Suppose $K(\sqrt{p^})=K(\sqrt{u})$. By definition $\tau$ fixes $\sqrt{u}$ thus it fixes $\sqrt{p^}$ and the congruence is symplectic; note that in this case $\sqrt{up^} \in K$. (b2) Suppose $K(\sqrt{p^}) \neq K(\sqrt{u})$. Then $K(\sqrt{p^})=K(\sqrt{-4u})$ and since $\tau(\sqrt{-4u}) = - \sqrt{-4u}$ the congruence is anti-symplectic; note that in this case $\sqrt{up^} \not\in K$. Recall from Proposition \[P:higherTwists\] there is also a $p$-congruence between $E$ and its quartic twist by $-4u$. Applying the previous argument to this congruence gives that it is symplectic if and only if $\sqrt{-4up^} \in K$. Thus the two congruences are of the same type if and only if $\sqrt{-1} \in K$ which is not the case by assumption. \(c) Suppose $n=6$. We have $\operatorname{Gal}(F/K) \simeq D_6$ and again there are exactly three quadratic sub-extensions of $F/K$. Furthermore, the fields $K'=K(\sqrt{-3})$, $K(\sqrt{p^})$, $K(\sqrt{u})$ are quadratic extensions of $K$ satisfying $K' \neq K(\sqrt{p^})$, $K(\sqrt{u})$. The rest of the argument follows similarly to (b). For the congruence conditions on $p$, which are only non-trivial when $n=4$ or $6$ (since $p>3$ implies $p\equiv\pm1\pmod6$) recall that when $n=4$ or $6$ the two quartic (respectively, sextic) twists are $2$-isogenous (respectively $3$-isogenous) to each other. When $\sqrt{p^}\in K$ these isogenies induce symplectic congruences, since both the twists of $E$ are symplectically congruent. By the isogeny criterion, this implies that $2$ (respectively $3$) is a quadratic residue, so $p\equiv\pm1\pmod{8}$ (respectively, mod $12$). When $\sqrt{p^}\notin K$ these isogenies induce anti-symplectic congruences, so $p\not\equiv\pm1\pmod{8}$ (respectively, mod $12$). We end this section with four families of examples. These include all possibilities for quartic and sextic twist congruences over ${\mathbb{Q}}$ with $p=3$ or $5$ in the quartic case and $p=5$ or $7$ in the sextic case, in the case where the elliptic curve has good reduction at $p$, so that the projective image is maximal, namely $D_{p-1}$ (respectively, $D_{p+1}$) in the split (respectively, non-split) case. \[Ex:j1728p3\] Let $K={\mathbb{Q}}$, $p=3$, $j=1728$ and write $E_a := E_{a,0}:\ Y^2=X^3+aX$, $a \in {\mathbb{Q}}^$. The mod $3$ projective image is the normaliser of a non-split Cartan, isomorphic to $D_4$, since the projective division field $F$ is obtained by adjoining the roots of the $3$-division polynomial $3X^4+6aX^2-a^2$, and $F={\mathbb{Q}}(\sqrt{-1},\root4\of u)$ where $u=-a^2/3$ and it is not hard to see that $a/\sqrt{-3}$ is not a square in ${\mathbb{Q}}(\sqrt{-1},\sqrt{-3})$. Following the proof of Proposition \[P:higherTwists\], one checks that up to $4$th powers, the subgroup of ${\mathbb{Q}}^/({\mathbb{Q}}^)^4$ which become $4$th powers in $F$ is generated by $u$ and $-4$: the maps ${\mathbb{Q}}^/({\mathbb{Q}}^)^4 \to {\mathbb{Q}}(\sqrt{-1})^/({\mathbb{Q}}(\sqrt{-1})^)^4$ and ${\mathbb{Q}}(\sqrt{-1})^/({\mathbb{Q}}(\sqrt{-1})^)^4 \to F^/(F^)^4$ have kernels generated by $-4$ and $u$ respectively. Hence, from Proposition \[P:higherTwists\] we expect a $3$-congruence between $E_a$ and either $E_{-1/(3a)}$ (twisting by $u$) or $E_{-3/a}$ (twisting by $u^{-1}=-3/a^2$). An explicit computation shows that only the first holds. Finally, we have $p^ = -3$ and $u=-a^2/3$, hence $\sqrt{p^u} = a \in {\mathbb{Q}}$ and the congruence is symplectic by Theorem \[T:typeHigher\]. Moreover, there is a congruence with the quartic twist by $-4u=4a^2/3$ which is antisymplectic. In summary, $E_a$ is symplectically $3$-congruent to $E_{-1/3a}$ and anti-symplectically $3$-congruent to both $E_{-4a}$ and $E_{4/3a}$. Note that in passing from the special case of the $3$-congruence between $E_1$ and $E_{-1/3}$ to the general case of the congruence between $E_{a}$ and $E_{-1/3a}$, we apply the quartic twist by $a$ to the first curve, but the inverse twist (by $a^{-1}$) to the second. \[Ex:j1728p5\] $K={\mathbb{Q}}$, $p=5$, $j=1728$. The projective image is the normaliser of a split Cartan, isomorphic to $D_4$. One can check that up to $4$th powers, the subgroup of ${\mathbb{Q}}^/({\mathbb{Q}}^)^4$ which become $4$th powers in the projective $5$-division field of $E_a$ is generated by $5/a^2$ and $-4$. Hence we expect $5$-congruences between $E_a$ and either $E_{5/a}$ (twisting by $u=5/a^2$) or $E_{a^3/5}$ (twisting by $u=a^2/5$). Only the first holds (by a computation similar to the previous example, though a little simpler since we are in the split case so do not need to extend scalars). Since $u=5/a^2$ is $+5$ times a square the congruence is symplectic. There is also a congruence with the quartic twist by $-4u=-20/a^2$, which is antisymplectic. \[Ex:j0p5\] $K={\mathbb{Q}}$, $p=5$, $j=0$. Denote the curve $E_{0,b}:\ Y^2=X^3+b$ simply by $E_b$. The projective image is the normaliser of a non-split Cartan, isomorphic to $D_6$. We expect a $5$-congruence between $E_b$ and $E_{bu}$ where $u\equiv5$ (modulo squares) and $u\equiv b/10$ or $10/b$ (modulo cubes), since one may check that $b/10$ is a cube in ${\mathbb{Q}}(E_b[5])$. Hence, modulo $6$th powers, we have either $u\equiv 4/5b^2$ or $u\equiv 5b^2/4$. The first works, hence $E_b$ and $E_{4/(5b)}$ are $5$-congruent. This congruence is symplectic; composing with the $3$-isogeny we also have an anti-symplectic congruence between $E_b$ and $E_{-108/(5b)}$. \[Ex:j0p7\] $K={\mathbb{Q}}$, $p=7$, $j=0$. The projective image is the normaliser of a split Cartan, isomorphic to $D_6$. We expect a $7$-congruence between $E_b$ and $E_{bu}$ where $u\equiv-7$ (modulo squares) and $u\equiv 7b/2$ or $2/(7b)$ (modulo cubes), since one may check that $7b/2$ is a cube in ${\mathbb{Q}}(E_b[7])$. Hence, modulo $6$th powers, we have either $u\equiv -28/b^2$ or $u\equiv -b^2/28$. The first works, hence $E_b$ and $E_{-28/b}$ are $7$-congruent. This congruence is symplectic; composing with the $3$-isogeny we also have an anti-symplectic congruence between $E_b$ and $E_{756/b}$. Finding congruences and determining their symplectic type {#S:statistics} ========================================================= In this section we discuss our systematic study of mod $p$ congruences between elliptic curves in the LMFDB database. As of September 2019, this database contains all elliptic curves defined over ${\mathbb{Q}}$ of conductor $N\le\numprint{500000}$, as computed by the first author using the methods of [@AMEC]; there are $\numprint{3064704}$ curves, in $\numprint{2164259}$ isogeny classes. Recall first that isogenous curves have mod $p$ representations which are isomorphic up to semisimplification, and actually isomorphic if the degree of the isogeny is not divisible by $p$. Secondly, two representations have isomorphic semisimplification if and only if they have the same traces, so that we can test this condition by testing whether $$a_{\ell}(E)\equiv a_{\ell}(E')\pmod{p} \quad \text{for all primes } \ell \nmid pNN',$$ where $N$ and $N'$ are the conductors of $E$ and $E'$ respectively. This test can very quickly establish rigorously that two curves do not* have isomorphic $p$-torsion up to semisimplification, by finding a single prime $\ell$ such that $a_{\ell}(E)\not\equiv a_{\ell}(E')\pmod{p}$. Moreover, it is possible to prove that two curves have isomorphic $p$-torsion up to semisimplification using this test for a finite number of primes $\ell$, as we explain in Step 2 below. We divide our procedure to determine all mod $p$ congruences between non-isogenous curves, and their symplectic type, for a fixed prime $p$, into five steps. We first outline the steps, and then consider each in detail in the following subsections. 1. Partition the set of isogeny classes of elliptic curves in LMFDB into subsets $S$, such that whenever two curves have mod $p$ representations with isomorphic semisimplifications, their isogeny classes belong to the same subset $S$, but not necessarily conversely. 2. For each subset $S$ prove that the curves in each isogeny class in $S$ really do have isomorphic mod $p$ representations up to semisimplification, if necessary further partitioning the subsets. Discard all “trivial” subsets of size $1$. 3. Separate the remaining subsets resulting from the previous step into those which have irreducible mod $p$ representations and the reducible ones. 4. For each irreducible subset $S$, and each pair of isogeny classes in $S$, pick curves $E$ and $E'$, one from each class in the pair; determine the symplectic type of the triple $(E,E',p)$; then use the isogeny criterion to partition the set of all the curves in all the isogeny classes in $S$ into one or two parts such that curves in the same part are symplectically isomorphic while those in different parts are antisymplectically isomorphic. 5. For each reducible subset $S$, determine whether, for each pair $E$, $E'$ chosen as in Step 4, there is an isomorphism between $E[p]$ and $E'[p]$ and not just between their semisimplifications, if necessary replacing $E'$ with the curve $p$-isogenous to it. If not, this means that $E[p]$ and $E'[p]$ are not in fact isomorphic. Thus we further partition each reducible set $S$ into subsets of isogeny classes of curves whose mod $p$ representations are actually isomorphic, not just up to semisimplification. For each of these new subsets, if nontrivial, proceed as in Step 4. Next we will explain each step in further detail. For the first three steps, $p$ is arbitrary, and we have carried these steps out for $7\le p\le97$. According to Theorem \[T:cong19\] below, no congruences (other than those induced by isogenies) exist for larger $p$. For the last two steps, we restrict to $p=7$ which is the most interesting case, as remarked in the Introduction. Sieving ------- In order that $E[p]\cong E'[p]$ up to semisimplification, it is necessary and sufficient that for all primes $\ell$ not dividing $pNN'$ we have $a_{\ell}(E)\equiv a_{\ell}(E')\pmod{p}$. In this step we may take one curve from each isogeny class, since isogenous curves have the same traces $a_\ell$, and have mod $p$ representations with isomorphic semisimplifications. Fix an integer $B\ge1$. Let ${\mathcal{L}}_B$ be the set of the $B$ smallest primes greater than $\numprint{500000}$. All curves in the database have good reduction at each prime in ${\mathcal{L}}_B$. Assume also $B$ is large enough that $p\notin{\mathcal{L}}_B$. Hence a necessary condition for two curves $E$ and $E'$ in the database to be congruent mod $p$ is that $a_{\ell}(E)\equiv a_{\ell}(E')\pmod{p}$ for all $\ell\in{\mathcal{L}}_B$. To each curve $E$ in the database we assign a “hash value” which is a simple function of the set $\{a_{\ell}(E)\pmod{p}\mid \ell\in{\mathcal{L}}_B\}$. For example we may enumerate ${\mathcal{L}}_B=\{\ell_0,\ell_1,\dots,\ell_{B-1}\}$ and use the integer value $\sum_{i=0}^{B-1}\overline{a}_{\ell_i}(E)p^i$, where for $a\in{\mathbb{Z}}$, $\overline{a}$ denotes the reduction of $a$ mod $p$ which lies in $\{0,1,\dots,p-1\}$. Curves whose mod $p$ representations are isomorphic up to semisimplification will have the same hash, and we may hope that clashes will be rare if $B$ is not too small. We proceed to compute this hash value for one curve in each isogeny class in the database, recording the curve’s label in a list indexed by the different hash values encountered. At the end of this step we can easily form a partition of the set of isogeny classes by taking these lists for each hash value. We then discard any such lists which are singletons. Using $B=40$, this process takes approximately 40 minutes for a single prime $p$. Note, however, that as most of the computation time taken is in computing $a_{\ell}(E)$ for all curves $E$ (up to isogeny), it is more efficient to compute the hash values for several primes in parallel. [Example.]{} After carrying out this step for $7\le p\le17$, using $B=50$, we find: $\numprint{23735}$ nontrivial subsets for $p=7$; $731$ for $p=11$; $177$ for $p=13$; and $8$ for $p=17$. There are no nontrivial subsets for any primes $p$ with $19\le p\le97$, so we can immediately conclude that there are no congruences in the database between non-isogenous curves modulo any prime in this range. See also Theorem \[T:cong19\] below. Proving isomorphism up to semisimplification -------------------------------------------- For each pair of isogeny classes within one subset obtained in the previous step, we use a criterion of Kraus–Oesterlé (see [@KO Proposition 4]), based on the Sturm bound and hence on the modularity of elliptic curves over ${\mathbb{Q}}$, to either prove isomorphism up to semisimplification, or reveal a “false positive”. The latter would happen if two curves which are not congruent mod $p$ have traces of Frobenius $a_{\ell}$ which are congruent modulo $p$ for all $\ell\in{\mathcal{L}}_B$. [Example (continued).]{} For $7\le p\le17$ we find no such false positives, so the curves within each subset do have mod $p$ representations which are genuinely isomorphic up to semisimplification. To avoid false positives, it is necessary to use a value of $B$ which is large enough. In our initial computations with conductor bound $\numprint{400000}$ we initially used 30 primes above $\numprint{400000}$. But the curves with labels [[](http://www.lmfdb.org/EllipticCurve/Q/25921a1)]{} and [[](http://www.lmfdb.org/EllipticCurve/Q/78400gw1)]{} have traces $a_{\ell}$ which are equal for all $\ell\in{\mathcal{L}}_{35}$, that is, for all $\ell$ with $400000\le \ell<400457$ (though not for $\ell=400457$). These curves have CM by the order of discriminant $-7$, and are quadratic twists by $230$; both have $a_\ell=0$ for all $\ell\equiv3,5,6\pmod{7}$, and $230$ is a quadratic residue modulo all other primes in ${\mathcal{L}}_{35}$. In our first computational runs (with $N\le\numprint{400000}$), we used $B=30$ and discovered this pair of curves giving rise to a false positive for every $p$. The sizes of the subsets of isogeny classes we find after the first two steps are as follows: for $p=7$ the $\numprint{23735}$ subsets have sizes between $2$ and $76$; for $p=11$, $p=13$ and $p=17$ they all have size $2$. Testing reducibility -------------------- For each set of isogeny classes of curves obtained in the previous step, we next determine whether the curves in the set have irreducible or reducible mod $p$ representations. To do this we apply a standard test of whether an elliptic curve admits a rational $p$-isogeny. For the curves in the database this information is already known. [Example (continued).]{} For $p=7$, of the $\numprint{23735}$ nontrivial sets from Step 2, we find that $\numprint{23448}$ are irreducible, [i.e.]{} consist of curves whose mod $7$ representations are irreducible, while $287$ are reducible. The irreducible sets have size at most $5$. In detail, there are $\numprint{21653}$ sets of size 2; $\numprint{1502}$ sets of size 3; 283 sets of size 4; and 10 sets of size 5. The reducible sets have size up to $80$. In Step 5 below we will further partition these sets after testing whether the curves are actually congruent mod 7 (not just up to semisimplification), after which the largest subset has only $4$ isogeny classes. For $p=11, 13$, and $17$, all the nontrivial subsets are of size $2$, and all are irreducible. Distinguishing symplectic from antisymplectic: irreducible case --------------------------------------------------------------- After the previous step we have a collection of sets of isogeny classes, such that for each pair of curves $E$, $E'$ taken from isogeny classes in each set, the $G_{{\mathbb{Q}}}$-modules $E[p]$ and $E'[p]$ are isomorphic and irreducible. Moreover, from Proposition \[P:conditionS\] we know that all isomorphisms $\phi : E[p] \simeq E'[p]$ have the same symplectic type. We wish to determine whether this type is symplectic or anti-symplectic. We may assume that $E$ and $E'$ are not isogenous, as otherwise we may simply apply the isogeny criterion. The local criteria of [@FKSym] suffice to determine the symplectic type for all the mod $p$ congruences found in the database for $p=7$ and $p=11$ (and also for $p=13, 17$), but this does not have to be the case as discussed in Section \[S:motivation\] (see [@FKSym Proposition 16] for an example with $p=3$ where the local methods fail). Therefore, we will now describe a procedure, using the modular curves $X_E(7)$, to obtain a method that works in all cases. We will use the modular parametrizations and explicit formulae of Kraus–Halberstadt [@Halberstadt-Kraus-XE7], Poonen–Schaefer–Stoll [@PSS], and as extended and completed by Fisher [@Fisher]. In [@Halberstadt-Kraus-XE7], Halberstadt and Kraus give an explicit model for the modular curve $X_E(7)$, for any elliptic curve $E$ defined over a field $K$ of characteristic not equal to $2$, $3$ or $7$. Recall that the $K$-rational points on $X_E(7)$ parametrize pairs $(E',\phi)$ where $E'$ is an elliptic curve defined over $K$ and $\phi:E[7]\to E'[7]$ is a symplectic isomorphism of $G_K$-modules; we identify two such isomorphisms $\phi$ when one is a scalar multiple of the other. The model for $X_E(7)$ given in [@Halberstadt-Kraus-XE7] is a plane quartic curve, a twist of the classical Klein quartic $X(7)$, given by an explicit ternary quartic form $F_{a,b}(X,Y,Z)$ in ${\mathbb{Z}}[a,b][X,Y,Z]$ where $E$ has equation $Y^2=X^3+aX+b$. The $24$ flexes on $X_E(7)$ are the cusps, that is, they are the poles of the rational function of degree $168$ giving the map $j:X_E(7)\to X(1)$. The base point $P_E=[0:1:0]\in X_E(7)(K)$ corresponds to the pair $(E,\operatorname{id})$. In [@Halberstadt-Kraus-XE7] one can also find explicit formulas for the rational function $j:X_E(7)\to X(1)$ and for the elliptic curve $E'$ associated with all but finitely many points $P=(x:y:z)\in X_E(7)$. More precisely, explicit polynomials $c_4, c_6 \in {\mathbb{Z}}[a,b][X,Y,Z]$ of degree $20$ and $30$, respectively, are given and the curve $E'$ associated with (all but finitely many) $P$ has model $$Y^2=X^3-27c_4(P)X-54c_6(P).$$ The finitely many common zeros of $c_4$ and $c_6$ are the exceptions, which Kraus and Halberstadt treat only incompletely. However, in [@Fisher] one may find formulas for four such pairs of polynomials $(c_4,c_6)$, of which the first is the pair in [@Halberstadt-Kraus-XE7], and such that at each point $P\in X_E(7)$ at least one pair $(c_4(P),c_6(P))\not=(0,0)$, thus supplying us with a model for the associated elliptic curve $E'$ at each point $P$. We make use of this model and formulas as follows, given curves $E$, $E'$ with $E[7]\cong E'[7]$. Using one curve $E$ we write down the model for $X_E(7)$. Then we find all preimages (if any) of $j'=j(E')$ under the map $X_E(7)\to X(1)$. While over an algebraically closed field there are $168$ distinct preimages of each $j'$, except that the ramification points $j=0$ and $j=1728$ have $56$ and $84$ preimages, over ${\mathbb{Q}}$, there are fewer: in the irreducible case there are at most $4$ by the results of Section \[S:cong-twist\]. If there are no preimages of $j'$, we conclude that the isomorphism $E[7]\cong E'[7]$ is not symplectic. Otherwise, for each preimage $P\in X_E(7)({\mathbb{Q}})$ we compute the curve associated to $P$, which may be a twist of $E'$, and test whether it is actually isomorphic to $E'$. If this holds for one such point $P$ in the preimage of $j(E')$, then the isomorphism between $E[7]$ and $E'[7]$ is symplectic. A similar method may be applied to test for antisymplectic isomorphisms, using another twist of $X(7)$ denoted $X_E^-(7)$, first written down explicitly in [@PSS], for which Fisher provides explicit formulae for the $j$-map and $c_4,c_6$ as above in [@Fisher]. We note that it is not necessary to apply both the symplectic and antisymplectic tests to a triple $(E,E',7)$ if we know already that $E[7]\cong E'[7]$ as $G_{\mathbb{Q}}$-modules, since one will succeed if and only if the other fails (by Proposition \[P:conditionS\]). However we did apply both tests in our computations with the curves in the database as a test of our implementation, verifying that precisely one test passes for each pair. We also checked that the results obtained for each pair using the local criteria are the same, so that we can be confident in the correctness of the results. These tests have only been carried out using a single curve in each isogeny class, since we know how to distinguish symplectic from antisymplectic isogenies. As a last step, we consider the full isogeny classes to obtain, for each elliptic curve $E$ in the database, the complete sets of all curves $E'$ (non-isogenous to $E$) which have symplectically and anti-symplectically isomorphic $7$-torsion modules to $E$. The output of this step consists of, for each of the subsets resulting from Steps 1–3, one or two sets of curves whose union is the set of all curves in the isogeny classes in the subset. All curves in the same set have symplectically isomorphic 7-torsion modules; when there are two sets, curves in different sets have antisymplectically isomorphic 7-torsion. [Example (continued).]{} Of the $\numprint{23448}$ non-trivial sets of isogeny classes with mutually isomorphic irreducible mod $7$ representations, we find that in $\numprint{16285}$ cases all the isomorphisms are symplectic, while in the remaining $\numprint{7163}$ cases antisymplectic isomorphisms occur. Using the local criteria of [@FKSym] for $p=11, 13, 17$ we find: for $p=11$, of the $731$ congruent pairs of isogeny classes, $519$ are symplectic and $212$ are antisymplectic; for $p=13$, of the $177$ congruent pairs of isogeny classes, $105$ are symplectic and $72$ are antisymplectic; for $p=17$, all of the $8$ congruent pairs of isogeny classes are antisymplectic. Let $p=7$. One of the subsets resulting from Steps 1–3 consists of the pair of isogeny classes $\{{\href{http://www.lmfdb.org/EllipticCurve/Q/344025bc1}{{\text{\rm344025bc1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/344025bd1}{{\text{\rm344025bd1}}}}\}$. Our test shows that ${\href{http://www.lmfdb.org/EllipticCurve/Q/344025bc1}{{\text{\rm344025bc1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/344025bd1}{{\text{\rm344025bd1}}}}$ are symplectically isomorphic. The isogeny class ${\href{http://www.lmfdb.org/EllipticCurve/Q/344025/bc}{\text{\rm344025bc}}}$ contains two $2$-isogenous curves, while class ${\href{http://www.lmfdb.org/EllipticCurve/Q/344025/bd}{\text{\rm344025bd}}}$ contains only one curve. Since $2$ is a quadratic residue mod 7, all three curves have symplectically isomorphic 7-torsion, and hence Step 4 returns a single set $$\{{\href{http://www.lmfdb.org/EllipticCurve/Q/344025bc1}{{\text{\rm344025bc1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/344025bc2}{{\text{\rm344025bc2}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/344025bd1}{{\text{\rm344025bd1}}}}\}.$$ Another subset resulting from Steps 1–3 is $\{{\href{http://www.lmfdb.org/EllipticCurve/Q/100800gw1}{{\text{\rm100800gw1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/100800hc1}{{\text{\rm100800hc1}}}}\}$. The same procedure results in the output of two sets of curves $$\{{\href{http://www.lmfdb.org/EllipticCurve/Q/100800gw1}{{\text{\rm100800gw1}}}}\}, \qquad \{{\href{http://www.lmfdb.org/EllipticCurve/Q/100800hc1}{{\text{\rm100800hc1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/100800hc2}{{\text{\rm100800hc2}}}}\},$$ since our tests show that ${\href{http://www.lmfdb.org/EllipticCurve/Q/100800gw1}{{\text{\rm100800gw1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/100800hc1}{{\text{\rm100800hc1}}}}$ are antisymplectically isomorphic, and the last two curves are $2$-isogenous. Consider the set of six elliptic curves $$\{{\href{http://www.lmfdb.org/EllipticCurve/Q/9225a1}{{\text{\rm9225a1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/9225e1}{{\text{\rm9225e1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/225a1}{{\text{\rm225a1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/225a2}{{\text{\rm225a2}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/11025c1}{{\text{\rm11025c1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/11025c2}{{\text{\rm11025c2}}}}\}$$ which form four complete isogeny classes. All have isomorphic mod 7 representations with image the normaliser of a split Cartan subgroup. The last four curves all have $j$-invariant $0$ and CM by $-3$. The first two are $-3$ quadratic twists of each other. The general methods of Step 4 of this section split this set into two subsets: $$\{{\href{http://www.lmfdb.org/EllipticCurve/Q/9225a1}{{\text{\rm9225a1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/225a1}{{\text{\rm225a1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/11025c1}{{\text{\rm11025c1}}}}\},\qquad \{{\href{http://www.lmfdb.org/EllipticCurve/Q/9225e1}{{\text{\rm9225e1}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/225a2}{{\text{\rm225a2}}}}, {\href{http://www.lmfdb.org/EllipticCurve/Q/11025c2}{{\text{\rm11025c2}}}}\}$$ Curves ${\href{http://www.lmfdb.org/EllipticCurve/Q/9225a1}{{\text{\rm9225a1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/9225e1}{{\text{\rm9225e1}}}}$ give a non-CM example of Proposition \[P:twist\]. Curves ${\href{http://www.lmfdb.org/EllipticCurve/Q/225a1}{{\text{\rm225a1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/11025c1}{{\text{\rm11025c1}}}}$ are sextic (but not quadratic or cubic) twists and illustrate Proposition \[P:higherTwists\]. Auxiliary results for the reducible case ---------------------------------------- Compared to the irreducible case, establishing reducible congruences requires extra work because when working with the semisimplifications $E[7]^{ss}$ and $E'[7]^{ss}$ important information is lost. The objective of this section is to establish Theorem \[T:reducible\] which will allow us to rigorously prove congruences in the reducible case. Let $B \subset {\operatorname{GL}}_2({\mathbb{F}}_p)$ be the standard Borel subgroup, i.e. the upper triangular matrices. Let $H \subset B$ be a subgroup of order divisible by $p$. We can write $H = D\cdot U$ where $D \subset B$ is a subgroup of diagonal matrices and $U$ is cyclic generated by $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$. Moreover, $U$ is a normal subgroup of $H$ and we write $\pi : H \to H/U \simeq D$ for the quotient map. \[P:inner\] Let $H = D \cdot U \subset B$ and $\pi$ be as above. Let $\phi$ be an automorphism of $H$. Assume that $\pi(x) = \pi(\phi(x))$ for all $x \in H$. Then $\phi$ is given by conjugation in $B$, i.e. there is $A \in B$ such that $\phi(x) = AxA^{-1}$. Let $x \in H$ and write it as $x = du$ with $d \in D, u \in U$. Note that $U$ is the unique normal subgroup with order $p$, so $\phi(U) = U$ and $\phi(D)=D$ for all $\phi \in \operatorname{Aut}H$. By hypothesis, we have $$\pi(x) = \pi(\phi(x)) = \pi(\phi(d))\pi(\phi(u)) \iff dU = \phi(d)U \iff d\phi(d)^{-1} \in U.$$ Since $d, \phi(d) \in D$ we have $d\phi(d)^{-1} \in D \cap U = \{ I_2\}$ therefore $d\phi(d)^{-1} = I_2$, where $I_2$ is the identity in ${\operatorname{GL}}_2({\mathbb{F}}_p)$. Thus $d = \phi(d)$ and we conclude that any $\phi$ as in the statement is the same as an automorphism of $U$ (extended to $H$ by $\phi(d)=d$ for all $d\in D$). Write $N_B(U)$ for the normaliser of $U$ in $B$ and $C_B(U)$ for its centralizer. We have $N_B(U) = B$ and $C_B(U)$ are the matrices of the form $\left(\begin{smallmatrix} \lambda & b \\ 0 & \lambda \end{smallmatrix} \right)$ with $\lambda \neq 0$. Thus $\# N_B(U)/C_B(U) = p-1$. Since we also have $$N_B(U)/C_B(U) \hookrightarrow \operatorname{Aut}U \simeq {\mathbb{F}}_p^$$ it follows that $\operatorname{Aut}U \simeq N_B(U)/C_B(U)$, as desired. \[P:fieldF\] Let $p > 2$ be a prime. Let $E/K$ be an elliptic curve such that ${{\overline{\rho}}}_{E,p}$ is reducible. Assume there is an element of order $p$ in the image of ${{\overline{\rho}}}_{E,p}$. Then, there is a field $F \supset K$ such that $[F : K] = p$ and $E$ acquires a second isogeny over $F$. We can choose a basis of $E[p]$ where $${{\overline{\rho}}}_{E,p} = \begin{pmatrix} \chi& h \\ 0 & \chi' \end{pmatrix}.$$ Let $H$ be the set of elements $\sigma \in G_K$ such that $h(\sigma) = 0$. Since ${{\overline{\rho}}}_{E,p}$ is a homomorphism it follows that $H$ is a subgroup of $G_K$. We let $F \subset K(E[p])$ to be the field fixed by $H$. Since there is an element of order $p$ in the image of ${{\overline{\rho}}}_{E,p}$ and $\left(\begin{smallmatrix} \chi & 0 \\ 0 & \chi' \end{smallmatrix} \right)$ has order coprime to $p$, it follows that $[F : K] =p$. \[T:reducible\] Let $p > 2$ be a prime. Let $E_1, E_2$ be elliptic curves over $K$ such that - ${{\overline{\rho}}}_{E_1,p}^{ss} \simeq {{\overline{\rho}}}_{E_2,p}^{ss} \simeq \chi \oplus \chi'$, where $\chi, \chi' : G_K \to {\mathbb{F}}_p^$ are characters; - both ${{\overline{\rho}}}_{E_1,p}$ and ${{\overline{\rho}}}_{E_2,p}$ have an element of order $p$ in their image. For $i=1,2$, let $F_i/K$ be a degree $p$ extension where $E_i$ acquires a second isogeny, as given by Proposition \[P:fieldF\]. After replacing $E_2$ by a $p$-isogenous curve if necessary, we have ${{\overline{\rho}}}_{E_1,p} \simeq {{\overline{\rho}}}_{E_2,p}$ if and only if $F_1 \simeq F_2$. Clearly, if ${{\overline{\rho}}}_{E_1,p} \simeq {{\overline{\rho}}}_{E_2,p}$ then $F_1 \simeq F_2$. We now prove the opposite direction. From (i) it follows that ${{\overline{\rho}}}_{E_i,p}$ is reducible and that, after replacing $E_2$ by a $p$-isogenous curve if necessary (to swap $\chi$ with $\chi'$), we have $${{\overline{\rho}}}_{E_i,p} = \begin{pmatrix} \chi & h_i \\ 0 & \chi' \end{pmatrix} \quad \text{ with } \quad h_i : G_K \to {\mathbb{F}}_p.$$ Let $L$ be the field cut out by $\chi \oplus \chi'$. It follows from (ii) that $h_i\|_{G_L} \neq 0$, and hence the matrix $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$ is in the image of ${{\overline{\rho}}}_{E_i,p}$ for $i=1,2$. Write $K_i = K(E_i[p])$. We have $[K_i : K] = [K_i : L][L : K] = p [L : K]$. Since the degree $[L : K]$ divides $(p-1)^2$ it is coprime to $p = [F_i : K]$ therefore we have $K_i = L F_i$. Suppose $F_1 \simeq F_2$. Since $K_1$ is Galois, we have $F_2 \subset K_1$ and therefore $K_p := K_1 = K_2$ is the field cut out by both ${{\overline{\rho}}}_{E_1,p}$ and ${{\overline{\rho}}}_{E_2,p}$, i.e. these representations have the same kernel. Write $G = \operatorname{Gal}(K_p / K)$. From now on we think of ${{\overline{\rho}}}_{E_i,p}$ as an injective representation of $G$. Note that the images of ${{\overline{\rho}}}_{E_1,p}$ and ${{\overline{\rho}}}_{E_2,p}$ is the same subgroup $H$ of the Borel. All the elements in $H$ are of the form ${{\overline{\rho}}}_{E_2,p}(\sigma)$ for $\sigma \in G$, so we can consider the map $\phi = {{\overline{\rho}}}_{E_1,p} \circ {{\overline{\rho}}}_{E_2,p}^{-1} : H \to H$. It is an automorphism of $H = D\cdot U$ satisfying the hypothesis of Proposition \[P:inner\], where $D$ are the matrices $\left(\begin{smallmatrix} \chi & 0 \\ 0 & \chi' \end{smallmatrix} \right)$. Then, $\phi$ is given by conjugation, that is $$\phi({{\overline{\rho}}}_{E_2,p}(\sigma)) = A {{\overline{\rho}}}_{E_2,p}(\sigma) A^{-1}.$$ Since we also have $$\phi({{\overline{\rho}}}_{E_2,p}(\sigma)) = {{\overline{\rho}}}_{E_1,p} \circ {{\overline{\rho}}}_{E_2,p}^{-1}({{\overline{\rho}}}_{E_2,p}(\sigma)) = {{\overline{\rho}}}_{E_1,p}(\sigma)$$ we conclude that ${{\overline{\rho}}}_{E_1,p}(\sigma) = A {{\overline{\rho}}}_{E_2,p}(\sigma) A^{-1}$, as desired. Distinguishing symplectic from antisymplectic: reducible case ------------------------------------------------------------- From Steps 1–3 we have (for certain pairs $(E,E')$ established that $E[7]^{\operatorname{ss}} \simeq E'[7]^{\operatorname{ss}}$ but this is insufficient to conclude $E[7] \simeq E'[7]$ when these are reducible $G_{\mathbb{Q}}$-modules. To achieve this we will apply Theorem \[T:reducible\] and its proof. Recall that $E[7]$ is reducible if and only if $E$ admits a rational $7$-isogeny. Over ${\mathbb{Q}}$ there is only ever at most one $7$-isogeny, since otherwise the image of the mod $7$ representation ${{\overline{\rho}}}_{E,7}$ attached to $E$ is contained in a split Cartan subgroup of ${\operatorname{GL}}(2,{\mathbb{F}}_7)$, and this cannot occur over ${\mathbb{Q}}$ (see [@GL Theorem 1.1]). Furthermore, it is well known that the size of the ${\mathbb{Q}}$-isogeny class of $E$ is either $2$, consisting of two $7$-isogenous curves, or $4$, consisting of two pairs of $7$-isogenous curves linked by $2$- or $3$-isogenies (but not both). Examples of these are furnished by the isogeny classes [[](http://www.lmfdb.org/EllipticCurve/Q/26/b)]{}, [[](http://www.lmfdb.org/EllipticCurve/Q/49/a)]{}, and [[](http://www.lmfdb.org/EllipticCurve/Q/162/b)]{} respectively. Fix an elliptic curve $E$ with $E[7]$ reducible. The image of ${{\overline{\rho}}}_{E,7}$ has the form $$\begin{pmatrix}\chi_1&\\0&\chi_2 \end{pmatrix},$$ where $\chi_1, \; \chi_2 : G_{{\mathbb{Q}}}\to{\mathbb{F}}_7^$ are characters and $$ (the upper right entry) is non-zero by the previous discussion. Moreover, the product $\chi_1\chi_2$ is the cyclotomic character, so in particular $\chi_1\not=\chi_2$. This last observation is valid over any field not containing $\sqrt{-7}$, so that the determinant is not always a square. Now let $E'$ be a second curve such that $E[7]^{\operatorname{ss}}\cong E'[7]^{\operatorname{ss}}$. The image of ${{\overline{\rho}}}_{E',7}$ has the form $$\begin{pmatrix}\chi_1'&'\\0&\chi_2' \end{pmatrix},$$ where $\{\chi_1,\chi_2\}=\{\chi_1',\chi_2'\}$ and $' \neq 0$ for the same reason as before. In particular, there is an element of order $7$ in the images of both ${{\overline{\rho}}}_{E,7}$ and ${{\overline{\rho}}}_{E',7}$. The next step in applying Theorem \[T:reducible\] is to decide if we need to replace $E'$ with its $7$-isogenous curve to obtain $\chi_1=\chi_1'$ and $\chi_2=\chi_2'$. For this we determine the “isogeny characters” characters $\chi_1$ and $\chi_1'$: the kernel of $\chi_1$ (respectively $\chi_1'$) cuts out the cyclic extension of ${\mathbb{Q}}$ of degree dividing $6$ generated by the coordinates of a point in the kernel of the unique $7$-isogeny from $E$ (respectively $E'$). In this way we can determine whether $\chi_1=\chi_1'$ and $\chi_2=\chi_2'$ or $\chi_1=\chi_2'$ and $\chi_2=\chi_1'$. In the second case, we replace $E'$ with its $7$-isogenous curve, which has the effect of interchanging $\chi_1'$ and $\chi_2'$ (as well as changing $'$). Now the image of ${{\overline{\rho}}}_{E',7}$ has the form $$\begin{pmatrix}\chi_1&'\\0&\chi_2 \end{pmatrix},$$ with the same characters, in the same order, as for ${{\overline{\rho}}}_{E,7}$. From Theorem \[T:reducible\] we have that $E[7]\cong E'[7]$ if and only if $F_1 \simeq F_2$, where $F_i$ are the fields in the statement of Theorem \[T:reducible\]. The field $F_1$ is the common field of definition of all of the other seven $7$-isogenies from $E$ (see also Proposition \[P:fieldF\]). The map from $X_0(7)$ to the $j$-line is given by the classical rational function (see Fricke) $$j = \frac{(t^{2} + 13t + 49) \cdot (t^{2} + 5t + 1)^{3}}{t},$$ where $t$ is a choice of Hauptmodul for the genus $0$ curve $X_0(7)$. Hence the roots of the degree $8$ polynomial $(t^{2} + 13t + 49) \cdot (t^{2} + 5t + 1)^{3} -t\cdot j(E)$ determine the fields of definition of the eight $7$-isogenies from $E$. In our setting, it has a single rational root (giving the unique $7$-isogeny from $E$ defined over ${\mathbb{Q}}$) and an irreducible factor of degree $7$, which defines $F_1$ as an extension of ${\mathbb{Q}}$. Similarly, starting from $E'$ we determine $F_2$; finally, we check whether $F_1$ and $F_2$ are isomorphic. In this way, for each pair $(E,E')$ whose mod $7$ representations are reducible with isomorphic semisimplifications, we may determine whether or not we do in fact have an isomorphism $E[7]\cong E'[7]$, possibly after replacing $E'$ by its unique $7$-isogenous curve. In most of the reducible cases encountered in the database, we found that there was no isomorphism between the $7$-torsion modules themselves. In those cases where there is such an isomorphism, we can determine whether or not it is symplectic using the same methods as in the irreducible case, noting that the test using the parametrizing curves $X_E(7)$ and $X_E^-(7)$ do not at any point rely on the irreducibility or otherwise of the representations. Finally, if there are also $2$- or $3$-isogenies present we can include these appropriately, since the former induce symplectic and the latter antisymplectic congruences. [Example (continued).]{} For $p=7$, after Steps 1–3, there are $287$ reducible sets of isogeny classes with isomorphic semisimplification, of size up to $80$. Step 5 refines these into smaller subsets which have actually isomorphic $7$-torsion modules, of which $384$ are nontrivial. Among these there are $38$ classes also admitting a $2$-isogeny and $22$ classes admitting a $3$-isogeny, making a total of $849$ curves, partitioned into mutually $7$-congruent subsets of size $2$, $3$ or $4$: there are $263$ sets of size $2$, of which the congruence is symplectic in $142$ cases and anti-symplectic in $121$ cases; $101$ of size $3$, of which all the congruences are symplectic in $56$ cases, and in the remaining $45$ cases, only one congruence is symplectic; and $20$ of size $4$, in which all congruences are symplectic in $8$ cases, there are two pairs of symplectically congruent curves (with congruences between curves in different pairs being anti-symplectic) in $2$ cases, and in $10$ cases there are $3$ curves mutually symplectically congruent and anti-symplectically congruent to the fourth curve. For $p\ge11$ there are no reducible cases to consider. Twists ------ If there is a mod $p$ congruence between two elliptic curves $E_1$ and $E_2$, then for any $d\in{\mathbb{Q}}^$ there will also be a congruence (with the same symplectic type) between their quadratic twists $E_1^d$ and $E_2^d$ (see [@FKSym Lemma 11]). Nevertheless, it is hard to say precisely how many congruences there in the database “up to twist”, since twisting changes conductor (in general), so we may have a set of mutually $7$-congruent elliptic curves in the database, but with one or more of their twists not in the database, so the twisted set in our data will be smaller. Instead, to have a measure of how many congruences we have found up to twist, we simply report on how many distinct $j$-invariants we found, excluding as before curves which are only congruent to isogenous curves. For $p=7$ there are $\numprint{11761}$ distinct $j$-invariants of curves with irreducible mod $7$ representations which are congruent to at least one non-isogenous curve, and $154$ distinct $j$-invariants in the reducible case. For $p=11$ there are $212$ distinct $j$-invariants and for $p=13$ there are $39$. For $p=17$, all $17$-congruent isogeny classes consist of single curves, the eight pairs are quadratic twists, and the $j$-invariants of the curves in each pair are $48412981936758748562855/77853743274432041397$ and $-46585/243$. One such pair of $17$-congruent curves consists of ${\href{http://www.lmfdb.org/EllipticCurve/Q/47775b1}{{\text{\rm47775b1}}}}$ and ${\href{http://www.lmfdb.org/EllipticCurve/Q/3675b1}{{\text{\rm3675b1}}}}$. Evidence for the Frey-Mazur conjecture in the database {#S:Frey-Mazur} ====================================================== The following theorem shows that the strong form of the Frey–Mazur conjecture with $C=18$ holds for the congruences available in the LMFDB database. We refer to [@Halberstadt-Kraus-FreyMazur] for one theoretical result towards this very challenging and still open conjecture. \[T:cong19\] Let $p \geq 19$ be a prime. Let $E/{\mathbb{Q}}$ and $E'/{\mathbb{Q}}$ be elliptic curves with conductors at most $\numprint{500000}$. Suppose that $E[p] \simeq E'[p]$ as $G_{\mathbb{Q}}$-modules. Then $E$ and $E'$ are ${\mathbb{Q}}$-isogenous. Let $p \geq 5$ be a prime. Let $N_E$ and $\Delta_E$ denote the conductor and the minimal discriminant of $E$, respectively. Write also $\tilde{N}_E$ to denote $N_E$ away from $p$ and let $N_p$ be the Serre level (i.e. the Artin conductor away from $p$) of ${{\overline{\rho}}}_{E,p}$. We have $N_p \mid \tilde{N}_E$. Recall that the conductor of an elliptic curve at primes $p \geq 5$ divides $p^2$. Moreover, from Kraus [@KrausThesis p. 30] it follows that, for each $\ell \neq p$, if ${\upsilon}_{\ell}(N_p) \neq {\upsilon}_{\ell}(N_E) = {\upsilon}_{\ell}(\tilde{N}_E)$ then ${\upsilon}_{\ell}(N_E) = 1$ and $p \mid {\upsilon}_{\ell}(\Delta_E)$. Therefore, we can find primes $q_i \nmid pN_p$ such that $$\label{E:condE} N_E = p^s \cdot N_p \cdot q_0 \cdot \ldots \cdot q_n, \qquad p \mid {\upsilon}_{q_i}(\Delta_E), \qquad 0 \leq s \leq 2$$ where the number of $q_i$ occurring is $\geq 1$ if and only if $\tilde{N}_E \neq N_p$. Now let $E'/{\mathbb{Q}}$ be another elliptic curve satisfying $E[p] \simeq E'[p]$ as $G_{\mathbb{Q}}$-modules. Write $N_{E'}$, $\tilde{N}_{E'}$, $\Delta_{E'}$, $N'_p$ and ${{\overline{\rho}}}_{E',p}$ to denote analogous quantities attached to $E'$. We have $N'_p \mid \tilde{N}_{E'}$. By assumption, we have ${{\overline{\rho}}}_{E',p} \simeq {{\overline{\rho}}}_{E,p}$ so these representations have the same Serre level, i.e. $N_p = N_p'$ and (similarly as for $E$) we can find primes $q_i' \nmid pN'_p$ such that $N_{E'}$ factors as $$\label{E:condE'} N_{E'} = p^{s'} \cdot N_p \cdot q'_0 \cdot \ldots \cdot q'_m, \qquad p \mid {\upsilon}_{q'_i}(\Delta_E'), \qquad 0 \leq s' \leq 2.$$ The representations ${{\overline{\rho}}}_{E,p}$ and ${{\overline{\rho}}}_{E',p}$ also have the same Serre weights $k$ and $k'$, respectively. Note that for $s=0$ ($E$ has good reduction at $p$) we have $k=2$ and for $s=1$ ($E$ has multiplicative reduction at $p$) we have $k=2$ if $p \mid {\upsilon}_p(\Delta_E)$ or $k=p+1$ otherwise (see for example [@KrausThesis p. 3]); moreover, for $s=2$ it follows from [@KrausThesis Théorème 1] that $k \not\in \{2, p+1\}$ for $p \geq 19$. Similar conclusions apply to $E'$, $s'$ and $k'$. Therefore, we have 2 cases: (i) if $s = 2$ or $s=1$ and $p \nmid {\upsilon}_{p}(\Delta_E)$ then $s'=s$; (ii) if $s=0$ or $s=1$ and $p \mid {\upsilon}_{p}(\Delta_E)$ then $s' \in \{0,1 \}$. Suppose $E'$ is a non-isogenous curve with the same conductor. Taking differences of traces of Frobenius at different primes shows that there are no congruence between any two of them for $p \geq 19$, otherwise $p$ needs to divide the differences (see (1) below). Thus $N_E \neq N_{E'}$. Suppose $\tilde{N}_E = \tilde{N}_{E'}$, so that the only difference in the conductors is at $p$. From the possibilities above for the Serre weights, after interchanging $E$ and $E'$ if needed, we can assume $s=1$ and $s'=0$ and we also know that $p \mid {\upsilon}_p(\Delta_E)$. On the other hand, if $\tilde{N}_E \neq \tilde{N}_{E'}$ then, after interchanging $E$ and $E'$ if needed, we have $N_p \neq \tilde{N}_E$ and so there is at least one prime $q_i \neq p$ appearing in the factorization , which in particular satisfies $p \mid {\upsilon}_{q_i}(\Delta_E)$. Let ${\mathcal{M}}_E$ be the set of pairs $(q,p)$ where $q$ is a multiplicative prime of $E$ and $p \geq 19$ is a prime satisfying $p \mid {\upsilon}_{q}(\Delta_E)$. Note that we can have $q=p$. Let ${\mathcal{M}}_{E'}$ be the analogous set for $E'$. From the previous paragraph we conclude that $p$ has to occurs in the second entry of one of the pairs $(q,p)$ in ${\mathcal{M}}_E$ or ${\mathcal{M}}_{E'}$. To complete the proof, we carried out the following computations on the LMFDB database of all elliptic curves defined over ${\mathbb{Q}}$ and conductor at most $\numprint{500000}$: 1. For each $N\le \numprint{500000}$ and each pair of non-isogenous curves $E_1,E_2$ of conductor $N$ (if there are at least two such isogeny classes), we computed $\gcd_{\ell\le B, \ell\nmid N}(a_{\ell}(E_1)-a_{\ell}(E_2))$ for increasing $B$ until the value of the $\gcd$ was ${}\le17$. The success of this computation shows that there are no congruences mod $p$ between non-isogenous curves of the same conductor for $p\ge19$. 2. For one curve $E$ in each isogeny class we computed the set ${\mathcal{M}}_E$ from the conductor and minimal discriminant. We found that the largest prime $p$ occurring in any ${\mathcal{M}}_E$ was $97$: in fact, all $p$ with $19\le p\le97$ occur except for $p=89$. Hence any mod $p$ congruence between non-isogenous curves in the database must have $p\le97$. In view of the computations of Section \[S:statistics\], there are no such congruences for $19\le p\le97$. 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[^1]: JEC was supported by EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, and the Horizon 2020 European Research Infrastructures project OpenDreamKit (\#676541) [^2]: NF was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skł[l]{}odowska-Curie grant agreement No. 747808 [^3]: Throughout the paper we use Cremona labels for elliptic curves over ${\mathbb{Q}}$; these curves may be found in the LMFDB (see [@lmfdb]). [^4]: when $j(E)=1728$ one of these is the quartic twist by $-4$ so not in fact quadratic.
--- abstract: 'We present the results of an X-ray analysis of 80 galaxy clusters selected in the 2500 deg$^2$ South Pole Telescope survey and observed with the Chandra X-ray Observatory. We divide the full sample into subsamples of $\sim$20 clusters based on redshift and central density, performing a joint X-ray spectral fit to all clusters in a subsample simultaneously, assuming self-similarity of the temperature profile. This approach allows us to constrain the shape of the temperature profile over $0 < r < 1.5R_{500}$, which would be impossible on a per-cluster basis, since the observations of individual clusters have, on average, 2000 X-ray counts. The results presented here represent the first constraints on the evolution of the average temperature profile from $z = 0$ to $z = 1.2$. We find that high-$z$ ($0.6 < z < 1.2$) clusters are slightly ($\sim$30%) cooler both in the inner ($r<0.1R_{500}$) and outer ($r>R_{500}$) regions than their low-$z$ ($0.3 < z<0.6$) counterparts. Combining the average temperature profile with measured gas density profiles from our earlier work, we infer the average pressure and entropy profiles for each subsample. Confirming earlier results from this data set, we find an absence of strong cool cores at high $z$, manifested in this analysis as a significantly lower observed pressure in the central $0.1R_{500}$ of the high-$z$ cool-core subset of clusters compared to the low-$z$ cool-core subset. Overall, our observed pressure profiles agree well with earlier lower-redshift measurements, suggesting minimal redshift evolution in the pressure profile outside of the core. We find no measurable redshift evolution in the entropy profile at $r\lesssim0.7R_{500}$ – this may reflect a long-standing balance between cooling and feedback over long timescales and large physical scales. We observe a slight flattening of the entropy profile at $r\gtrsim R_{500}$ in our high-$z$ subsample. This flattening is consistent with a temperature bias due to the enhanced ($\sim$3$\times$) rate at which group-mass ($\sim$2keV) halos, which would go undetected at our survey depth, are accreting onto the cluster at $z\sim1$. This work demonstrates a powerful method for inferring spatially-resolved cluster properties in the case where individual cluster signal-to-noise is low, but the number of observed clusters is high.' author: - 'M. McDonald B. A. Benson, A. Vikhlinin, K. A. Aird, S. W. Allen, M. Bautz, M. Bayliss, L. E. Bleem, S. Bocquet, M. Brodwin, J. E. Carlstrom, C. L. Chang, H. M. Cho, A. Clocchiatti, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, R. J. Foley, W. R. Forman, E. M. George, M. D. Gladders, A. H. Gonzalez, N. W. Halverson, J. Hlavacek-Larrondo, G. P. Holder, W. L. Holzapfel, J. D. Hrubes, C. Jones, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, J. Liu, M. Lueker, D. Luong-Van, A. Mantz, D. P. Marrone, J. J. McMahon, S. S. Meyer, E. D. Miller, L. Mocanu, J. J. Mohr, S. S. Murray, S. Padin, C. Pryke, C. L. Reichardt, A. Rest, J. E. Ruhl, B. R. Saliwanchik, A. Saro, J. T. Sayre, K. K. Schaffer, E. Shirokoff, H. G. Spieler, B. Stalder, S. A. Stanford, Z. Staniszewski, A. A. Stark, K. T. Story, C. W. Stubbs, K. Vanderlinde, J. D. Vieira, R. Williamson, O. Zahn, A. Zenteno' title: \| The Redshift Evolution of the Mean Temperature, Pressure,\ and Entropy Profiles in 80 SPT-Selected Galaxy Clusters --- Introduction ============ Galaxy clusters, despite what the name implies, consist primarily of matter that is not in galaxies. A typical cluster is well-modeled by a central dark matter halo ($\sim$85% by mass) and a diffuse, optically-thin plasma ($\sim$15% by mass). The response of this hot ($\gtrsim10^7$ K) plasma, known as the intracluster medium (ICM), to the evolving gravitational potential is one of our best probes of the current state and evolution of galaxy clusters. X-ray imaging and spectroscopy of the ICM allow estimates of the cluster mass profile via the spectroscopic temperature and gas density [e.g., @forman82; @nevalainen00; @sanderson03; @arnaud05; @kravtsov06; @vikhlinin06a; @arnaud07], the enrichment history of the cluster via the ICM metallicity [e.g., @deyoung78; @matteucci86; @deplaa07; @bregman10; @bulbul12], the cooling history via the cooling time or entropy [e.g., @white97; @peres98; @cavagnolo08; @mcdonald13b], the feedback history via the presence of X-ray bubbles [e.g., @rafferty06; @mcnamara07; @rafferty08; @hlavacek12], and the current dynamical state and merger history of the cluster via the X-ray morphology [e.g., @jones79; @mohr95; @roettiger96; @schuecker01; @jeltema05; @nurgaliev13]. While there is much diversity in the ICM from cluster to cluster, it is valuable to determine if there are broad similarities in clusters of a given mass and redshift. The construction of a “Universal” pressure profile, for example, can allow comparisons to simulated galaxy clusters, as well as provide a functional form for matched-filtering algorithms, such as those that are used to select galaxy clusters using the Sunyaev-Zel’dovich [SZ; @sunyaev72] effect. Much effort has been made to quantify the average temperature [e.g., @loken02; @vikhlinin06a; @pratt07; @leccardi08a; @baldi12], entropy [e.g., @voit05; @piffaretti05; @cavagnolo09; @pratt10], and pressure [e.g. @arnaud10; @sun11; @bonamente12; @planck13] profiles for low-redshift galaxy groups and clusters based on both X-ray and SZ selection. In all cases, the average profiles have a substantial amount of scatter at $r\lesssim0.2R_{500}$, due to the presence (or lack) of a cool, dense core [e.g., @vikhlinin06a; @cavagnolo09; @arnaud10], but collapse onto the self-similar expectation at larger radii. This suggests that non-gravitational processes (e.g., cooling, AGN feedback) are important in the central region of the cluster while gravity is the dominant force in the outer region. While the aforementioned studies have made significant progress in quantifying the average temperature, entropy, and pressure profiles of galaxy groups and clusters, they have focused almost entirely on low-redshift ($z\lesssim0.2$) systems. This is in part due to the relative ease with which one can measure the temperature profile in nearby systems, but also due to the fact that, until recently, large, well-selected samples of galaxy clusters at high redshift did not exist. This has changed, with the recent success of large SZ surveys from the Atacama Cosmology Telescope [ACT; @act11; @act13], Planck [@planck11; @planck13b], and the South Pole Telescope [SPT; @vanderlinde10; @reichardt13]. These surveys have discovered hundreds of new galaxy clusters at $z>0.3$, allowing the study of galaxy cluster evolution for the first time out to $z>1$ using large, homogeneous data sets. In this paper, we present a joint-fit spectroscopic analysis of 80 SPT-selected galaxy clusters in the SPT-XVP sample [@mcdonald13b Benson [et al. ]{}in prep]. Utilizing uniform-depth X-ray observations of these clusters we can, for the first time, constrain the redshift evolution of the average ICM temperature, entropy, and pressure profiles. We present the details of this analysis in §2, including the resulting projected and deprojected temperature profiles in §2.2 and §2.3, respectively. In §3 we infer the average pressure (§3.1) and entropy (§3.2) profiles. In §4 we discuss the implications of the observed evolution, specifically in the inner $\sim$100kpc and outskirts ($r\gtrsim R_{500}$) of the mean pressure and entropy profiles, and assess any potential biases in our analysis. Finally, we summarize these results in §5 before suggesting future applications of these data. Throughout this work, we assume H$_0$=70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_M$ = 0.27, and $\Omega_{\Lambda}$ = 0.73. Data and Analysis ================= Sample ------ The majority of the observations for this program were obtained as part of a Chandra X-ray Visionary Project to observe the 80 most massive SPT-selected galaxy clusters at $0.4 < z < 1.2$ (PI: B. Benson). This survey is described in more detail in [@mcdonald13b] and Benson [et al. ]{}(in prep). We begin our sample definition by identifying 91 galaxy clusters that are detected in the SPT 2500 deg$^2$ survey and have been observed with Chandra. We first exclude four clusters which are detected with the SPT at $S/N < 6$ (SPT-CLJ0236-4938, SPT-CLJ0522-5818, SPT-CLJ2011-5725, SPT-CLJ2332-5053), which gives our sample a uniform SZ selection at $S/N \sim 6.5$. We exclude two additional clusters (SPT-CLJ0330-5228, SPT-CLJ0551-5709) which suffer from severe projection effects due to extended foreground sources (i.e., nearby, low-mass groups). In the remaining 85 systems, we identify a tight distribution of X-ray photon counts at $r>0.2R_{500}$ (see Figure \[fig:counts\]), and exclude four clusters with exceptionally high signal-to-noise (SPT-CLJ0658-5556 (the Bullet Cluster), SPT-CLJ2248-4431, SPT-CLJ0232-4421, SPT-CLJ0102-4915), which could dominate the stacking analysis, and one cluster with very low counts (SPT-CLJ0037-5047), which will not contribute meaningful signal. What remains is a sample of 80 clusters that occupy a tight sequence in signal-to-noise at large radii. These 80 clusters define our sample of massive (M$_{500} \gtrsim 3\times10^{14}$ E($z$)$^{-1}$ M$_{\odot}$) galaxy clusters with uniform-depth X-ray imaging, spanning the redshift range $0.3 \lesssim z \lesssim 1.2$. We note that, in the outermost annulus, the combined Galactic and extragalactic background is roughly an order of magnitude brighter than the source emission. However by combining $\sim$20 spectra ($\sqrt{20}$ improvement) and joint-fitting the background in the on- and off-source regions ($\sqrt{2}$ improvement), we improve the signal-to-noise from $\sim$3 per spectrum to $\sim$20 – enough to constrain the spectroscopic temperature to within $\sim$10%. This sample of 80 clusters was divided into six subsamples, based on individual cluster redshift and the presence (or lack) of a cool core, in order to probe the redshift evolution and cooling-dependence of the universal temperature, pressure, and entropy profiles. For simplicity, and so that subsamples are of equivalent signal-to-noise, we divide the low-redshift and high-redshift subsamples in half, with the 50% “cuspiest” clusters, where cuspiness ($\alpha$) is defined as the slope of the gas density profile at 0.04R$_{500}$ [@vikhlinin07], making up the cool core (CC) subsample and the 50% least cuspy clusters defining the non-cool core (NCC) subsample. This yields the six subsamples summarized in Table \[table:subsamples\]. The choice of $z=0.6$ as a dividing line was motivated by the desire to have an equal number of clusters in both the high-$z$ and low-$z$ bins. The mean redshift for the two redshift bins are $\left<z\right>_{low} = 0.46$ and $\left<z\right>_{high} = 0.82$, which provides a broad baseline for comparison to previous studies at $\left<z\right> \sim 0.1$ (see §1). [c c c c c c]{}\[htb\] low-$z$ & $z<0.6$ & – & 40 & 0.46 & $5.5 \pm 0.3$\ low-$z$, CC & $z<0.6$ & $\alpha > 0.39$ & 19 & 0.48 & $5.3 \pm 0.5$\ low-$z$, NCC & $z<0.6$ & $\alpha < 0.39$ & 21 & 0.45 & $5.7 \pm 0.4$\ \ high-$z$ & $z>0.6$ & – & 40 & 0.82 & $4.2 \pm 0.2$\ high-$z$, CC & $z>0.6$ & $\alpha > 0.39$ & 20 & 0.80 & $3.9 \pm 0.3$\ high-$z$, NCC & $z>0.6$ & $\alpha < 0.39$ & 20 & 0.84 & $4.4 \pm 0.3$\ \[table:subsamples\] ![image](sample.ps){width="90.00000%"} Global cluster properties (e.g., M$_{500}$, kT$_{500}$) used in this work are derived in Benson [et al. ]{}(in prep), following closely the procedures described in [@andersson11]. Briefly, we estimate R$_{500}$, the radius within which the average enclosed density is 500 times the critical density, by iteratively adjusting R$_{500}$ until the measured Y$_X$ ($\equiv M_g \times kT$) satisfies the Y$_{X,500}-$M$_{500}$ relation [@vikhlinin09a], which assumes a purely self-similar evolution, M$_{500}$ $\propto$ Y$_{X,500}$ $E(z)^{-2/5}$. Once the radius converges, kT$_{500}$ is measured in a core-excised annulus from 0.15R$_{500}$ $<$ $r$ $<$ R$_{500}$. Further details on the derivation of global X-ray properties can be found in [@andersson11]. We point out that we have also examined the effects of using an M$_{gas}$-derived temperature to normalize the temperature profiles, following [@vikhlinin06a], and confirm that the results we will present below are independent of our choice of normalization. In Figure \[fig:sample\] we show the distributions of redshift ($z$), mass (M$_{500}$), and total X-ray counts for the full sample of 80 clusters, as well as the low- and high-redshift subsamples. Here M$_{500}$ is computed assuming the Y$_X$–M relation from [@vikhlinin09a], as described in [@mcdonald13b] and Benson [et al. ]{}(in prep). This figure demonstrates that the low-$z$ and high-$z$ subsamples are comprised of similar clusters, in terms of their total mass, and have been observed to similar depths, allowing a fair comparison. Joint Spectral Fitting ---------------------- For each cluster, we extract spectra in 12 annuli spanning $0 < r < 1.5R_{500}$. At $r>0.3R_{500}$, we use logarithmically-spaced bins, while interior to $0.3R_{500}$, we choose a binning scheme which achieves fine sampling while also maintaining suitable signal-to-noise per annulus. The bin edges are tabulated in Table \[table:data\]. This two-part binning scheme yields roughly equal signal-to-noise in all bins, without requiring overly narrow/wide bins at any radius. The total number of annuli was chosen, through trial and error, so that the cluster-to-cluster scatter in $kT/kT_{500}$ within a given annulus is similar to the fit uncertainty (i.e., there is negligible improvement in scatter from widening the bins). The number of counts per bin, for each cluster, is shown in Figure \[fig:counts\]. This figure demonstrates that, even in our outermost bin ($1.2R_{500} < r < 1.5R_{500}$), we have $\sim$100 X-ray counts per cluster spectrum. ![image](univT.eps){width="99.00000%"} For each cluster, we extract spectra in each of these annuli, using <span style="font-variant:small-caps;">ciao</span> v4.6 and <span style="font-variant:small-caps;">caldb</span> v4.6.1.1, along with accompanying background (both blank-sky and adjacent to source) and response files. Blank-sky background spectra are rescaled based on the observed 9.5–12keV flux and subtracted from both on- and off-source spectra, which are then simultaneously modeled in the procedures outlined below. All spectra are binned in energy with wide bins at low ($\lesssim$1keV) and high ($\gtrsim$4keV) energy where the signal-to-noise is low. The average binning was $\Delta(\log_{10}E) = 0.1$. Given a subsample of $N$ clusters (e.g., 40 low-$z$ clusters), we randomly draw $N$ clusters, allowing repeats (i.e., bootstrap analysis). This means that, in some cluster realizations, the contributions from cluster $i$ may be double-counted, while cluster $j$ will be excluded. The spectra for these $N$ clusters are simultaneously fit over the energy range 0.5–10.0 keV with <span style="font-variant:small-caps;">xspec</span> [v12.8.0; @arnaud96], using a combination of a single-temperature plasma [<span style="font-variant:small-caps;">apec</span>; @smith01], a soft X-ray background contribution (<span style="font-variant:small-caps;">apec</span>, $kT=0.18$ keV), a hard X-ray background contribution (<span style="font-variant:small-caps;">bremss</span>, $kT=40$ keV), and a Galactic absorption model (<span style="font-variant:small-caps;">phabs</span>)[^1]. These additional soft (Galactic foreground) and hard (unresolved CXB) emission models account for any residual emission after the blank-sky backgrounds are subtracted. The various free parameters of the plasma model are constrained as described in Table \[table:xspec\]. [c c c c]{}\[htb\] N$_H$ & H column density & constrained$^1$ & constrained$^1$\ $kT$ & plasma temperature & free ($kT_0$) & $kT_0\left(\frac{kT_{500,i}}{kT_{500,0}}\right)$\ z & redshift & constrained$^2$ & constrained$^2$\ Z & metal abundance & free$^3$ & free$^3$\ N & normalization$^4$ & free$^{ ~}$ & free$^{ ~}$ \[table:xspec\] [c c c c c c c]{}\[htb\] 0.00–0.04 & 1.03$_{-0.14}^{+0.21}$ & 1.07$_{-0.19}^{+0.19}$ & 1.06$_{-0.12}^{+0.16}$ & 0.79$_{-0.07}^{+0.11}$ & 0.74$_{-0.04}^{+0.09}$ & 1.32$_{-0.35}^{+0.35}$\ 0.04–0.08 & 1.07$_{-0.07}^{+0.06}$ & 1.05$_{-0.10}^{+0.06}$ & 1.13$_{-0.11}^{+0.17}$ & 1.02$_{-0.07}^{+0.09}$ & 0.97$_{-0.05}^{+0.08}$ & 1.21$_{-0.12}^{+0.13}$\ 0.08–0.13 & 1.12$_{-0.05}^{+0.06}$ & 1.18$_{-0.08}^{+0.10}$ & 1.02$_{-0.07}^{+0.06}$ & 1.08$_{-0.08}^{+0.12}$ & 1.02$_{-0.07}^{+0.12}$ & 1.31$_{-0.13}^{+0.17}$\ 0.13–0.20 & 1.21$_{-0.05}^{+0.06}$ & 1.21$_{-0.05}^{+0.06}$ & 1.23$_{-0.10}^{+0.08}$ & 1.19$_{-0.05}^{+0.08}$ & 1.16$_{-0.14}^{+0.10}$ & 1.25$_{-0.10}^{+0.10}$\ 0.20–0.28 & 1.09$_{-0.05}^{+0.05}$ & 1.11$_{-0.05}^{+0.12}$ & 1.08$_{-0.06}^{+0.05}$ & 1.17$_{-0.06}^{+0.07}$ & 1.28$_{-0.07}^{+0.09}$ & 1.08$_{-0.07}^{+0.08}$\ 0.28–0.36 & 1.12$_{-0.06}^{+0.10}$ & 1.12$_{-0.09}^{+0.19}$ & 1.10$_{-0.05}^{+0.08}$ & 1.05$_{-0.06}^{+0.07}$ & 1.01$_{-0.08}^{+0.07}$ & 1.15$_{-0.10}^{+0.09}$\ 0.36–0.46 & 0.91$_{-0.04}^{+0.05}$ & 1.03$_{-0.10}^{+0.12}$ & 0.87$_{-0.05}^{+0.05}$ & 0.97$_{-0.08}^{+0.04}$ & 0.90$_{-0.08}^{+0.08}$ & 0.98$_{-0.08}^{+0.09}$\ 0.46–0.58 & 0.94$_{-0.04}^{+0.08}$ & 0.89$_{-0.07}^{+0.05}$ & 1.04$_{-0.11}^{+0.12}$ & 0.83$_{-0.03}^{+0.03}$ & 0.84$_{-0.05}^{+0.08}$ & 0.88$_{-0.03}^{+0.04}$\ 0.58–0.74 & 0.81$_{-0.04}^{+0.09}$ & 0.69$_{-0.07}^{+0.10}$ & 0.93$_{-0.04}^{+0.08}$ & 0.80$_{-0.08}^{+0.07}$ & 0.85$_{-0.15}^{+0.14}$ & 0.81$_{-0.10}^{+0.13}$\ 0.74–0.95 & 0.67$_{-0.06}^{+0.07}$ & 0.82$_{-0.10}^{+0.12}$ & 0.57$_{-0.06}^{+0.08}$ & 0.54$_{-0.03}^{+0.04}$ & 0.52$_{-0.08}^{+0.07}$ & 0.55$_{-0.05}^{+0.07}$\ 0.95–1.20 & 0.64$_{-0.07}^{+0.10}$ & 0.68$_{-0.08}^{+0.09}$ & 0.60$_{-0.11}^{+0.20}$ & 0.50$_{-0.10}^{+0.12}$ & 0.41$_{-0.05}^{+0.09}$ & 0.48$_{-0.10}^{+0.14}$\ 1.20–1.50 & 0.66$_{-0.24}^{+0.33}$ & 1.11$_{-0.26}^{+0.39}$ & 0.25$_{-0.09}^{+0.19}$ & 0.45$_{-0.14}^{+0.07}$ & 0.45$_{-0.16}^{+0.16}$ & 0.42$_{-0.18}^{+0.10}$ \[table:univT\] This method, which has $2N+1$ free parameters per annulus (see Table \[table:xspec\]), makes the assumption that, in the subsample of $N$ clusters, there is a universal temperature profile of the form $kT/kT_{500}$ = $f$($r/R_{500}$). We do not make any further assumptions about the form of this function. We simultaneously fit spectra for all $N$ clusters, including multiple observations where available, along with off-source background spectra (at $r\gtrsim3R_{500}$) unique to each cluster. Goodness-of-fit was determined using the $\chi^2$ parameter, with weighting based on [@churazov96], which has been shown to yield unbiased parameter estimates for spectra containing as few as $\sim$50 total counts. We repeat this process 100 times for each annulus in order to assess the cluster-to-cluster variation in $kT/kT_{500}$. We note that, while the metallicity (Z$_i$) is left as a free parameter, we do not have sufficiently deep data to put meaningful constraints on the shape of Z($r/R_{500}$). We leave this parameter free so that, in the few clusters with a strong Fe <span style="font-variant:small-caps;">xxv</span> emission line, the temperature is not skewed high in order to improve the fit. This method was found to yield the smallest residuals in temperature for simulated clusters with low counts and unknown metallicity[^2]. ![image](univT_model.eps){width="99.00000%"} In Figure \[fig:univT\] and Table \[table:univT\] we show the result of this joint-fit analysis for each of the 6 subsamples described in §2.1. This figure demonstrates our ability to constrain the projected temperature to within $\sim$10% over 100 realizations, despite the fact that the average individual cluster contributes only $\sim$100–200 X-ray counts to each of these bins (Figure \[fig:counts\]). The uncertainty range shown in Figure \[fig:univT\] represents the cluster-to-cluster scatter in the $kT/kT_{500}$ profile, which dominates over the statistical uncertainty in the joint fit. Overall the general shapes are as expected – centrally-concentrated clusters have cool cores, while their counterparts do not. The agreement with previous average projected temperature profiles [e.g., @pratt07; @baldi12] is qualitatively good. In particular, when the results from [@baldi12] are scaled to the same normalization ($r/R_{500}$, $kT/kT_{500}$), they agree at the 1$\sigma$ level with our low-$z$ subsample. In general, the ICM in the outskirts of high-$z$ clusters appears to have a steeper temperature gradient than the ICM in their low-$z$ counterparts. We note, however, that the temperature profiles shown in Figure \[fig:univT\] are projected – in order to determine the true radial distribution of ICM temperature, entropy, and pressure, we must first perform a three-dimensional deprojection. Deprojecting Mean Temperature Profiles -------------------------------------- To determine the three-dimensional average temperature profile, we project an analytic function onto two dimensions and fit this projected model to the data. Our three-dimensional temperature model is inspired by Eq. 6 from [@vikhlinin06a]: $$\frac{k\rm{T}}{k\rm{T}_{500}} = \textrm{T}_0\frac{((x/r_c)^{a_{cool}}+(\textrm{T}_{min}/\textrm{T}_0))}{(1+(x/r_c)^{a_{cool}})}\frac{(x/r_t)^{-a}}{(1+(x/r_t)^b)^{c/b}}$$ where $x=r/R_{500}$. This equation models the temperature profile in two parts: 1) the core region, which has a temperature decline parametrized by a minimum temperature ($T_{min}$), scale radius ($r_c$), and shape ($a_{cool}$) [@allen01] and 2) a broken power law with a characteristic inner slope ($a$), transition steepness ($b$), and outer slope ($c$), and a transition radius ($r_t$). Since we only sample the cool core ($r\lesssim0.1R_{500}$) with 2–3 data points (Figure \[fig:univT\]), we have removed two degrees of freedom from the more general parametrization presented in [@vikhlinin06a], fixing $a_{cool}=2$ and $a=0$, as otherwise the fit would be completely unconstrained in the inner parts. To project this function onto two dimensions, we follow the procedures described in detail by [@vikhlinin06b], which require two additional ingredients, aside from the analytic temperature profile: the three-dimensional electron density and metallicity profiles. For the latter, we assume the average metallicity profile from [@leccardi08b], but confirm that radically different metallicity profiles (i.e., flat, inverted) result in $\lesssim$5% differences in the deprojected temperature, and then only at low temperatures ($\lesssim2$ keV). For the electron density profile, we utilize the deprojected density profiles for each cluster from [@mcdonald13b]. Since each bootstrapped temperature profile is actually the weighted average of $N$ clusters, we compute the appropriate three-dimensional gas density profile as follows: $$\left<\,\frac{\rho_g(r)}{\rho_{crit}}\,\right> = \frac{\sum\limits_{i=1}^N C_i (r) \times \frac{\rho_{g,i}(r)}{\rho_{crit}}}{\sum\limits_{i=1}^N C_i (r)} ~~,$$ where $C_i(r)$ is the number of X-ray counts for cluster $i$ at radius $r$. This produces a mean gas density profile, weighted in approximately the same way as the mean temperature profile. For each bootstrap realization, we use the same $N$ clusters in the calculation of both the mean temperature and density profiles. From these profiles, we project the temperature profile along a given line of sight, following [@vikhlinin06b]. This procedure accounts for different contributions from continuum and line emission along the line integral, providing an accurate estimate of the projected single-temperature model. To correctly factor in the temperature-sensitive detector response, we convert to absolute temperature units (from $kT/kT_{500}$) using the average $kT_{500}$ for the subsample of N clusters. For each radial bin, the projected temperature was computed by numerically integrating along the line of sight over $-4R_{500} < r < 4R_{500}$ as well as along the bin in the radial direction. Once complete, this procedure yielded a projected temperature profile which was fit to the data using a least-squares $\chi^2$ minimization routine. This process was repeated for each realization of the projected temperature profile (Figure \[fig:univT\]), allowing an estimate of the uncertainty in the deprojected model. [c c c c c c]{}\[htb\] low-$z$ & 0.10 & 0.77 & 0.40 & 2.79 & 0.64\ low-$z$, CC & 0.13 & 0.69 & 0.30 & 2.19 & 0.57\ low-$z$, NCC & 0.08 & 0.81 & 0.96 & 2.74 & 2.42\ \ high-$z$ & 0.10 & 0.49 & 0.38 & 3.26 & 0.94\ high-$z$, CC & 0.11 & 0.47 & 0.41 & 3.30 & 1.08\ high-$z$, NCC & 0.05 & 0.82 & 0.46 & 3.41 & 1.02\ \[table:ktfits\] The resulting deprojected temperature profiles, along with the 1$\sigma$ uncertainty regions, are shown in Figure \[fig:univT\_model\] and Tables \[table:univT\] and \[table:data\]. These results suggest that high-redshift clusters tend to be, on average, $\sim$30–40% cooler at large radii ($r\sim 1.5\textrm{R}_{500}$) than their low-redshift counterparts. However, when non-cool cores are considered on their own, this trend is reversed (albeit at low significance). This may be a result of, on average, more significant “clumping” in the outskirts of high-$z$ clusters [@nagai11]. Infalling subhalos will tend to have lower temperature and higher density than the surrounding ICM, leading to a bias towards low temperatures in the emission measure-weighted spectra if they are unresolved. We will discuss this possibility further in §4, in the context of the average entropy profile. Similar to previous works, we find that the peak temperature for relaxed (cool core) systems is reached at $\sim$0.2–0.3$R_{500}$. The cores of high-z cool core clusters appear to be cooler than their low-z counterparts at the $\sim$30% level, an effect still pronounced in the the full ensemble. Given that our typical 1-$\sigma$ uncertainty in the central bin is $\sim$15%, this is only marginally significant. Mean Pressure and Entropy Profiles ================================== ![image](univP.eps){width="98.00000%"} -0.1 in Previous studies (see §1) have measured the “Universal” pressure and/or entropy profiles by simply taking the mean or median of a large number of individual profiles, which are individually calculated as follows: $$P(r) = n_e(r) \times kT(r), ~~~K(r) = n_e(r)^{-2/3} \times kT(r) .$$ This approach is unfeasible for this sample, given that each individual cluster only has $\sim$2000 X-ray counts and, thus, the individual $kT(r)$ profiles are unconstrained. However, if we define $P_{500}$ and $K_{500}$ as follows [from @nagai07]: $$P_{500} = n_{g,500} \times kT_{500}, ~~~K_{500} = \frac{kT_{500}}{n_{e,500}^{2/3}}$$ where $n_{g,500} = (\mu_e/\mu)n_{e,500} = 500f_b\rho_{crit}/(\mu m_p)$, and we assume that $\mu = 0.59$ is the mean molecular weight and $f_b=\Omega_b/\Omega_M \sim 0.14$ [@gonzalez13] is the universal baryon fraction, then we can express the scaled pressure and entropy profiles as follows: $$\frac{P}{P_{500}} = 0.0073\left(\frac{kT}{kT_{500}}\right)\left(\frac{\rho}{\rho_{crit}}\right) ,$$ $$\frac{K}{K_{500}} = 17.2\left(\frac{kT}{kT_{500}}\right)\left(\frac{\rho}{\rho_{crit}}\right)^{-2/3} .$$ These expressions allow us to measure $P/P_{500}$ and $K/K_{500}$ as a function of radius, without actually measuring $P(r)$ or $K(r)$ for any given cluster, as is usually done. Instead, we simply combine the average $\rho(r)/\rho_{crit}$ profiles [@mcdonald13b] with the average $kT(r)/kT_{500}$ profile from our joint-fit analysis (see §2). The average pressure and entropy profiles, computed using Eqs. 6 and 7, are provided in Table \[table:data\]. Pressure -------- In Figure \[fig:univP\] we show the average pressure profile, $P(r)/P_{500}$, from this work. We compare our joint-fit results to previous studies by [@vikhlinin06a] (V06; $\left<z\right> = 0.13$), [@arnaud10] (A10; $\left<z\right> = 0.11$), [@planck13] (P13; $\left<z\right> = 0.17$), after normalizing all profiles by $f(M) = (M_{500}/3\times10^{14}h_{70}^{-1}M_{\odot})^{0.12}$, following [@sun11] and [@planck13], in order to account for small differences in mass range between samples. We apply an additional scaling to profiles derived based on XMM-Newton data, which accounts for a 16% difference in temperature normalization between Chandra and XMM-Newton for massive clusters [@schellenberger14]. Both A10 and P13 define $P_{500} \propto M_{500}^{2/3} \propto kT_{500}^{0.37}$, so a 16% normalization error in temperature corresponds to a 10% normalization error in $P/P_{500}$. Likewise, since R$_{500} \propto M_{500}^{1/3} \propto kT_{500}^{0.19}$, we apply a 3% renormalization in R$_{500}$ to account for differences between the two X-ray telescopes. Figure \[fig:univP\] demonstrates our ability to reach larger radii than [@arnaud10], who used data from XMM-Newton, due to the fact that the Chandra ACIS-I field of view represents a larger physical size at high redshift. While we are unable to probe as deep into the outskirts as Planck, the unmatched angular resolution of Chandra allows us to sample a factor of $\sim$10 finer than the native Planck resolution in the cluster core. Thus, this work bridges the gap between A10, who primarily samples the inner pressure profile, and P13, who primarily samples the cluster outskirts. For our low-$z$ subsample, there is good overall agreement between our results and previous work. Specifically, at $0.2R_{500} < r < 1.5R_{500}$, our results are fully consistent at the $\sim1\sigma$ level with both A10 and P13. At $r<0.2R_{500}$, the cool cores from A10 appear to have higher pressure. This may be due to their different cool core classification, based on central density, which also varies as a function of total mass [@vikhlinin06a]. Alternatively, the difference could be due to a difference in centering – we use the large-area centroid, while A10 use the X-ray peak – but these should both agree for relaxed, centrally-concentrated systems. The difference in the core pressure is less dramatic when the full samples are considered. \ -0.1 in \[fig:centroid\] [c c c c c c c]{}\[htb\] low-$z$ & $\bm{ 4.33_{- 1.66}^{+ 3.90}$} & $ & $\bm{ 0.26_{- 0.26}^{+ 0.22}$} & $ & $\bm{ 3.30_{- 0.57}^{+ 0.86}$} & \bf 1.070\\ $ & $\bm{ 1.38_{- 0.11}^{+ 0.06}$} & $ & $\bm{ 0.99_{- 0.01}^{+ 0.19}$} &\bf 5.00$\^$\\ A10 & 8.40 & 1.18 & 0.31 & 1.05 & 5.49$\^$\\ P13 & 6.41 & 1.81 & 0.31$\^$ & 1.33 & 4.13\\\\ $ & $\bm{ 1.19_{- 0.21}^{+ 0.09}$} & $ & $\bm{ 0.73_{- 0.04}^{+ 0.23}$} &\bf 5.00$\^$\\ \bf low-$z$, CC & $ & $\bm{ 3.42_{- 0.74}^{+ 0.78}$} & $ & $\bm{ 2.31_{- 1.04}^{+ 3.58}$} & $ & 1.064\ A10, CC & 3.25 & 1.13 & 0.77 & 1.22 & 5.49$^{\dagger}$\ P13, CC & 11.8 & 0.60 & 0.31$^{\dagger}$ & 0.76 & 6.58\ \ low-$z$, NCC & $\bm{ 5.10_{- 2.15}^{+ 0.02}$} & $ & $\bm{ 0.00_{- 0.00}^{+ 0.17}$} & $ & $\bm{ 7.58_{- 3.16}^{+ 0.42}$} & \bf 1.076\\ $ & $\bm{ 1.59_{- 0.10}^{+ 0.06}$} & $ & $\bm{ 1.48_{- 0.04}^{+ 0.32}$} &\bf 5.00$\^$\\ A10, NCC & 3.20 & 1.08 & 0.38 & 1.41 & 5.49$\^$\\ P13, NCC & 4.72 & 2.19 & 0.31$\^$ & 1.82 & 3.62\\ \\ \bf high-$z$ & $ & $\bm{ 2.59_{- 0.38}^{+ 0.37}$} & $ & $\bm{ 2.27_{- 0.40}^{+ 0.89}$} & $ & 1.034\ high-$z$, CC & $\bm{ 3.70_{- 0.86}^{+ 2.17}$} & $ & $\bm{ 0.21_{- 0.21}^{+ 0.13}$} & $ & $\bm{ 3.34_{- 0.42}^{+ 1.01}$} & \bf 1.026\\ $ & $\bm{ 1.70_{- 0.09}^{+ 0.07}$} & $ & $\bm{ 1.33_{- 0.04}^{+ 0.20}$} &\bf 5.00$\^$\\ \bf high-$z$, NCC & $ & $\bm{ 1.50_{- 0.40}^{+ 0.74}$} & $ & $\bm{ 1.70_{- 0.17}^{+ 0.99}$} & $ & 1.043\ \[table:gnfw\]************ Non-cool core clusters (both high- and low-$z$) appear to have a deficit of pressure in their cores relative to A10 and P13, both at the 1–2$\sigma$ level. Again, this may be a result of different centering choices: A10 and, presumably, P13 choose the X-ray peak as the cluster center, while we have measured the cluster center at larger radii in order to minimize the effects of core sloshing [see e.g., @zuhone10]. Our choice of centering allows for the X-ray peak to be offset from the chosen center, which would lead to lower central pressure. An example of the difference between the X-ray peak and the “large-radius centroid” is shown in Figure \[fig:centroid\] for SPT-CLJ0014-4952. At large radii, the cluster is clearly circular on the sky. However, the X-ray peak, presumably representing a sloshing cool core, is located at a distance of $\sim0.3R_{500}$ from the large-radius centroid. It is most likely these off-center emission peaks that are leading to the small differences between our stacked pressure profile and those of A10 and P13, specifically at $r \lesssim 0.1$R$_{500}$ and $r \gtrsim R_{500}$. We would argue that, for non-cool core clusters, our choice of the cluster center is more representative of the true dark matter distribution. ![image](univK.eps){width="99.00000%"} We find that high-$z$ cool core clusters have lower central pressure than those at intermediate- and low-$z$, by factors of $\sim$3 and $\sim$6, respectively, consistent with our earlier work which showed a rapidly-evolving central gas density between $z\sim0$ and $z\sim1$, with the central, low-entropy core being less massive at $z\gtrsim0.6$ [@mcdonald13b]. This drop in central pressure with increasing redshift is a result of both lower central temperature (see Figure \[fig:univT\_model\]) and lower central density [@mcdonald13b] in relaxed, high-$z$ clusters. We note that, under the assumption of hydrostatic equilibrium, increasing the gas mass of the core ought to decrease the central temperature (assuming dark matter dominates the mass budget). The fact that we observe the opposite implies that some additional form of heating, either gravitational (e.g., increased dark matter density, adiabatic compression) or feedback-related, is raising the central temperature of low-$z$ clusters. The observed change in central pressure would seem to suggest that high-$z$ X-ray cavities, inflated by radio-mode AGN feedback, should be a factor of $3^{1/3}$ larger in radius in order to maintain the same energetics (PdV) as low-$z$ cavities. This factor of $\sim$40% in size evolution is currently smaller than the typical scatter in cavity size for high-$z$ clusters [@hlavacek12 Hlavacek-Larrondo [et al. ]{}in prep]. To allow a more direct comparison to A10 and P13, we fit the stacked pressure profiles with a generalized NFW (GNFW) profile following [@nagai07]: $$\frac{P}{P_{500}}\frac{1}{f(M)} = \frac{P_0}{(c_{500}x)^{\gamma}[1+(c_{500}x)^{\alpha}]^{(\beta-\gamma)/\alpha}}$$ where $x=r/R_{500}$ and $f(M)=(M_{500}/3\times10^{14} h_{70}^{-1} M_{\odot})^{0.12}$. This generalized version of the NFW profile [@nfw] is a broken power law with a characteristic inner slope ($\gamma$), outer slope ($\beta$), curvature ($\alpha$), and scale radius ($c_{500}$). The results of this fit, which is constrained over the radial range $0.01R_{500} \lesssim r \lesssim 1.5R_{500}$, are provided in Table \[table:gnfw\]. At low-$z$, these fits are consistent with earlier work by A10 and P13. At high-$z$, these represent the first constraints on the shape of the average pressure profile, allowing a comparison to simulations over 8 Gyr of cosmic time (see §4). Entropy ------- In Figure \[fig:univK\] we show the evolution of the stacked entropy profile. Here, we compare our data to the non-radiative simulations of [@voit05], rescaled to $\Delta=500$ by [@pratt10]. This represents the “base” entropy due to purely gravitational processes. The solid purple line corresponds to the median entropy profile for nearby REXCESS clusters [$0.05 < z < 0.18$; @pratt10], rescaled to include the cross-calibration differences between XMM-Newton and Chandra [see §3.1; @schellenberger14]. We are unable to compare to [@cavagnolo09] due to the lack of a normalized ($r/R_{500}$, $K/K_{500}$) median profile, or the appropriate quantities to do such a scaling ourselves. The agreement between our low-$z$ average entropy profile and that of [@pratt10] is excellent, suggesting that there has been little evolution in the entropy structure of clusters since $z\sim0.6$. At large radii ($r>R_{500}$), the low-$z$ average entropy profiles are consistent with the gravity-only simulations [@voit05], with the exception of the very last NCC data point, suggesting that the process responsible for injecting excess entropy at $r<R_{500}$ is unimportant at larger radii. On average, the NCC clusters have $\sim2.5\times$ higher entropy in the central bin, corresponding to a factor of $\sim$4 increase in central cooling time [@cavagnolo09]. The average CC central entropy of $\sim$0.085K$_{500}$ corresponds to $K\sim85$ keV cm$^2$ (assuming $\left<kT_{500}\right> = 6.5$ and $\left<z\right>=0.45$), or a cooling time of $\sim3\times10^9$ yrs [@cavagnolo09]. We note that our CC/NCC separation is such that the CC class will contain both “strong” and “weak” cool cores, leading to a higher average cooling time [see e.g., @hudson10]. Consistent with [@mcdonald13b], we find no significant evolution in the core entropy of CC clusters over the full redshift range studied here, despite the fact that both the central density [@mcdonald13b] and temperature (Figure \[fig:univT\_model\]) have evolved significantly ($\geq2\sigma$) over the same timescale. The universality of the average CC entropy profile at $r\lesssim0.7R_{500}$, shown more clearly in Figure \[fig:Kevol\], may be indicative of a long-standing balance between cooling and feedback processes on large scales (see also Hlavacek-Larrondo [et al. ]{}in prep). ![This plot compares the low- and high-$z$ entropy profiles for three different subsamples. Here, the y-axis is the ratio of the measured entropy for the high-$z$ subsample to the low-$z$ subsample at a given radius. Error bars represent 1$\sigma$ uncertainty. All three profiles are consistent with negligible evolution at $r<0.7R_{500}$ and significant evolution at $r>0.7R_{500}$.[]{data-label="fig:Kevol"}](Kevol.eps){width="49.00000%"} ![image](univP+sims.eps){width="99.00000%"} At $z>0.6$, we measure a distinct entropy decrement at $r>0.5R_{500}$ in high-$z$ clusters when compared to their low-$z$ counterparts. This flattening of the entropy profile is apparent in the high-$z$ CC subsample as well as the combined sample (at $>90\%$ significance in all cases), indicating that it is not being driven by one or two extreme clusters. Figure \[fig:Kevol\] shows that this flattening becomes significant at $r\gtrsim0.7R_{500}$. Such a flattening could plausibly be due to clumping of the intracluster medium [e.g., @nagai11] – we will discuss this possibility and the inferred clumping required to explain the measurements in the next section. In summary, we find no significant evolution in the average entropy profile from $z\sim0.1$ [@pratt10] to $z\sim1$ within $r\lesssim0.7R_{500}$, suggesting that the balance between cooling and feedback is exceptionally well-regulated over long periods of time ($\sim8$Gyr). Discussion ========== Below, we discuss the results of §2 and §3, comparing these to previous observations and simulations in order to aid in their interpretation. In addition, we investigate potential systematic errors and/or biases which may conspire to influence our conclusions. Comparison to Simulations ------------------------- In Figure \[fig:univP+sims\] we compare our average pressure profiles to those from simulations, as presented in [@battaglia12] and Dolag [et al. ]{}(in prep). For the latter simulations, pressure profiles are computed and presented in Liu [et al. ]{}(in prep). For each simulation, we show three redshift slices, similar to our low- and high-$z$ subsamples as well as a $z\sim0$ subsample for comparison to P13, and have made mass cuts similar to the SPT 2500 deg$^2$ survey selection function. All samples have been normalized by $\left<f(M)\right>$ (see §3.1). We do not show comparisons for CC and NCC subsamples here, since i) we do not have cuspiness measurements for the simulated clusters, and ii) it is unlikely that the simulated clusters will span the full range of properties from cool core to non-cool core clusters. In general, simulations struggle to get the complicated balance between cooling and feedback right in the cores of clusters ($r<0.1R_{500}$), but perform well outside of the core where gravity dominates. As shown in Figure \[fig:univP+sims\], our measured pressure profiles and the pressure profiles from both sets of simulations agree reasonably well at $r > 0.1R_{500}$ for all redshift slices. At $r<0.1R_{500}$, the difference between the simulations and the data becomes worse with increasing redshift. The simulated clusters appear to have massive cool cores in place already at $z\sim1$, while the observed clusters are becoming more centrally concentrated over the past $\sim$8Gyr [@mcdonald13b]. At large radii, the best-fit profile is consistent with Dolag [et al. ]{}(in prep), and slightly steeper than that predicted by [@battaglia12], but all profiles are consistent at the 1$\sigma$ level with the data. We stress that any steepening of the pressure profile may be artificial, indicative of a bias due to clumping of the ICM at higher redshift, a point we will address below. We can also compare our observations of an unevolving entropy core (Figure \[fig:univK\]) to simulations, this time by [@gaspari12] who focus on the delicate balance between AGN feedback and cooling in the cores of simulated galaxy clusters. These simulations demonstrate that, while at the very center ($<$10kpc) of the cluster the entropy can fluctuate significantly (factors of $\sim$2–3) on short (Myr) timescales, the entropy at $\gtrsim$20 kpc is relatively stable of $\sim$5Gyr timescales. These simulations, which reproduce realistic condensation rates of cool gas from the ICM, suggest that a gentle, nearly-continuous injection of mechanical energy from the central AGN is sufficient both to offset the majority of the cooling (preventing the cooling catastrophe) and to effectively “freeze” the entropy profile in place. Overall, the agreement between observations and simulations is encouraging. The primary difference between the two occurs at $r\lesssim0.1R_{500}$, with excess pressure in the simulated cores. As these are the radii where the complicated interplay between ICM cooling, bulk ICM motions, and AGN feedback is most important, it is perhaps unsurprising that the deviations between data and model are most severe in this regime. Cluster Outskirts: Halo Accretion? ---------------------------------- In recent years, a number of different studies have observed a flattening of the entropy profile for a number of different galaxy clusters at the virial radius [e.g., @bautz09; @walker13; @reiprich13; @urban14]. This flattening, while not observed in all clusters [e.g., @eckert13], has been attributed to clumping in the intracluster medium [see e.g., @simionescu11; @nagai11; @urban14]. If a substantial fraction of the ICM beyond the virial radius is in small, overdense clumps, the measured electron density ($n_e$) over a large annulus will be biased high, due to surface brightness being proportional to $n_e^2$. These clumps are thought to be the halos of infalling galaxies or small groups. Due to their low mass, they ought to be cooler than the ambient ICM, which could also lead to the measured temperature being biased low. Given that we measure, on average, lower temperatures (and entropies) at large radii ($r\gtrsim R_{500}$) in high-$z$ clusters, it is worth discussing whether this result could be driven by clumping and, specifically, how massive these clumps could be. If we assume that an extended source with $<$20 X-ray counts would go undetected against the diffuse cluster emission, we can estimate a limiting X-ray luminosity at $z=0.8$ of $L_X\sim2\times10^{43}$ erg s$^{-1}$, corresponding to a halo mass of $M_{500} \sim 8\times10^{13}$ M$_{\odot}$ and temperature of $\sim$2 keV [@vikhlinin09a]. Thus, it is quite possible that the measured temperature in the outskirts of clusters at $z>0.6$ is biased low due to our inability to detect and mask group-sized halos which are in the process of accreting onto the massive cluster. The entropy flattening that we measure in Figure \[fig:univK\] is driven primarily by the evolution in the temperature profile (Figures \[fig:univT\]–\[fig:univT\_model\]), with only a small, insignificant evolution measured in the outer part of the gas density profile [@mcdonald13b]. This makes sense, if the temperature profile is in fact biased by infalling $>10^{13}$ M$_{\odot}$ groups at $\sim$R$_{500}$. Figure \[fig:cartoon\] illustrates this scenario, showing the density and temperature profiles for a typical SPT-selected cluster (M$_{500} = 6\times10^{14}$ M$_{\odot}$, $kT_{500} = 6.5$ keV), and an infalling group-sized system (M$_{500} = 6\times10^{13}$ M$_{\odot}$, $kT_{500} = 1.5$ keV). For simplicity, we assume that the infalling group is isothermal and constant density, with $\rho_g = M_{g,500}/\frac{4}{3}\pi R_{500}^3$, where both M$_{g,500}$ and $R_{500}$ can be derived from the group mass, assuming a gas fraction of 0.12. This simple test shows that, at $r\gtrsim 1.7R_{500}$, group-sized halos will significantly bias the measured density high, while at $r\lesssim 1.7R_{500}$ they will bias the measured temperature low. At $\sim$R$_{500}$, where we measure a flattening of the entropy profile, the density of the infalling group and the ambient ICM are roughly equal, with a factor of $\sim$3 difference in temperature. This temperature contrast would result in an artificial steepening of the temperature profile, as we observe in Figures \[fig:univT\]–\[fig:univT\_model\]). Following [@vikhlinin06b], we estimate that the group-sized halos would need to contribute $\sim$30–40% of the total X-ray counts in the outer annuli to bias the temperature low by the observed 40%, with the exact fraction depending on the relative temperature of the cluster and group. ![Idealized depiction of a group-sized (M$_{500} = 6\times10^{13}$ M$_{\odot}$; blue lines) halo falling into a massive (M$_{500} = 6\times10^{14}$ M$_{\odot}$; red lines) galaxy cluster. The infalling group is assumed to be isothermal and constant density, with the density equal to $\rho_g = 0.12$M$_{500}/\frac{4}{3}\pi R_{500}^3$ and temperature taken from the M–T$_X$ relation [@vikhlinin09a]. This figure demonstrates that, as a group-sized halo falls into a massive cluster, it will first significantly bias the density high at $r\gtrsim 1.7R_{500}$ (right of dashed vertical line), and then bias the temperature low at $r\lesssim 1.7R_{500}$ (left of dashed vertical line). The latter effect may be driving the steep temperature profile (Figures \[fig:univT\]–\[fig:univT\_model\]) and entropy flattening (Figure \[fig:univK\]) that we observe in high-$z$ clusters. []{data-label="fig:cartoon"}](cartoon.eps){width="49.00000%"} Simulations suggest that at $z\gtrsim1$ there is significantly more massive substructure in the outskirts of galaxy clusters. For example, [@tillson11] find that the accretion rate onto massive halos evolves by a factor of $\sim$3.5 from $z\sim1.5$ to $z\sim0$, while [@fakhouri10] find that 10$^{14}$ M$_{\odot}$ halos are accreting 10$^{13}$ M$_{\odot}$ subhalos at a rate $\sim$3 times higher at $z\sim1$ than at $z\sim0$. These results suggest that the entropy flattening which we measure (Figure \[fig:univK\]) is consistent with a temperature bias due to our inability to detect (and mask) large substructures in the outskirts of SPT-selected clusters. We stress that this “superclumping” is qualitatively different than the “clumping” inferred in nearby clusters [e.g., @simionescu11; @nagai11; @urban14], which is commonly interpreted as large numbers of small subhalos raining onto clusters at the virial radius, where group-sized halos would be detected and masked. Cool Core Evolution ------------------- In an earlier analysis of this dataset [@mcdonald13b], we saw evidence for evolution in the central gas density of cool cores over the past 8 Gyr but no evidence that the minimum ICM entropy in the central $\sim$10 kpc had evolved since $z \sim 1$, maintaining a floor at $\sim$10 keV cm$^2$. Now, with a more rigorous joint-fit analysis to constrain the central temperature, providing a more accurate estimate of the central entropy, we revisit this result. From Figure \[fig:univK\], we see no measureable evolution in the central entropy bin ($0<r<0.04R_{500}$), from $K/K_{500} = 0.095_{-0.02}^{+0.02}$ at low-$z$ to $K/K_{500} = 0.102_{-0.01}^{+0.02}$ at low-$z$. Indeed, the average cool core entropy profile shows no evidence for evolution interior to $r<0.7R_{500}$ since $z\sim1$ (Figure \[fig:Kevol\]). In the absence of feedback or redistribution of entropy, one would expect the average entropy to drop rapidly in the cores of these clusters, on Gyr or shorter timescales. Given the 5 Gyr spanned by this sample, and the consistency with the $z\sim0$ work by [@pratt10], we can argue that some form of feedback is precisely offsetting cooling between $z\sim1$ and $z\sim0$. Specifically, as the central gas density increases, the core temperature also increases. This trend is contrary to what one would expect from simple hydrostatic equilibrium in a dark matter-dominated halo, but is consistent with the expectation for adiabatic compression of the gas. A likely culprit for this heat injection is radio-mode feedback [e.g., @churazov01; @fabian12; @mcnamara12], which has been shown to be operating steadily over similar timescales [@hlavacek12]. Indeed, [@gaspari11] demonstrate that the immediate result of a burst of AGN feedback is to increase the core temperature of the gas, while leave the large-scale ($r\gtrsim0.1R_{500}$) distribution of temperatures unchanged. We finish by stressing that this work and that of [@mcdonald13b] refer to the entropy in the inner $\sim$40 kpc as the “central entropy”. This annulus, which contains all of the lowest entropy gas falling onto the central cluster galaxy, is limited in size by our relatively shallow exposures. Indeed, [@panagoulia13] show that with improved angular resolution the entropy continues to drop toward the central AGN. Our discussion of an “entropy floor” is always referring to a fixed radius, within which the mean entropy is not evolving. Systematic Biases/Uncertainties ------------------------------- Below we briefly address three potential issues with our data analysis: whether the low signal-to-noise in cluster outskirts is driving the entropy flattening, whether joint spectral fitting yields the same results as averaging individual fits, and whether the average temperature profile is mass-dependent. ### X-ray Spectrum Signal-to-Noise While our observing program was designed to obtain 2000 X-ray counts per cluster, a variety of effects conspired to create the scatter in the observed number of counts per cluster (see Figure \[fig:sample\]). These factors include uncertainties in the $\xi$–L$_X$ relation, uncertainties on early redshift measurements, and the presence or lack of a cool core. Here, we investigate how strongly the measured average entropy profile depends on the S/N of the included observations. ![Joint-fit entropy profile for both the low- and high-$z$ subsamples (see also Figure \[fig:univK\]). The red and blue points correspond to the joint-fit profiles for low- and high-S/N subsamples, respectively, as described in §4.4.1. We find that, at large radii, the flattening of the entropy profile correlates with both increasing redshift and decreasing S/N. The most significant flattening is present in the high-$z$, low-S/N subsample, which contains 7 of the 8 clusters at $z>1$ and all four $z>1.1$ clusters. Given that the low-$z$ and high-$z$ low-S/N subsamples have similar S/N but different degrees of flattening, we propose that the observed flattening is driven primarily by increasing redshift. []{data-label="fig:univK_bycounts"}](univK_bycounts.eps){width="49.00000%"} In Figure \[fig:univK\_bycounts\] we have divided the low-$z$ and high-$z$ subsamples by the S/N in the three outermost bins ($r>0.75R_{500}$), specifically so that we can test whether the observed entropy flattening is a function of S/N. For the low-S/N subsamples, there are a total of $\sim$1000 X-ray counts in each of the three outermost bins and $\sim$2700 counts per radial bin over the full radial range, compared to $\sim$2800 (outer) and $\sim$4600 (full radial range) per bin for the high-S/N subsamples. For the low-$z$ clusters, the measured entropy profile appears to be independent of the S/N – the difference of a factor of $\sim$2 in the total number of X-ray counts used in the spectral modeling does not appear to have a significant affect on the resulting entropy profile. For the high-$z$ clusters, the low- and high-S/N profiles are identical at $r < 0.6 R_{500}$, with more flattening at larger radii in the low-S/N clusters. Since the low-S/N clusters also tend to be higher redshift (the high-$z$, low-S/N subsample contains 7 of the 8 clusters at $z>1$ and all 4 clusters at $z>1.1$), it is not clear which effect is most responsible for the flattening. In general, there is a trend of more flattening going to both higher redshift and lower S/N. We do not expect a significant bias from low cluster counts, due to our background modeling on an observation-by-observation basis (§2.2), but we can not rule out this possibility. Given that the low-$z$, low-S/N clusters have equally low S/N to the high-$z$, low-S/N clusters, we suggest that the flattening is more significantly driven by redshift evolution. ### Joint-Fitting Versus Profile Averaging To test whether our joint-fitting technique is introducing a systematic bias, we compute individual temperature profiles for our low-$z$ subsample (Figure \[fig:allfits\]). Given that each annulus has on the order of $\sim100$ X-ray counts, these individual fits are poorly constrained. However by averaging $\sim$40 profiles (unweighted), we can constrain the average temperature profile for this subsample. For comparison, we show the results of our joint-fit analysis for the same clusters. We find that the joint-fit method and the averaging method yield consistent results. Since the uncertainty on the joint-fit analysis is really the scatter in the mean for a number of realizations (black points), we have shown the standard error on the mean (standard deviation divided by $\sqrt{N}$) in the average profile (red points) in order to make a fair comparison. This simple test confirms that our method of joint-fitting multiple spectra is largely unbiased with respect to the true average profile. Naively, one might expect a joint-fit analysis to be biased towards the highest signal-to-noise spectra, since each cluster is essentially weighted by its total X-ray counts, while each cluster is weighted equally in the averaging method. However, this test shows that any bias that would be imparted by joint-fitting spectra of varying signal-to-noise is offset by randomly drawing and fitting subsamples of spectra. ### Mass Bias [@vikhlinin06a] show that, for a sample of relaxed, low-redshift clusters, low-mass systems tend to have higher central temperatures than their high-mass counterparts. We explore this idea in Figure \[fig:univT\_Mg\] by dividing our high-$z$ subsample by total gas mass, $M_{g,500}$, rather than by cuspiness (the following results hold for the low-$z$ subsample as well). This figure confirms that the temperature profiles of galaxy clusters are not self-similar at $r\lesssim0.3R_{500}$. We find that low mass systems have temperatures $\sim$20–30% higher in their cores, consistent with work by [@vikhlinin06a] which covered a larger mass range. At $r>0.3R_{500}$ there appear to be no deviations from self-similarity, suggesting that non-gravitational processes are most likely driving the differences in the core. This figure demonstrates how important a well-selected sample is for such a joint-fit analysis to be successful and yield results representative of the true population. We expect that, given the similar mass distribution of our low- and high-$z$ subsamples (see Figure \[fig:sample\]), this mass bias is not driving any of the trends discussed in §3. ![This figure demonstrates the similarity in the average temperature profile (red) and the “joint-fit” profile (black; see §2.2). Individual cluster profiles are shown as red dashes, while the average of these profiles is shown as thick red points. The uncertainty shown for the average profile is the standard error on the mean (standard deviation divided by $\sqrt{N}$) to allow a better comparison to the joint-fit uncertainties, which are measuring the scatter in the mean temperature for a large number of realizations. The joint-fit result, which is fully consistent with the average profile, is shown in black. This figure demonstrates that our joint-fit analysis is not strongly affected by combining spectra of varying signal-to-noise.[]{data-label="fig:allfits"}](univT_allfits.eps){width="49.00000%"} ![Average temperature profiles for high-$z$ clusters. We show the combined fits in grey, low-mass systems in blue, and high-mass systems in red. This figure demonstrates that the deviation from self-similarity interior to $0.3R_{500}$, consistent with earlier work by [@vikhlinin06a], is present out to $z\sim1$. Beyond $0.3R_{500}$, there is no evidence for a mass bias.[]{data-label="fig:univT_Mg"}](univT_Mg.eps){width="49.00000%"} It is also possible that our use of the $Y_{X,500}$–M$_{500}$ relation to infer R$_{500}$ could impart a bias in these results, if the assumed evolution on this relation is incorrect. To investigate this potential bias, we re-extracted spectra using R$_{500}$ estimates based on the M$_{gas}$–M$_{500}$ relation, and repeated the analysis described in §2.2. The resulting temperature profiles were consistent with what we have presented here, suggesting that our assumed evolution on the $Y_{X,500}$–M$_{500}$ relation is appropriate out to $z\sim1.2$. Summary ======= We have presented a joint-fit analysis of X-ray spectra for 80 SPT-selected galaxy clusters spanning redshifts $0.3 < z < 1.2$. These spectra, which individually only contain $\sim$2000 X-ray counts, are divided into subsamples of $\sim$20 clusters each, and the spectra in each subsample are simultaneously modeled assuming a self-similar temperature profile. This allows us to constrain the redshift evolution of the temperature, pressure, and entropy profiles for massive clusters. Our major results are summarized below: - We are able to constrain the average temperature profile out to $\sim1.5R_{500}$ for both low-$z$ ($0.2 < z<0.6$) and high-$z$ ($0.6 < z < 1.2$) clusters. The temperature profile for our low-$z$ subsample is consistent with earlier works by [@vikhlinin06a], [@pratt07], and [@baldi12]. Combined with density profiles from [@mcdonald13b], we constrain the pressure and entropy profiles over $0.01R_{500} < r < 1.5R_{500}$, providing the first constraints on the redshift evolution of the Universal pressure profile. - The cores of high-$z$ cool core galaxy clusters appear to be marginally ($\sim$2$\sigma$) cooler than those of their low-$z$ counterparts by $\sim$30%. This is precisely what is needed to maintain constant central entropy since $z\sim1$, given the observed evolution in the central electron density, as reported by [@mcdonald13b]. - The average temperature profile in the outskirts of high-$z$ cool core clusters is steeper than in the outskirts of low-$z$ cool core clusters. This results in a steepening of the outer pressure profile and a flattening of the outer entropy profile. These data are consistent with an increase in the number of group-mass ($\sim$1.5 keV) halos falling into the cluster at $\gtrsim$R$_{500}$ which our relatively shallow exposures are unable to detect. This “superclumping” should be a factor of $\sim$3 times more common at $z\sim1$ than it is today. Failure to mask these massive subhalos can bias the temperature at $\gtrsim$R$_{500}$ low by the observed amount ($\sim$40%). - The cores of low-$z$ clusters have significantly higher pressure than those of high-$z$ clusters, increasing by a factor of $\sim$10 between $z\sim1$ and $z=0$. This is driven primarily by the increase in central density with decreasing redshift [@mcdonald13b], but is also boosted by the increasing central temperature with decreasing redshift. - We find good overall agreement between our low-$z$ average pressure profile and those of [@arnaud10] and [@planck13]. - Simulated clusters from [@battaglia12] and Dolag [et al. ]{}(in prep) reproduce the evolution of the observed pressure profile at $r\gtrsim0.1R_{500}$. The growth of cool cores, resulting in a factor of $\sim$10 increase in the central pressure over the past $\sim$8Gyr is not reproduced in simulations. - We measure no significant redshift evolution in the entropy profile for cool cores at $r\lesssim0.7R_{500}$, suggesting that the average entropy profile for massive clusters is stable on long timescales and over a large range of radii. This may be a result of a long-standing balance between ICM cooling and AGN feedback.\ This work demonstrates that a joint-spectral-fit X-ray analysis of low signal-to-noise cluster observations can be used to constrain the average temperature, pressure, and entropy profile to large radii. This has proven to be a powerful method for analyzing high-redshift clusters, where obtaining $>$10,000 X-ray counts per cluster is unfeasible. These techniques will add additional power to future surveys by, for example, eRosita, or serendipitous surveys like ChaMP [@barkhouse06], XCS [@mehrtens12], and XXL [@pierre11], where the number of clusters with data is high, but the data quality per cluster is low. $ $ Acknowledgements {#acknowledgements .unnumbered} ================ We thank M. Voit and N. Battaglia for helpful discussions and, along with K. Dolag, for sharing their simulated galaxy cluster pressure profiles. M. M. acknowledges support by NASA through a Hubble Fellowship grant HST-HF51308.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. The South Pole Telescope program is supported by the National Science Foundation through grants ANT-0638937 and PLR-1248097. Partial support is also provided by the NSF Physics Frontier Center grant PHY-0114422 to the Kavli Institute of Cosmological Physics at the University of Chicago, the Kavli Foundation, and the Gordon and Betty Moore Foundation. Support for X-ray analysis was provided by NASA through Chandra Award Numbers 12800071, 12800088, and 13800883 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA. Galaxy cluster research at Harvard is supported by NSF grant AST-1009012 and at SAO in part by NSF grants AST-1009649 and MRI-0723073. 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[B[ö]{}hringer]{}, G. W. [Pratt]{}, A. [Finoguenov]{}, & P. [Schuecker]{}, 48 , A., [Kravtsov]{}, A., [Forman]{}, W., [et al.]{} 2006, , 640, 691 , A., [Burenin]{}, R. A., [Ebeling]{}, H., [et al.]{} 2009, , 692, 1033 , G. M., [Kay]{}, S. T., & [Bryan]{}, G. L. 2005, , 364, 909 , S. A., [Fabian]{}, A. C., [Sanders]{}, J. S., [Simionescu]{}, A., & [Tawara]{}, Y. 2013, , 432, 554 , D. A., [Jones]{}, C., & [Forman]{}, W. 1997, , 292, 419 , J. A., [Markevitch]{}, M., & [Johnson]{}, R. E. 2010, , 717, 908 Average Gas Density, Temperature, Entropy, and Pressure Profiles ================================================================ Below, we provide all of the three-dimensional deprojected quantities used in this work. Details on the derivation of these quantities, which are based on the two-dimensional temperature profiles (Table \[table:univT\]) and the three-dimensional gas density profiles from [@mcdonald13b], can be found in §2 and §3. All uncertainties are 1$\sigma$.\ [c \| c c c \| c c c \| c c c ]{} & & &\ r/R$_{500}$ & $k$T$/k$T$_{500}$ & P/P$_{500}$ & K/K$_{500}$ & $k$T$/k$T$_{500}$ & P/P$_{500}$ & K/K$_{500}$ & $k$T$/k$T$_{500}$ & P/P$_{500}$ & K/K$_{500}$\ \ \ 0.00-0.04 & 1.02$^{+0.12}_{-0.25}$ & 10.37$^{+3.44}_{-2.44}$ & 0.13$^{+0.03}_{-0.03}$ & 1.00$^{+0.17}_{-0.19}$ & 19.28$^{+5.12}_{-4.23}$ & 0.09$^{+0.02}_{-0.02}$ & 0.99$^{+0.23}_{-0.23}$ & 4.68$^{+1.51}_{-1.33}$ & 0.22$^{+0.07}_{-0.05}$\ 0.04-0.08 & 1.04$^{+0.09}_{-0.07}$ & 6.58$^{+0.94}_{-1.09}$ & 0.20$^{+0.02}_{-0.03}$ & 1.05$^{+0.10}_{-0.07}$ & 9.42$^{+2.00}_{-1.74}$ & 0.16$^{+0.02}_{-0.02}$ & 1.05$^{+0.11}_{-0.16}$ & 4.53$^{+0.82}_{-0.76}$ & 0.26$^{+0.03}_{-0.04}$\ 0.08-0.13 & 1.13$^{+0.06}_{-0.08}$ & 5.17$^{+0.61}_{-0.52}$ & 0.26$^{+0.02}_{-0.02}$ & 1.14$^{+0.07}_{-0.08}$ & 6.40$^{+0.72}_{-0.86}$ & 0.23$^{+0.02}_{-0.02}$ & 1.09$^{+0.11}_{-0.09}$ & 4.04$^{+0.50}_{-0.52}$ & 0.30$^{+0.03}_{-0.03}$\ 0.13-0.20 & 1.18$^{+0.07}_{-0.06}$ & 3.81$^{+0.28}_{-0.23}$ & 0.35$^{+0.03}_{-0.02}$ & 1.18$^{+0.08}_{-0.08}$ & 4.12$^{+0.43}_{-0.35}$ & 0.33$^{+0.03}_{-0.03}$ & 1.13$^{+0.08}_{-0.09}$ & 3.19$^{+0.33}_{-0.37}$ & 0.37$^{+0.03}_{-0.03}$\ 0.20-0.28 & 1.17$^{+0.07}_{-0.05}$ & 2.60$^{+0.20}_{-0.15}$ & 0.44$^{+0.03}_{-0.02}$ & 1.18$^{+0.09}_{-0.09}$ & 2.69$^{+0.27}_{-0.21}$ & 0.44$^{+0.04}_{-0.04}$ & 1.11$^{+0.10}_{-0.07}$ & 2.38$^{+0.22}_{-0.21}$ & 0.44$^{+0.03}_{-0.04}$\ 0.28-0.36 & 1.13$^{+0.06}_{-0.04}$ & 1.73$^{+0.07}_{-0.05}$ & 0.55$^{+0.03}_{-0.03}$ & 1.13$^{+0.07}_{-0.07}$ & 1.69$^{+0.15}_{-0.14}$ & 0.56$^{+0.04}_{-0.04}$ & 1.12$^{+0.04}_{-0.06}$ & 1.69$^{+0.12}_{-0.08}$ & 0.55$^{+0.02}_{-0.03}$\ 0.36-0.46 & 1.08$^{+0.04}_{-0.03}$ & 1.14$^{+0.04}_{-0.05}$ & 0.67$^{+0.03}_{-0.03}$ & 1.07$^{+0.07}_{-0.06}$ & 1.09$^{+0.09}_{-0.08}$ & 0.68$^{+0.05}_{-0.03}$ & 1.07$^{+0.05}_{-0.04}$ & 1.14$^{+0.07}_{-0.04}$ & 0.66$^{+0.03}_{-0.03}$\ 0.46-0.58 & 0.99$^{+0.03}_{-0.03}$ & 0.69$^{+0.03}_{-0.03}$ & 0.81$^{+0.03}_{-0.03}$ & 0.97$^{+0.04}_{-0.05}$ & 0.63$^{+0.04}_{-0.03}$ & 0.83$^{+0.06}_{-0.05}$ & 1.01$^{+0.06}_{-0.06}$ & 0.72$^{+0.04}_{-0.04}$ & 0.82$^{+0.05}_{-0.05}$\ 0.58-0.74 & 0.90$^{+0.03}_{-0.03}$ & 0.43$^{+0.02}_{-0.02}$ & 0.95$^{+0.05}_{-0.05}$ & 0.89$^{+0.03}_{-0.04}$ & 0.40$^{+0.02}_{-0.03}$ & 0.98$^{+0.05}_{-0.05}$ & 0.92$^{+0.07}_{-0.06}$ & 0.45$^{+0.04}_{-0.03}$ & 0.96$^{+0.08}_{-0.07}$\ 0.74-0.95 & 0.78$^{+0.04}_{-0.04}$ & 0.22$^{+0.02}_{-0.01}$ & 1.17$^{+0.07}_{-0.05}$ & 0.78$^{+0.06}_{-0.05}$ & 0.22$^{+0.02}_{-0.01}$ & 1.19$^{+0.11}_{-0.08}$ & 0.76$^{+0.05}_{-0.05}$ & 0.22$^{+0.02}_{-0.02}$ & 1.13$^{+0.09}_{-0.08}$\ 0.95-1.20 & 0.67$^{+0.05}_{-0.05}$ & 0.11$^{+0.01}_{-0.01}$ & 1.48$^{+0.11}_{-0.14}$ & 0.70$^{+0.06}_{-0.06}$ & 0.11$^{+0.01}_{-0.01}$ & 1.52$^{+0.13}_{-0.16}$ & 0.55$^{+0.06}_{-0.06}$ & 0.08$^{+0.01}_{-0.01}$ & 1.23$^{+0.12}_{-0.12}$\ 1.20-1.50 & 0.59$^{+0.07}_{-0.05}$ & 0.06$^{+0.01}_{-0.01}$ & 1.78$^{+0.22}_{-0.14}$ & 0.61$^{+0.08}_{-0.05}$ & 0.06$^{+0.01}_{-0.01}$ & 1.76$^{+0.22}_{-0.14}$ & 0.41$^{+0.08}_{-0.08}$ & 0.04$^{+0.01}_{-0.01}$ & 1.28$^{+0.32}_{-0.22}$\ \ \ 0.00-0.04 & 0.71$^{+0.08}_{-0.08}$ & 5.69$^{+0.86}_{-0.90}$ & 0.11$^{+0.02}_{-0.02}$ & 0.68$^{+0.07}_{-0.07}$ & 7.34$^{+1.49}_{-1.65}$ & 0.09$^{+0.02}_{-0.01}$ & 1.09$^{+0.49}_{-0.56}$ & 4.58$^{+2.68}_{-2.42}$ & 0.26$^{+0.12}_{-0.13}$\ 0.04-0.08 & 0.86$^{+0.09}_{-0.09}$ & 4.30$^{+0.53}_{-0.51}$ & 0.19$^{+0.02}_{-0.02}$ & 0.82$^{+0.07}_{-0.07}$ & 4.95$^{+1.16}_{-0.83}$ & 0.16$^{+0.02}_{-0.02}$ & 1.14$^{+0.36}_{-0.26}$ & 4.08$^{+1.45}_{-1.11}$ & 0.31$^{+0.09}_{-0.07}$\ 0.08-0.13 & 1.04$^{+0.09}_{-0.10}$ & 3.92$^{+0.40}_{-0.57}$ & 0.28$^{+0.03}_{-0.03}$ & 1.01$^{+0.10}_{-0.09}$ & 4.56$^{+0.63}_{-0.63}$ & 0.24$^{+0.03}_{-0.03}$ & 1.26$^{+0.16}_{-0.20}$ & 3.89$^{+0.61}_{-0.64}$ & 0.38$^{+0.05}_{-0.06}$\ 0.13-0.20 & 1.17$^{+0.14}_{-0.06}$ & 3.32$^{+0.32}_{-0.30}$ & 0.39$^{+0.04}_{-0.03}$ & 1.14$^{+0.11}_{-0.10}$ & 3.55$^{+0.53}_{-0.43}$ & 0.34$^{+0.05}_{-0.04}$ & 1.26$^{+0.12}_{-0.12}$ & 3.26$^{+0.37}_{-0.36}$ & 0.43$^{+0.04}_{-0.05}$\ 0.20-0.28 & 1.23$^{+0.11}_{-0.07}$ & 2.57$^{+0.23}_{-0.19}$ & 0.49$^{+0.05}_{-0.03}$ & 1.22$^{+0.10}_{-0.12}$ & 2.62$^{+0.34}_{-0.17}$ & 0.46$^{+0.05}_{-0.05}$ & 1.25$^{+0.07}_{-0.08}$ & 2.55$^{+0.21}_{-0.22}$ & 0.50$^{+0.04}_{-0.04}$\ 0.28-0.36 & 1.21$^{+0.04}_{-0.07}$ & 1.81$^{+0.10}_{-0.11}$ & 0.60$^{+0.02}_{-0.04}$ & 1.21$^{+0.12}_{-0.11}$ & 1.80$^{+0.15}_{-0.19}$ & 0.60$^{+0.05}_{-0.05}$ & 1.20$^{+0.04}_{-0.06}$ & 1.83$^{+0.14}_{-0.10}$ & 0.58$^{+0.03}_{-0.04}$\ 0.36-0.46 & 1.11$^{+0.04}_{-0.03}$ & 1.16$^{+0.05}_{-0.04}$ & 0.69$^{+0.03}_{-0.02}$ & 1.09$^{+0.07}_{-0.06}$ & 1.11$^{+0.08}_{-0.07}$ & 0.70$^{+0.04}_{-0.04}$ & 1.13$^{+0.05}_{-0.07}$ & 1.27$^{+0.10}_{-0.07}$ & 0.67$^{+0.04}_{-0.04}$\ 0.46-0.58 & 0.96$^{+0.04}_{-0.03}$ & 0.67$^{+0.03}_{-0.03}$ & 0.79$^{+0.03}_{-0.03}$ & 0.96$^{+0.04}_{-0.06}$ & 0.64$^{+0.04}_{-0.03}$ & 0.80$^{+0.04}_{-0.04}$ & 0.98$^{+0.08}_{-0.06}$ & 0.75$^{+0.06}_{-0.05}$ & 0.76$^{+0.06}_{-0.04}$\ 0.58-0.74 & 0.83$^{+0.03}_{-0.03}$ & 0.40$^{+0.02}_{-0.02}$ & 0.87$^{+0.04}_{-0.03}$ & 0.82$^{+0.06}_{-0.06}$ & 0.38$^{+0.03}_{-0.02}$ & 0.86$^{+0.07}_{-0.06}$ & 0.85$^{+0.04}_{-0.05}$ & 0.44$^{+0.03}_{-0.03}$ & 0.83$^{+0.08}_{-0.05}$\ 0.74-0.95 & 0.66$^{+0.04}_{-0.02}$ & 0.19$^{+0.01}_{-0.01}$ & 1.00$^{+0.05}_{-0.05}$ & 0.64$^{+0.05}_{-0.05}$ & 0.19$^{+0.01}_{-0.01}$ & 0.95$^{+0.06}_{-0.08}$ & 0.68$^{+0.04}_{-0.03}$ & 0.19$^{+0.03}_{-0.02}$ & 1.04$^{+0.08}_{-0.09}$\ 0.95-1.20 & 0.53$^{+0.04}_{-0.04}$ & 0.08$^{+0.01}_{-0.01}$ & 1.14$^{+0.12}_{-0.08}$ & 0.50$^{+0.06}_{-0.07}$ & 0.08$^{+0.01}_{-0.01}$ & 1.06$^{+0.13}_{-0.13}$ & 0.55$^{+0.05}_{-0.04}$ & 0.08$^{+0.01}_{-0.02}$ & 1.35$^{+0.21}_{-0.18}$\ 1.20-1.50 & 0.44$^{+0.06}_{-0.06}$ & 0.04$^{+0.01}_{-0.01}$ & 1.37$^{+0.22}_{-0.16}$ & 0.40$^{+0.05}_{-0.07}$ & 0.04$^{+0.01}_{-0.01}$ & 1.21$^{+0.17}_{-0.18}$ & 0.45$^{+0.07}_{-0.09}$ & 0.03$^{+0.01}_{-0.01}$ & 1.69$^{+0.52}_{-0.34}$\ \[table:data\] [^1]: http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/manual/\ XspecModels.html [^2]: https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSfakeit.html
--- abstract: \| We present a variational study of the 2D and 3D Wigner crystal phase of large polarons. The method generalizes that introduced by S. Fratini,P. Qu[é]{}merais \[Mod. Phys. Lett. B [12]{} 1003 (1998)\]. We take into account the Wigner crystal normal modes rather than a single mean frequency in the minimization procedure of the variational free energy. We calculate the renormalized modes of the crystal as well as the charge polarization correlation function and polaron radius. The solid phase boundaries are determined via a Lindemann criterion, suitably generalized to take into account the classical-to-quantum cross-over. In the weak electron-phonon coupling limit, the Wigner crystal parameters are renormalized by the electron-phonon interaction leading to a stabilization of the solid phase for low polarizability of the medium. Conversely, at intermediate and strong coupling, the behavior of the system depends strongly on the polarizability of the medium. For weakly polarizable media, a density crossover occurs inside the solid phase when the renormalized plasma frequency approaches the phonon frequency. At low density, we have a renormalized polaron Wigner crystal, while at higher densities the electron-phonon interaction is weakened irrespective of the [bare]{} electron-phonon coupling. For strongly polarizable media, the system behaves as a Lorentz lattice of dipoles. The abrupt softening of the internal polaronic frequency predicted by Fratini and Quemerais is observed near the actual melting point only at very strong coupling, leading to a possible liquid polaronic phase for a wider range of parameters. author: - 'G. Rastelli and S. Ciuchi' title: Wigner crystallization in a polarizable medium --- Introduction ============ As it was first proposed by Wigner last century [@Wigner] the long range Coulomb interaction is able to stabilize a crystal of electrons, which eventually melts upon increasing the density at a quantum critical point. Experiments done on heterostructures [@2DWC] and quantum Monte Carlo simulations confirm this scenario [@Ceperley_3D; @Ceperley_2D]. The presence of impurities is known to stabilize the crystal phase in two dimensions [@Tanatar]. Another mechanism which could help the stabilization of the crystal phase is the effect of a polar material. As a single electron moves in a polar crystal, it polarizes its environment creating a new quasi-particle: a Fröhlich large polaron, with an enlarged effective mass [@Kuper; @Devreese]. One expects then an enlargement of the Wigner crystal phase. Interesting properties of the liquid phase in polar doped semiconductors arise also due to the interaction with the polarization, as for example, the mixing between plasmons and longitudinal optical (LO) phonons [@Mooradian]. Such a mixing can be explained by assuming a long range interaction between the carriers and optical lattice vibrations of the Fröhlich type [@Varga]. The resulting coupled LO-phonon-plasmon modes (CPPM´s) are found in polar semiconductors ([n]{}-type GaP or [p]{}-type GaAs) [@Irmer]. Another interesting playground for this kind of physics is the surface-polaron, i.e. electron close to the surface of a polar crystal, which have been intensively studied especially for intermediate electron-phonon coupling $\alpha$ as in InSb where $\alpha \sim 4.5$ [@Sun_Gu] or in AgCl where $\alpha \sim 3$ [@Zhang_Xiao]. It has been also observed that the gate materials (SiO$_2$,Al$_2$ O$_3$ in organic thin films in field transistors are polar dielectrics and the interaction between the electrons and the surface phonons of the polar dielectric is relevant [@Kirova_Bussac]. The aim of this work is to study the stabilization of the Wigner Crystal phase and its properties in the the presence of a polarizable medium. We consider a general model in which the key feature is the presence of long range interaction which arise from direct Coulomb interactions between electrons and from the polarizable medium. The presence of long range interactions, high polarizability and low carrier density is also a common feature of high-temperature superconductors. Of course, in these materials, short range interactions and lattice effects play an important role. Nonetheless, polarons have been detected by optical measurements in the antiferromagnetic insulating phase of both superconducting and parent cuprates [@Kim; @Taliani; @Calvani]. Moreover, some evidence of strong electron-phonon coupling effects has been given recently in the underdoped regime [@Lanzara]. A new interesting physics is introduced when studying these materials by the fact that the carrier concentration can be varied from very low to sufficiently high density. Prediction on optical properties and more specifically the behavior of the so called Mid Infrared Band (MIR) by varying the doping has been proposed according polaronic models [@Simone_1; @Kastner; @Lorenzana] as well as its interpretation as charge ordering in stripes [@Calvani_stripes]. A similar behavior has been also found in the optical properties of potassium doped Barium Bismutate[@BaBiO_timusk]. When we consider a system composed of many interacting large polarons we are faced with the problem of screening of [both]{} electron-electron (e-e) and electron-phonon (e-ph) interactions as we increase the carrier density. A density crossover is therefore expected when the doping concentration is varied so that the plasma frequency approaches the optical longitudinal phonon frequency $\omega_{LO}$. At high density, phonons cannot follow the much faster plasma oscillations of the electron gas and therefore they do not contribute to the screening of the e-e interactions. On the other hand, the electronic density fluctuations screen the e-ph interaction leading to the undressing of the electrons from their polarization clouds. As a consequence, the polaronic mass renormalization is hugely reduced [@Mahan]. In this case, the plasma frequency of the pure electron gas $\omega^2_P= 4 \pi e^2 \rho / m$ is renormalized by the high frequency dielectric constant ${\varepsilon_{\infty}}$ $$\begin{aligned} \label{eqn:omega_ren_high} \omega^2_{P,H}= \frac{\omega^2_P}{{\varepsilon_{\infty}}} \end{aligned}$$ In the high density region, the self energy has been studied by a perturbative approach for weak e-ph coupling ($m_{pol} \simeq m$) in the metallic phase[@Mahan]. The validity of this approach is ruled by the condition of $\omega^2_{LO} \ll \omega^2_P/ {\varepsilon_{\infty}}$, since in this regime $\omega^2_P/ {\varepsilon_{\infty}}$ is representative of electron density fluctuations. In this case, the electron screening weakens the [effective]{} e-ph coupling constant and it is argued that the perturbative approach is suitable also for semiconductors which have intermediate values of the [bare]{} e-ph coupling in the low doping phase. The same results have been obtained at weak and intermediate couplings by a ground state study [@Devreese_Lemmens]. An approach which is able to span the strong e-ph coupling regime has been presented in ref. [@Iadonisi_Capone], where an untrapping transition is found by increasing the density via the plasmon screening of the e-ph interaction. There it is concluded that there is no polaron formation at high density, irrespectively of the strength of the bare e-ph coupling constant. At low density, the phonon energy scale (phonon-frequency) exceeds the electronic energy scale (plasma frequency). In this limit, the phonons can follow the oscillations of the slower electrons and they screen the e-e interaction. Thus, the frequency of the electron collective modes is renormalized by the static dielectric constant ${\varepsilon_0}$. Moreover, in the case of intermediate and strong e-ph coupling, polarons are formed [@Bozovic] so that the appropriate expression for the general renormalized plasma frequency becomes $$\label{eqn:omega_ren_low} \omega^2_{P,L} = \frac{m}{m_{pol}} \frac{\omega^2_P}{{\varepsilon_0}}$$ where $m_{pol}$ is the polaron mass. In the case of $GaAs$ the mass renormalization due to the e-ph interaction is negligible, and usually eq.(\[eqn:omega\_ren\_low\]) is used to interpret the experimental data with $m$, the band mass of the carriers, in place of $m_{pol}$ [@Irmer]. In ref.[@DeFilippis_Cataudella] an approximation is developed which allows to study a system of many interacting large polarons in the intermediate/low density regime for weak and intermediate coupling strengths. The phonon degrees of freedom are eliminated by a generalized Lee-Low-Pines transformation [@LeeLowPines] obtaining an effective pair potential between electrons which is non-retarded, with a short range [attractive]{} term and a long range Coulomb repulsive term, statically screened by ${\varepsilon_0}$. The role of the inverse polarizability parameter $\eta={\varepsilon_{\infty}}/{\varepsilon_0}$ is evident in ref. [@DeFilippis_Cataudella]. In the case of $\eta \ll 1$,which is hereafter reported as the high polarizability regime, repulsive interaction and the retarded phonon-mediated attractive interaction are comparable leading to a softening of the energy of the collective modes at a finite value of the wave vector $k$, signaling a charge density wave instability. The attractive interaction term between the electrons plays a crucial role also at very low density where the ground state can be bi-polaronic below a certain value of the polarizability parameter [@Verbist_Smondyrev], or can undergo a solid/liquid phase transition similar to the Wigner Crystallization (WC) [@Quemerais]. In ref. [@Simone_meanfield; @Simone], a Large Polaron Crystal (LPC) is studied using a path-integral scheme. In ref [@Simone], for $\eta =1/6$ (high polarizability regime), the authors conclude that in the weak and intermediate e-ph coupling regimes at $T=0$ the LPC melts toward a polaron liquid, but in the strong coupling regime a phonon instability appears near the melting. The authors argue this behavior from the softening of a long wavelength collective mode due to the e-e dipolar-interaction. A study which shows that the presence of long range order is not necessary for this kind of scenario has been presented in ref. [@Lorenzana] for a simplified model of a classical liquid of interacting dipoles, which are the polarons treated [à la ]{} Feynman. The dipolar mode (internal frequency) is renormalized by the mean field of the other dipoles and it is shown to soften as the density increases, leading to the dissociation of the dipole (polaron). The present work generalizes the approach of ref. [@Simone] using a formalism which allows to span from high ($\eta \ll 1$) to low polarizability regime ($\eta \simeq 1$). We calculate the boundaries of the solid phase in three as well as in two dimensions. We also calculate, within the solid phase, the correlation function between the electron density and the charge polarization density. Our results confirm the relevant role of the parameter $\eta$ in the strong e-ph coupling regime. According to the values of this parameter two distinct behaviors are found: i\) the high polarizability regime in which we found a scenario similar to that of ref. [@Simone] i.e. the melting of the crystal is driven by the instability of the internal polaronic mode. Interestingly our more quantitative prediction push the instability-driven melting toward very strong couplings leaving the possibility of a liquid polaronic phase for a wider range of parameters. ii\) the low polarizability regime, studied here even at strong coupling, in which we found that the undressing transition argued in the liquid phase [@Iadonisi_Capone] occurs also in the solid phase. Nonetheless e-ph interaction is able to stabilize the crystal against the liquid phase even for moderately polarizable mediums. This paper is organized as follows: in the first section we illustrate the model and the approximations used, we introduce the quantities of interest, and we also discuss the Lindemann criterion used to determine the transition temperature. In the second section, we present the results in the three dimensional case. In the third section, the results of the two dimensional case are compared to the those in 3D. The conclusions are reported in the last section. Appendices contain technical details of the calculations. I - The model and the method ============================= a) The model ------------ The model describes a system of N interacting electrons in a $D$-dimensional space, which are coupled to longitudinal (undispersed) optical phonons. The Hamiltonian of the model is a generalization of that introduced by Fr[ö]{}hlich for a single large polaron [@Evrard] to N-large polarons [@Mahan]. We consider electrons as distinguishable particles. This approximation is justified inside the solid phase, where the overlap between the wavefunctions of different localized electrons is negligible [@Carr]. Using the Path Integral technique [@Feynman_Hibbs] phonons can be easily traced out taking advantage of their gaussian nature and we end up with the following partition function [@Simone_meanfield]: $$\label{eqn:Z_eff} \mathcal{Z} = \oint \prod_{\imath} \mathcal{D} \left[\vec{r}_{\imath}(\tau) \right] e^{- \frac{1}{\hbar} \mathcal{S}_{eff}}$$ where $\oint$ means the functional integration over all cyclic space-time paths of the particles $\vec{r}_{\imath}(\tau)$ between zero and $\beta = \hbar / k_B T$. The effective electron action reads $$\label{eqn:S_eff} \mathcal{S}_{eff} = \mathcal{S}_{K} + \mathcal{S}_{e-e} + \mathcal{S}^{self}_{e-ph-e} + \mathcal{S}^{dist}_{e-ph-e} + \mathcal{S}_{J}$$ where $$\begin{aligned} \label{eqn:S_eff1} \mathcal{S}_{K} &= & \int_0^{\beta} \!\!\!\! d\tau \sum_{\imath} \frac{1}{2} m {\left\| \dot{\vec{r}}_{\imath} (\tau) \right\|}^2 \\ \mathcal{S}_{e-e} & = & \frac{e^2}{2 {\varepsilon_{\infty}}} \int_0^{\beta} \!\!\!\! d\tau \sum_{\imath \neq \jmath} \frac{1}{\left\| \vec{r}_{\imath}(\tau)-\vec{r}_{\jmath}(\tau) \right\|} \label{eqn:S_dist_e-e} \\ \mathcal{S}^{self}_{e-ph-e} & = & - \frac{\omega_{LO} (1-\eta) e^2}{4 {\varepsilon_{\infty}}} \int_0^{\beta} \!\!\!\! d\tau \!\! \int_0^{\beta} \!\!\!\! d\sigma \sum_{\imath} \frac{D_o \left( \tau-\sigma \right) } {\left\| \vec{r}_{\imath} (\tau)-\vec{r}_{\imath} (\sigma) \right\|} \nonumber \\ & & \label{eqn:S_self_e-ph-e} \\ \mathcal{S}^{dist}_{e-ph-e} & = & - \frac{\omega_{LO} (1-\eta) e^2}{4 {\varepsilon_{\infty}}} \int_0^{\beta} \!\!\!\! d\tau \!\! \int_0^{\beta} \!\!\!\! d\sigma \sum_{\imath \neq \jmath} \frac{D_o \left( \tau-\sigma \right) } {\left\| \vec{r}_{\imath} (\tau)-\vec{r}_{\jmath} (\sigma) \right\|} \nonumber \\ & & \label{eqn:S_dist_e-ph-e} \\ \mathcal{S}_{J} & = & \beta \frac{{\left(e \rho_J\right)}^2}{2 {\varepsilon_0}} V \!\! \int \!\!\! \frac{d\vec{r}}{r} \! - \! \int_0^{\beta} \!\!\!\! d\tau \sum_{\imath} \!\! \int \!\!\! d\vec{r} \frac{e^2 \rho_{J} / {\varepsilon_0}} {\left\| \vec{r}_{\imath}(\tau) - \vec{r} \right\|} \label{eqn:S_e-J} \end{aligned}$$ Here $e^2$ is the electron charge, $m$ is the electron band mass and $V$ is the volume. $(e \rho_J)$ is the static jellium charge density. The integration of phonons leads to the appearance of retarded e-e interaction terms –eqs.(\[eqn:S\_self\_e-ph-e\],\[eqn:S\_dist\_e-ph-e\])–, where the phonon propagator is $$\label{eqn:propagator_o} D_o (\tau) = \frac{\cosh \left( \omega_{LO}[\beta/2 - \tau]\right)} { \sinh \left( \beta \omega_{LO} /2 \right)}$$ Using polaronic units (p.u.) ($\hbar \omega_{LO}$ for energy, $1 / \omega_{LO}$ for imaginary time $\tau$ and $\sqrt{\hbar / m \omega_{LO}}$ for lengths) $\mathcal{S}_{e-ph-e}$ becomes proportional to the dimensionless e-ph coupling constant $\alpha$ defined as $$\label{eqn:alpha} \alpha = \frac{e^2}{\sqrt{2}} \frac{1 - \eta}{{\varepsilon_{\infty}}} \sqrt{ \frac{m}{\hbar^3 \omega_{LO} } }$$ while $\mathcal{S}_{e-e}$ will be proportional to the e-e coupling constant $$\label{eqn:alpha_e} \alpha_e = \frac{\sqrt{2} e^2}{{\varepsilon_{\infty}}} \sqrt{ \frac{m}{\hbar^3 \omega_{LO} } }$$ the ratio $ \alpha_e / \alpha = 2 / \left(1 - \eta \right)$ is thus solely determined by $\eta = {\varepsilon_{\infty}}/ {\varepsilon_0}$ : when $\eta \simeq 1$ the Coulomb repulsion overwhelms the attraction mediated by phonons, while they become comparable for $\eta \ll 1$. Therefore, in the Fröhlich model, the inverse polarizability parameter rules the relative weight between the repulsive and attractive (phonon-mediated) interactions. This attraction can lead to a bi-polaronic ground state as $\alpha > \alpha_c(\eta)$ [@Verbist_Smondyrev]. Roughly speaking this condition implies strong couplings $\alpha > \alpha_c$ [and]{} high polarizability $\eta < \eta_c$ where $\alpha_c = 9.3$ and $\eta_c = 0.131$ in 3D, $\alpha_c = 4.5$ and $\eta_c = 0.158$ in 2D case [@Verbist_Smondyrev]. We have investigated the system for two values of $\eta$, representative respectively of the high and low polarizability regimes, and several values of the e-ph coupling $\alpha$. We choose respectively $\eta= 1/6$ as in ref.[@Simone], which gives $\alpha_e / \alpha = 2.4$, and $\eta= 0.90519$ so that the coupling $\alpha_e / \alpha$ is increased by a factor of ten [@footnote_epsinf]. For these values of $\eta$, no bipolaron ground state exists. b) The harmonic variational approximation in the solid phase ------------------------------------------------------------ We generalize the harmonic variational approach originally introduced in ref.[@Simone_meanfield] to study the model eq.(\[eqn:S\_eff\]). First of all we recall here the variational theory in the path integral formalism. Let us consider a suitable trial action $\mathcal{S}_T$ which depends on some variational parameters. Substituting $\mathcal{S}_{eff}$ with $\mathcal{S}_{T}$ in eq.(\[eqn:Z\_eff\]) we obtain the partition function $\mathcal{Z}_T$ for the trial action and the free energy associated to it $\mathcal{F}_T = - k_B T \ln \mathcal{Z}_T$. Then the exact free energy can be expressed as $$\label{eqn:F} \mathcal{F} = \mathcal{F}_T - k_B T \ln {\left< e^{- \frac{1}{\hbar} \Delta \mathcal{S} }\right>}_T$$ where $\Delta \mathcal{S} = \mathcal{S}_{eff} - \mathcal{S}_{T}$ and the mean value ${\left< \dots \right>}_T$ is $$\label{eqn:mean} {\left< \dots \right>}_T = \frac{1}{\mathcal{Z}_T} \oint \prod_{\imath} \mathcal{D} \left[\vec{r}_{\imath}(\tau) \right] \left( \dots \right) e^{- \frac{1}{\hbar} \mathcal{S}_{T}}$$ The variational free energy is obtained by a cumulant expansion of the logarithm appearing in eq. (\[eqn:F\]). At first order in $\Delta \mathcal{S}$ it reads: $$\label{eqn:F_var} \mathcal{F}_{V} = \mathcal{F}_T + \frac{1}{\beta} {\left< \Delta \mathcal{S} \right>}_T$$ where $\mathcal{F}_{V} \geq \mathcal{F}$. To define a suitable trial action we proceed in two steps as in ref.[@Simone_meanfield]. First we treat the self interaction term $\mathcal{S}^{self}_{e-ph-e}$ of eq. (\[eqn:S\_self\_e-ph-e\]) [a la Feynman]{} [@Feynman; @Schultz]. Therefore we substitute $\mathcal{S}^{self}_{e-ph-e}$ with $\mathcal{S}_{Feyn}$ $$\label{eqn:S_Fey} \mathcal{S}_{Feyn} = \!\!\! \frac{(v^2 - w^2) m w}{8} \sum_{\imath} \!\! \int_0^{\beta} \!\!\!\!\! d\tau \!\! \int_0^{\beta} \!\!\!\!\! d\sigma D_{V} (\tau-\sigma) {\left\|\vec{r}_{\imath}(\tau)-\vec{r}_{\imath}(\sigma) \right\|}^{2}$$ $v$ and $w$ are the two variational parameters. The variational propagator $D_V (\tau)$ is given by eq. (\[eqn:propagator\_o\]) with $w$ replacing $\omega_{LO}$. We remind that $S_{Feyn}$ eq.(\[eqn:S\_Fey\]) can be obtained by integrating out an action where each electron interacts elastically ($K_T~=~m \left( v^2 - w^2 \right)$) with a fictitious particle of mass $M_T~=~m \left[ (v^2/w^2) - 1 \right]$. Then $v$ is the internal frequency and $1/\mu= 1/m + 1/M_T$ is the reduced mass of the two particle system As a second step, we treat the $\mathcal{S}_{e-e}$, $\mathcal{S}_{J}$ eqs.(\[eqn:S\_dist\_e-e\],\[eqn:S\_e-J\]) and the distinct part $(\mathcal{S}^{dist}_{e-ph-e})$ eq.(\[eqn:S\_dist\_e-ph-e\]) of $\mathcal{S}_{eff}$ in eq.(\[eqn:S\_eff\]) by means of a harmonic approximation. Expressing the position of the electrons around the Wigner lattice points as $\vec{r}_{\imath} = \vec{u}_{\imath} + \vec{X}_{\imath}$ where $\vec{X}_{\imath}$ are the vectors of the Bravais lattice (b.c.c. in 3D, hexagonal in 2D) and omitting the constant terms of the solid phase potential energy, we obtain the following harmonic variational action: $$\begin{aligned} \label{eq:Svar} \mathcal{S}_{T} &=& \mathcal{S}_{K} + \mathcal{S}_{Feyn} + \mathcal{S}^{H}_{J} + \mathcal{S}^{H}_{e-e} + \mathcal{S}^{H,dist}_{e-ph-e} \label{eqn:S_T_Harm} \end{aligned}$$ where $$\begin{aligned} \mathcal{S}_K & = & \int_0^{\beta} \!\!\!\! d \tau \sum_{\imath} \frac{1}{2} m {\|\dot{\vec{u}}_{\imath}(\tau)\|}^2 \label{eq:SK} \\ \mathcal{S}^{H}_{e-J} + \mathcal{S}^{H}_{e-e} &=& \int_0^{\beta} \!\!\!\! d\tau \sum_{\imath} \frac{1}{2} m \frac{\omega^2_W}{{\varepsilon_0}} {\|\vec{u}_{\imath}(\tau)\|}^2 \nonumber \\ &+& \int_0^{\beta} \!\!\!\! d\tau \frac{e^2}{2 {\varepsilon_{\infty}}} \sum_{\imath \neq \jmath} \vec{u}_{\jmath}(\tau) \overline{\mathcal{I}_{\imath \jmath}} \vec{u}_{\imath}(\tau) \label{eqn:S_H_dist_e-e}\end{aligned}$$ $$\label{eqn:S_H_dist_e-ph-e} \mathcal{S}^{H,dist}_{e-ph-e} = - \frac{\omega_{LO} e^2}{4 {\bar{\varepsilon}}} \sum_{\imath \neq \jmath} \int_0^{\beta} \!\!\!\! d\tau \!\! \int_0^{\beta} \!\!\!\! d\sigma D_o(\tau-\sigma) \vec{u}_{\jmath}(\sigma) \overline{\mathcal{I}_{\imath \jmath}} \vec{u}_{\imath}(\tau)$$ In eq.(\[eqn:S\_H\_dist\_e-e\]), the Wigner frequency is defined as usual in 3D as $\omega^2_{W,3D} = \omega^2_P / 3$ (for the 2D case see eq.(\[eqn:w\_Wig\_2D\]) in Appendix B). The force constants ${\left[ \overline{\mathcal{I}_{\imath \jmath}} \right]}_{\alpha \beta}$ are obtained through a harmonic expansion for the Coulomb potential (see appendix B). In our calculations, we neglect the anharmonic terms in $\Delta \mathcal{S}$ of eq. (\[eqn:F\_var\]), therefore we get $$\begin{aligned} \label{eqn:F_harm_var} \mathcal{F}_{V} & = & \mathcal{F}_{T} + \frac{1}{\beta} {\left< \mathcal{S}^{self}_{e-ph-e} -\mathcal{S}_{Feyn} \right>}_T \end{aligned}$$ We have minimized $\mathcal{F}_{V} / N$ varying $w,v$ at given density and temperature keeping $\alpha$ and $\eta$ fixed. Minimization is constrained by a convergence condition on the gaussian integrals appearing in $\mathcal{F}_{V}$. The constrained minimization procedure is described in appendix C. So far, the discussed scheme appears very similar to the one of ref.[@Simone_meanfield]. However, we stress that the $\mathcal{F}_{V}$, which we have minimized to obtain the variational parameter $v$ and $w$ , contains the hetero-interaction terms $\mathcal{S}^{H}_{e-e}$ and $\mathcal{S}^{H,dist}_{e-ph-e}$, which are not included in the minimization procedure of ref. [@Simone_meanfield]. Moreover, we have also used the [whole]{} trial action $\mathcal{S}_{T}$ eq.(\[eq:Svar\]) to calculate the mean electronic fluctuation which we have used in the Lindemann rule, as explained in the following section. c) Lindemann rule and phase diagrams ------------------------------------ To determine the solid-liquid transition we use the phenomenological Lindemann criterion, suitably generalized to take into account the classical-to-quantum cross-over [@4_Hansen_Mokovotich]: $$\label{eqn:del_L} \frac{ {\left< {\left\| \vec{u} \right\|}^2 \right>}_{eff} }{d^2_{n.n.}} = \gamma^2 \left( \eta_q \right)$$ in the l.h.s. of eq. (\[eqn:del\_L\]) we have the Lindemann ratio between the mean fluctuation of the electrons around its equilibrium position and the nearest neighbors distance $d_{n.n.}$. When it exceeds a critical value (r.h.s. of eq. (\[eqn:del\_L\])), the solid melts. In eq.(\[eqn:del\_L\]) ${\left< \dots \right>}_{eff}$ is the average taken over $\mathcal{S}_{eff}$ eq.(\[eqn:S\_eff\]). The average is carried out at the zeroth order in the cumulant expansion as an average over $\mathcal{S}_T$ eq.(\[eq:Svar\]). Contrary to the classical liquid-solid transition, where the Lindemann rule predicts the full melting line using a constant $\gamma = \gamma_{cl}$, in the case of a quantum crystal an interpolating formula for $\gamma$ is necessary to determine the melting line as obtained by comparing the free-energies of the two phases calculated using quantum simulations [@Ceperley_3D]. Hence the analytic expansion of the quantum corrections to the classical free energy respect to the quantum parameter $\eta_q$ and the zero-temperature melting density provides the interpolating function (r.h.s. of eq.\[eqn:del\_L\]) for $\gamma \left( \eta_q \right)$ [@4_Hansen_Mokovotich]. $\eta_q$ is defined for the pure electron gas as the ratio between zero point and thermal activation energies as: $$\begin{aligned} \label{eqn:eta_q} \eta_q &=& \frac{\hbar \omega_p}{2 k_B T}.\end{aligned}$$ We have chosen for the function $\gamma(\eta_q )$ the form of refs. [@Nagara; @Ceperley]: $$\begin{aligned} \label{eqn:gamma} \gamma(T,r_s) & = & \gamma_{q} - \frac{\gamma_{q} - \gamma_{cl}}{1 + A \eta^2_q} \end{aligned}$$ Formula (\[eqn:gamma\]) has a single interpolation parameter $A$ which we take as $A=1.62~\cdot~10^{-2}$ in 3D [@Nagara] and $A=3~\cdot~10^{-2}$ in 2D [@Ceperley]. The chosen value of $\gamma_{cl} = 0.155$ is such that the classical transition lines ($T=2 / \Gamma_c r_s$ a.u.) are recovered in both the 3D ($\Gamma_c=172$ from ref. [@Nagara]) and 2D ($\Gamma_c=135$ from ref. [@Ceperley]) cases. The value $\gamma_{q}=0.28$ is chosen to reproduce the zero temperature quantum transition in 3D ($r_s =100$ a.u. from ref. [@Nagara] ) and 2D ($r_s = 37$ a.u. from ref. [@Ceperley]). Roughly speaking, the transition curve is limited by the classical line $T = \left(2 / \Gamma_c\right) 1/r_s$ and the quantum melting $1/r_s = 1 / r^c$. The actual transition curve is a smooth interpolation between these two limiting behaviors. Of course, the precise knowledge of the interpolation formula (i.e. the knowledge of parameters appearing in it) is critical only for the determination of the transition line at high temperatures (see fig.\[fig:phasediagram\]). We notice that the particular values of the parameters entering in eq.(\[eqn:gamma\]) depend on the kind of statistics (boson, fermion) and on the system parameters only via the ratio $\eta_q$ [@footnote_gamma]. This parameter depends on the mass of the particles via $\omega_p$, which measures the zero point energy of the oscillator which eventually melts [@nota_m_pol]. Therefore, to generalize the Lindemann criterion to the interacting large polaron system we are left with the alternative of choosing between the electron and the polaron effective mass in eqs. (\[eqn:eta\_q\],\[eqn:gamma\]). The polaron exists as a well defined quasiparticle when [both]{} $k_B T \ll \hbar \omega_{LO}$ [@CiuchiPierleoni] and $\hbar \bar{\omega}_P \ll \hbar \omega_{LO}$. The second condition relies on the effectiveness of the e-ph interaction, as explained in the introduction. Therefore, if both conditions are fulfilled, we have to replace $\omega_P$ in eq. (\[eqn:eta\_q\]) by $\bar{\omega}_P$ given by eq. (\[eqn:omega\_ren\_low\]). In this case, between the classical $(\eta_q \simeq 0)$ and and quantum melting $(\eta_q \rightarrow \infty)$, we have a polaronic Wigner Crystal. This is the case of the high polarizability $(\eta~=~0.17)$. For low polarizability $(\eta~=~0.9)$, a cross-over occurs inside the solid phase when $\hbar \bar{\omega}_P \sim \hbar \omega_{LO}$ and the coupling is intermediate or strong, as it will be discussed in details later on. In this case, we still have a classical melting of polaronic quasi-particles, but the quantum melting involves the undressed electrons. In the classical regime (low density), the transition line does not depend appreciably on the quantum parameter, as $\gamma$ attains its classical limit ($\eta_q \rightarrow 0$). In the quantum regime at high density and low temperatures ($\eta_q \rightarrow \infty$) the function $\gamma$ eq. (\[eqn:gamma\]) saturates to its quantum value $\gamma_{q}$and the density $r_c$ of the quantum melting does not depend of the choice for the quantum parameter $\eta_q$. Instead, a pronounced dependency on the actual value of the quantum parameter is expected in the calculation of the melting line at high temperatures and intermediate densities. For $\eta=0.9$ we choose the high density estimate $\omega_P / \sqrt{{\varepsilon_{\infty}}}$ as the plasma frequency entering in eq. (\[eqn:eta\_q\]). This choice produces, in the intermediate temperature/density region, an upward deviation (fig. \[fig:phasediagram\] lower panel) from the classical slope. This is a drawback of our approximation, which is however correct at low temperatures for both low and high density. We finally discuss to which extent we use the Lindemann criterion in 2D, and more generally on the applicability of the harmonic theory in 2D. This is related to the well known problem of the existence of two dimensional crystalline long-range order at finite temperature [@Mermin]. In a pure electron gas, for $T=0$, this problem does not arise and the properties of the system in the harmonic approximation have been studied extensively [@Bonsall; @Maradudin]. The general statement for the classical impossibility of 2D crystalline long-range order was first pointed out by Peierls [@Peierls]. Landau [@Landau] gave a general argument according to which fluctuations destroy crystalline order possessing only a one or two dimensional periodicity. The first microscopic treatment of the problem (not valid in case of Coulomb interaction) is due to Mermin [@Mermin]: his proof is based on Bogolyubov’s inequality that leads to the conclusions that the Fourier component of the mean density is zero for every vector $k$ in the thermodynamic limit. Motivated by the interest of the 2D electron gas, Mermin’s proof was critically re-examined for the long range potential [@Baus; @Alastuey] . We discuss here the argument of Peierls for the 2D electron crystal. The mean square thermal fluctuations of a generic classical particle diverges in two dimensions for an infinite harmonic crystal. At low density, we have $\eta_q \simeq 0$ and the mean electronic fluctuation can be approximated by the classical value $$\begin{aligned} \left< u^2 \right>_{Cl,WC} &=& \frac{D k_B T}{2 m \omega^2_P} \mathcal{M}_{-2} \label{eqn:u_classic} \\ \mathcal{M}_{-2} &=& \int \!\!\! d \omega \rho \left( \omega \right) \frac{\omega^2_P}{\omega^2} \label{eqn:M_inv_2}\end{aligned}$$ where $\mathcal{M}_{-2}$ is the dimensionless second inverse moment of the density of the states (DOS) of charge fluctuation normal modes in the pure WC ($\rho (\omega)$). Since long-wavelength acoustical vibrational modes scale as $\omega = c_s k$, the DOS is given at low energies by $\rho (\omega) \sim \omega$ for $\omega \rightarrow 0$ [@Crandall] and the integral eq.(\[eqn:u\_classic\]) diverges logarithmically. However, a lower cut-off in the frequency spectrum, which exists for a large but finite system studied in laboratory [@Crandall] or in a computer simulation [@Ceperley_2D; @Gann], removes the logarithmic divergence. We have chosen a cut-off frequency which corresponds to a fixed number of particles $N \simeq 5 \cdot 10^{5}$. The dependence of the cut-off is discussed in appendix A. There and later on it is shown that our results are cut-off independent for low temperatures and density near the quantum critical point. Therefore we will discuss 2D case only in this region. d) Correlation functions and polaron radius ------------------------------------------- We now introduce the correlation functions between the electron and the polarization densities for a system with N electrons, and a measure of the polaron radius. The polarization density vector of the medium is associated to the optical phonon modes $Q_{\vec{q}}$ through the relation [@Devreese]: $$\label{eqn:P} \vec{P} \left( \vec{r} \right) = \sum_{\vec{k}} {\dot{\imath}}\frac{\omega_{LO}}{\sqrt{4 \pi {\bar{\varepsilon}}V}} \frac{\vec{k}}{\left\| k \right\|} e^{{\dot{\imath}}\vec{k} \vec{r}} Q_{\vec{k}}.$$ The induced charge density is defined by [@Devreese] $$\label{eqn:n_i} n_{i}\left( \vec{r} \right) = - \frac{1}{e} \vec{\nabla} \cdot \vec{P}\left( \vec{r} \right)$$ Correlation between a given electron and the induced charge density can be defined as: $$\label{eqn:n_imath} C_1 \left( \vec{r'} ,\vec{r} \right) = \frac{ \left< \rho_1 (\vec{r}) n_{\imath} (\vec{r'})\right>} {\left< \rho_1 (\vec{r}) \right>}$$ with $\rho_{1} (\vec{r}) = \delta \left( \vec{r} - \vec{r}_{1} \right)$. In eq.(\[eqn:n\_imath\]) we have chosen the appropriate normalization for the correlation function between [one]{} electron and the polarization. Integrating out the phonons we arrive at the following expression, in which we express all quantities in terms of averages weighted by the effective action eqs.(\[eqn:S\_eff\]) $$\label{eqn:rhodef} C_1 \left( \vec{r'} ,\vec{r} \right) = \!\! \frac{1}{{\bar{\varepsilon}}} \int^{\beta}_{0} \!\!\!\! d \tau \frac{\omega_{LO}}{2} D_o(\tau) \frac{ {\left< \rho_1 (\vec{r}) \rho (\vec{r'},\tau) \right>}_{eff} } {{\left< \rho_1 (\vec{r}) \right>}_{eff}}$$ In eq.(\[eqn:rhodef\]) $\rho (\vec{r}',\tau)$ is the path density defined by: $$\label{eqn:def-ro-self} \rho (\vec{r'},\tau) = \delta \left( \vec{r'} - \vec{r}_1 (\tau) \right) + \sum_{\imath \neq 1} \delta \left( \vec{r'} - \vec{r}_{\imath} (\tau) \right)$$ where we have explicitly separated the contribution $\rho_1(\vec{r'},\tau)$ due to the electron $1$ from the remainder. The first contribution in r.h.s. of eq. (\[eqn:def-ro-self\]) give rise to a self term in the correlation function eq. (\[eqn:rhodef\]) given by $$\begin{aligned} \label{eqn:C_self} C^{self}_1 &=& \frac{1}{{\bar{\varepsilon}}} \int^{\beta}_{0} \!\!\! d \tau \frac{\omega_{LO}}{2} D_o(\tau) \frac{ {\left< \rho_1 (\vec{r}) \rho_1 (\vec{r'},\tau) \right>}_{eff} } {{\left< \rho_1 (\vec{r}) \right>}_{eff}} \end{aligned}$$ Notice that in the limit of a single isolated polaron, this correlation function reduces to the one evaluated in ref. [@CiuchiPierleoni]. Assuming an electron at origin $(\vec{r}=0)$, we have $C^{self}_1$ depending only on $\vec{r'}$. The radial induced charge density $g(r)$ can be defined as $$\label{eqn:g_r} g (r) = r^{D-1} \int d^D \Omega \;\;\; C^{self}_1 \left( \vec{r} \right)$$ Using this function, we can define, as a measure of the polaronic radius, the square root of the second moment of $g(r)$ $$\label{eqn:R_p} R_p = {\left( \int^{\infty}_0 \!\!\! d r \; r^2 g(r) \right)}^{1/2}$$ The actual calculation for the mean values appearing in eq.(\[eqn:C\_self\]) are carried out at the zeroth order of the variational cumulant expansion. Explicit calculations are reported in Appendix E. II - Results in 3D ================== Here we compare the low $(\eta=0.9)$ and high $(\eta=1/6)$ polarizability cases in 3D. ![ Phase diagrams for a 3D LPC for $\eta=0.17$ (upper panel) and $\eta=0.9$ (lower panel). Atomic units (a.u.) are used for temperature and $r_s$ (see text). Solid phase is enclosed below transition lines. In both the upper and the lower panels continuous bold curve is the pure WC transition line and solid line gives the classical melting. In the upper panel dashed line is the renormalized classical melting.[]{data-label="fig:phasediagram"}](fig1a.eps "fig:") ![ Phase diagrams for a 3D LPC for $\eta=0.17$ (upper panel) and $\eta=0.9$ (lower panel). Atomic units (a.u.) are used for temperature and $r_s$ (see text). Solid phase is enclosed below transition lines. In both the upper and the lower panels continuous bold curve is the pure WC transition line and solid line gives the classical melting. In the upper panel dashed line is the renormalized classical melting.[]{data-label="fig:phasediagram"}](fig1b.eps "fig:") For each polarizability, the electron-phonon coupling constant $\alpha$ spans from weak to strong coupling regime: $\alpha= 1,3,5,7,9,11,13,15$. Phase diagrams obtained through the Lindemann criterion are shown in figs. \[fig:phasediagram\] where the solid-liquid transition lines of the LPC are compared to that of the pure Wigner crystal. Density is expressed in term of the adimensional parameter $r^3_s = a^3_o / [(4 \pi / 3) \rho]$, where $a_o$ is the Bohr radius with $(m=m_e,{\varepsilon_{\infty}}=1)$. A common feature of both the low and high polarizability cases is the enlargement [in density scale]{} of the solid phase as far as e-ph coupling increases. However, in both cases, the solid phase cannot be stabilized for any density by increasing the e-ph interaction, and the quantum melting point saturates at a maximum value when the e-ph coupling is very strong. To illustrate this different behavior it is worth to introduce a simplified model. a) A simplified model --------------------- In the simplified model, introduced in ref. [@Simone], the electrons interact with each other and with [all]{} the fictitious particles $(\{ \vec{R}_{\imath} \})$ with mass $M_T$ which represent the polarization of the medium. After integration of the fictitious particles, we obtain the effective electronic lagrangian $\mathcal{L}_{eq}$. The effective harmonic lagrangian $\mathcal{L}_{eq}$ generated by the simplified model corresponds exactly to the lagrangian of the action $\mathcal{S}_{T}$ eq.(\[eq:Svar\]) with the parameter $w=\omega_{LO}$. This approximation restricts the space of variational parameters, and therefore gives rise to a worse estimate for the free energy. Nonetheless, it allows to describe the physics of the system in a simplified fashion. Each WC’s branch is splitted in two branches for the LPC and the frequencies of the system are given by the two roots $\Omega^2_{\pm} (\omega_{s,\vec{k}})$ (eqs.(24,25) of the work [@Simone]), where $\omega_{s,\vec{k}}$ are the WC frequencies with wave vector $\vec{k}$ and branch index $s$. The expression for the mean fluctuation ${\left< u^2 \right>}_{eq}$ of electrons around their equilibrium value in the simplified model is easily obtained by inserting $w = \omega_{LO}$ in the variational expression ${< u^2 >}_T$ (see Appendix D, eqs.\[eqn:sigma2\],\[eqn:del2\_+\_eq\],\[eqn:del2\_-\_eq\]). The $\Omega_{\pm}$ branches give rise to a natural splitting of contributions to the fluctuation $$\label{eqn:del2_eq} \frac{{\left< u^2 \right>}_{eq}}{d^2_{n.n.}} = \frac{{\left< u^2 \right>}_{+}}{d^2_{n.n.}} + \frac{{\left< u^2 \right>}_{-}}{d^2_{n.n.}}$$ In the low density regime of the simplified model [@Simone], i.e. when phonons are much faster than density fluctuations, the spectrum can be decomposed into the renormalized WC frequencies $\tilde{\Omega}_{-} (\omega_{s,\vec{k}})$ and the polaronic optical frequencies $\tilde{\Omega}_{+} (\omega_{s,\vec{k}})$, which can be obtained by expanding the general solutions $\Omega^2_{\pm} (\omega_{s,\vec{k}})$ with respect to the parameter $\epsilon_{s,\vec{k}}$ defined as $$\label{eqn:eps_s_k} \epsilon_{s,\vec{k}} = \omega^2_{s,\vec{k}} / \left( {\varepsilon_0}v^2 \right)$$ which is small for [all]{} frequencies $\omega_{k,s}$ of WC normal modes at low density regime. The first part of the spectrum represents the [low]{} frequencies associated to the oscillation of the center of mass $(m_{pol} = m + M_T)$ of the two-particle system, i.e. the electron and its relative fictitious particle (polarization), while the second part of the spectrum describes the dipolar modes associated to the internal motion of oscillating electron-fictitious particle system (fig.1 of ref.[@Simone]). Dipolar modes are weakly dispersed around the frequency $\omega_{pol}$ (eq.(25) of ref.[@Simone]) defined as the $k=0$ mode of the polaronic branches. It represents the internal frequency of oscillation of the electron inside its polarization well. b) Classical and renormalized quantum melting --------------------------------------------- Now let us consider the classical transition. This transition is located in the low density regime of the simplified model. Using the low density expansion for the spectrum $( \tilde{\Omega}_{-} (\omega_{s,\vec{k}}),\tilde{\Omega}_{+} (\omega_{s,\vec{k}}) )$, it is possible to associate each term, ${\left< u^2 \right>}_{+}$ and ${\left< u^2 \right>}_{-}$, to a definite degree of freedom of the two-particle system, i.e. to the fluctuation of the center of mass $$\label{eqn:del2_-low_eq} {\left< u^2 \right>}_{-} \simeq \int \! d \omega \; \rho \left( \omega \right) \frac{\hbar D \coth \left[ \hbar \left( \sqrt{\frac{m}{{\varepsilon_0}m_{pol}}} \right) \omega/ 2 k_B T \right] } {2 \; m_{pol} \; \left( \sqrt{\frac{m}{{\varepsilon_0}m_{pol}}} \right) \omega }$$ and to the fluctuation associated to the internal dipolar mode with $\vec{\rho} = \vec{u} - \vec{R}_T$ and reduced mass $\mu$ $$\label{eqn:del2_+low_eq} {\left< u^2 \right>}_{+} \simeq {\left( \frac{M_T}{m + M_T}\right)}^2 \frac{\hbar D}{2 \mu \omega_{pol}} \coth \left( \frac{\hbar \omega_{pol}}{2 k_B T} \right)$$ In this case we can easily estimate the ratio between electronic fluctuations in LPC and in WC by taking into account only the renormalized WC spectrum, i.e. the fluctuation associated to the center of mass eq.(\[eqn:del2\_-low\_eq\]). Using eq.(\[eqn:u\_classic\]) we have $<u^2>_{LPC}/<u^2>_{WC}=1/\epsilon_0$ then by Lindemann criterion eq. (\[eqn:del\_L\]) and by eq. (\[eqn:u\_classic\]) at a given density, the critical temperature ratio also equals $1/\epsilon_0$ $$\label{eq:ratioclass} \frac{ T^{Cl}_{LPC} }{ T^{Cl}_{WC} } = \frac{1}{{\varepsilon_0}}.$$ Therefore, the slope of the classical transition line is lowered by the same factor, as can be seen in fig. \[fig:phasediagram\] (upper panel), where ${\varepsilon_0}$ is appreciably large. The quantum melting is ruled by the zero point fluctuations of the electronic oscillations. A zero temperature estimate for the pure WC gives $$\begin{aligned} \label{eqn:QuantMelt} \left< u^2 \right>_{WC} &=& \frac{\hbar D}{2 m \omega_P} \mathcal{M}_{-1} \\ \mathcal{M}_{-1} &=& \int \!\!\! d \omega \rho \left( \omega \right) \frac{\omega_P}{\omega}\end{aligned}$$ where $\mathcal{M}_{-1}$ is the dimensionless inverse moment of the WC DOS. If we consider only the renormalized WC spectrum eq.(\[eqn:del2\_-low\_eq\]), and we take into account eq.(\[eqn:QuantMelt\]), we get for the LPC $$\label{eq:ratioquant} \frac{ \left< u^2 \right>_{Q,LPC} } {\left< u^2 \right>_{Q,WC}} = \left ( \frac{m{\varepsilon_0}}{m_{pol}} \right )^{1/2} {\left( \frac{r_s}{r_s(WC)} \right)}^{3/2}.$$ then using Lindemann criterion we obtain at the quantum critical point ($r_s=r_c$) $$\label{eqn:rc_scale} \frac{r_c(WC)}{r_c}=\frac{m_{pol}}{m{\varepsilon_0}}.$$ eq.(\[eqn:rc\_scale\]) generalizes the result of the ref.[@Simone_meanfield] where the Lindemann rule was discussed within a mean field approach. ![Zero temperature phase diagram in the 2D (open symbols) and 3D (solid symbols) cases. In 2D $\alpha$ has been scaled according the zero density limit. Circles are the scaled quantum melting $r_c$ [vs]{} e-ph coupling constant $\alpha$. Dashed line is the renormalized quantum melting transition curve from eq.(\[eqn:rc\_scale\]). Upper panel: $\eta=0.17$. Triangles locates the softening of $\omega_{pol}$. Lower panel: $\eta=0.9$. The shaded area encloses the cross-over region inside the solid phase.[]{data-label="fig:3D2D_summary"}](fig2.eps) At high polarizability ${\left< u^2 \right>}_{-}$ eq.(\[eqn:del2\_-low\_eq\]) is the leading term in the mean electronic fluctuation ${\left< u^2 \right>}_{eq}$ eq.(\[eqn:del2\_eq\]) near the quantum melting for small and intermediate couplings $\alpha \le 7$. In this case, the quantum melting density scales as eq.(\[eqn:rc\_scale\]). Notice that at weak coupling the mass renormalization is weak, but phonon screening through ${\varepsilon_0}$ dominates, leading to quantum melting at lower densities than in a purely electronic Wigner Crystal (upper panel figs.\[fig:phasediagram\],\[fig:3D2D\_summary\]). At low polarizability eq.(\[eqn:rc\_scale\]) is valid up to $\alpha\simeq 3$ (fig.\[fig:3D2D\_summary\]). Upon increasing the coupling, $m_{pol}$ scales as $\sim \alpha^4$ in strong coupling and eq.(\[eqn:rc\_scale\]) predicts a divergence of the quantum melting density. As shown in figs. \[fig:phasediagram\],\[fig:3D2D\_summary\], the quantum melting density saturates to an $\alpha$-independent value at strong coupling, and the prediction of eq. (\[eqn:rc\_scale\]) is no longer valid. We will see in the next subsection that deviation from the prediction of eq. (\[eqn:rc\_scale\]) arise from different reasons in low and high polarizability cases. c) High polarizability: softening of internal mode -------------------------------------------------- This is the case in which the polarization gives a large contribution to the total interaction energy of the system. The system can be thought as being composed by interacting dipoles which are made by electrons surrounded by their polarization. ![ The Lindemann ratio (solid line) and the function $\gamma$ (dashed line) for $\eta=0.17$ and $\alpha=13$, $T=1. 10^{-5}$ (a.u.). Contributions $\delta^2_\pm=<u^2>_\pm/ d^2_{n.n.}$ of the simplified model eq.(\[eqn:del2\_eq\]) are also shown. The inset shows the abrupt slope increase of the term $\delta^2_+$. []{data-label="fig:fluctuations_02_a9"}](fig3.eps) In the case of strong e-ph coupling we observe a saturation of the critical quantum melting density. In fig. \[fig:fluctuations\_02\_a9\] the electronic fluctuation is reported for $\alpha=13$. Contrary to the small/intermediate coupling case, ${\left< u^2 \right>}_{+}$ eq.(\[eqn:del2\_+low\_eq\]) is now the leading term near the quantum melting. The melting density given by eq.(\[eqn:rc\_scale\]) is not a good estimate due to the contribution of polaronic optical modes which is now important at the quantum melting. The same scenario of ref.[@Simone] is recovered: the optical polaronic frequencies drive the melting at strong coupling and high polarizability. Moreover we notice that ${\left< u^2 \right>}_{+} \sim (1 / \omega_{pol})$ and, as density approaches a critical value, $\omega_{pol}$ softens inducing an abrupt increase of electron fluctuation which is dominated by the term ${\left< u^2 \right>}_{+}$ (see fig.\[fig:fluctuations\_02\_a9\]). Same behavior for $\omega_{pol}$ is reported in ref.[@Simone] and explained in term of the attractive interaction between the polarons (polarization catastrophe). We stress however that employing a more quantitative Lindemann criterion together with a self-consistent variational calculation of all Feynmans’ parameters we get quantum melting in a region in which $\omega_{pol}$ do not actually soften. As a result the softening of internal polaronic frequency approaches quantum melting only asymptotically for very large $\alpha$ (fig.\[fig:3D2D\_summary\]).. ![ Polaron radius in polaronic units (upper panel) and polaron radius scaled with $r_s$ (lower panel) [vs]{} $(1 / r_s)$ (a.u.) for different $\alpha$ and $\eta=0.17$ at low temperature ($T=5 \; 10^{-3}$ p.u.). Filled points refer to the solid phase.[]{data-label="fig:radius_02"}](fig4.eps) Saturation occurs to value of $r_c$ which seems to lie in the high density regime where our approach could be questionable. We must stress however that in the pure electron gas, the parameter $r_s$ is a measure of [both]{} coupling and density. Indeed $r_s$ can be obtained from the ratio of the Fermi energy to the mean Coulomb interaction, even if scaled with the band mass and the dielectric constant of the host medium, which are anyway fixed, i.e. no-density dependent. $\alpha$ $r_c$ $r^{}_c$ ---------- ------- ----------- 1 510 99.8 3 334 99.3 5 168 99.5 7 46 99.8 9 20 197.5 11 15 452.2 : The critical value at the melting of the density ($r_c$) and coupling ($r^_c$) parameters as function of $\alpha$ for high polarizability $\eta=0.16$.[]{data-label="tab:r_c"} If we introduce another coupling in the system, as the e-ph interaction, the two concepts are distinct. Global interaction is not only a function of the density but it is also a function of the e-ph coupling $\alpha$. Now in the high polarizability case polarons are well defined as quasi-particles and we can use $m_{pol}(\alpha)$ as effective mass while the repulsive interactions is screened by ${\varepsilon_0}$ in the low density regime. Only in this case we can introduce a measure of the [coupling]{} through the parameter $r^{}_s = (m_{pol} / {\varepsilon_0}m ) r_s$. For the low polarizability case the last assumption is not valid as explained onward. The values of $r^{}_s$ at the quantum melting ($r^{}_c$) are reported in tab.\[tab:r\_c\]. When the coupling $\alpha < 7$ the quantum melting can be extimated thru eq.(\[eqn:rc\_scale\]) which means $r^{}_c \simeq r_c(WC) = 100$ that is the [coupling]{} parameter $r^_c$ tends to the value of the Wigner crystal melting of a 3D electron-gas. On the contrary in the strong e-ph coupling the values of the effective coupling parameter $r^{}_c$ are much bigger than those of the density $r_c$ due to the huge enhancement of the polaron mass. Of course the exchange effects [at]{} the crystal melting are relevant and they can be taken into account only phenomenologically in our harmonic approximation (see footnote [@footnote_gamma] and discussion in sec. II.C). However in the solid phase we must notice that these effects are ruled in LPC by the parameter $r^_s$ rather than $r_s$ making them much more negligible that those at the same density in the pure electron gas. To realize this fact we assume that the localized electronic wave function is a gaussian of variance $\sigma$ then the overlap between two of these functions at distance $r_s$ is proportional to $\exp(-r^2_s/4\sigma^2)$. Now $\sigma$ in the harmonic approximation can be extimated as $\sigma^2=1/2 m_{pol}\omega_W$ where $\omega^2_W=\omega^2_{P,L}/3$ is the LPC Wigner frequency and $\omega^2_{P,L}$ is given by eq. (\[eqn:omega\_ren\_low\]). Then is immediate to see that $r^2_s/4\sigma^2=\sqrt{r^_s}/2$ a results that can be compared with the same for electron-gas [@Carr] in which appears $r_s$ and a different coefficient due to a more elaborate variational procedure. Taking into account data of table \[tab:r\_c\] we have that exchange effects are [a fortiori]{} negligible in a first approximation in the case of the strong e-ph coupling where the quantum melting occurs at huge coupling parameter $r^{}_c$. In fig. \[fig:radius\_02\] we show the behavior of the polaron radius as a function of density. While in the solid phase it remains almost constant, when approaching the melting density it suddenly increases. This behavior can be understood by taking into account that the polaron radius is essentially determined by the diffusion in imaginary time of the electron path defined in eq.(\[eqn:d\_tau\]) of Appendix C (see also eqs.\[eqn:R\_p\_T\],\[eqn:defll\] in Appendix E). Its maximum value occurs at $\tau = \beta/2$ which diverges at the softening of the polaronic frequency $(\omega_{pol} \sim 0)$. Polaronic clouds tends to overlap (fig.\[fig:radius\_02\] lower panel). However, the polaronic nature of each particle of the LPC is preserved up to quantum melting. d) Low polarizability: cross-over in solid phase ------------------------------------------------- In this regime $(\eta \sim 1)$, the repulsive interactions among electrons overwhelm the attractive interactions due to the polarizability of the background, as can be seen by the relative weight of e-e and e-ph interaction coupling constant eqs.(\[eqn:alpha\],\[eqn:alpha\_e\]). However, self-trapping effects are still present at least at strong coupling and at low density, where electrons are localized. Eq.(\[eqn:rc\_scale\]) quantitatively describes quantum melting in the low polarizability case only at weak coupling $(\alpha \leq 3)$. When $\alpha$ exceeds this value, a cross-over between a polaronic and a non polaronic phase is found inside the solid phase and the estimate of eq. (\[eqn:rc\_scale\]) no longer describes quantum melting. The low density regime, introduced in the previous subsection, is found only for the classical part of the crystal phase, where the polarization follows adiabatically the electron and the solid phase is a Wigner crystal made of polarons with an effective mass determined by the e-ph interaction, in the way discussed for high polarizability. ![ Frequencies of the system in the simplified model as a function of $\vec{k}$ along the direction $(100)$, for $\alpha = 5$ and $\eta=0.9$, at $1/r_s = 2. 10^{-2}$ and at $T = 1.8 \; 10^{-5}$ (a.u.). Density is close to the quantum melting. $(\Omega_{-}[\omega_{acu}],\Omega_{+}[\omega_{acu}])$ result from the splitting of $\omega_{acu}(s,k)$ the acoustical WC branch. $(\Omega_{-}[\omega_{opt}],\Omega_{+}[\omega_{opt}])$ result from the splitting of the high frequency WC optical branch (eqs.(\[eqn:Om2\_-high\],\[eqn:Om2\_+high\])). For comparison are shown the pure WC frequencies (dotted lines). []{data-label="fig:frequency_high"}](fig5.eps) As far as the density increases (inside the solid phase), we observe that the two energy scale, $\omega_{LO}$ of phonons and $\omega_{P,L}$ and eq.(\[eqn:omega\_ren\_low\]) of the renormalized WC frequencies, come close and we found a cross-over region [inside the solid phase]{}, where the electrons and the polarization modes are mixed as in the liquid phase CPPM (fig.1 of ref.[@Varga] and fig.1 of ref.[@Irmer]). Example of the general situation is given in fig.\[fig:frequency\_high\]. To estimate the density dependence of the LPC frequencies $\Omega_{\pm}(\omega_{s,k})$, let us substitute $\omega_{s,\vec{k}}$ by the plasma frequency $\omega_P$. Results are reported in fig.\[fig:ren\_plasma\_frequency\], which illustrates the density cross-over. ![Filled points are the typical frequencies of the simplified model obtained with $\omega _{s,k}=\omega_P$ for $\alpha = 5$ and $\eta=0.9$. Solid line is the low density renormalized plasma frequency eq.(\[eqn:omega\_ren\_low\]). Dashed line the high density renormalized plasma frequency eq.(\[eqn:omega\_ren\_high\]). Arrows mark the crossover region (see text). []{data-label="fig:ren_plasma_frequency"}](fig6.eps) In the low density limit $(r_s \rightarrow \infty)$ $\Omega_{-} = \omega_{P,L}$ eq.(\[eqn:omega\_ren\_low\]) while $\Omega_+$ converges to $\omega_{pol} \simeq v$, the internal frequency for an single polaron. In this case, the electrons are far apart, and the external harmonic field generated by the surrounding electrons of the crystalline array is weak $(K_e \sim e^2/r^3_s)$. Therefore the frequency of electron oscillation $(\omega^2_P \sim K_e /m)$ can be lower than that of the phonon $(\omega_{LO})$, and the polarization follows the electron oscillation. The polaron vibrates as a whole with a lower frequency $K_e / m_{pol} \sim {\omega}^2_P \left( m / m_{pol} {\varepsilon_0}\right)$. The polarization charge distribution is undisturbed as a first approximation, so that the value of the internal polaronic frequency ($\sim v$) of an electron inside its polarization well doesn’t change . By increasing the density, we approach the opposite limit of strong external field. Now the frequency associated to this field is too large and the polarization cannot follow the electron oscillation, so that each electron becomes undressed from its polarization cloud. In this case $\Omega_+$ approaches $\omega_P / \sqrt{{\varepsilon_{\infty}}}$, the high density renormalized plasma frequency eq.(\[eqn:omega\_ren\_high\]), while $\Omega_{-} \simeq \sqrt{{\varepsilon_{\infty}}/{\varepsilon_0}} \omega_{LO} = \omega_{TO}$ is the characteristic renormalized frequency of the polarization. We notice that at low density $\Omega_-$ gives a measure of the frequency of carrier density fluctuations, while in the opposite limit of high density, the same role is played by $\Omega_+$. As we can see from fig.\[fig:ren\_plasma\_frequency\], the cross-over amplitude is determined by the conditions $\omega_{P,H} \simeq \omega_{TO}$ and $\omega_{P,L} \simeq v$. It is interesting to compare our fig.\[fig:ren\_plasma\_frequency\] with the figure 1 of ref. [@Varga]. We notice that the asymptotic boundary given there by the phonon frequency $\omega_{LO}$ here it is played by the internal frequency. The cross-over of the renormalization of the plasma frequency from low to high density regime doesn’t imply the melting of the crystal. Indeed, it is observed within the boundary of the solid phase estimated by Lindemann criterion. This behavior is even more clear once we consider the fluctuations of the position of the electrons which enter in the Lindemann criterion. The leading term for the Lindemann ratio at the classical melting is $\delta^2_{-}={\left< {\left\| u \right\|}^2 \right> }_-/ d^2_{n.n.}$ which is associated to the fluctuation of the center of mass eq.(\[eqn:del2\_-low\_eq\]). Of course, in the classical region quantum fluctuations are ineffective, the electrons and its polarization cloud behave as a single classical particle with mass $m_{pol}$. The term $\delta^2_{+}={\left< {\left\| u \right\|}^2 \right> }_+/ d^2_{n.n.}$ associated to the internal polaronic frequencies eq.(\[eqn:del2\_+low\_eq\]) is indeed negligible. To analyze the high density region where we meet eventually the Lindemann criterion for quantum melting, we notice that the condition $\epsilon_{s,\vec{k}} \gg 1 $, where $\epsilon_{s,\vec{k}}$ is defined in eq. (\[eqn:eps\_s\_k\]), can be fulfilled by the majority of normal modes at high density. Of course, long wavelength acoustical and even optical modes in 2D have vanishing energies, but their spectral weight is low enough to be neglected in the following considerations. Expanding $\Omega_{\pm}(\omega_{s,k})$ in $1/\epsilon_{s,\vec{k}}$ we get $$\begin{aligned} \Omega_{-} & \simeq & \sqrt{\frac{{\varepsilon_{\infty}}}{{\varepsilon_0}}} \omega_{LO} \label{eqn:Om2_-high} \\ \Omega_{+} & \simeq & \frac{\omega_{s,\vec{k}}} { \sqrt{{\varepsilon_{\infty}}}} \label{eqn:Om2_+high}\end{aligned}$$ In fig.\[fig:frequency\_high\] the general solutions $\Omega_{\pm}(\omega_{s,k})$ are shown for all the branches of the simplified model near the quantum melting. The branches $(\Omega_{-}[\omega_{opt}],\Omega_{+}[\omega_{opt}])$ which results as splitting of optical model of the Wigner crystal $\omega_{opt}(s,k)$ are well described by approximation of eqs.(\[eqn:Om2\_-high\],\[eqn:Om2\_+high\]). The frequency dispersions $(\Omega_{-}[\omega_{acu}],\Omega_{+}[\omega_{acu}])$ of the modes which originate from the splitting of acoustical branches of the Wigner crystal is also reported. While at short wavelength, the dispersion approaches the estimates given in eqs.(\[eqn:Om2\_-high\],\[eqn:Om2\_+high\]) the long wavelength part of the spectrum is conversely described by the low density expansion $\tilde{\Omega}_{\pm}$. Thus we have that at the quantum melting the low energy part of the spectrum still behaves as in the low density regime. The modes depicted in the lower part of fig.\[fig:frequency\_high\] belongs to this part of the spectrum. A measure of the wave-vector below which we have this behavior can be obtained by the condition $\epsilon_{s,\vec{k}}=1$. The associated energy scale is given by $\omega^2_{c} = m \omega^2_{LO} / \left( {\varepsilon_0}m_{pol} \right)$. Contrary to low density regime eqs.(\[eqn:del2\_-low\_eq\],\[eqn:del2\_+low\_eq\]), it is not possible to associate to each term of the fluctuation eqs.(\[eqn:del2\_+\_eq\],\[eqn:del2\_-\_eq\]) a definite degree of freedom. However, expanding the electron fluctuation with respect to the parameter $\epsilon_{s,\vec{k}}$ for the frequencies $\omega_{s,k} < \omega_c$ and with respect to the parameter $1 / \epsilon_{s,\vec{k}}$ for the frequencies $\omega_{s,k} > \omega_c$ and using eq. (\[eqn:del2\_eq\]) the electron position fluctuations can be approximated by: $$\begin{aligned} {\left< u^2 \right>}_{-} &\simeq& \int^{\omega_c}_{0} \!\!\!\!\! d \omega \rho \left( \omega \right) \frac{\hbar D \coth \left[ \hbar \left( \sqrt{\frac{m}{m_{pol} {\varepsilon_0}}} \right) \omega / 2 k_B T \right] } {2 m_{pol} \left( \sqrt{\frac{m}{m_{pol} {\varepsilon_0}}} \right) \omega} \label{eqn:del2_-high_eq}\\ {\left< u^2 \right>}_{+} &\simeq& {\left( \frac{M_T}{m + M_T}\right)}^2 \frac{\hbar D}{2 \mu \omega_{pol}} \coth \left( \frac{\hbar \omega_{pol}}{2 k_B T}\right) \int^{\omega_c}_{0} \!\!\!\!\! d \omega \rho \left( \omega \right) \nonumber \\ &+& \;\; \int_{\omega_c}^{\infty} \!\!\!\!\! d \omega \rho \left( \omega \right) \frac{\hbar D}{2 m \frac{\omega}{\sqrt{{\varepsilon_{\infty}}}} } \coth \left( \hbar \frac{\omega}{\sqrt{{\varepsilon_{\infty}}}} / 2 k_B T \right) \label{eqn:del2_+high_eq}\end{aligned}$$ Notice that the interpretation of the fluctuations associated to electronic motion in this case is different from that valid at low density. In particular the high energy contribution (the second term of eq. (\[eqn:del2\_+high\_eq\]) represents a Wigner crystal-like fluctuation with a low energy cutoff. This is the largest contribution to the fluctuation at quantum melting and does not depend on e-ph interaction. Indeed the leading term of fluctuations at quantum melting is ${\left< u^2 \right>}_{+}$. This is due to the vanishing of the spectral weight associated to the low frequencies $\omega < \omega_c $ at high density (eq.(\[eqn:del2\_-high\_eq\])). The saturation of the quantum melting point can be seen in the phase diagram of fig. \[fig:phasediagram\] (lower panel). Two comments are needed. First, in the case of very low e-ph coupling, the density crossover does not occur inside the solid phase. Therefore, these arguments do not apply. The quantum melting point depends on the e-ph coupling as we have discussed in the previous section. However a saturation of the quantum melting density is observed clearly in fig.\[fig:phasediagram\] for intermediate and strong coupling. As a second point we have to emphasize that the quantum melting density [is not]{} that of a purely electronic Wigner crystal. This fact can be explained by writing the total electron fluctuation as the sum of the two terms $<u^2>_{Hi}$ and $<u^2>_{Low}$ where $<u^2>_{Hi}$ is the contribution to fluctuations of modes having energies higher (lower) than $\omega_c$. We notice from eq (\[eqn:del2\_+high\_eq\]) that in both LPC and WC case $<u^2>_{Hi}$ are the same. But while in the WC case the two terms are of the same order $<u^2>_{Low} \simeq <u^2>_{Hi}$, in the LPC case $<u^2>_{Low} \ll <u^2>_{Hi}$ as far as the density increases. This is due to the renormalization of the low energy frequencies. Therefore, the electronic fluctuation in the LPC increases more slowly with density than those of the WC. At given density, the electronic fluctuation of the WC is greater than those of the LPC and this fact explains the shifting of the quantum melting toward higher densities. The cross-over is also evident in the polaron radius. In the upper panel of fig. \[fig:radius\_09\] we plot the polaron radius as defined by the eq.(\[eqn:R\_p\]). We see that for any value of the e-ph coupling, the polaron radius tends to decrease as far as the density is increased. We recall that as far as the renormalized plasma frequency eq.(\[eqn:omega\_ren\_low\]) exceeds the phonon frequency, we enter in a region in which the polarization is adiabatically slow compared to the electronic motion. Therefore, the electronic charge appears as a static distribution whose radius decreases upon increasing the density and the polaron radius follows this trend. The crossover is evident by scaling the polaron radius with $r_s$, as reported in the lower panel of fig. \[fig:radius\_09\] at intermediate and strong $\alpha$. Notice that as in the high polarizability case at the transition $R_p/r_s \simeq 0.475$. ![ Polaron radius in polaronic units (upper panel) and polaron radius scaled with $r_s$ (lower panel) [vs]{} $(1 / r_s)$ (a.u.) for different $\alpha$ in the case $\eta=0.9$ at low temperature ($T =5 \; 10^{-2}$ p.u.). Filled points refers to the solid phase. []{data-label="fig:radius_09"}](fig7.eps) It is possible to estimate the high density limit of the radial distribution of the induced charge. Using the condition $\omega_o \ll \omega/\sqrt{{\varepsilon_{\infty}}}$ we have the following expression valid at low temperature ($k_B T \ll \hbar \omega_o$) (for details see Appendix E) $$\label{eqn:gr_approx} \tilde{g}(r) \simeq \frac{1}{{\bar{\varepsilon}}} \left[ \frac{r}{\frac{\hbar}{m \omega_{LO}}} \left( 1 - \mbox{erf} \sqrt{\frac{r^2}{ \left< u^2 \right> } } \right) + \frac{2 r^2}{\left< u^2 \right>} \frac{e^{- \frac{r^2}{2 \left< u^2 \right>} }} {{ \left(2 \pi \left< u^2 \right> \right)}^{3/2}} \right].$$ The first term of eq. (\[eqn:gr\_approx\]) takes into account quantum charge fluctuations which are relevant at small distances, while the remaining term is a classical contribution coming from the static charge distribution. Notice that only the first term depends on the e-ph interaction. Therefore, the polaron radius tends to the same high density asymptotic value for different values of the e-ph coupling $\alpha$ (see upper panel of fig.\[fig:radius\_09\]). As a last point we notice that the cross-over condition, roughly estimated as $\omega_P \sim \omega_{LO} \sim \ (1-\eta)^2/\alpha^2 \sim 0.01/\alpha^2$, shifts toward higher densities as the e-ph coupling constant $\alpha$ is reduced. In the weak coupling regime it lies in the liquid phase where RPA approaches in both 3D refs.[@Mahan; @daCosta] and in 2D case [@Xiaoguang] can be applied. It is also worth to remark that for high polarizability and for all coupling the polaronic crossover is located in the liquid phase according to the highest value of $\omega_{LO} \sim 0.7/\alpha^2$ . III - 2D case ============= The results obtained in the 2D case are qualitatively similar to the 3D case. Both the cross-over phenomenon in the low polarizability case and the softening of the polaronic frequency in the high polarizability case are observed. Results are reported in the zero temperature phase diagram of fig. \[fig:3D2D\_summary\]. In this figure, we compare the phase diagrams in 2D and 3D by scaling appropriately the 2D e-ph coupling constant following the single polaron results of ref.[@scaling_Devreese]: $\alpha_{3D} = (3 \pi / 4) \alpha_{2D}$. 2D and 3D melting curves scale well according to the zero density scaling for all studied cases. A discrepancy is found in the the high polarizability strong e-ph coupling softening of $\omega_{pol}$. Let us first discuss the scaling at finite density. In our variational scheme, the DOS of the WC is the peculiar difference between the 2D and 3D cases. To see this explicitly let us compare the e-ph interaction terms $\mathcal{S}^{self}_{e-ph-e}$. Assuming polaronic units we get: $$\begin{aligned} \frac{1}{\beta} \frac{ {\left< \mathcal{S}^{self}_{e-ph-e} \right>}_{T,3D} } {3 N} & = & - (\alpha) \frac{\sqrt{2}}{6} \int_0^{\frac{\beta}{2}} \!\!\! d\tau \frac{D_o(\tau)}{ \sqrt{ \frac{\pi}{2} d_{3D}(\tau) } } \label{eqn:scaling_I} \\ \frac{1}{\beta} \frac{ {\left< \mathcal{S}^{self}_{e-ph-e} \right>}_{T,2D} } {2 N} & = & - \left( \frac{3 \pi}{4} \alpha \right) \frac{\sqrt{2}}{6} \int_0^{\frac{\beta}{2}} \!\!\! d\tau \frac{D_o(\tau)}{ \sqrt{ \frac{\pi}{2} d_{2D}(\tau) } } \nonumber \\ & & \label{eqn:scaling_II}\end{aligned}$$ where the imaginary-time diffusion $d(\tau)$ eq.(\[eqn:d\_tau\]) is itself a functional of the DOS. We notice from eqs.(\[eqn:scaling\_I\],\[eqn:scaling\_II\]) that the free energy functional scales explicitly as in the single polaron case [@scaling_Devreese] by scaling the coupling constant $\alpha$. Related to the different 2D and 3D DOS we remark the different behavior of the frequencies of the normal modes. Noticeably, the “optical” branches go to zero as $\sim \sqrt{k}$ at long wavelengths [@Crandall]. As in the 3D case, the frequencies of the LPC are splitted in four branches (fig.\[fig:2D\_frequenze\_09\]) $\Omega_{\pm} [\omega_{acu}(s,\vec{k})]$ and $\Omega_{\pm} [\omega_{opt}(s,\vec{k})]$ according to the same equation of 3D (see fig.\[fig:2D\_frequenze\_09\]), where the 2D value for $\omega_W$ is given in appendix B eq. (\[eqn:w\_Wig\_2D\]). Let us discuss the deviation from the the scaling at strong coupling, which we see from fig.\[fig:3D2D\_summary\] in the density of the softening of the polaronic frequency $\omega_{pol}$. Actually we observe that a steep fall of the variational parameter $v(r_s)$ occurs as density increases determining the softening of $\omega_{pol}$. Peculiar features of the DOS enters in the variational determination of $v(r_s)$ as can achieved by the following argument. First of all we assume that $w$ is very close to the value $\omega_{LO}$ at strong coupling. Then we notice that as in 3D high polarizability case the renormalized plasma frequency is much less than the phonon frequency and the discussion which follows eq. (\[eqn:eps\_s\_k\]) holds for all densities lower than critical density of the softening. In this case the spectrum is composed by the low energy branches (renormalized WC) and the by polaronic branches weakly dispersed around $v$ (see also fig.\[fig:2D\_frequenze\_09\]). Using this results at low temperatures $(\beta \rightarrow \infty)$, the condition for the extrema of $\mathcal{F}_{V}$ reads: $$\label{eqn:v_self_eq} 1 - \frac{1}{v^{3/2}} \sqrt{\frac{\omega^2_P}{w \varepsilon_o}} \mathcal{M}_2 \; + \; g(\alpha, \eta,r_s,v) = 0$$ where the first and second terms are the derivative of $\mathcal{F}_T$ eq.(\[eqn:F\_T\]) and $g$ is the derivative of eqs.(\[eqn:mean\_S\_self\_3D\],\[eqn:mean\_S\_self\_2D\],\[eqn:mean\_S\_T\]) from Appendix C. As $v(r_s) \rightarrow 0$ for $r_s \simeq r_c$ the second term acquires importance and DOS enters in the second moment $\mathcal{M}_2$. However there are other terms which are divergent as $v \rightarrow 0$ coming from the explicit form of the function $g$. ![ 2D case. The eigenfrequencies of system along the direction (10) for $\eta=0.9$ and $\alpha=2.12$. $1/r_s = 8 \; 10^{-3}$ (a.u.), $T=2.5 \; 10^{-5}$ (a.u.). Density is close to the classical liquid-solid transition. Upper panel: The frequencies of polaronic branch weakly dispersed around the polaronic frequency $\omega_{pol}\simeq v$ (dashed line). Lower panel: The renormalized Wigner crystal branches (points) and the pure WC branches (dashed lines).[]{data-label="fig:2D_frequenze_09"}](fig8.eps) Conclusions =========== We have studied the behavior of a low density electron gas in the presence of a polarizable medium, where polaronic effects may play a relevant role. To determine the transition line, we have used a generalized Lindemann criterion which reproduces correctly the pure electron gas quantum melting. Because the amplitude of quantum fluctuations depends on the e-ph renormalized plasma frequency, the Lindemann rule has been critically re-examined and adapted to the polaron crystal. This procedure allows to determine quantitatively the phase diagram of the model and to extend the study of the model to the low polarizability case, which was not studied before. We have also studied the 2D case, showing that the dimensional dependence is not crucial to determine the nature of the quantum melting within in our harmonic variational scheme. The scaling predicted for the e-ph coupling constant at zero density do apply as well at non zero density up to quantum melting. A noticeable difference instead is in the position of the density of the softening of the polaronic frequency which in 3D is much closer to the melting than in 2D case. This suggests that the hetero-interactions are less effective to destabilize the dipolar crystal in 2D. However other possible mechanism, (lattice-effects,structural disorder or impurities) can cooperate to the localization together with the interactions between the electrons and lead to the formation of a pinned Wigner Crystal. In this case the melting can not be predicted by the Lindemann rule but a similar dipolar instability due to the long-range interaction between the electrons can still drives the melting. While the weak e-ph coupling regime is similar for both low and high polarization case, the strong coupling scenario is qualitatively different. In the high polarizability regime, we have recovered the incipient instability which was found in previous studies near the solid phase [@Simone_meanfield; @Simone] and also in the liquid phase [@Lorenzana]. In comparison with previous work, we have found that this regime is restricted to very large values of the coupling $\alpha>10$ leaving an interesting intermediate region of coupling in which polarons may exist in the liquid phase. This region can in principle be explored with non perturbative numerical techniques as e.g. Path Integral Monte Carlo. Work along this direction is currently in progress. In the low polarizability regime, a crossover occurs inside the solid phase when the renormalized plasma frequency approaches the phonon frequency. At low density, we still have a LPC, while at higher densities the electron-phonon interaction is weakened irrespective of the [bare]{} electron-phonon coupling. In this case, polaron clouds overlap as and the polaron feature of the crystal is lost. The crossover from polaronic $(\omega_P < \omega_{LO})$ to non polaronic $(\omega_P > \omega_{LO})$ character has been observed in weakly coupled systems such as GaAs in the liquid phase and analyzed in term RPA [@Mahan; @Xiaoguang]. In this system it occurs around $r_s \sim 0.6-0.7$ while for ZnO $\alpha$ is larger shifting the crossover to $r_s \sim 7$. Finding a system with low polarizability and larger e-ph coupling is difficult since it implies very low $\omega_{LO}$: $\alpha \sim (1 - \eta) / \omega_{LO}$ from eq.(\[eqn:alpha\]). However in surfaces of InSb an $\alpha = 4.5$ has been predicted together with $\eta= 0.88$ [@Sun_Gu] leading to the possible observation of the crossover inside the solid phase. We notice also that our low polarizability scenario of density crossover inside the solid phase bear some resemblance to that found for ripplonic polaron systems [@Andrei_book]. Though the electron-ripplon interaction in these systems is different from the Fröhlich type, resonances in the absorption spectrum observed by Grimes and Adams [@Grimes_Adams], their explanation at low density [@Fisher_Halperin] relays on the same qualitative arguments developed in the present work. Recent works on high density ripplonic polaron systems realized on a helium bubbles predicts also in this case a mixing between plasmon and polaron modes [@DeVreese-multiripplons]. Finally we remark that we have obtained an appreciable stabilization of the crystal phase even for intermediate regime $\alpha \sim 3-5$ in low polarizability cases. We conclude that the general result that e-ph interaction effects can stabilize the Wigner crystal phase could motivate experimental studies on two dimensional electronic devices involving polarizable media. To this aim a layered configuration is advised even with some warnings [@Fomin_Smondyrev]. In 2D heterostructure the use of a perpendicular electric field [@Fomin_Smondyrev] could not only increase the polaron effect but also tune it as it was shown in the case of charged helium surfaces [@Helium_surface]. ACKNOWLEDGMENTS =============== Authors acknowledge S. Fratini and P. Quemerais for useful discussions and critical reading of the manuscript. We thanks also J. Lorenzana for useful suggestions. One of us (G.R.) thanks the also the kind hospitality of CNRS-LEPES Grenoble were a part of this work has been done. This work was supported by MIUR-Cofin 2001 and MIUR-Cofin 2003 matching funds programs. APPENDIX A: The low energy cut off in 2D ======================================== The 2d DOS function can be defined as $$\rho(\omega) = \sum_{s=1,2} \int_{V_ {B}} \!\!\!\! d^2 k \; \delta (\omega - \omega_{s,k})$$ where $V_ {B}$ is the volume of the first Brillouin‘s zone $(1BZ)$. Let us consider a small fraction $\varepsilon$ of the plasma frequency $\omega_P$. At long wavelength $(k=0)$, we have the 2D dispersion laws for the acoustical mode is $\omega_1 (k) \simeq c_1 k $ while the “optical” $\omega_2 (k) \simeq c_2 \sqrt{k}$ [@Bonsall]. As a consequence the behavior of the DOS for $\omega \simeq 0$ is $$\begin{aligned} \int^{2 \pi}_0 \!\!\!\! d \theta \int^{k(\varepsilon)}_0 \!\!\!\!\!\!\! d k \; k \; \delta (\omega - c_1 k) &=& \frac{2 \pi}{c^2_1} \omega \\ \int^{2 \pi}_0 \!\!\!\! d \theta \int^{k(\varepsilon)}_0 \!\!\!\!\!\!\! d k \; k \; \delta (\omega - c_2 \sqrt{k}) &=& \frac{4 \pi}{c^4_2} \omega^3 \end{aligned}$$ Introducing the scaled frequency $x$ defined as $\omega = \omega_P x$ and the quantum parameter $\eta_q$ eq.(\[eqn:eta\_q\]), the thermal electronic fluctuation is expressed as an average on the DOS as $$\label{eqn:u2DOS} \left< u^2 \right> = \frac{\hbar}{m} {\left< \frac{\coth (\eta_q x)}{x} \right>}_{DOS}$$ $$\label{eqn:u2_expansion} = \frac{\hbar}{m} \int \!\!\! d x \; \rho(\omega_P x) \left[ \frac{1}{\eta_q x^2} + \eta_q \left( \alpha_0 + \alpha_2 {\left( \eta_q x \right)}^{2} + \dots \right) \right]$$ Since $\rho(\omega_P x) \sim x$ for $x \rightarrow 0$, the average in eqs. (\[eqn:u2DOS\]) converges for any of $x^{2n}$ with $n \ge 0$ in the the expansion eq.(\[eqn:u2\_expansion\]). In the $n=-1$ term we consider infrared cut-off $x_c$ giving $$\begin{aligned} {\left< \frac{1}{x^2} \right>}_{DOS} &\simeq& \int^{\varepsilon}_{x_c} \!\!\!\!\! d x \; \frac{(2 \pi / c^2_1) \omega_P}{x} + \int_{\varepsilon} \!\!\! d x \; \frac{\rho(\omega_P x)}{x^2} \nonumber \\ &\simeq & \frac{2 \pi \omega_P}{c^2_1} \ln \left( \frac{\varepsilon}{x_c} \right) + \int_{\varepsilon} \!\!\! d x \; \frac{\rho(\omega_P x)}{x^2}\end{aligned}$$ This term diverges logarithmically as $x_c \rightarrow 0$. However $\eta_q \rightarrow \infty$ as we approach the quantum region. The electronic fluctuation turns out to be cut-off independent if $$\left< \frac{1}{x^2} \right> \ll \eta^2_q \left< \alpha_0 + \alpha_2 {\left( \eta_q x \right)}^{2} + \dots \right>. \label{eqn:mom_inv2_vs_quantum_corr}$$ We have chosen for the cut-off frequency $x_c = \omega_{min} / \omega_P \simeq 5 \cdot 10^{-5}$ so that the condition eq.(\[eqn:mom\_inv2\_vs\_quantum\_corr\]) is fulfilled around $\eta_q(T,r_s) \ge 10$ which corresponds to a large region inside to the solid phase. By the relation for acoustical long wave excitation $\omega_{min} = c_1 k_{min}$ and $k_{min} = 2 \pi / \left( r_s \sqrt{N} \right)$, the number of electrons is $N= 5.24 \cdot 10^{6}$. Our inverse second moment of DOS is $\mathcal{M}_{-2} = 12.5$ (cfr ref. [@Gann] $\mathcal{M}_{-2}=8.16$ for $N=1024$). APPENDIX B: The harmonic variational approximation ================================================== We expand in the harmonic approximation the terms $\mathcal{S}_{e-e},\mathcal{S}_{e-J},\mathcal{S}^{dist}_{e-ph-e}$ (eqs.\[eqn:S\_dist\_e-e\],\[eqn:S\_e-J\],\[eqn:S\_dist\_e-ph-e\]). Let $\vec{r}_{\imath} = \vec{R}_{\imath} + \vec{u}_{\imath}$, where $\vec{R}_{\imath}$ is the lattice point of the crystal and $\vec{u}_{\imath}$ is the electronic displacement from $\vec{R}_{\imath}$, and set $\Delta \vec{u}_{\imath,\jmath}(\tau,\sigma) = \vec{u}_{\jmath}(\sigma) -\vec{u}_{\imath}(\tau)$ and $\vec{R}_{\jmath,\imath} = \vec{R}_{\jmath} - \vec{R}_{\imath}$. The static terms give $$\mathcal{S}^o_{e-e}(\{ \vec{R}_{\imath} \}) + \mathcal{S}^o_{e-J}(\{ \vec{R}_{\imath} \}) + \mathcal{S}^{o,dist}_{e-ph-e}(\{ \vec{R}_{\imath} \}) = \frac{\mathcal{S}^{o}_{WC}(\{ \vec{R}_{\imath} \})}{{\varepsilon_0}}$$ the e-ph interaction does not change the equilibrium positions of the pure electronic crystal (WC) which corresponds to the minimum of the potential energy.\ The sum of the dynamical parts in the harmonic approximation gives $$\begin{aligned} \mathcal{S}^{H}_{e-e} + \mathcal{S}^{H}_{e-J} + \mathcal{S}^{H,dist}_{e-ph-e} = \nonumber \\ = \int^{\beta}_{0} \!\!\!\! d \tau \sum_{\imath} \left[ V_{J} (\vec{u}_{\imath}(\tau)) + V_{e-e} (\vec{u}_{\imath}(\tau)) \right] \label{eqn:S_dyn}\end{aligned}$$ where $$\begin{aligned} V_{J} (\vec{u}_{\imath}(\tau)) & = & - \frac{e^2 \rho_J}{{\varepsilon_0}} \int \!\!\! d^D \! r \left( \frac{1}{\|\vec{u}_{\imath}(\tau) - \vec{r}\|} - \frac{1}{r} \right) \label{eqn:V_J}\\ V_{e-e} (\vec{u}_{\imath}(\tau)) &=& \frac{e^2}{2} \int^{\beta}_{0} \!\!\!\! d \sigma F(\tau -\sigma) \sum_{\jmath \neq \imath} U_{\jmath,\imath}(\tau,\sigma) \label{eqn:V_H_ij} \\ U_{\jmath,\imath}(\tau,\sigma) &=& \frac{1}{2} \Delta \vec{u}_{\imath,\jmath}(\tau,\sigma) \cdot \overline{\mathcal{I}}(\vec{R}_{\jmath,\imath}) \Delta \vec{u}_{\imath,\jmath}(\tau,\sigma) \label{eqn:U_H_ij} \\ F(\tau -\sigma) &=& \frac{\delta(\tau-\sigma)}{{\varepsilon_{\infty}}} - \frac{\omega_{LO}}{2 {\bar{\varepsilon}}} D_o(\tau - \sigma) \\ {\left[ \overline{\mathcal{I}_{\imath \jmath}} \right]}_{\alpha \beta} &=& \frac{\delta_{\alpha \beta}}{{\left\| \vec{R}_{\jmath,\imath} \right\|}^3} - 3 \frac{{\left[ \vec{R}_{\jmath,\imath} \right]}_{\alpha} {\left[\vec{R}_{\jmath,\imath}\right]}_{\beta}} {{\left\| \vec{R}_{\jmath,\imath} \right\|}^5} \label{eqn:I_ij}\end{aligned}$$ >From now we drop on the double $\sigma,\tau$ indexes in $\Delta \vec{u}_{\imath,\jmath}$. To evaluate the integral in eq.(\[eqn:V\_J\]) and the sums on index $\jmath$ in eq.(\[eqn:V\_H\_ij\]), we consider a sphere $S_R$ of radius $R_s$ (a disk in 2D) centered on site $\imath$. We first sum on index $\jmath$ and then we perform the limit $R_s \rightarrow \infty$. Finally we sum on index $\imath$ in eq. (\[eqn:S\_dyn\]). 3D case ------- By Gauss’s law (with the condition $V_{J}(0)=0$), we have $$\label{eqn:Wigner_3D} V_{J} (\vec{u}_{\imath}(\tau)) = \frac{1}{2} m \frac{\omega^2_W}{{\varepsilon_0}} {\left\|\vec{u}_{\imath}(\tau)\right\|}^2$$ where the Wigner frequency is $\omega^2_W = \omega^2_P / 3$. Because of $V_{J}$ is independent of the size of $S_R$, eq.(\[eqn:Wigner\_3D\]) does not change in the limit $R_s \rightarrow \infty$.\ To evaluate the sum in eq.\[eqn:V\_H\_ij\] with the condition $R_{\jmath} < R_s$ we remind that we have two self terms ($(i,i)$ and $(j,j)$) and two distinct terms ($(i,j)$ and $(j,i)$) in eq. (\[eqn:U\_H\_ij\]). The two self terms give the same contribution ,as can be easily check if we firstly we carry on the limit $R_s \rightarrow \infty$ and then the sum on index $\jmath$ and $\imath$. They vanishes because of cubic symmetry of the lattice. When the two distinct terms ($(i,j)$ and $(j,i)$) of eq. (\[eqn:U\_H\_ij\]) are inserted in eq.(\[eqn:V\_H\_ij\]) and the limit $R_s \rightarrow \infty$ is taken, we obtain the term $V_{e-e}(u_{\imath})$ of eq.\[eqn:S\_dyn\] $$\label{eqn:U_H_ij_R_sum_III} V_{e-e} (\vec{u}_{\imath}(\tau)) = \frac{e^2}{2} \sum_{\jmath \neq \imath} \int^{\beta}_0 \!\!\!\! d \sigma F(\tau - \sigma) \vec{u}_{\jmath}(\sigma) \overline{\mathcal{I}}(\vec{R}_{\jmath,\imath}) \vec{u}_{\imath}(\tau)$$ Summing on index $\imath$ and integrating on variable $\tau$ the eqs.(\[eqn:Wigner\_3D\],\[eqn:U\_H\_ij\_R\_sum\_III\]), we obtain the terms $\mathcal{S}^H_{e-J} , \mathcal{S}^{H}_{e-e}, \mathcal{S}^{H,dist}_{e-ph-e}$ eqs.(\[eqn:S\_H\_dist\_e-e\],\[eqn:S\_H\_dist\_e-ph-e\]). 2D case ------- In 2D the interaction potential $V^{R}_{J}(u)$ of a uniform positive charged disk of radius $R_s$ eq. (\[eqn:V\_J\]) is $$V^{R}_{J}(u) = - \frac{e^2 \rho_J}{{\varepsilon_0}} \int^{2 \pi}_0 \!\!\!\!\!\! d \theta F(\theta) \\$$ where $$\begin{aligned} & F(\theta) = \sqrt{R^2_s + u^2 -2 R_s u \cos(\theta)} - u - R_s &\\ &+ u \cos(\theta) \ln \frac{R_s - u \cos(\theta) + \sqrt{R^2_s + u^2 -2 R_s u \cos(\theta)}}{u(1-\cos(\theta))}&\end{aligned}$$ In the limit $R_s \rightarrow \infty$ $$\lim_{R \rightarrow \infty} V^{R}_{J}(\vec{u}_\imath) = \lim_{ \frac{u}{R} \rightarrow 0} \frac{e^2}{{\varepsilon_0}} \rho_J \frac{\pi}{R_S} u^2_{\imath} = 0$$ since the total electric field of an infinite charged disk is perpendicular to the disk.\ Then we have to evaluate the sums eq.(\[eqn:U\_H\_ij\]). The two distinct terms ($(i,j)$ and $(j,i)$) gives the identical result eq.\[eqn:U\_H\_ij\_R\_sum\_III\] of the 3D case while the self term $(\imath,\imath)$ is written as $$\frac{1}{2} \vec{u}_{\imath} \left( \sum_{ \stackrel{\vec{R}_{\jmath} < R}{\jmath \neq \imath} } \overline{\mathcal{I}_{\imath \jmath}} \right) \vec{u}_{\imath} = \vec{u}_{\imath} \overline{\mathcal{D}} \vec{u}_{\imath}$$ the matrix $\mathcal{D}$ in 2D is defined as sum of the matrices $\overline{\mathcal{I}}(\vec{R}_\jmath)$ eq.(\[eqn:I\_ij\]) on hexagonal lattice points $\vec{R}_\jmath$. Contrary to the 3D case, the matrix $\mathcal{D}$ is not zero in 2D case. By the lattice symmetry, the off-diagonal elements are zero while the diagonal terms are equal to the local potential which acts on each electrons $$\label{eqn:Wigner_2D} \frac{e^2}{2}\int^{\beta}_{0} d \sigma F(\tau - \sigma) \overline{\mathcal{D}}_{\alpha \alpha} {\left\|\vec{u}_{\imath}(\tau)\right\|}^2 = \frac{1}{2} m \frac{\omega^2_W}{{\varepsilon_0}} {\left\|\vec{u}_{\imath}(\tau)\right\|}^2$$ where we use as definition 2D Wigner frequency $$\label{eqn:w_Wig_2D} \omega^2_{W} = \frac{e^2}{m} \lim_{R \rightarrow \infty} \sum_{\stackrel{\jmath \neq \imath}{R_{\jmath,\imath}< R}} \frac{1}{2 R^3_{\jmath,\imath}} \qquad \mbox{(2D)}$$ For an hexagonal lattice of nearest neighbor distance $d_{n.n.}$, we have $\sum_{\jmath \neq \imath}(1/2 R^3_{\jmath,\imath}) = 5.51709 / d^3_{n.n.}$.\ Summing on index $\imath$ eqs.(\[eqn:U\_H\_ij\_R\_sum\_III\],\[eqn:Wigner\_2D\]) we obtain the terms $\mathcal{S}^H_{e-J} , \mathcal{S}^{H}_{e-e}, \mathcal{S}^{H,dist}_{e-ph-e}$ eqs.(\[eqn:S\_H\_dist\_e-e\],\[eqn:S\_H\_dist\_e-ph-e\]). Normal modes ------------ The WC normal modes are defined as $$\label{eqn:q_ks} \vec{u}_{\imath} = \frac{1}{\sqrt{N}} \sum_{\vec{k},s} \hat{\varepsilon}_{\vec{k},s } q_{\vec{k},s} e^{{\dot{\imath}}\vec{k} \vec{R}_{\imath}}$$ where the vectors $\vec{k}$ belongs to the $1BZ$ of the reciprocal lattice, $\hat{\varepsilon}_{\vec{k},s}$ are eigenvector with eigenvalue $\omega^2_{\vec{k},s}$ of the dynamical matrix $\overline{\mathcal{M}}$ which is defined as $$\label{eqn:M_dyn} {\overline{\mathcal{M}}}_{\alpha \beta} = \delta_{\alpha \beta} \omega^2_W + \frac{e^2}{m} \sum_{\vec{R}_{\imath} \neq 0} {\overline{\mathcal{I}}}_{\alpha \beta} (\vec{R}_{\imath}) e^{{\dot{\imath}}\vec{k} \cdot \vec{R}_{\imath}}$$ Inserting the WC normal modes eq.(\[eqn:q\_ks\]) in eqs.(\[eqn:S\_Fey\],\[eq:SK\],\[eqn:S\_H\_dist\_e-e\],\[eqn:S\_H\_dist\_e-ph-e\]), we express the harmonic variational action $\mathcal{S}_T$ as $$\label{eqn:S_T_q} \mathcal{S}_{T} (\{ q_{s,\vec{k}}(\tau) \})= \sum_{s,\vec{k}} \int_0^{\beta} \!\!\! d \tau L_{s,k} \left( \tau \right)$$ where the Lagrangian is $$\label{eqn:L_sk} \begin{array}{ll} L_{s,k} &= \frac{1}{2} m {\left\| \dot{q}_{\vec{k},s} (\tau) \right\|}^{2} + \frac{1}{2} m \frac{\omega^2_{\vec{k},s}}{{\varepsilon_0}} {\left\|q_{\vec{k},s} (\tau) \right\|}^2 \\ &+ \frac{m w \left(v^2 - w^2\right)}{8} \displaystyle{\int_0^{\beta} } \!\!\!\! d\sigma D_V(\tau-\sigma) {\left\| q_{\vec{k},s} (\tau) - q_{\vec{k},s} (\sigma) \right\| }^2 \\ &+ \frac{m \omega_{LO} \left(\omega^2_{\vec{k},s} -\omega^2_W \right)}{8 {\bar{\varepsilon}}} \displaystyle{\int_0^{\beta} } \!\!\!\! d\sigma D_o(\tau-\sigma) {\left\|q_{\vec{k},s} (\tau)- q_{\vec{k},s} (\sigma) \right\|}^2 \\ \end{array}$$ APPENDIX C: The variational free energy $\mathcal{F}_V$ ======================================================= The first term of the variational free energy $\mathcal{F}_V$ eq. (\[eqn:F\_harm\_var\]) is the free energy $\mathcal{F}_T$ associated to the partition function of the trial action $\mathcal{Z}_T$. This is calculated as the functional integral eq.(\[eqn:Z\_eff\]) where $\mathcal{S}_{eff}$ eq. (\[eqn:S\_eff\]) is replaced by $\mathcal{S}_T$ eq. (\[eq:Svar\]). The second term of $\mathcal{F}_V$ is the mean value eq. (\[eqn:mean\]) of the difference between $\mathcal{S}^{self}_{e-ph-e}$ eq. (\[eqn:S\_self\_e-ph-e\]) and $\mathcal{S}_{Feyn}$ eq. (\[eqn:S\_Fey\]).\ We start by changing the dynamical variables of integration from $\{ \vec{u}_\imath (\tau) \}$ to $\{ q_{s,\vec{k}} (\tau) \}$. By reality condition we have $q_{-\vec{k},s} = q^{\ast}_{\vec{k},s}$ and $\hat{\varepsilon}_{-\vec{k},s }=- \hat{\varepsilon}_{\vec{k},s}$, we must sum only $\vec{k}$ vectors in the upper half space ($k_z > 0$) of $1BZ$ $$\label{eqn:q_ks_II} \vec{u}_{\imath} = \frac{1}{\sqrt{N}}\sum_{s,\vec{k},k_z > 0} \hat{\varepsilon}_{\vec{k},s } \left[ q_{\vec{k},s} e^{{\dot{\imath}}\vec{k} \vec{R}_{\imath}} - q^{\ast}_{\vec{k},s} e^{- {\dot{\imath}}\vec{k} \ \vec{R}_{\imath}} \right]$$ Therefore the real and imaginary part of $q_{s,\vec{k}}$ for all $k$ with $(k_z >0)$ of the $1BZ$ are the actual independent variables and the Jacobian of canonical transformation is $J=2^{DN}$ $$\label{eqn:Z_T} \mathcal{Z}_T = \int \!\! J \!\! \prod_{s,\vec{k},k_z > 0} \mathcal{D} [q^{Re}_{s,\vec{k}} (\tau)] \mathcal{D} [q^{Im}_{s,\vec{k}} (\tau)] e^{- \mathcal{S}_{T} \left[ \{ q_{s,\vec{k}}(\tau) \} \right] }$$ Using the periodicity condition $(q_{s,\vec{k}}(0)~=~q_{s,\vec{k}} (\beta))$, we have the following Fourier expansion $(\omega_n = (2 \pi/\beta) n)$ $$\begin{aligned} q_{s,\vec{k},n} &=& \frac{1}{\beta} \int^{\beta}_0 \!\! d \tau \; q_{s,\vec{k}}(\tau) e^{- {\dot{\imath}}\omega_n \tau} \nonumber \\ q_{s,\vec{k}}(\tau) & = & q_{s,\vec{k},c} + \delta q_{s,\vec{k}}(\tau) \\ q_{s,\vec{k},c} & = & \frac{1}{\beta} \int^{\beta}_0 d \tau \; q_{s,\vec{k}}(\tau) \label{eqn:centroid}\\ \delta q_{s,\vec{k}}(\tau) & = & \sum^{\infty}_{\stackrel{n= - \infty}{n \neq 0}} q_{s,\vec{k},n} e^{{\dot{\imath}}\omega_n \tau} \label{eqn:fluctua}\end{aligned}$$ where we have separated the mean value of path on the imaginary time eq.(\[eqn:centroid\]) (centroid) from the fluctuation around it eq.(\[eqn:fluctua\]). The action $\mathcal{S}_{T}(\{ q_{s,\vec{k}}(\tau) \})$ is quadratic in $\{ q_{s,\vec{k},n} \}$ therefore we can separate eq.(\[eqn:Z\_T\]) in two gaussian integrals $$\mathcal{Z}_T = \mathcal{Z}_{T,c} \mathcal{Z}_{T, \delta q}$$ $$\begin{aligned} \mathcal{Z}_{T,c} &=& \!\! \int \!\! \prod_{s,\vec{k},k_z > 0} \frac{d q^{Re}_{\vec{k},s,c} d q^{Im}_{\vec{k},s,c}}{\pi \hbar^2 / m k_B T} \;\; e^{- \mathcal{S}^{c}_{T} \{ q_{s,\vec{k},c} \}} \nonumber \\ & = & \int \prod_{s,\vec{k},k_z > 0} \frac{d q^{Re}_{\vec{k},s,c} d q^{Im}_{\vec{k},s,c}}{\pi \hbar^2 / m k_B T} \;\; e^{-\frac{m}{k_B T} \!\!\! \frac{{\left\| q_{\vec{k},s,c} \right\|}^2}{\omega^2_{s,\vec{k}}/ {\varepsilon_0}} } \nonumber \\ &=& \prod_{s,\vec{k}} \frac{k_B T} {\hbar \; \omega_{s,\vec{k}} / \sqrt{{\varepsilon_0}} } \label{eqn:Z_T_C}\end{aligned}$$ hence after we omit the classic term $\mathcal{Z}_{T,c}$ eq.(\[eqn:Z\_T\_C\]). $$\begin{aligned} \mathcal{Z}_{T, \delta q} &=& \!\! \int \!\! \prod_{\stackrel{n \neq 0}{s,\vec{k},k_z > 0}} \frac{d q^{Re}_{\vec{k},s,n} d q^{Im}_{\vec{k},s,n} }{\pi k_B T / m \omega^2_n} \;\; e^{- \delta \mathcal{S}_{T} \{ \delta q_{s,\vec{k}}(\tau) \} } \nonumber \\ &=& \int \prod_{\stackrel{n \neq 0}{s,\vec{k},k_z > 0}} \frac{d q^{Re}_{\vec{k},s,n} d q^{Im}_{\vec{k},s,n} }{\pi k_B T / m \omega^2_n} \;\; e^{-\frac{m}{k_B T} \frac{{\left\| q_{\vec{k},s,n} \right\|}^2}{\lambda_{s,\vec{k},n}} } \nonumber \\ &=& \prod_{\stackrel{n \neq 0}{s,\vec{k},k_z > 0}} \omega^2_n \lambda_{s,\vec{k},n} \label{eqn:Z_del_A}\end{aligned}$$ where $$\begin{aligned} \lambda_{s,\vec{k},0} &=& \frac{1}{\omega^2_{\vec{k},s} / {\varepsilon_0}} \\ \lambda_{s,\vec{k},n} & = & \sum^{3}_{\gamma=1} \frac{A_{\gamma}}{\omega^2_n +\Omega^2_{\gamma}} \\ A_{1} &=& \frac{(\Omega^2_1 - \omega^2_{LO}) (\Omega^2_1 - w^2_T)} {(\Omega^2_1 - \Omega^2_2)(\Omega^2_1-\Omega^2_3)} \quad (\mbox{cyclic perm.} \; \gamma=1,2,3) \nonumber \\ & & \label{eqn:A_factors} \end{aligned}$$ the frequencies $\Omega^2_{\gamma} \; (\gamma=1,2,3)$ are the opposite of the roots of cubic $$\begin{aligned} \mathcal{P}_3 (z) & = & z^3 + a_2 z^2 + a_1 z + a_0 \label{eqn:pol3}\\ a_2 & = & v^2 + \omega^2_{LO} + \frac{\omega^2_{\vec{k},s}}{{\varepsilon_0}} + \frac{\omega^2_{\vec{k},s} - \omega^2_W}{{\bar{\varepsilon}}} \nonumber \\ a_1 & = & \omega^2_{LO} v^2 + \frac{\omega^2_{\vec{k},s}}{{\varepsilon_0}} (\omega^2_{LO} + w^2 ) + w^2 \frac{\omega^2_{\vec{k},s} - \omega^2_W}{{\bar{\varepsilon}}} \nonumber \\ a_0 & = & \frac{\omega^2_{LO} w^2 \omega^2_{\vec{k},s}}{{\varepsilon_0}} \nonumber\end{aligned}$$ The gaussian integrals eq.(\[eqn:Z\_del\_A\]) are convergent if $\lambda_{s,\vec{k},n}$ are positive numbers $\forall (s,\vec{k},n)$. This condition is fulfilled if $\Omega^2_{\gamma}$ are [all]{} positive. The numerical minimization of the variational free energy has been made enforcing this constraint. Performing the infinite product in eq.(\[eqn:Z\_del\_A\]) we have $$\begin{aligned} \mathcal{Z}_{T, \delta q} & = & {\left( \frac{\sinh(\hbar \omega_{LO} /2 k_B T) }{\hbar \omega_{LO} /2 k_B T } \right)}^{DN} {\left( \frac{\sinh(\hbar w /2 k_B T) }{\hbar w /2 k_B T } \right)}^{DN} \nonumber \\ &\cdot& \prod_{s,\vec{k},\gamma} \frac{\hbar \Omega_{\gamma,s,\vec{k}} /2 k_B T } {\sinh(\hbar \Omega_{\gamma,s,\vec{k}} /2 k_B T) } \nonumber\end{aligned}$$ and finally we substitute the sum on $(\vec{k}_{\imath},s)$ with the integral on the WC DOS $\rho(\omega)$ in the free energy $\mathcal{F}_T$ $$\begin{aligned} \label{eqn:F_T} \frac{\mathcal{F}_T}{D N} &=& - k_B T \ln \left[\sinh\left( \frac{\hbar \omega_{LO}}{2 k_B T} \right) \sinh\left( \frac{\hbar w}{2 k_B T} \right) \right] \nonumber \\ & + & k_B T \int d \omega \rho(\omega) \sum^{3}_{\gamma=1} \ln \left[ \sinh \left( \frac{\hbar \Omega_{\gamma} (\omega)}{2 k_B T} \right) \right]\end{aligned}$$ To calculate the mean value of $\mathcal{S}^{self}_{e-ph-e}$ eq. (\[eqn:S\_self\_e-ph-e\]) in $3D$ we use the following identity [@footnote_D_o] $$\begin{aligned} \iint_0^{\beta} \!\!\! d\tau d\sigma D_o(\tau-\sigma) \int \!\! \frac{d^3 q}{ {\left( 2 \pi \right)}^3 } \frac{4 \pi}{q^2} {\left< e^{{\dot{\imath}}\vec{q} \cdot \left[ \vec{u}_{\imath}(\tau)-\vec{u}_{\imath}(\sigma) \right] } \right>}_T &=& \nonumber \\ = - 2 \beta \int_0^{\frac{\beta}{2}} \!\!\! d\tau \frac{D_o(\tau)}{ \sqrt{ \frac{\pi}{2} d_{3D}(\tau) } } & & \label{eqn:mean_S_self_3D}\end{aligned}$$ while in $2D$ $(q^2 = q^2_{\perp} + q^2_z)$ $$\begin{aligned} \iint_0^{\beta} \!\!\! d\tau d\sigma D_o(\tau-\sigma) \int \frac{d^2 q_{\perp}}{ {\left( 2 \pi \right)}^2 } \frac{2 \pi}{q_{\perp}} e^{- \frac{1}{2} d_{2D}(\tau - \sigma) q^2_{\perp}} &=& \nonumber \\ = - 2 \beta \left( \frac{\pi}{2} \right) \int_0^{\frac{\beta}{2}} \!\!\! d\tau \frac{D_o(\tau)}{ \sqrt{ \frac{\pi}{2} d_{2D}(\tau) } } & & \label{eqn:mean_S_self_2D}\end{aligned}$$ where $d_D (\tau)$ is the imaginary time diffusion in the LPC defined as (3D or 2D) $$\label{eqn:d_tau} d_D (\tau) =\frac{ {\left< {\left\| \vec{u}(\tau) - \vec{u}(0) \right\|}^2 \right>}_T } {D}$$ The mean value of $\mathcal{S}_{Feyn}$ eq. (\[eqn:S\_Fey\]) is $$\begin{aligned} & & {\left< \mathcal{S}_{Feyn} \right>}_T / N = \nonumber \\ &=& - D \frac{m w \left(v^2 - w^2\right)}{8} \iint_0^{\beta} \!\! d\tau d\sigma D_T(\tau-\sigma) d_D (\tau - \sigma) \nonumber \\ & & \label{eqn:mean_S_T}\end{aligned}$$ To obtain eqs.(\[eqn:mean\_S\_self\_3D\],\[eqn:mean\_S\_self\_2D\],\[eqn:mean\_S\_T\]) we have used $$\label{eqn:Y_tau} {\left< e^{{\dot{\imath}}\vec{q} \cdot \left[ \vec{u}_{\imath}(\tau)-\vec{u}_{\imath}(\sigma) \right] } \right>}_{T} = e^{- \frac{1}{2} d_D (\tau - \sigma) q^2 }$$ We will demonstrate eq.(\[eqn:Y\_tau\]) in the next subsection. Calculation of ${\left<\exp({\dot{\imath}}\vec{q} \cdot \left[ \vec{u}_{\imath}(\tau)-\vec{u}_{\imath}(\sigma) \right]) \right>}_T$ ----------------------------------------------------------------------------------------------------------------------- >From eqs.(\[eqn:q\_ks\],\[eqn:fluctua\]) we have $$\label{eqn:mean_exp_I} {\dot{\imath}}\vec{q} \cdot \left[ \vec{u}_{\imath}(\tau)-\vec{u}_{\imath}(\sigma) \right] = \sum_{\stackrel{s,k_z>0}{n \neq 0}} \left[ q_{\vec{k},s,n} J^{\ast}_{s,k,n}(\tau-\sigma,\vec{q}) + \mbox{c.c.} \right]$$ $$\label{eqn:mean_exp_II} J^{\ast}_{s,k,n}(\tau-\sigma,\vec{q}) = \frac{{\dot{\imath}}}{\sqrt{N}} \vec{q} \cdot \hat{\varepsilon}_{\vec{k},s } \left( e^{{\dot{\imath}}\omega_n \tau} - e^{{\dot{\imath}}\omega_n \sigma} \right) e^{{\dot{\imath}}\vec{k} \vec{R}_{\imath}}$$ then we have $$\begin{aligned} & & {\left<\exp({\dot{\imath}}\vec{q} \cdot \left[ \vec{u}_{\imath}(\tau)-\vec{u}_{\imath}(\sigma) \right]) \right>}_T = \nonumber \\ &=& \frac{1}{\mathcal{Z}_{T, \delta q}} \!\!\! \int \!\!\!\!\!\! \prod_{\stackrel{n \neq 0}{s,\vec{k},k_z > 0}} \!\!\! \frac{d q^{Re}_{\vec{k},s,n} d q^{Im}_{\vec{k},s,n} }{\pi k_B T / m \omega^2_n} \;\; e^{-\frac{m}{k_B T} \frac{{\left\| q_{\vec{k},s,n} \right\|}^2}{\lambda_{s,\vec{k},n}} + q_{\vec{k},s,n} J^{\ast}_{\vec{k},s,n} + \mbox{c.c.} } \nonumber \\ &=& \prod_{\stackrel{s,k_z>0}{n \neq 0}} e^{- \frac{k_B T}{m} \lambda_{s,\vec{k},n} {\left\| J_{\vec{k},s,n} \right\|}^{2} } = e^{- \frac{1}{2} \frac{1}{N} \sum_{s,k} {\left\| \hat{q} \cdot \hat{\varepsilon}_{\vec{k},s} \right\|}^{2} d_{\omega_{s,k}}(\tau -\sigma) q^2 } \nonumber \\ &=& e^{- \frac{1}{2} \frac{1}{N D} \sum_{s,k} d_{\omega_{s,k}}(\tau - \sigma) q^2 } = e^{- \frac{1}{2} d_D (\tau - \sigma) q^2 } \label{eqn:mean_exp_III}\end{aligned}$$ where the component of frequency $\omega_{s,k}$ of the imaginary time diffusion $d_D(\tau)$ is ($A_{\gamma}=A_{\gamma}(\omega_{s,k}), \Omega_{\gamma}= \Omega_{\gamma}(\omega_{s,k})$) $$d_D(\tau) = \frac{1}{N D} \sum_{s,k} d_{\omega_{s,k}}(\tau) \nonumber$$ $$= \frac{1}{N D} \sum_{s,k} \frac{\hbar}{m} \sum_{\gamma} \frac{A_{\gamma}}{\Omega^2_{\gamma}} \frac{ \cosh\left( \beta \Omega_{\gamma} /2 \right) - \cosh \left( \Omega_{\gamma} [\beta/2 -\tau] \right)} { \sinh \left( \beta \Omega_{\gamma} \right) } \label{eqn:d_tau_II}$$ APPENDIX D: MEAN ELECTRONIC FLUCTUATION ======================================= The relation between the mean electronic fluctuation and the imaginary time diffusion $d_D (\tau)$ eq.(\[eqn:d\_tau\]) is $$\label{eqn:dif_sigma} d_D \left( \tau \right)= \frac{2}{D} \left[ < {\left\| \vec{u}(0) \right\|}^2 > - < \vec{u}(\tau) \cdot \vec{u}(0) > \right]$$ comparing eq.(\[eqn:dif\_sigma\]) and eq.(\[eqn:d\_tau\_II\]) for $d_D (\tau)$ and inserting the DOS function, we have $$\label{eqn:sigma2} \sigma^2_T = \frac{< {\left\| \vec{u} \right\|}^2 >}{D} = \int \!\!\! d \omega \rho(\omega) \sum^{3}_{\gamma=1} \frac{\hbar A_{\gamma}(\omega)}{2 m \Omega^2_{\gamma}(\omega)} \coth \left( \frac{\beta \Omega_{\gamma}(\omega)}{2} \right)$$ If we fix $w=\omega_o$, we have $\Omega_3=\omega_o$ for one solution of the cubic polynomial eq.(\[eqn:pol3\]) and by eq.(\[eqn:A\_factors\]) we have also $A_3=0$. The other two terms give $$\label{eqn:del2_+_eq} {\left< u^2 \right>}_{+} = \!\!\!\! \int \!\! d \omega \rho \left( \omega \right) \frac{\Omega^2_1 - \omega^2_{LO}}{\Omega^2_1 - \Omega^2_2} \frac{\hbar D}{2 m \Omega_1} \coth \left( \frac{\hbar \Omega_1}{2 k_B T} \right)$$ $$\label{eqn:del2_-_eq} {\left< u^2 \right>}_{-} = \!\!\!\! \int \!\! d \omega \rho \left( \omega \right) \frac{\Omega^2_2 - \omega^2_{LO}}{\Omega^2_2 - \Omega^2_1 } \frac{\hbar D}{2 m \Omega_2} \coth \left( \frac{\hbar \Omega_2}{2 k_B T} \right)$$ Notice that if we take a single Wigner frequency being representative of the electronic spectrum ($\rho (\omega)=\delta(\omega-\omega_W)$) we recover the results of ref. [@Simone_meanfield]. APPENDIX E: POLARON RADIUS ========================== We now calculate the density-density correlation function of the eq.(\[eqn:C\_self\]) for the variational harmonic action $\mathcal{S}_T$. We assume that the equilibrium position of the reference electron $\imath=1$ is the origin. With the same method to obtain eqs.(\[eqn:mean\_exp\_III\]), we performed the following Gaussian integrals for the density distribution $\rho_1 (\vec{r})$ $${\left< \hat{\rho}_1 (\vec{r}) \right>}_{T} = \int \!\! \frac{d^D q}{ {\left(2 \pi \right)}^D} e^{{\dot{\imath}}\vec{q} \vec{r}_e } {\left< e^{{\dot{\imath}}\vec{u}_1 \vec{q}} \right>}_{T} = \frac{e^{- r^2 / 2 \sigma^2_T }} { {\left[2 \pi \sigma^2_T \right]}^{D/2} }$$ and $$\label{eqn:2rela} {\left< e^{- {\dot{\imath}}\vec{q} \vec{u}_1 } e^{{\dot{\imath}}\vec{q'} \left( \vec{u}_1 (\tau) - \vec{u}_1 \right) } \right>}_{T} = e^{- \frac{\sigma^2_T}{2} q^2} e^{- \frac{d(\tau)}{2} \vec{q'} \cdot \left[\vec{q'} + \vec{q} \right]}$$ Inserting eq.(\[eqn:2rela\]) in eq.(\[eqn:C\_self\]), we have the density-density correlation function in the imaginary-time for the $\imath=1$ electron $$\label{eqn:rho_rho} {\left< \rho_1 (\vec{r}) \rho_1 (\vec{r'},\tau) \right>}_T = {\left< \hat{\rho}_1 (\vec{r}) \right>}_T \frac{ e^{ - \frac{{\left\| \vec{r} - \vec{r'} + \frac{ d(\tau)}{2 {\sigma}^{2}} \vec{r} \right\|}^{2}} {2\ell^2(\tau) } } } { {\left(2 \pi \ell^2(\tau) \right) }^{D/2}}$$ where $$\label{eqn:defll} \ell^2(\tau) = d_D(\tau) \left[ 1 - \frac{d_D(\tau)}{4 \sigma^2_T} \right]$$ We notice that the function of eq.(\[eqn:rho\_rho\]) does not depend only on the relative distance ${\vec{r}}^{'} - {\vec{r}}$ but also from the distance of electron from its localization position in the crystal.Then the eq.(\[eqn:C\_self\]) becomes $$\label{eqn:C_self_T} C^{self}_{1,T} = \frac{1}{{\bar{\varepsilon}}} \int^{\beta}_{0} \!\!\! d \tau \frac{\omega_{LO}}{2} D_o(\tau) \frac{ {\left< \rho_1 (\vec{r}) \rho_1 (\vec{r'},\tau) \right>}_{T} } {{\left< \rho_1 (\vec{r}) \right>}_{T}}$$ We assume $\vec{r} = 0$ (electron in its lattice point) and then we obtain the variational radial induced charge density $$\label{eqn:g_r_T} g_T(r) = \frac{\pi \omega_{LO}}{2{\bar{\varepsilon}}} {\left( 2 r \right)}^{D-1} \int^{\beta}_{0} \!\!\! d \tau D_o(\tau) \frac{ e^{ - r^2 / 2 \ell^2 (\tau)} } { {\left(2 \pi \ell^2(\tau) \right)}^{3/2} }$$ by eq.(\[eqn:R\_p\]) we obtain the variational polaron radius $$\label{eqn:R_p_T} R_{p,T} = {\left( \mbox{D} \; \frac{\omega_{LO}}{2} \int^\beta_0 \!\!\! d\tau D_o (\tau) \ell^2(\tau) \right)}^{1/2}$$ High density limit ------------------ The characteristic length $\ell^2(\tau)$ defined in eq.(\[eqn:defll\]) is expressed in term of $\tau$-dependent positional fluctuations $d_D(\tau)$, eq.(\[eqn:d\_tau\]), which is an integral of a function $d_{\omega_{s,k}} (\tau)$ weighted by the DOS $\rho (\omega)$ of the Wigner lattice, eq.(\[eqn:d\_tau\_II\]).To have an estimate of this integral we replace the integration by inserting an average frequency in the function $d_{\omega_{s,k}}$. We choose $\omega_P / \sqrt{{\varepsilon_{\infty}}}$ because it is the typical frequency of the electronic fluctuation in the crystal for the high density regime eq. (\[eqn:del2\_+high\_eq\]). Moreover we consider the low temperature limit ($k_B T \ll \hbar \omega_{P} / \sqrt{{\varepsilon_{\infty}}}$). Then from eq.(\[eqn:defll\]) we get the following estimate for $\ell^2(\tau)$ $$\label{eqn:ell2_high} \ell^2 \left(\tau \right) \simeq \frac{\hbar}{m \omega_P/ \sqrt{{\varepsilon_{\infty}}}} \left( 1 - e^{- 2 \frac{\omega_P}{\sqrt{{\varepsilon_{\infty}}}} \tau} \right)$$ The characteristic time scale of electronic diffusion in imaginary time is $\tau_{el} = {( \omega_P / \sqrt{{\varepsilon_{\infty}}} )}^{-1}$. The rising-time is $\tau_{el} = 1/ (2 \omega_P /\sqrt{{\varepsilon_{\infty}}})$. Therefore we have approximately $$\begin{aligned} \ell^2 (\tau) &\simeq& \frac{\hbar}{m} \tau \quad (\tau \ll \tau_{el}) \\ \ell^2 (\tau) &\simeq& \frac{\hbar}{2 m \frac{\omega_P}{\sqrt{{\varepsilon_{\infty}}}} } \quad (\tau \gg \tau_{el})\end{aligned}$$ Now in the variational polaron radius $R_{p,T}$ of eq.(\[eqn:R\_p\_T\]) another time scale appears $\tau_{ph}=\omega_{LO}^{-1}$ but at high density $\tau_{ph}>>\tau_{el}$. Now we can separate the lowest time scale $\tau_{el}$ contribution in the imaginary time integral so that we can approximate the integral in eq.(\[eqn:R\_p\_T\]) as $$\begin{aligned} \int^{\tau_{el}}_{0} \!\!\! d \tau D_o(\tau) \frac{ e^{ - \frac{r^2}{ 2 \ell^2 (\tau)} } } { {\left(2 \pi \ell^2(\tau) \right)}^{3/2} } &\simeq& D_o(0) \int^{\tau_{el}}_{0} \!\!\! d \tau \frac{ e^{ - \frac{r^2}{ 2 \frac{\hbar}{m} \tau } } } { {\left(2 \pi \ell^2(\tau) \right)}^{3/2} } \\ & = & \frac{m}{2 \pi \hbar} \frac{1}{r} \left( 1 - \mbox{erf} \sqrt{ \frac{r^2}{\left< u^2 \right>} } \right) \\ \int^{\beta}_{\tau_{el}} \!\!\! d \tau D_o(\tau) \frac{ e^{ - \frac{r^2}{ 2 \ell^2 (\tau)} } } { {\left(2 \pi \ell^2(\tau) \right)}^{3/2} } &\simeq& \frac{ e^{ - \frac{r^2}{ \frac{\hbar}{m \omega_P / \sqrt{{\varepsilon_{\infty}}} }} } } { {\left(2 \pi \left< u^2 \right> \right) }^{3/2} } \int^{\beta}_{\tau_{el}} \!\!\! d \tau D_o(\tau) \\ &\simeq& \frac{ e^{ - r^2 / 2 \left< u^2 \right> } } { { \left(2 \pi \left< u^2 \right> \right)}^{3/2} }.\end{aligned}$$ Collecting these results we get eq. 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--- abstract: 'We experimentally investigate the non-equilibrium steady-state distribution of the work done by an external force on a mesoscopic system with many coupled degrees of freedom: a colloidal crystal mechanically driven across a commensurate periodic light field. Since this system mimics the spatiotemporal dynamics of a crystalline surface moving on a corrugated substrate, our results show general properties of the work distribution for atomically flat surfaces undergoing friction. We address the role of several parameters which can influence the shape of the work distribution, e.g. the number of particles used to locally probe the properties of the system and the time interval to measure the work. We find that, when tuning the control parameters to induce particle depinning from the substrate, there is an abrupt change of the shape of the work distribution. While in the completely static and sliding friction regimes the work distribution is Gaussian, non-Gaussian tails show up due to the spatiotemporal heterogeneity of the particle dynamics during the transition between these two regimes.' address: - '$^1$2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany' - '$^2$Max-Planck-Institute for Intelligent Systems, Heisenbergstrasse 3, 70569 Stuttgart, Germany' author: - 'Juan Ruben Gomez-Solano$^{1,2}$, Christoph July$^1$, Jakob Mehl$^1$, and Clemens Bechinger$^{1,2}$' title: 'Non-equilibrium work distribution for interacting colloidal particles under friction' --- [Keywords]{}: non-equilibrium work fluctuations, stochastic thermodynamics of interacting particles, friction, colloidal crystals Introduction ============ ![(a) Schematic illustration of the sectional view of the experimental setup. (b) Snapshot of the equilibrium crystalline structure of the colloidal monolayer at $B=0.5$ mT, light potential depth $30k_B T$ and $v = 0$. The blue bar represents $20\, \mu\mathrm{m}$. See text for explanation.](figure_0.eps "fig:"){width="100.00000%"} \[fig:Fig0\] A basic concept for a system driven in a non-equilibrium process by the change of some external parameters is that of work. For mesoscopic systems, e.g. colloidal particles or biomolecules, the work spent in such a process becomes a fluctuating quantity which follows a probability distribution of finite width [@sekimoto_a; @seifert_a]. During the last two decades, various non-equilibrium work relations, above all the Jarzynski [@jarzynski_a] and the Crooks relations [@crooks], have been established and shown to restrict the shape of this probability distribution depending on the underlying specific features of both the system and the non-equilibrium process. From a more fundamental point of view, these relations refine the second law of thermodynamics at the mesoscopic scale. Further exact statements involving the applied work are rare, even though it has been demonstrated in the framework of stochastic thermodynamics [@sekimoto_a; @seifert_a; @jarzynski_b] that the aforementioned relations as well as different ones for other thermodynamic quantities [@evans; @gallavotti; @kurchan; @lebowitz; @hatano], namely entropy production and dissipated heat, can be derived from a broader perspective [@seifert_a; @seifert_b]. Experimental tests of non-equilibrium work relations have been carried out for a variety of different systems: for single colloidal particles in time-dependent harmonic [@wang; @imparato; @gomez_a] and non-harmonic potentials [@blickle_a; @imparato_b], biomolecules in folding-unfolding assays [@hummer; @liphardt; @collin; @gupta], mechanical torsion pendulums coupled to a heat bath [@douarche], and charge transitions in electronic devices [@garnier; @saira]. Common to all experimental systems studied so far is that they only consist of a small number of degrees of freedom, whose internal interactions are irrelevant. From an experimental point of view the reason for this is straightforward: controlling all external forces acting on a system of many coupled degrees of freedom during a non-equilibrium process in a well-defined way, i. e., measuring the applied work, is a huge challenge [@lander]. On the other hand, a non-equilibrium phenomenon where the concept of work plays a prominent role is friction. For atomically flat surfaces sliding against each other, friction results from the interplay between externally applied forces and the nonlinear interaction of a large number of degrees of freedom making up extended contacts at the interface. Inspired by simplistic models [@frenkel], a 2D system which has attracted much attention in recent years in the field of tribology in order to investigate in a controlled manner the spatiotemporal dynamics of crystalline surfaces under friction consists of a monolayer of interacting particles suspended in a fluid and mechanically driven through a periodic potential. Since experimental realizations [@bohlein] and numerical simulations [@vanossi; @hasnain] of this system have successfully shed light on the mechanisms behind friction, it represents also an appropriate model to investigate the statistical properties of the non-equilibrium work done by a well-controlled external force on a system composed of many interacting degrees of freedom. So far, this kind of analysis has only been numerically carried out to characterize plastic depinning of interacting particles within a stochastic thermodynamic context [@drocco]. Here, we experimentally study the fluctuations of the work done by an external force on a mesoscopic system with many coupled degrees of freedom: a crystalline monolayer of magnetically interacting colloidal particles moving on a periodic light field under commensurate conditions. The dynamics of this system mimics the transition from static to sliding friction, where a solid surface (the colloidal monolayer) is driven across a corrugated substrate (the periodic light potential) by an external force. We investigate the role of several parameters which can influence the shape of the work distribution, e.g. the number of particles used to locally probe the properties of the system and the integration time to measure the work. We find that, when tuning the control parameters to induce particle depinning from the substrate, there is an abrupt change of the shape of the work distribution. While in the completely static and sliding friction regime, the work distribution is Gaussian, non-Gaussian tails show up due to the spatiotemporal heterogeneity of the particle dynamics during the transition between these two regimes. Finally, we discuss the asymmetry of the work distribution within the context of the non-equilibrium fluctuation theorems. Experimental description ======================== ![(a) Trajectories of 50 neighboring particles measured over 200 s at constant potential depth $V_{max} = 30k_B T$ and different applied forces $f$. From left to right: $f=12.4$ fN, 18.6 fN and 24.8 fN. (b) Trajectories of 50 particles measured over 200 s at constant applied force $f = 12.4$ fN and different potential depths $V_{max}$. From left to right: $V_{max} = 30 k_BT$, $20 k_BT$ and $15k_BT$.](figure_1.eps "fig:"){width="100.00000%"} \[fig:Fig1\] Parts of the experimental setup have been described in detail elsewhere [@bohlein] and will be discussed only briefly. A schematic illustration of the experimental setup is shown in Fig. \[fig:Fig0\](a). The system consists of a monolayer of superparamagnetic colloidal particles with $2r = 4.5\,\mu\mathrm{m}$ in diameter (Dyna M-450 Epoxy, Life Technologies), suspended in a $2.3\,\mathrm{g/l}$ sodium dodecyl sulfate aqueous solution and situated in a sample cell of $10\,\mu\mathrm{m}$ height. The total number of particles forming the monolayer is $N \approx 5000$. Using video microscopy, we simultaneously track the center of mass of approximately 500 particles in the full field of view at 3.3 frames per second with a spatial accuracy of $40\,\mathrm{nm}$ [@grier]. The viscous drag coefficient of the particles in the solvent under this confinement, measured from their equilibrium mean-square displacement, is $\gamma = 6.2 \times 10^{-8}\, \mathrm{kg}\, \mathrm{s}^{-1}$. A coupling between the particles is obtained by a static homogeneous magnetic field ${\bf{B}} = B{\bf{e}}_z$ applied perpendicular to the sample plane. This field induces a repulsive dipole-dipole interaction $U(d)=\mu_0(\chi B)^2/(4\pi d^3)$ with $\mu_0$ the magnetic constant, $d$ the particle separation distance and $\chi\simeq3.1\times10^{-11}\mathrm{A}\, \mathrm{ m}^2\,\mathrm{T^{-1}}$ a constant, which allows to control the stiffness of the colloidal monolayer. By interference of three laser beams ($\lambda=1064\,\mathrm{nm}$) a light field with hexagonal symmetry inside the sample cell is generated, corresponding to a 2D periodic potential landscape $V$, whose maximum depth is $V_{max} = 30{k_B T}$. The potential profile is described by the function $$\label{eq:potentialprofile} V(x,y) = -\frac{2}{9} V_{max}\left[ \frac{3}{2} + 2 \cos \left( \frac{2 \pi x}{a} \right) \cos \left( \frac{2 \pi y}{\sqrt{3}a} \right) + \cos\left( \frac{4 \pi y}{\sqrt{3}a} \right) \right].$$ The lattice constant $a$ and the depth $V_{max}$ can be tuned by the intensities and the angles of incidence of the laser beams [@burns]. Before a measurement is performed, the colloidal monolayer is allowed to equilibrate at room temperature $T=298 \pm 0.5$ K for at least one hour in presence of a magnetic field of $B=0.5\,\mathrm{mT}$. The resulting homogeneous crystalline state with hexagonal symmetry exhibits a lattice constant of approximately $10\,\mu\mathrm{ m}$ with an interaction potential of $U(10\,\mu\mathrm{ m}) \simeq 5 {k_B T}$, as shown in Fig. \[fig:Fig0\](b). The lattice constant $a$ of the light potential $V$ is adjusted to the same value, i.e. we focus on commensurate conditions, in order to resolve the transition from static to sliding friction of the colloidal monolayer [@bohlein]. Dynamical response under applied force ====================================== ![Mean work done by the external force as a function on the number $N$ of particle trajectories counted in the monolayer for two different frictional regimes (a) $f = 24.8$ fN and $V_{max} = 30k_B T$ (stick-slip motion) and (b) $f = 12.4$ fN and $V_{max} = 15k_B T$ (complete sliding), computed over different time intervals: $\tau = 0.6$ s ($\circ$), 1.2 s ($\Box$), 2.4 s ($\diamond$) , 4.8 s ($\triangleleft$), 9.6 s ($\triangleright$), 19.2 s ($\times$), and 38.4 s ($+$). The dotted lines represent Eq. (\[eq:meanwork\]), with no fitting parameters.](figure_n.eps "fig:"){width="100.00000%"} \[fig:Fign\] The whole sample cell is displaced in the x-direction with velocity $v$ by use of a piezo table in order to move the particles across $V$, as sketched in Fig. \[fig:Fig0\](a). The movement allows to create a controlled homogeneous flow ${\bf v}=v{\bf e}_x$ and to drive the particles into non-equilibrium steady states (NESS). By tuning the value of $v$ we can mimic the response of the colloidal crystal to the effective external force ${\bf f} \equiv \gamma {\bf v}$. The reason of this choice is that many tribological processes take place under these conditions [@carpick; @lee; @vanossi2], where an external force is applied to a crystalline surface with the purpose of moving it against a substrate, and where the surrounding fluid, in our case the solvent, only plays the passive role of a thermal bath to keep the system at fixed temperature. In order to induce a transition of the monolayer in response to $\bf{f}$ from static to sliding friction, we perform two different experimental protocols. In the first, we keep the depth of the light potential at its maximum value $V_{max} = 30k_B T$ and then we displace the cell at different velocities to tune the magnitude of the applied force $f = \|{\bf{f}}\|$. In Fig. \[fig:Fig1\](a) we plot some trajectories of 50 neighboring particles moving according to this protocol. At small velocities, corresponding to values of $f$ much smaller than the maximum restoring force exerted by the light potential (\[eq:potentialprofile\]), $\max \{ - \nabla V \} = 8\pi V_{max}/(9a) \approx 34\,\mathrm{fN} $, the particles remained pinned by the potential wells, as shown in the left panel of Fig \[fig:Fig1\](a) for $f = 12.4\,\mathrm{fN}$ ($v = 200\, \mathrm{nm}\,\mathrm{s}^{-1}$). Note that, even when immobile in average, the position of each individual particle fluctuates due to the collision of the solvent molecules. As $f$ approaches values comparable to or larger than $8\pi V_{max}/(9a) $, i.e. when overcoming static friction, the particles are able to move across the potential barriers, thus resulting in collective motion. For example, at $f = 18.6\,\mathrm{fN}$ ($v = 300\, \mathrm{nm}\,\mathrm{s}^{-1}$), the monolayer is partially depinned and some of the particles start to move in the direction of ${\bf{f}}$, as shown in the central panel of Fig. \[fig:Fig1\](a). For $f = 24.8\,\mathrm{fN}$ ($v = 400\, \mathrm{nm}\,\mathrm{s}^{-1}$, right panel of Fig. \[fig:Fig1\](a)), all the particles in the monolayer are able to move in the direction of ${\bf f}$. Under these conditions, the particles undergo stick-slip motion because their mobility is hindered by the presence of the potential barriers [@bohlein; @lee], resulting in a mean particle velocity $105\,\mathrm{nm}\,\mathrm{s}^{-1} < v$. In the second protocol, we fix the velocity of the sample at $v = 200\, \mathrm{nm}\,\mathrm{s}^{-1}$, i.e. at constant $f = 12.4 \,\mathrm{fN}$, and we vary the depth $V_{max}$ of the light potential. The response of the particles to this protocol with decreasing values of $V_{max}$ is qualitatively similar to that observed when increasing $f$ at constant $V_{max}$ and also has a transition from a static to a sliding friction regime. This effect can be observed in Fig. \[fig:Fig1\](b) for potentials depths $30\,k_BT$, $20\,k_BT$ and $15\,k_BT$, at which the maximum restoring force $8\pi V_{max}/(9a)$ has the values 34 fN, 23 fN and 17 fN, respectively. We point out that, although qualitatively similar, the particle dynamics resulting from these two protocols are not completely equivalent. Indeed, close inspection of the trajectories in Fig. \[fig:Fig1\](a) and \[fig:Fig1\](b) shows that in the second protocol, Brownian motion is more significant and the particle mobility is higher because the local confinement created by the substrate is reduced when decreasing $V_{max}$. Therefore, the second protocol reproduces the effect of changing the roughness of the substrate, which in turn results in higher particle velocities compared to the first protocol for the same values of the parameter $9fa/(8\pi V_{\max})$. For instance, at $9fa/(8\pi V_{\max}) = 0.72$, the mean particle velocity obtained by means of the first protocol is only 26% of the velocity $v$ of the sample cell (right panel of Fig. \[fig:Fig1\](a)), whereas in the second protocol, it almost reaches free sliding at 93% of $v$ (right panel of Fig. \[fig:Fig1\](b)). Stochastic Thermodynamics of the monolayer under applied force ============================================================== We focus on the work done on the colloidal monolayer of $N$ interacting particles driven across the corrugation potential $V$ by an external force $\bf{f}$, which is the common situation encountered in many tribological problems [@carpick; @lee; @vanossi2]. We fist present the equations of motion for our specific experimental protocol under flow $\bf{v}$, which allows to mimic in a controlled manner the dynamics under applied force $\bf{f}$. Then, we derive the corresponding stochastic-thermodynamic quantities of the latter tribological process. When the $i$-th particle ($i=1,...,N$) moves at instantaneous position ${\bf{r}}_i = (x_i,y_i)$ and velocity $\dot{{\bf{r}}}_i = (\dot{x}_i, \dot{y}_i)$ in presence of flow $\bf{v}$, the viscous drag force relative to the flow is $\gamma(\dot{{\bf{r}}}_i - {\bf v})$. In our system, there is no actual external force but only conservative forces derived from the magnetically-induced repulsive interactions and the periodic light field. In addition, each particle is subject to the random thermal collisions of the solvent molecules. Therefore, the dynamics of the $i$-th particle is described by the Langevin equation $$\label{eq:langevin} \gamma(\dot{{\bf r}}_i -{\bf v}) = -{ \nabla_i}E+{\bf{\xi}}_i,$$ where $E$ is the total potential energy of the system, which includes the light potential $V$, the pair-interaction potential $U$ of all the particles and the confining potential exerted by the sample cell, $V_{conf}$, which maintains the monolayer in a packed configuration and prevents the particles at the boundaries from escaping from the monolayer due to the repulsive interactions $$\label{eq:potential} E = \sum_{i=1}^N V({\bf{r}}_i) + \frac{1}{2} \sum_{i = 1}^N \sum_{j \neq i} U(\|{\bf{r}}_i - {\bf{r}}_j\|) + V_{conf},$$ whereas the fast interactions with the surrounding solvent molecules are modeled by a Gaussian white noise ${ \xi_i}$ of zero mean and correlations $\langle{ \xi_i}(t){ \xi_j}^T(t')\rangle=2\gamma k_B T\delta(t-t') \delta_{ij}$. Because of the structure of Eq. (\[eq:langevin\]), the dynamics of every particle in response to the flow and in absence of an external force [@mehl] is equivalent to the dynamics in response to an external uniform force, ${\bf{f}} \equiv \gamma {\bf v}$, and without external flow[^1] $$\label{eq:langevinforce} \gamma\dot{\bf r}_i = {\bf{f}} -{ \nabla_i}E+{ \xi_i}.$$ Therefore, hereafter we will focus only on Eq. (\[eq:langevinforce\]) in order to study the stochastic thermodynamics of a the monolayer under external constant force and without flow. In the context of stochastic thermodynamics, the first law for the potential energy variation along a single stochastic realization of the dynamics of the system can be written as [@sekimoto_a; @sekimoto_b] $$\begin{aligned} \label{eq:firstlaw} \mathrm{d}E & = & \sum_{i=1}^N \nabla_i E \cdot \mathrm{d} {\bf{r}}_i, \nonumber\\ & = & \mathrm{d}W - \mathrm{d}Q,\end{aligned}$$ where $\mathrm{d}Q$ and $\mathrm{d}W$ are the heat dissipated into the solvent and the work done by ${\bf{f}}$, respectively, and are given by $$\begin{aligned} \label{eq:workheat} \mathrm{d}Q & = & \sum_{i=1}^N [{\bf{f}} - \nabla_i E ] \cdot \mathrm{d} {\bf{r}}_i, \nonumber\\ \mathrm{d}W & = & \sum_{i=1}^N {\bf{f}} \cdot \mathrm{d} {\bf{r}}_i.\end{aligned}$$ Then, from Eq. (\[eq:workheat\]) the work done by the uniform force ${\bf f}$ on the colloidal monolayer, normalized by $k_B T$, over the time interval $[0,\tau]$ reads $$\begin{aligned} \label{eq:workforce} w_{\tau} & = & \frac{1}{k_B T}\int_0^{\tau}\sum_{i=1}^N {\bf{f}} \cdot \dot{{\bf{r}}}_i\,\mathrm{d}t \,, \nonumber \\ & = & \frac{f}{k_B T}\sum_{i=1}^N[{x}_i(\tau) -x_i(0) ].\end{aligned}$$ It should be noted that the expression of the work in Eq. (\[eq:workforce\]) only involves the value of the force $f$, which can be tuned experimentally by means of $v$, and the instantaneous values of the x-coordinates of each particle, which are determined by videomicroscopy. Consequently, the work can be directly determined from the particles’ trajectories without the need to measure the pair interactions. From Eq. (\[eq:workforce\]), we can conclude that, regardless of the nature of the pair interactions, the mean value of the NESS work done by $\bf{f}$ over a time interval of duration $\tau$ can be expressed as $$\label{eq:meanwork} \mu_{\tau} \equiv \langle w_{\tau}\rangle = \frac{Nf\langle \dot{x} \rangle \tau}{k_B T} .$$ where the brackets stand for an ensemble average over $N$ particle trajectories and $\langle \dot{x} \rangle = \frac{1}{N} \sum_{i=1}^N \dot{x}_i$ is the drift particle velocity in response to $\bf{f}$. We check that Eq. (\[eq:meanwork\]) is valid in all the frictional regimes investigated in our experiments [^2]. For instance, it is trivially satisfied for static friction, where $\langle \dot{x} \rangle = 0$ yields $\mu_{\tau} = 0$ because in average no mechanical work is done by $\bf{f}$ on the monolayer. On the other hand, for $\langle \dot{x} \rangle > 0$ the linearity of $\mu_{\tau}$ with respect to $N$ and $\tau$ predicted by Eq. (\[eq:meanwork\]) is also verified. For example, in Figs. \[fig:Fign\](a) and \[fig:Fign\](b) we plot for different integration times $\tau$ the value of the mean work $\mu_{\tau}$ for stick-slip motion and free sliding, respectively, as a function of the number $N$ of particle trajectories used in the computation of $w_{\tau}$. In this case, $\mu_{\tau}$ is determined by taking the average over all possible values of $w_{\tau}$ at fixed $N$ and $\tau$. We also plot as dotted lines the values of $\mu_{\tau}$ computed by means of Eq. (\[eq:meanwork\]), where the NESS drift velocity $\langle \dot{x} \rangle$ is independently determined from the particle dynamics. We observe that the agreement between both kinds of calculations is excellent. Therefore, from the validity of Eq. (\[eq:meanwork\]) we conclude the mean work mirrors the bulk frictional properties of the monolayer, namely a smooth transition from $\mu_{\tau} = 0$ (static friction with zero mobility at small $f$) to $\mu_{\tau} \propto f^2$ (sliding friction with constant finite mobility at sufficiently large $f$) [@bohlein]. ![Probability density function of the work done by a constant force $f = 12.4$ fN over $\tau=3.6$ s on subsystems composed of different number $n$ of particles across two potentials of depths (a) $V_{max} = 30k_BT$ and (b) $V_{max} = 15k_ BT$. From top to bottom in \[fig:Fig2\](a) and from left to right in \[fig:Fig2\](b): $n=1$ (dark blue), 5 (light blue), 10 (cyan), 25 (dark green), 50 (light green), 100 (orange), and 250 (red). Insets: work distribution rescaled according to Eq. (\[eq:rescaling\]). The symbols correspond to $n=1$ ($+$), 5 ($$), 10 ($\circ$), 25 ($\times$), 50 ($\Box$), 100 ($\bigtriangledown$), and 250 ($\triangleleft$). The solid lines represent the rescaled Gaussian distributions for non-interacting particles given by Eqs. (\[eq:case1\]) and (\[eq:case2\]), respectively.](figure_2.eps "fig:"){width=".8\textwidth"} \[fig:Fig2\] Non-interacting particles {#subsect:Noninteracting} ------------------------- In principle, the fluctuations of $w_{\tau}$ depend on the strength of the repulsive interactions, the number of particles $N$, the integration time $\tau$, the force ratio $9fa/(8\pi V_{max})$ and the depth $V_{max}$ of the substrate potential. Nevertheless, using the Langevin model of Eq. (\[eq:langevinforce\]), we can gain some insight into the statistical properties of the work by analyzing two limit ideal cases which bear resemblance to static and sliding friction, respectively. The first case corresponds to $N$ non-interacting particles moving under the influence a very weak force $f \ll 8\pi V_{max}/(9a)$ in presence of a very high potential barrier $V_{max} \gg {k_B T}$, such that the inverse Kramers rate of each particle becomes much larger than the other characteristic time-scales of the system. In this case, which resembles static friction conditions, the system is in a quasi-equilibrium state, where the particles are pinned by the potential wells at an average distance $9fa^2/(16 \pi^2 V_{max})$ from the minima in order to balance the external force $f$. The probability density function of the work $w_{\tau}$ is Gaussian, i.e. $P(w_{\tau}) = \frac{1}{\sqrt{2\pi\sigma_{\tau}^2}} \exp\left[-\frac{(w_{\tau} - \mu_{\tau})^2}{2\sigma_{\tau}^2}\right]$, with mean $\mu_{\tau}$ and variance $\sigma_{\tau}^2$ given by $$\begin{aligned} \label{eq:case1} \mu_{\tau} & = & 0 \nonumber,\\ \sigma_{\tau}^2 & = & \frac{2N f^2 }{ k_B T k} \left[ 1 - \exp\left( -\frac{ k \tau}{\gamma } \right) \right],\end{aligned}$$ where $k = [4\pi/(3a)]^2V_{max}$ is the effective stiffness of the restoring force exerted by a periodic light potential with hexagonal symmetry (\[eq:potentialprofile\]). Note that, while $\mu_{\tau} = 0$ because no mechanical work is done in average by ${\bf{f}}$, $\sigma_{\tau}^2$ is non-zero. This is due to the thermal fluctuations of the solvent molecules, which can promote either positive or negative work fluctuations by randomly moving the particles with or against the applied force. The second ideal case is when $N$ non-interacting particles are driven by a sufficiently large force $f \gg 8\pi V_{max}/(9a)$, such that they move at the highest possible average velocity $\langle \dot{x} \rangle = f/\gamma$, where the influence of the periodic potential is negligible, similar to free sliding friction. In such a case, the probability density function of $w_{\tau}$ is also Gaussian, with mean and variance $$\begin{aligned} \label{eq:case2} \mu_{\tau} & = & \frac{N f^2}{k_B T\gamma} \tau \nonumber,\\ \sigma_{\tau}^2 & = & \frac{2 N f^2}{k_B T\gamma} \tau = 2 \mu_{\tau},\end{aligned}$$ respectively. We point out that only in this particular case, the non-equilibrium work trivially satisfies the detailed steady-state fluctuation theorem [@seifert_a] $$\label{eq:FT} \ln \frac{P(w_{\tau} = w)}{P(w_{\tau} = -w)} = \frac{2 \mu_{\tau}}{\sigma_{\tau}^2} w = w,$$ because $w_{\tau}$ is actually equal to the total entropy production of the system, normalized by $k_B$. In the following, we discuss how the previous ideal expressions for $P(w_{\tau})$ compare to the experimental work distributions for interacting particles in the static and sliding friction regimes. Furthermore, we also investigate the work distribution in the intermediate regime when tuning the control parameters to induce a transition from static to sliding friction of the colloidal monolayer. Work distribution for interacting particles under applied force =============================================================== ![(a) Probability density function of the work done by a constant force $f = 12.4$ fN on $n=50$ particles against a periodic light potential of depth $V_{max} = 30k_BT$ over different time intervals: $\tau = 0.6$ s ($+$), 1.2 s ($\ast$), 2.4 s ($\bullet$), 4.8 s ($\times$), 9.6 s ($\Box$), 19.2 s ($\bigtriangledown$), and 38.4 s ($\triangleleft$). The solid line represents the case of non-interacting particles under the same $V_{\max}$ and $f$. Inset: Standard deviation of $w_{\tau}$ as a function of $\tau$ for non-interacting (solid line) and magnetically coupled (dashed line) particles (b) Probability density function of the work done by a constant force $f = 12.4$ fN on $n=50$ particles against a light potential of depth $V_{max} = 15 k_BT$ over different time intervals. From left to right: $\tau = 0.6$ s, 1.2 s, 2.4 s, 4.8 s, 9.6 s, 19.2 s, and 38.4 s. Inset: work distribution measured over different $\tau$ and rescaled according to Eq. (\[eq:rescaling\_time\]). Same symbols as in Fig. \[fig:Fig3\](a). The solid line represents the case of non-interacting particles.](figure_3.eps "fig:"){width=".75\textwidth"} \[fig:Fig3\] Subsystem size {#subsect:size} -------------- Since only a portion of the complete monolayer of $N \approx 5000$ particles is accessible for data analysis, a frequent problem encountered in spatially extended systems [@ayton; @shang; @michel], we first investigate the effect of measuring the work done on a smaller subsystem composed of $n < N$ particles, thus ignoring its coupling with the $N - n$ degrees of freedom of the rest of the system. A possible way to probe the role of such a coupling is by means of the differences between the statistical properties of the work applied on a subsystem of $n$ interacting particles with those observed in a subsystem of the same size $n$ of non-interacting ones, where there is no coupling. Note that in absence of interactions, the work is a Gaussian variable with mean and variance proportional to the number of components $n$ of the subsystem for the two limit cases described by Eqs. (\[eq:case1\]) and (\[eq:case2\]), i.e. the width of the distribution scales in both cases as $\sigma_{\tau} \propto \sqrt{n}$. Therefore, upon translating the work distribution to the origin by an amount $\mu_{\tau}$ and then squeezing it by its width $$\label{eq:rescaling} w^_{\tau} = \frac{w_{\tau} - \mu_{\tau}}{\sqrt{n}}, \,\,\, P^(w^_{\tau}) = \sqrt{n} P(\sqrt{n} w^_{\tau} + \mu_{\tau})$$ any subsystem composed of $n$ non-interacting particles exhibits a $n$-independent profile $ P^(w^_{\tau})$. This means that the statistical properties of the work done on the whole system can be probed by measuring the work done on any subsystem of arbitrary size. This situation can change drastically in presence of particle interactions, though. As discussed in [@michel; @mehl], because of the spatio-temporal correlations created by the interactions between the subsystem and the surroundings, the undercount of slow degrees of freedom can give rise to strong modifications of the statistical properties of the thermodynamic quantities of the subsystem with respect to those of the complete system. Then, it is not expected that the variance scales as $\sigma_{\tau}^2 \propto n$ for sufficiently small $n$ in presence of interactions. The effect of the coupling with the surroundings only vanishes when the size of the sampling subsystem spans a length-scale larger than the typical correlation length induced by the interactions, thus recovering the actual statistical properties of the complete system [@michel]. Indeed, in presence of repulsive interactions we observe this kind of non-trivial dependence of the work distribution on the number $n$ of NESS trajectories used to compute $w_{\tau}$ from Eq. (\[eq:workforce\]) in both static and free sliding frictional regimes. In Fig \[fig:Fig2\](a) we plot the probability density function $P(w_{\tau})$ of the work computed over $\tau = 3.6$ s for subsystems composed of different number $n$ of particles ($n=1,5,10,25,50,100, 250$) at $f = 12.4\,\mathrm{fN}$ and $V_{max} = 30\,k_B T$, for which all the particles in the monolayer are pinned by the potential wells over the observation times accessible in the experiment, see left panel of Fig. \[fig:Fig1\](a). Each subsystem is chosen in such a way that $n$ neighboring particles cover an approximately square area $\approx n a^2$. We find that for all the values of $n$, $P(w_{\tau})$ is symmetric and peaked around $w_{\tau} = 0$ because no work is done in average in this quasi-equilibrium state, whereas its width increases with increasing $n$. In the inset of Fig. \[fig:Fig2\](a) we plot the work distribution rescaled according to Eq. (\[eq:rescaling\]). We observe that $P^(w_{\tau}^)$ has a Gaussian profile, and unlike the case of non-interacting pinned particles, its width increases with increasing $n$. This implies that the variance $\sigma_{\tau}^2$ grows faster than $n$ in presence of repulsive interactions for sufficiently small $n$, an indication that the correlations between the subsystem and the $N-n$ surrounding particles are significant. Nevertheless, for sufficiently large values of $n$, we find that $P^(w_{\tau}^)$ seems to converge to a size-independent profile, as shown in the inset of Fig. \[fig:Fig2\](a) for $n \ge 100$. The convergence demonstrates that finite-size effects due to the spatial correlations between the sampling subsystem and rest of the particles in the monolayer become negligible compared to the global behavior of $w_{\tau}$ for sufficiently large $n$. However, the effect of the particle interactions on the fluctuations of $w_{\tau}$ persists even for sufficiently large $n$. As a matter of fact, when comparing the experimental $P^(w_{\tau}^)$ for $n=100$ and $250$ with that computed from Eq. (\[eq:case1\]) with $k=2.2\times10^{-8}\,\mathrm{N}\,\mathrm{m}^{-1}$ for non-interacting particles (solid line in the inset of Fig. \[fig:Fig2\](a)), we find that the former are much wider than the latter. This suggest that, with increasing numbers of particles $n$, the randomness created by the strongly non-linear coupling accumulate, giving rise to fluctuations of $w_{\tau}$ larger than those that would be otherwise observed in absence of interactions. A different behavior is observed for free sliding, where all the particles are able to move across the potential landscape at a mean velocity close to $v = f/\gamma$, as those shown in the right panel of Fig. \[fig:Fig1\](b). An example of such a behavior is shown in Fig. \[fig:Fig2\](b) where we plot the probability density function $P(w_{\tau})$ of the work done by a force $f = 12.4$ fN on subsystems formed by different number of particles, $n = 1,5,10,25,50,100, 250$, across a potential of depth $V_{max} = 15 k_B T$. In this case, the work distribution is Gaussian, whose maximum is located at positive values of $w_{\tau}$, because the applied force is able to perform mechanical work by moving the monolayer. The mean work, which coincides with the location of the maximum of $P(w_{\tau})$, increases linearly with increasing $n$, in quantitative agreement with Eq. (\[eq:meanwork\]), as shown in Fig. \[fig:Fign\](b). On the other hand, the presence of interactions affects the behavior of the fluctuations of $w_{\tau}$ compared to the ideal sliding case described by Eq. (\[eq:case2\]). In order to highlight these differences, in the inset of Fig. \[fig:Fig2\](b) we plot the rescaled work distribution $P^(w^_{\tau})$ defined in Eq. (\[eq:rescaling\]). Once more, the effect of the correlation between the subsystem and the rest of the monolayer can be observed for small values of $n$, for which the width of the rescaled distribution increases with $n$. However, for $n > 25$, $P^(w^_{\tau})$ converges to a $n$-independent profile, thus probing the actual statistical properties of $w_{\tau}$ for the complete system. This convergence implies that the variance of the work scales as $\sigma_{\tau}^2 \propto n$ for sufficiently large $n$. In the inset of Fig. \[fig:Fig2\](b) we also plot as a solid line the rescaled work distribution of non-interacting sliding particles, described by Eq. (\[eq:case2\]). Interestingly, we find that the width of the rescaled work distribution in the presence of interactions is smaller than that of the non-interacting case. We can interpret this narrowing as a reduction of the work fluctuations due to the repulsive interactions, which give rise to an effective stiffening of the monolayer, thus preventing large random excursion of the particles induced by the thermal fluctuations around the drift imposed by $\bf{f}$. This is consistent with the fact that for a perfectly stiff colloidal crystal, which can be realized in the limit of infinitely strong repulsive interactions, thermal fluctuations are suppressed [@hasnain], which gives rise to a complete sharpening of the work distribution around the mean value of Eq. (\[eq:meanwork\]). Integration time {#sect:time} ---------------- We now focus on the dependence of the probability density function of the work on the integration time $\tau$. We point out that for values of $\tau$ smaller than the relaxation time-scales of the system, time-correlations can affect also the statistical properties of the work, because the expression of $w_{\tau}$ in Eq. (\[eq:workforce\]) involves differences at distinct times of the particle positions. Nonetheless, for sufficiently large values of $\tau$, such time-correlations vanish and therefore the shape of the work distribution must converge to a single profile upon time rescaling. We first show in Fig. \[fig:Fig3\](a) the results for the case of a pinned colloidal monolayer, where in average no mechanical work is done. Here we plot the probability density function of the work $w_{\tau}$ done on $n=50$ particles by a force $f = 12.4$ fN against a light potential of depth $V_{max}=30k_B T$ over different integration times, $0.6\,\mathrm{s}\le \tau \le 38.4 \, \mathrm {s}$. We observe that, for all the values of $\tau$. $P(w_{\tau})$ is Gaussian and centered around $w_{\tau}=0$, whose width increases with increasing $\tau$. However, for $\tau > 9.6$ s, the width of the distribution levels off and all curves collapse onto a master curve regarless of $\tau$. This is further verified in the inset of Fig. \[fig:Fig3\](a), where we plot as a dashed line the dependence of the standard deviation $\sigma_{\tau}$ of the work on $\tau$, observing a saturation to a constant value at sufficiently large $\tau$. The dependence of $\sigma_{\tau}$ on $\tau$ is qualitatively similar to that for non-interacting particles, shown as a solid line in the inset of Fig. \[fig:Fig3\](a). In this case, according to Eq. (\[eq:case1\]), the variance of the Gaussian work distribution levels off exponentially for integration times larger than the viscous relaxation time of the particles in the potential wells, $\gamma/k = 2.8$ s. This behavior of $w_{\tau}$ can be actually understood at the single-particle level. For $\tau \ll \gamma/k$, the particle motion is strongly auto-correlated in time due to the energy stored by the confining light potential, which translates into a very narrow distribution $P(w_{\tau})$. The motion becomes less and less correlated when $\tau$ approaches $\gamma/k$, and therefore each particle is able to perform larger Brownian displacements within the potential wells both with and against the applied force, thus resulting in a broadening of $P(w_{\tau})$. Nevertheless, the fluctuations of $w_{\tau}$ cannot grow indefinitely with increasing $\tau$ because the single-particle motion is always bounded to the potential wells, giving rise to a saturation of $\sigma_{\tau}$ for $\tau >\gamma/k$ . Although qualitatively similar as a function of $\tau$, we observe a quantitative difference at $\tau \gg \gamma/k$ between the standard deviation of $w_{\tau}$ for interacting particles with respect to that in the non-interacting case, as shown in the inset Fig. \[fig:Fig3\](a). This difference is due to the strong coupling between the particles, which gives rise to a complex non-linear particle dynamics within the potential wells. In Fig. \[fig:Fig3\](b) we illustrate the dependence of the work distribution $P(w_\tau)$ on the integration time $\tau$ for the sliding friction regime of $n=50$ particles driven at $f=12.4$ fN and $V_{max}=15k_BT$. We find that $P(w_\tau)$ is Gaussian and the location of the maximum increases linearly with increasing $\tau$ in accordance with Eq. (\[eq:meanwork\]). Once more, inspired by the comparison with non-interacting particles, where the width of the distribution scales as $\sigma_{\tau} \propto \sqrt{\tau}$ (see Eq. (\[eq:case2\])), we can test a scaling with respect to $\tau$ similar to Eq. (\[eq:rescaling\]) $$\label{eq:rescaling_time} w^_{\tau} = \frac{w_{\tau} - \mu_{\tau}}{\sqrt{\tau}}, \,\,\, P^(w^_{\tau}) = \sqrt{\tau} P(\sqrt{\tau} w^_{\tau} + \mu_{\tau}).$$ Interestingly, in the inset of Fig. \[fig:Fig3\](b) we show that the work distributions, rescaled according to Eq. (\[eq:rescaling\_time\]), collapse onto a master curve for all $\tau$. This essentially means that in this frictional regime the variance of the work scales as $\sigma_{\tau}^2 \propto \tau$ even in presence of particle interactions. Note that in this case, the particles are not confined to move in the potential wells, and consequently there is no intrinsic relaxation time in the dynamics, which explains the very fast convergence of $P^(w^_{\tau})$ to the master curve. Quantitative differences are observed between the experimental $P^(w^_{\tau})$ and the case without interactions (solid line in the inset of Fig. \[fig:Fig3\](b)), though. This occurs due to the narrowing of the work distribution due to the effective stiffening of the monolayer. Depinning transition -------------------- We now show how the shape of the work distribution changes between the two very distinct cases previously studied, i.e. when changing the experimental parameters to induce a transition from the regime where all the particles are pinned on the substrate, to the depinning of the colloidal monolayer and subsequent free sliding. We recall that in the two extreme regimes of static and sliding friction, the work distribution is Gaussian, even though the mean and variance behave differently as a function of $n$ and $\tau$. While in static friction these quantities also depend strongly on both the elastic stiffness $k$ exerted by the substrate potential and the repulsive pair-interactions, they are only affected by the strength of the interactions for sliding friction. In Fig. \[fig:Fig4\](a) we illustrate the effect on the shape of the work distribution for $n=50$ particles, computed over $\tau=9.6$ s, when increasing the value of the applied force $f$ at constant potential depth $V_{max} = 30k_B T$ in order to induce particle depinning. Interestingly, we observe that the work distribution becomes asymmetric with respect to the maximum with increasing $f$, as can be observed for $f=18.6$ fN and $f=24.8$ fN. In particular, non-Gaussian tails appear at positive values of $w_{\tau}$, as highlighted in the semilog plot of the inset of Fig. \[fig:Fig4\](a). For these values of $f$, the spatio-temporal dynamics of the monolayer becomes heterogeneous, as can be observed from the particle trajectories of Fig. \[fig:Fig1\](a). For example, for $f=18.6$ fN there are regions where the particles are still confined by the potential wells, because the external force is still smaller than the maximum conservative force exerted by the light field: $9fa/(8\pi V_{max})=0.54$. However, the combination of thermal fluctuations and non-linear repulsive interactions can promote hops to the neighboring potential wells, thus creating collective motion of clusters of particles. The collective motion is in turn facilitated by the symmetry beaking induced by the external force. While the stagnant particles do not contribute to the mean value of the work but only to the fluctuations around $w_{\tau} =0$, the non-Gaussian tails originate from the work done on the sliding particles. This regime persists even when the complete monolayer can slide across the periodic substrate, as observed at $f=24.8$ fN and $V_{max} = 30k_B T$, where the particles undergo stick-slip motion. In this case, the spatial heterogeneity is induced by the corrugation potential, creating zones around the potential minima where the particles slow down, whereas they move faster when overcoming the potential barriers, as illustrated by the trajectories in the right panel of Fig. \[fig:Fig1\](a). Note that this heterogeneity results also in a mean particle velocity $\langle \dot{x} \rangle = 105\,\mathrm{nm}\,\mathrm{s}^{-1}$ much smaller than the maximum velocity that could be achieved in presence of a completely flat substrate ($v = 400\, \mathrm{nm}\,\mathrm{s}^{-1}$). Gaussianity of $w_{\tau}$ is recovered at sufficiently large $9fa/(8\pi V_{max})$, though, with a respective narrowing of $P(w_{\tau})$. This can be observed in Fig. \[fig:Fig4\](b), where we plot $P(w_{\tau})$ for the second experimental protocol with which we can reach more easily the free sliding regime. A reduction of only $5k_B T$ in the potential depth $V_{max}$, from $20k_B T$ to $15k_BT$, is enough to observe a prominant change of the shape of the work distribution, as shown in the inset of Fig. \[fig:Fig4\](b). For this values of $V_{max}$, the mean particle velocity changes from $74\,\mathrm{nm}\,\mathrm{s}^{-1}$ to $185\,\mathrm{nm}\,\mathrm{s}^{-1}$, whereas the maximum velocity that could be achieved for this value of $f$ on a completely flat surface is $200\, \mathrm{nm}\,\mathrm{s}^{-1}$. This dramatic change in the shape of $P(w_{\tau})$ reveals that not only the average tribological properties of the monolayer [@bohlein] but also the properties of the fluctuations of the work done on it become very sensitive when tuning the experimental parameters close to the depinning transition. Asymmetry of the non-equilibrium work distribution ================================================== ![(a) Probability density function of the work done by different external forces $f$ on $n=50$ particles against a potential of depth $V_{max}= 30k_BT$, computed over $\tau=9.6$ s. From left to right: $f=12.4$ fN, 18.6 fN, and 24.8 fN. (b) Probability density function of the work done by an external force $f = 12.4$ fN on $n=50$ particles against potentials of different depth $V_{max}$, computed over $\tau=9.6$ s. From left to right: $V_{max} = 30k_BT, 25k_BT, 20k_BT$, and $15k_BT$. The insets are semi-logarithmic representations of the same plots in the main figures.](figure_4.eps "fig:"){width=".8\textwidth"} \[fig:Fig4\] Finally, we investigate the work distribution $P(w_{\tau})$ within the context of non-equilibrium work relations. More specifically, we focus on the evaluation of quantity $\ln \frac{P(w_{\tau} = +w)}{P(w_{\tau} = - w)}$, which quantifies the asymmetry of the probability of observing positive work fluctuations, where the monolayer moves in the direction of the applied force, with respect to the probability of observing rare negative fluctuations, where the monolayer moves against the force. We point out that, although empirically satisified in many steady-state complex systems [@drocco; @aumaitre; @zamponi; @majumdar; @joubaud2; @hayashi; @gradenigo; @naert; @jimenez; @kumar], a simple linear relation such as Eq. (\[eq:FT\]) for the asymmetry function, $\ln \frac{P(w_{\tau} = +w)}{P(w_{\tau} = - w)} \propto w$, is not expected to hold generally for the NESS system we study. Indeed, for this kind of frictional processes, such a linear asymmetry relation is strictly valid only for the work done by an external force on a collection of non-interacting particles freely sliding on a perfectly flat substrate. In such a case, Eq. (\[eq:FT\]) is a direct consequence of the detailed Fluctuation Theorem, which only applies to the total entropy production of the system, and which in that specific case equals the work done by an external force. On the other hand, since the work defined by Eq. (\[eq:workforce\]) has a definite parity under time-reversal, it must satisfy a generalized Fluctuation Theorem in presence of the corrugation potential, the pair-interactions and the global confining potential of the sample cell [@seifert_a] $$\label{eq:detailedFT} \ln \frac{P(w_{\tau} = +w)}{P(w_{\tau} = - w)} = w- \ln \langle e^{\Delta e_{\tau} - \Delta s_{\tau}} \| w \rangle.$$ In Eq. (\[eq:detailedFT\]), $\Delta e_{\tau} = [E(\tau) - E(0)]/(k_B T)$ is the variation of the total potential energy of the system during a time interval $\tau$, given by Eq. (\[eq:potential\]), $\Delta s_{\tau}$ is the stochastic entropy change over $\tau$ [@seifert_b], and the brackets denote a conditional average over the stochastic realizations for which $w_{\tau}$ equals the value $w$. The last term on the right-hand side of Eq. (\[eq:detailedFT\]) is in general non-zero in presence of particle interactions, and therefore $P(w_{\tau})$ does not necessarily satisfy the exact linear relation of Eq. (\[eq:FT\]). Note that if $P(w_{\tau})$ is Gaussian, the asymmetry function can still be proportional to $w$, i.e. $$\label{eq:asymmetryGaussian} \ln \frac{P(w_{\tau} = +w)}{P(w_{\tau} = - w)} = \alpha w,$$ where the parameter $\alpha$, i.e. the slope of the linear relation, is given by $$\label{eq:slope} \alpha =\frac{ 2\mu_{\tau} } {\sigma_{\tau}^2}.$$ However, unlike the ideal case of non-interacting particle described by Eq. (\[eq:FT\]), the parameter $\alpha$ is in general different from 1 because the second term on the right-hand side of Eq. (\[eq:detailedFT\]), which involves the particle interactions and the substrate potential, is non-zero. For instance, we observe that Eq. (\[eq:asymmetryGaussian\]) holds in the static friction regime (completely pinned monolayer) and in the sliding regime, as shown in Fig. \[fig:Fig5\](a). In the static friction regime, the parameter $\alpha$ is equal to $0$ for all $n$ and $\tau$, because the system is in a quasi-equilibrium state, with equal probabilities $P(w_{\tau} = +w)$ and $P(w_{\tau} = -w)$. On the other hand, we find $\alpha \approx 2$ for all $\tau$ in the free sliding regime, as illustrated by the symbols around the dashed line in Fig. \[fig:Fig5\](a). This implies that in this case the second term on the right hand side of Eq. (\[eq:detailedFT\]) is non-zero: $\ln \langle e^{\Delta e_{\tau} - \Delta s_{\tau}} \| w \rangle \approx -w$. It should be noted that the direct computation of the asymmetry function from $P(w_{\tau})$ is restricted to rather small values of $n$ and $\tau$, because negative work fluctuations are difficult to sample with increasing values of such parameters. However, taking into account that $P(w_{\tau})$ is Gaussian, $\alpha$ can be estimated from the mean and the variance by means of Eq. (\[eq:slope\]). Surprisingly, in the inset of Fig. \[fig:Fig5\](a) we show that the value $\alpha \approx 2$ holds even for $n$ and $\tau$ as large as $500$ and $40$ s, respectively, thus demonstrating that the particle interactions give rise to a robust behavior of the term $\ln \langle e^{\Delta e_{\tau} - \Delta s_{\tau}} \| w \rangle \approx -w$ in Eq. (\[eq:detailedFT\]). This unconventional behavior of the asymmetry function can be traced back to the strong coupling between the particles forming the crystalline monolayer. Indeed, with increasing strength of repulsive interactions, which is fixed in our experiment by the magnetic field $\bf{B}$, the width of the work distribution decreases because of the increasing stiffness of the colloidal crystal. Note that in the limit of a perfectly stiff colloidal crystal, i.e. created by infinitely large repulsive interactions, the fluctuations of $w_{\tau}$ are completely suppressed. In this case the work distribution becomes a delta function, $P(w_{\tau}) = \delta (w_{\tau} - \mu_{\tau})$, which gives rise to $\alpha \rightarrow \infty$. Hence, for sliding friction $\alpha$ must be an increasing function of the pair interaction strength, bounded by the values $\alpha = 1$ (no interactions) and $\infty$ (infinitely large repulsions). The value $\alpha \approx 2$, specific to our experimental conditions, clearly illustrates that repulsive pair interactions reduce the fluctuations of $w_{\tau}$ compared to the value $\alpha= 1$ of Eq. (\[eq:FT\]) in absence of interactions. The intermediate frictional regime, where $P(w_{\tau})$ exhibits non-Gaussian tails due to the heterogeneous spatio-temporal dynamics of the monolayer, is particularly interesting. In this case, we observe that the asymmetry function is not even linear in $w$, as shown in Fig. \[fig:Fig5\](b) for $f=12.4$ fN and $V_{\max}= 25k_BT$, at which only partial depinning from the substrate is achieved. The asymmetry function is approximately linear for small values of $w$, with slope $\alpha \ll 1$ due to large negative work fluctuations on the stagnant particles. Nevertheless, significant deviations from this linear behavior show up at larger work fluctuations, $w>5$, when probing values of $w_{\tau}$ on the non-Gaussian tails plotted in Fig. \[fig:Fig4\](b). Once more, the behavior of the asymmetry functions seems to be robust, as shown in Fig. \[fig:Fig5\](b), where all the data points collapse to a master curve for different values of $\tau$ and $n$. We point out that in general, such a non-linear behavior of the asymmetry function is not easily observed in systems described by a small number degrees of freedom, because in such a case large negative fluctuations are difficult to sample [@mehl]. In our experimental system we are able to achieve this because of the existence of strong negative work fluctuations, which originate from the heterogenous dynamics of the coupled degrees of freedom of the system during the depinning transition of the colloidal monolayer. Summary and conclusion ====================== We have investigated the statistical properties of the work done by an external force on a monolayer of magnetically interacting particles driven across a periodic potential, which mimics friction between crystalline atomic surfaces. We have studied the influence of the number of particles used to probe these properties, the integration time, and the control parameters that are tuned to induce a transition from a pinned state (static friction) to complete depinning from the substrate potential (sliding friction). We have shown that, in the static and free sliding regimes, the work distribution converges to a Gaussian master curve for sufficiently large number of particles and integration times upon rescaling of these parameters. We have found that the mean and variance of such work distributions depend on the strength of the repulsive interactions, which in particular give rise to a stiffening of the monolayer for free sliding. Interestingly, we have also found that in the intermediate friction regime, where the monolayer undergoes a depinning transition, the work distribution becomes non-Gaussian because of the heterogeneity of the particle dynamics, e.g. due to partial depinning and stick-slip motion. We have shown that in general, the work distribution exhibits unconventional asymmetry properties within the context of non-equilibrium fluctuations relations. We have demonstrated that such a behavior originates from the presence of repulsive particle interactions. Thus, we provide the first experimental measurements of a stochastic thermodynamic quantity with non-trivial properties for a mesoscopic system with many coupled degrees of freedom. ![(a) Asymmetry function of the work distribution $P(w_{\tau})$ for: static ($+$), and sliding friction regime measured for $n=10$ particles at $f = 12.4$ fN, $V_{max} = 15k_BT$, over $\tau= 0.6$ s ($\circ$), $\tau = 1.2$ s ($\Box$), $\tau = 2.4$ s ($\diamond$). The solid circles are measurements under the same conditions for $n=25$ particles during $\tau = 1.2$ s. The dashed line is a guide to the eye with slope $\alpha = 2$. Inset: dependence of the parameter $\alpha$ on $\tau$ computed by means of Eq. (\[eq:slope\]) for $n=100$ ($\triangleright$) and $500$ (solid line) particles. (b) Asymmetry function of the work distribution $P(w_{\tau})$ measured for $n=25$ particles moving with a heterogeneous dynamics under $f=12.4$ fN and $V_{\max}= 25k_BT$ over $\tau=0.6$ s ($+$), $\tau=1.2$ s ($\ast$), $\tau=2.4$ s ($\circ$), $\tau=4.8$ s ($\times$), $\tau=9.6$ s ($\Box$), $\tau=19.2$ s ($\triangleright$), and $\tau=38.4$ s ($\diamond$). The solid circles are measurements under the same conditions for $n=50$ particles over $\tau = 38.4$ s. The dashed line is a guide to the eye with slope $\alpha = 0.15$.](figure_5.eps "fig:"){width="100.00000%"} \[fig:Fig5\] Acknowledgments {#acknowledgments .unnumbered} =============== We thank Udo Seifert for helpful discussions. We acknowledge financial support of the Deutsche Forschungsgemeinschaft, BE 1788/10-1. References {#references .unnumbered} ========== [9]{} Sekimoto K 2010 [Stochastic Energetics]{} (Springer-Verlag Berlin, Heidelberg) Seifert U 2012 [Rep. Prog. Phys.]{} [75]{} 126001 Jarzynski C 1997 [Phys. Rev. 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Mech.]{} [2013]{} P02027 [^1]: This equivalence is only valid at sufficiently low Reynolds number, where the flow field around the particle is Stokesian and therefore the resulting drag force can be written as $\gamma(\dot{{\mathrm{r}}}_i - {\bf v})$. In our experiments this assumption is fully justified because the Reynolds number is $\mathrm{Re} < 10^{-4}$. [^2]: The linear relation $\mu_{\tau} \propto \tau$ is not necessarily fulfilled for systems with many coupled degrees of freedom driven by time dependent forces, see for example [@lacoste].
--- abstract: 'Using the Hopkins Ultraviolet Telescope and Hubble Space Telescope, observers have now obtained UV spectra with sufficient signal to noise and resolution to allow quantitative spectroscopic analyses of the WDs in several DNe. In the “cleanest” DNe, such as U Gem, the observations are allowing the basic physical parameters of the WD – temperature, radius, gravity, rotation rate, and surface abundances – to be established. A second component also exists in these systems, which may either be the disk or may be related to the WD itself. Here I summarize the current state of the observations and our understanding of the data, highlighting some of the uncertainties in the analyses as well the prospects for fundamentally advancing our understanding of DNe and WDs with future observations.' address: \| Space Telescope Science Institute\ 3700 San Martin Drive\ Baltimore, MD 21218, United States author: - 'Knox S. Long' title: What We Learn from Quantitative Ultraviolet Spectroscopy of Naked White Dwarfs in Cataclysmic Variables --- Introduction ============ White dwarfs in cataclysmic variables are important since they are the gravitational engines which power CVs. Furthermore, compared to most WDs, conditions on the surface of the WDs in CVs are extreme, making these WDs basic laboratories for WD physics. With sensitive UV spectrographs in space, it has become possible to carry out detailed UV spectroscopy of WDs in some DNe. The first UV spectra of WDs in CVs were obtained with IUE. Panek & Holm \[1985\] showed that for U Gem in quiescence the depth of the , the shape of the UV spectrum, and the overall flux were consistent with a 1.2 WD at a distance of 90 pc with a temperature of 30,000 K. Mateo & Szkody \[1984\] interpreted the deep profile of VW Hyi as arising from a 20,000 K WD. Similarly, Holm \[1988\] argued that the spectrum of WZ Sge far from outburst resembled a DA WD. Some other systems, such as EK TrA and ST L Mi, and recently SW UMa and CU Vel [@GK99], also show absorption line profiles. However, in these systems it is less clear that the WD was dominates the spectrum, since models of optically thick accretion disks also have prominent profiles. Furthermore, the IUE observations also showed that the majority of DNe have spectra, even in quiescence, that exhibit none of the signatures expected of a WD. There are various ways to observe WDs at UV wavelengths in CVs. In eclipsing DNe such as OY Car, the WD can sometimes be observed as a sharp jump in the flux due to occultation of the WD by the secondary star. From the magnitude of the flux change and the shape of the spectrum in such eclipsing systems, one can estimate the WD temperature. On the other hand, a detailed analysis of the photometric spectrum is usually impossible, because of the effects of the so-called “Fe-curtain” on the WD spectrum [@horne94]. And in polars like AM Her, the UV continuum, at least in some orbital phases, is likely dominated by emission from the WD photosphere [@gansicke_amher]. However, detailed analysis is again difficult because radiation from the accretion column heats and alters the structure of the the photosphere. Here we will concentrate on WD in systems in which the WD dominates the UV emission, i.e. we will concentrate on “naked WDs” in DNe. U Geminorum =========== A qualitative improvement in the UV data on a WD in a CV was made with the observation of U Gem using the Hopkins Ultraviolet Telescope [@Long_ugem93]. The spectrum obtained about 11 days after U Gem had returned to quiescence from a normal outburst is shown in Figure 1. The spectrum shows not only several members of the Lyman series in absorption, but also a significant number of narrow absorption lines expected from a hot WD atmosphere salted with metal-rich material from an accretion disk. The overall spectrum is consistent with a WD surface temperature of 38,000 K and normal abundances, although Long et al. \[1993\] argued that a better fit to the data could be obtained if the 85% of the WD surface were 30,000 K and 15% were 57,000 K. Long et al. associated the higher temperature region with an accretion belt left over from the previous outburst, and suggested that the cooling of the accretion belt accounted for the decline in UV flux which U Gem exhibits through the interoutburst interval (and which was known from IUE observations [@KSS]). ![A comparison between the spectra of U Gem obtained on Astro 1 and Astro 2. The difference in flux reflects that fact that the second observation occurred 185 days into the outburst cycle.](fig1.ps){width="90.00000%"} ![A comparison between the spectra of U Gem obtained on Astro 1 and Astro 2. The difference in flux reflects that fact that the second observation occurred 185 days into the outburst cycle.](fig2.ps){width="90.00000%"} Subsequent observations with HST and HUT have confirmed both that the effective temperature of U Gem, as observed in the FUV, drops through early portion of the interoutburst period and that flux decline is less than expected if the entire WD cools. For example, Long et al. \[1994\], using the FOS on HST, found that the mean temperature of the WD dropped from 39,400 K 11 days after an outburst to 32,100 K 70 days after the same outburst. Figure 2 shows a comparison between an observation with HUT on Astro 2 in 1994 of U Gem 185 days from an outburst and the spectrum obtained on Astro-1 in 1990 11 days from outburst [@Long_ugem95]. The difference in temperature is evident in the relative fluxes at 950 Å compared to 1450 Å, the depth of the Lyman lines, and the specific ions in the spectrum. The nature of the “second source” in U Gem is not resolved. It is not known whether it is a result of a structure left over on the WD surface from the previous outburst, or whether it is a result of ongoing accretion after the outburst. At 1400 Å, the flux from the “second source” could be produced by an optically thick accretion disk with an accretion rate of . Meyer & Meyer-Hofmeister \[1994\] have suggested the accretion rate in quiescence gradually declines after an outburst as the inner accretion disk is eaten away. It is also not known how best to model the second source. Long et al.used a non-rotating WD; Cheng et al. \[1997\] considered an “accretion ring”. The difficulties are that higher S/N HST observations can only be carried out longward of 1150 Å where the “second source” fraction of emission is small, that it is not very clear what the spectrum of an accretion ring should be, and that the number of parameters can be large if abundances are allowed to vary (as Cheng et al. assumed). The best hope to resolve these difficulties is likely an intensive survey of U Gem with HST, or possibly FUSE, with multiple observations in a single outburst interval. ![Constraints on the mass and radius of the WD in U Gem based on recent analyses of the UV spectrum of the WD and the astrometric distance.](fig3.ps){width="90.00000%"} ![Constraints on the mass and radius of the WD in U Gem based on recent analyses of the UV spectrum of the WD and the astrometric distance.](fig4.ps){width="90.00000%"} The HUT and early HST studies of U Gem assumed normal abundances. However, independent analyses of two different high S/N GHRS datasets of U Gem far from outburst show clear evidence of CNO processing in the material in the photosphere of U Gem. Sion et al. \[1998\] found C to be 0.05$\times$ solar, N 4$\times$ solar and Si 0.4$\times$ solar. Similarly, Long & Gilliland \[1999\] found C to be 0.1$\times$ solar, N 4$\times$ solar, Si 0.4$\times$ solar, and Al 0.4$\times$ solar. A comparison between the model obtained by Long & Gilliland and the data is shown in Figure 3. At this point, systematic errors in the calibration of GHRS data, errors due to the “second source” in the actual model spectra, and atomic parameters dominate the errors. The N abundance is really due to one line complex at 1184 Å. (Note however that there are additional N features in the HUT spectra and these too indicate a significant overabundance of N relative to C.) These and other high S/N spectra have also been used to derive very accurate measurements of the rotation velocity ($<100 \VEL$ [@Sion_ugem94], $\gamma$ velocity, and K$_1$ velocity of the WD in U Gem. The low rotation velocity may account for the fact that for U Gem in outburst the boundary layer and disk luminosity are comparable, unlike SS Cyg and VW Hyi [@Long_euve]. Figure 4 is a graphical representation of the current constraints on the mass and radius of U Gem, an update of a figure presented by Long & Gilliland \[1999\] to reflect the astrometric distance determination for U Gem [@harrison99]. The solid blue lines bound the allowed region established by the gravitational redshift of the WD; this region is broad primarily because of the uncertainty in the $\gamma$ velocity of the secondary.The solid black curves bound the region allowed by UV measurements of the temperature and flux and the Bailey relation for the distance. The red lines bound the region allowed by the preferred astrometric distance and the temperature and flux. The magenta curve represents a standard WD mass-radius relation. Assuming this applies, the mass of the WD in U Gem is $1.02\pm0.04$ and the radius is . The inclination for U Gem is $\sim70^{o}$, somewhat greater than usually assumed. ![The HUT spectrum of VW Hyi in quiescence compared to a model with contributions from WD and an optically thick accretion disk.](fig5.ps){width="90.00000%"} ![The HUT spectrum of VW Hyi in quiescence compared to a model with contributions from WD and an optically thick accretion disk.](fig6.ps){width="90.00000%"} VW Hydri ======== Like U Gem, VW Hyi has been the subject of a variety of UV studies using instrumentation on HST and HUT. Like U Gem, there appears to be at least two sources of UV emission. In VW Hyi, the “second source” is somewhat more prominent. Figures 5 and 6 show comparisons between models and FOS and HUT data, respectively. Sion et al \[1996\] compared the FOS data to a WD or alternatively a WD and an accretion belt (by which they mean a rapidly rotating ring which radiates like a rotating stellar atmosphere). They found that WD had a temperature of 22,500 K, N:C ratios which suggest CNO processing, and a v sin(i) of 300 $\VEL$. They found the ring had a somewhat greater temperature 26,000 K and v sin(i) of 3350 $\VEL$, close to that expected from Keplerian motion. Long et al. \[1996b\] favored a model consisting of a relatively normal abundance WD with a temperature of 18,700 K and an optically thick accretion disk with an accretion rate of . Sion et al. \[1997\] have also obtained the GHRS spectrum of VW Hyi shown in Figure 7. These data were taken 30 days after a normal outburst. The data show a WD with v sin(i) of 400 $\VEL$, K$_{1}$ of 69 $\VEL$ and a gravitational redshift of 25-61 $\VEL$. The implied mass is 0.86 and the radius is . Thus in VW Hyi, as in U Gem, the physical parameters are tightly constrained. However, the most intriguing feature in the GHRS spectrum is the absorption line near 1250 Å. Sion et al., after considering several alternatives, conclude that the line is most likely due to PII, and that P abundance $\sim900\times$ solar is required. If correct, this would provide the very strong evidence that material in the photosphere of VW Hyi has been processed in the thermonuclear runaway of a nova explosion. Since the diffusion time scale on the surface of a WD is rather short, this material would have to be maintained in the photosphere by some kind of mixing process or by accretion from the secondary star. This observational claim is significant and requires strong verification. There are features in the spectra of some B3 stars observed with IUE, e.g. $\eta$ UMa, which, if convolved to simulate a system with v sin (i) of VW Hyi, look rather like what is observed in the VW Hyi spectrum. However, it is not clear these would be expected in the metal-enriched photosphere of a WD. If the abundance of P is high, then it should be possible to verify the result with additional observations since there are other features of PII of comparable strength which should appear at other UV wavelengths. ![FOS spectra of WZ Sge compared to non-rotating and rotating WD models.](fig7.ps){width="90.00000%"} ![FOS spectra of WZ Sge compared to non-rotating and rotating WD models.](fig8.ps){width="90.00000%"} WZ Sagittae =========== The third relatively clean WD in a DN which has been studied with HST is WZ Sge [@Cheng97b]. This system is interesting because it has the largest outburst amplitude (7 m), the longest interoutburst period (33 years), and of the shortest orbital periods (81 m) of any DN. The spectrum indicates a temperature of 14,800 K, and v sin(i) of 1200 $\VEL$ and a periodicity of $\sim$28 s, a periodicity that had been observed previously at optical wavelengths. The fact that the periodicity is observed in the UV suggests to Welsh et al. \[1997\] that the WD in WZ Sge is magnetic. The large rotational velocity implies that radius of the WD in WZ Sge would be expected to be significantly greater than for a non-rotating WD of the same mass. From the value of v sin(i), a periodicity in the data near 28 s, and estimates of the inclination of the system, Cheng et al. estimate a WD radius of . From the spectral analysis, log(g) appears to be $\sim8.0$ and in that case $M_{WD}\sim0.3$ , much lower than VW Hyi or U Gem. The large value of v sin (i) may complicate the abundance analyses, but the indication in WZ Sge is that C is more abundant than N. Summary and Future Prospects ============================ The decade of the 90s has been the decade of the first quantitative spectroscopy of WDs. For U Gem, VW Hyi, and WZ Sge, the observations are in fact sufficient to determine the basic physical properties of WDS and to begin to challenge our understanding of the physics of WDs in DNe. Much remains to be done however. Observers need to study more systems at high S/N, with good resolution, and covering simultaneously a large wavelength range to disentangle the various sources in the system. Multiple observations especially in a single outburst interval are needed to understand the response of the WD to the outburst and ongoing accretion. And accurate parallaxes are need to tie down the distances. Theorists need to develop a better understanding of the structure quiescent disks so that modelers will be able to calculate the spectra of this contaminating component of the the WD spectrum. 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--- abstract: 'The evolution of a generally covariant theory is under-determined. One hundred years ago such dynamics had never before been considered; its ramifications were perplexing, its future important role for all the fundamental interactions under the name gauge principle could not be foreseen. We recount some history regarding Einstein, Hilbert, Klein and Noether and the novel features of gravitational energy that led to Noether’s two theorems. Under-determined evolution is best revealed in the Hamiltonian formulation. We developed a covariant Hamiltonian formulation. The Hamiltonian boundary term gives covariant expressions for the quasi-local energy, momentum and angular momentum. Gravity can be considered as a gauge theory of the local Poincaré group. The dynamical potentials of the Poincaré gauge theory of gravity are the frame and the connection. The spacetime geometry has in general both curvature and torsion. Torsion naturally couples to spin; it could have a significant magnitude and yet not be noticed, except on a cosmological scale where it could have significant effects.' address: \| $^1$Department of Physics, National Central University, Chungli 32001, Taiwan\ $^2$Center for Mathematical and Theoretical Physics, National Central University\ Chungli 32001, Taiwan\ $^3$Graduate Institute of Astronomy, National Central University,\ Chungli 32001, Taiwan\ $^4$Leung Center for Cosmology and Particle Astrophysics,\ National Taiwan University, Taipei 10617, Taiwan\ $^5$nester@phy.ncu.edu.tw\ $^6$cmchen@phy.ncu.edu.tw author: - 'James M. Nester$^{1,2,3,4,5}$ and Chiang-Mei Chen$^{1,2,6}$' title: 'Gravity: a gauge theory perspective' --- Introduction ============ There have long been disputes about all of the principles used by Einstein for his gravity theory. Kretschmann in 1917 argued that general covariance has no real physical content and no connection to an extension of the principle of relativity.[@Norton93; @Norton03] From a different perspective general covariance has deep fundamental ramifications. GR with general covariance is the premier gauge theory. The consequences of this, especially regarding gravitational energy and under-determined evolution, were long perplexing. The Hamiltonian approach clarifies these issues. Gravity can be understood as a gauge theory of the local Poincaré symmetries of spacetime. Some Historical Background ========================== In 1915 Einstein made presentations to the Prussian Academy of Sciences on Nov. 4, 11, 18 and 25 (published one week later). The last had his generally covariant equations along with energy conservation. Hilbert presented his “[Foundations of physics]{}” on November 16 and 20; he submitted his first note on Nov. 19 (published in March 1916).[@Sauer99] His first theorem is central to our concerns: Theorem I ([‘Leitmotiv’]{}[^1]). In the system of $n$ Euler-Lagrange differential equations in $n$ variables obtained from a generally covariant variational integral such as in Axiom I, 4 of the $n$ equations are always a consequence of the other $n - 4$ in the sense that 4 linearly independent combinations of the $n$ equations and their total derivatives are always identically satisfied. Some discussions of Hilbert’s work have appeared recently,[@Sauer99; @Rowe99; @Corry04] often it has been viewed from the Einstein GR perspective (e.g., Ref. ). An alternative view[@BR08] of Hilbert’s agenda argues that a main aim was to reconcile the tension between general covariance and its inevitable consequence: a lack of unique determinism[^2]—this is the essence of gauge theory. Dynamical equations obtained from a variational principle had formerly had deterministic Cauchy initial value problems, but for GR there was a differential identity connecting the evolution equations; they were not independent and could not give uniquely determined evolution. Later it was found that this is best addressed using the Hamiltonian approach.[@Dirac64; @Earman03] We note some excerpts from Einstein’s correspondence, from Vol. 8 in Ref. . “In your paper everything is understandable to me now except for the energy theorem. Please do not be angry with me that I ask you about this again. …How is this cleared up? It would suffice, of course, if you would charge Miss Noether with explaining this to me.” (Doc. 223 to Hilbert 30 May 1916) “The only thing I was unable to grasp in your paper is the conclusion at the top of page 8 that $\varepsilon^\sigma$ was a vector.” (Doc. 638 to Klein 22 Oct 1918) “…Meanwhile, with Miss Noether’s help, I understand that the proof for the vector character of $\varepsilon^\sigma$ [from “higher principles” as I had sought was already given by Hilbert on pp. 6, 7 of his first note]{}, …” (Doc. 650 from Klein 10 Nov 1918) Briefly, after a couple of years Klein clarified Hilbert’s energy-momentum “vector”; he related it to Einstein’s pseudotensor, but he disagreed with Einstein’s physical interpretation of divergenceless expressions.[^3] Enlisted by Hilbert and Klein, it was Emmy Noether who resolved the primary puzzle regarding gravitational energy. Automatic Conservation of the Source and Gauge Fields ===================================================== In 1916 Einstein showed that local coordinate invariance plus his field equations gives material energy momentum conservation, without using the matter field equations (see Doc. 41 in Vol. 6 of Ref. ). This is referred to as automatic conservation of the source (see section 17.1 in Ref. ); it uses a [Noether second theorem]{} local (gauge) symmetry type of argument to obtain current conservation. Hermann Weyl argued in this way for the electromagnetic current in his papers of 1918 (the name gauge theory comes from this work) and 1929,[^4] whereas modern field theory generally uses Noether’s first theorem for current conservation.[@Brad02]. The essence of gauge theory is a local symmetry, consequently: (i) a differential identity, (ii) under-determined evolution, (iii) restricted type of source coupling, (iv) automatic conservation of the source. [Yang-Mills is only one special type]{}. Our gauge approach to gravity does not try to force it into the Yang-Mills mold, but rather simply recognizes the natural symmetries of spacetime geometry. Noether’s 1918 Contribution =========================== One word well describes 20th century physics: symmetry. Most of the theoretical physics ideas involved symmetry—essentially they are applications of Noether’s two theorems.[@KS-Noether] The first associates conserved quantities with global symmetries. The second concerns local symmetries: it is the foundation of the modern gauge theories. Why did Noether make her investigation? Klein was looking into the relationship between Einstein’s pseudotensor and Hilbert’s energy vector. He published a paper based on his correspondence with Hilbert. We quote some excerpts:[@KS-Noether] Klein: “You know that Miss Noether advises me continually regarding my work, and that in fact it is only thanks to her that I have understood these questions.” Hilbert: “I fully agree in fact with your statements on the energy theorems: Emmy Noether, on whom I have called for assistance more than a year ago to clarify this type of analytical questions concerning my energy theorem, found at that time that the energy components that I had proposed—as well as those of Einstein—could be formally transformed, …into expressions whose divergence vanishes identically,…Indeed I believe that in the case of general relativity, i.e., in the case of the general invariance of the Hamiltonian function, the energy equations which in your opinion correspond to the energy equations of the theory of orthogonal invariance do not exist at all; I can even call this fact a characteristic of the general theory of relativity.” This is why Noether wrote her paper. After presenting her two famous theorems she uses them to draw the conclusion that clarifies the situation:[@KS-Noether] “Given $I$ invariant under the group of translations, then the energy relations are improper if and only if $I$ is invariant under an infinite group which contains the group of translations as a subgroup. …As Hilbert expresses his assertion, the lack of a proper law of energy constitutes a characteristic of the “general theory of relativity.” For that assertion to be literally valid, it is necessary to understand the term “general relativity” in a wider sense than is usual, and to extend it to the aforementioned groups that depend on $n$ arbitrary functions.” Her result regarding the lack of a proper law of energy applies not just to Einstein’s GR, but to all geometric theories of gravity. The modern view is that energy-momentum is quasi-local, associated with a closed 2 surface.[@Sza09]. Energy-momentum Pseudotensors and the Hamiltonian ================================================= The Einstein Lagrangian differs from Hilbert’s by a total divergence: $$\begin{aligned} 2\kappa{\cal L}_{\rm E}(g_{\alpha\beta}, \partial_\mu g_{\alpha\beta}):= -\sqrt{-g}g^{\beta\sigma} \Gamma^\alpha{}_{\gamma\mu} \Gamma^\gamma{}_{\beta\nu}\delta^{\mu\nu}_{\alpha\sigma}%\nonumber \label{EinsteinL} \equiv \sqrt{-g}R-\hbox{div}.\end{aligned}$$ The Einstein pseudotensor is the associated canonical energy-momentum density: $$\label{EpseudoT} \mathfrak{t}_{\rm E}^\mu{}_\nu := \delta^\mu_\nu{\cal L}_{\rm E} - \frac{\partial{\cal L}_{\rm E}}{\partial \partial_\mu g_{\alpha\beta}} \partial_\nu g_{\alpha\beta}.$$ Using $\sqrt{-g}G^\mu{}_\nu=\kappa\mathfrak{T}^\mu{}_\nu$ one gets a conserved total energy-momentum: $$\partial_\mu (\mathfrak{T}^\mu{}_\nu+\mathfrak{t}_{\rm E}^\mu{}_\nu)=0, \qquad\Longleftrightarrow\qquad \sqrt{-g}G^\mu{}_\nu+\kappa\mathfrak{t}_{\rm E}^\mu{}_\nu=\partial_\lambda \mathfrak{U}^{[\mu\lambda]}{}_\nu.$$ The superpotential was found by Freud in 1939:[@Freud] $\mathfrak{U}_{\rm F}^{\mu\lambda}{}_\nu:=-\mathfrak{g}^{\beta\sigma} \Gamma^\alpha{}_{\beta\gamma}\delta^{\mu\lambda\gamma}_{\alpha\sigma\nu}%\label{UF} $. Other pseudotensors likewise follow from different superpotentials. They are all inherently reference frame dependent. Thus there are two big problems: (1) which pseudotensor? (2) which reference frame? The Hamiltonian approach has answers. With constant $Z^\mu$, the energy-momentum within a region is $$\begin{aligned} -Z^\mu P_\mu(V) &:=& - \int_V Z^\mu ({{\mathfrak{T}}^\nu{}_\mu+{\mathfrak{t}}^\nu{}_\mu}) \sqrt{-g}d^3\Sigma_\nu \nonumber\\ &\equiv& \int_V \left[ Z^\mu \sqrt{-g} \left( \frac1{\kappa} G^\nu{}_\mu - T^\nu{}_\mu \right) - \frac1{2\kappa} \partial_\lambda \left( Z^\mu {\mathfrak{U}^{\nu\lambda}{}_\mu} \right) \right] d^3\Sigma_\nu \nonumber\\ &\equiv& \int_V Z^\mu {\cal H}^{\rm GR}_\mu + \oint_{S=\partial V} {\cal B}^{\rm GR}(Z) \equiv H(Z, V). \label{basicHam}\end{aligned}$$ ${\cal H}^{\rm GR}_\mu$ is the covariant expression for the [Hamiltonian density]{}. The [boundary term]{} 2-surface integral is determined by the superpotential. The value of the pseudotensor/Hamiltonian is quasi-local, from just the boundary term, since by the initial value constraints the spatial volume integral vanishes. The Hamiltonian Approach ======================== Noether’s work can be combined with the Hamiltonian formulation. In Hamiltonian field theory, the conserved currents are the generators of the associated symmetry. For spacetime translations (infinitesimal diffeomorphisms), the associated current expression (i.e., the energy-momentum density) is the Hamiltonian density—the canonical generator of spacetime displacements. Because it can be varied one gets a handle on the conserved current ambiguity. The Hamiltonian variation gives information that tames the ambiguity in the boundary term—namely boundary conditions. Pseudotensor values are values of the Hamiltonian with certain boundary conditions.[@CNC99] Thus Problem (1) is under control. The Hamiltonian approach reveals certain aspects of a theory. The constrained Hamiltonian formalism was developed by Dirac[@Dirac64] and by Bergmann and coworkers. It was applied to GR by Pirani, Schild and Skinner[@PSS] and by Dirac[@Dirac58]. Later the ADM approach[@ADM] became dominant. For the Poincaré gauge theory of gravity (PG) the Hamiltonian approach was developed by Blagojević and coworkers.[@BN83] The Covariant Hamiltonian and its Boundary Term =============================================== From a first order Lagrangian formulation, ${\cal L}=d\varphi\wedge p-\Lambda$, which gives pairs of first order equations for an $f$-form $\varphi$ and its conjugate $p$, we developed a 4D-covariant Hamiltonian formalism.[@CNT95; @CN99; @CNC99; @CN00; @CNT05; @GR100] The Hamiltonian generates the evolution of a spatial region along a vector field. The Hamiltonian density is the first order translational Noether current 3-form, it is linear in the displacement vector plus a total differential: $${\cal H}(Z) := \pounds_Z \varphi \wedge p - i_Z {\cal L} =: Z^\mu {\cal H}_\mu + d {\cal B}(Z), \label{5:HN}$$ and is a conserved “current” [on shell]{} (i.e., when the field equations are satisfied): $$\label{5:IdH} - d {\cal H}(Z) \equiv \pounds_Z \varphi \wedge \frac{\delta {\cal L}}{\delta \varphi} + \frac{\delta {\cal L}}{\delta p} \wedge \pounds_Z p.$$ Furthermore, from [local]{} diffeomorphism invariance, it follows that ${\cal H}_\mu$ is linear in the Euler-Lagrange expressions. Hence the translational Noether current conservation reduces to a differential identity. This an instance of Noether’s 2nd theorem, exactly the case to which Hilbert’s “lack of a proper energy law” remark applies. The value of the Hamiltonian is quasi-local (associated with a closed 2-surface): $$-P(Z, V) =H(Z,V) := \int_V {\cal H}(Z) = \oint_{\partial V} {\cal B}(Z). \label{5:EN}$$ The Hamiltonian boundary term has two important roles: (i) it gives the quasi-local values, (ii) it gives the boundary conditions. The boundary term can be adjusted to match suitable boundary conditions. We were led to a set of general boundary terms which are linear in $\Delta\varphi:=\varphi-\bar\varphi$, $\Delta p:=p-\bar p$, where $\bar\varphi,\bar p$ are reference values:$${\cal B}(Z) := i_Z \left\{ \begin{array}{c} {\varphi} \\ {\bar\varphi} \end{array} \right\} \wedge \Delta p - (-1)^f \Delta\varphi \wedge i_Z \left\{ \begin{array}{c} p \\ \bar p \end{array} \right\}.\label{genB}$$ The associated variational Hamiltonian boundary term is $$\delta{\cal H}(Z) \sim d\left[ \left\{ i_Z \delta\varphi \wedge \Delta p \atop - i_Z \Delta\varphi \wedge \delta p \right\} + (-1)^f \left\{ -\Delta\varphi \wedge i_Z \delta p \atop \delta\varphi \wedge i_Z \Delta p \right\} \right]. \label{deltaHZbound}$$ Here for each bracket independently one may choose either the upper or lower term, which represent essentially a choice of Dirichlet (fixed field) or Neumann (fixed momentum) boundary conditions for the space and time parts of the fields separately.[^5] For asymptotically flat spaces the Hamiltonian is well defined, i.e., the boundary term in its variation vanishes and the quasi-local quantities are well defined at least on the phase space of fields satisfying Regge-Teitelboim like asymptotic conditions: $$\Delta \varphi \approx {\cal O}^+(1/r) + {\cal O}^-(1/r^2), \qquad \Delta p \approx {\cal O}^-(1/r^2) + {\cal O}^+(1/r^3). \label{5:asymptotics}$$ Also the formalism has natural boundary term related energy flux expressions.[@CNT05] Gauge and Geometry ================== For the history of gauge theory see Ref. . Gravity as a gauge theory was pioneered by Utiyama (1956, 1959), Sciama (1961) and Kibble (1961). For accounts of gravity as a spacetime symmetry gauge theory, see Hehl and coworkers[@HHKN; @Hehl80; @HHMN95; @GFHF96], Mielke[@MieE87] and Blagojević[@Blag02]. A comprehensive reader with summaries, discussions and reprints has recently appeared.[@BlagHehl] For the observational constraints on torsion see Ni.[@Ni10] GR can be seen as the original gauge theory: the first physical theory where a local gauge freedom (general covariance) played a key role. Although the electrodynamics potentials with their gauge freedom were known long before GR yet this gauge invariance was not seen as having any important role in connection with the nature of the interaction, the conservation of current, or a differential identity—until the seminal work of Weyl, which post-dated (and was inspired by) GR. We also note the developments of the concept of a connection in geometry by Levi-Civita, Weyl, Schouten, Cartan, Eddington, and others. [Riemann-Cartan geometry]{} (with a metric and a metric compatible connection, having both curvature and torsion) is the most appropriate for a dynamic spacetime geometry theory: [its local symmetries are just those of the local Poincaré group]{}. The conserved quantities, energy-momentum and angular momentum/center-of-mass momentum are associated with the Minkowski spacetime symmetry, i.e., the Poincaré group. Riemann-Cartan Geometry and PG Dynamics ======================================= It is natural to consider gravity as a gauge theory of the local Poincaré group. The spacetime geometry that suits this perspective is Riemann-Cartan geometry, which has a (Lorentz signature) metric and a metric compatible connection: $Dg_{\mu\nu}\equiv0$. The translation and Lorentz gauge potentials are, respectively, the coframe $\vartheta^\alpha=e^\alpha{}_k dx^k$ and connection $\Gamma^\alpha{}_\beta=\Gamma^\alpha{}_{\beta k}dx^k$ one-forms. The associated field strengths are the torsion and curvature 2-forms: $$\begin{aligned} T^\alpha := D \vartheta^\alpha := d \vartheta^\alpha + \Gamma^\alpha{}_\beta \wedge \vartheta^\beta &=& \frac12 T^\alpha{}_{ij} dx^i \wedge dx^j, \qquad\\ \label{tor2} % % R^\mu{}_\nu := d \Gamma^\mu{}_\nu + \Gamma^\mu{}_\lambda \wedge \Gamma^\lambda{}_\nu&=&\frac12 R^\mu{}_{\nu ij} dx^i \wedge dx^j. \qquad \label{curv2} %\end{aligned}$$ The first and second [Bianchi identities]{} are $DT^\alpha\equiv R^\alpha{}_\beta\wedge\vartheta^\beta$ and $DR^\alpha{}_\beta\equiv0$. The [Ricci identity]{}, $\left[ \nabla_\mu, \nabla_\nu \right] V^\alpha = R^\alpha{}_{\beta\mu\nu} V^\beta - T^\gamma{}_{\mu\nu} \nabla_\gamma V^\alpha$, reflects the holonomy and the Lorentz and translational field strengths. For an orthonormal frame $g_{\mu\nu}=\hbox{const}$ and $\Gamma^{\alpha\beta}$ is antisymmetric. The PG dynamics has been discussed in detail in Ref. including (i) the Lagrangian, both 2nd and 1st order, (ii) the Noether symmetries, conserved currents and differential identities, (iii) the covariant Hamiltonian including the generators of the local Poincaré gauge symmetries, (iv) our [preferred Hamiltonian boundary term]{}, (v) the quasi-local energy-momentum and angular momentum/center-of-mass moment obtained therefrom, and (vi) the [choice of reference]{} in the boundary term. Preferred Hamiltonian Boundary Terms and Reference ================================================== For the PG and GR our preferred Hamiltonian boundary terms are $${\cal B}_{\rm PG}(Z) = i_Z \vartheta^\alpha \tau_\alpha + \Delta \Gamma^\alpha{}_\beta \wedge i_Z \rho_\alpha{}^\beta + {\bar {D}}_\beta Z^\alpha \Delta \rho_\alpha{}^\beta, \label{Bpref(Z)}$$ $${\cal B}_{\rm GR}(Z) = \frac{1}{2\kappa} (\Delta\Gamma^{\alpha}{}_{\beta} \wedge i_Z \eta_{\alpha}{}^{\beta} + \bar D_{\beta} Z^\alpha \Delta\eta_{\alpha}{}^\beta), \qquad \eta^{\alpha\beta\dots} := * (\vartheta^\alpha \wedge \vartheta^\beta \wedge \cdots). \label{BprefGR}$$ Like many other choices, at spatial infinity the latter gives the standard values for energy-momentum and angular momentum/center-of-mass momentum. Our preferred GR expression has some special virtues: (i) at null infinity it gives the Bondi-Trautman energy and the Bondi energy flux, (ii) it is covariant, (iii) it is positive—at least for spherical solutions and large spheres, (iv) for small spheres it is a positive multiple of the Bel-Robinson tensor, (v) first law of thermodynamics for black holes, (vi) for spherical solutions it has the hoop property, (vii) for a suitable choice of reference it vanishes for Minkowski space. Regarding the second ambiguity inherent in our quasi-local energy-momentum expressions: the choice of reference. Minkowski space is the natural choice, but one needs to choose a specific Minkowski space. Recently we proposed (i) 4D isometric matching on the boundary,[^6] and (ii) energy optimization as criteria for selecting the “best matched” reference on the boundary of the quasi-local region. A detailed discussion of our covariant Hamiltonian boundary terms and our reference choice proposal was presented in the MS parallel session[@CLNS15] and in Ref. . They have been tested on spherically symmetric and axisymmetric spacetimes.[@SCLN14] The Poincaré Gauge Theory of Gravity ==================================== The standard PG Lagrangian density has a quadratic field strength form:[^7] $$\begin{aligned} \label{quadraticL} {\cal L}_{\rm PG} \sim \frac{1}{\kappa}\left(\Lambda+ \text{curvature} +\text{torsion}^2\right) + \frac{1}{\varrho}\,\text{curvature}^2\,.\end{aligned}$$ Varying $\vartheta,\Gamma$ gives quasi-linear 2nd order dynamical equations for the potentials: $$\begin{aligned} \kappa^{-1}(\Lambda+ \hbox{curv} + D \hbox{ tor} + \hbox{tor}^2)+ \varrho^{-1}\hbox{curv}^2&=& \hbox{energy-momentum},\qquad\\ \kappa^{-1}\hbox{tor}+\varrho^{-1} D\hbox{ curv}&=& \hbox{spin}.\end{aligned}$$ The general theory has 11 scalar plus 7 pseudoscalar parameters, but there is one even parity and two odd total differentials; effectively 15 “physical” parameters.[@BHN; @BH] Torsion couples to intrinsic spin, not orbital angular momentum.[@HOP13] But highly polarized spin density is practically nonexistent in the present day universe. So on ordinary scales matter hardly excites or responds directly to torsion. Torsion could have a significant magnitude and yet be hardly observable: “dark torsion”. At very high densities it a different story; at around $10^{52}$ gm/cm$^3$ the nucleon spin density is comparable to the material energy density, [and beyond that the spin-torsion interaction dominates gravity in the PG]{}. So one can expect major effects in the early universe. But even in the present day, while being hardly noticeable on the lab, solar system, or galactic scale, the gravitational effects of torsion (like $\Lambda$) could well have measurable effects on the cosmological scale. General PG Homogeneous and Isotropic Cosmologies ================================================ The general PG homogeneous and isotropic cosmology has been considered recently.[@HCNY15] For such cosmologies the general PG has an effective Lagrangian. From this with $\dot a=aH$, 6 first-order equations for $a,H$, the scalar and pseudoscalar curvatures $R,X$ and the two “scalar” torsion components $u,x$ were obtained: $$\begin{aligned} -\frac{w_{4+6}}2\dot R-\frac{\mu_{3-2}}4 \dot X&=&-\left[-3 \tilde a_2-w_{4+6}R-\frac{\mu_{3-2}}2 X\right]u %\nonumber\\ %&& +\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]x\nonumber\\ &&+w_{4-2}[2X-24(H-u)x]x, \label{dotRdotX'} %\label{Epsi} \\ % % -\frac{\mu_{3-2}}4 \dot R+\frac{w_{2+3}}2\dot X&=&-\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]u %\nonumber\\ %&& +\left[12\tilde a_3+w_{4+6}R+\frac{\mu_{3-2}}2 X\right]x\nonumber\\ &&-w_{4-2}(2R-12[(H-u)^2-x^2+k a^{-2}])x, \label{dotXdotR'}\\ %\label{Echi} % \dot H -\dot u&=&\frac{R}6-2H^2+3Hu-u^2+x^2-ka^{-2},\label{Hudot'} % \\ % a_2\dot u &=&\frac13( -a_0R-\tilde\sigma_2X+\rho-3p+4\Lambda) %\nonumber\\ %&& +a_2(u^2-3Hu)-4a_3x^2, \\ \qquad\dot x&=&\frac{X}6-3Hx+2xu\label{xdot'}. \label{udotrho3p'}\end{aligned}$$ Here the material energy density satisfies a generalized Friedmann relation: $$\begin{aligned} \rho &=&-\Lambda+3a_0[(H-u)^2-x^2+ka^{-2}]\nonumber\\&&-\frac32 a_2(u^2-2Hu) +6 a_3x^2+6\tilde\sigma_2x(H-u)\nonumber\\ &&\qquad+\frac{w_{4+6}}{24}\left[R^2-12R\left\{(H-u)^2-x^2+ka^{-2}\right\}\right] \nonumber\\ % &&\qquad+\frac{\mu_{3-2}}{24}\left[RX-6X\left\{(H-u)^2-x^2+ka^{-2}\right\}-12Rx(H-u)\right]\nonumber\\ % &&\qquad -\frac{w_{2+3}}{24}\left[X^2-24Xx(H-u)\right]\label{density}.\end{aligned}$$ The above equations are the most general—they include all the quadratic PG cosmologies. Generically the model has, in addition to the usual metric scale factor, effectively two dynamical “scalar” torsion components carrying spin 0$^+$ and 0$^{-}$. Typically they, and the Hubble expansion rate, show damped oscillations. But the model has other types of behavior in the various degenerate special cases, including GR.[@HCNY15] Acknowledgments {#acknowledgments .unnumbered} =============== This was prepared while J.M.N. was visiting the Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing 100190, China. C.M.C. was supported by the Ministry of Science and Technology of the R.O.C. under the grant MOST 102-2112-M-008-015-MY3. [00]{} J. D. Norton, Rep. Prog. Phys. [56]{} (1993) 791. J. D. Norton, General covariance, gauge theories and the Kretschmann objection, in Symmetries in Physics: Philosophical Reflections, eds. K. Brading and E. Castellani (Cambridge University Press, Cambridge, 2003), pp. 110–123. T. 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Peutzfeld, Phys Lett A [377]{} (2013) 1775. F.-H. Ho, H. Chen, J. M. Nester and H.-J. Yo: Chinese J. Phys. [53]{} (2015) 110109. [^1]: guiding theme [^2]: Einstein had struggled with this in connection with his hole argument. [^3]: Things were not as easy then; in particular the Bianchi identity and its contracted version were not known to these people;[@Pais82; @Rowe02] they each had to rediscover an equivalent identity on their own. [^4]: An English translation of Weyl’s seminal papers can be found in Ref. . [^5]: There are more complicated possibilities, “mixed” choices involving some linear combination of the upper and lower expressions. [^6]: The hardest part of 4D isometric matching is the embedding of the 2D surface $S$ into Minkowski space; Yau and coworkers have extensively investigated this.[@WangYau; @CWY15] [^7]: $\kappa:=8\pi G/c^4$ and $\varrho^{-1}$ has the dimensions of action. $\Lambda$ is the cosmological constant.
--- author: - Evandro Konzen - Cláudia Neves - Philip Jonathan bibliography: - 'NonStatRefereces2020.bib' title: \| Modelling non-stationary extremes of storm severity:\ a tale of two approaches --- [[KEY WORDS AND PHRASES:]{} [Circular statistics, direction, endpoint, estimation, kernel smoothing, parametric, peaks over threshold, extreme quantile, semi-parametric, significant wave height, threshold selection.]{}\ ]{} Introduction and Motivation {#Sct:Int} =========================== Statistical inference for extreme values has been a dynamic and rapidly developing field over the last decade or so, and offers considerable scope for practical application in science and engineering. Curiously, in the midst of this active development, two seemingly divergent camps of statistical thought have emerged, proposing different approaches to extreme value modelling, yielding inferences which are not always obviously in agreement. The work presented in this paper seeks to identify potential points of contact between these so-called parametric and semi-parametric frameworks for extreme value inference, to encourage better common understanding and convergence of at least some practices, in particular for tackling non-stationary extremes. Non-stationarity is commonplace in environmental extremes; physical processes generate extreme values which typically vary systematically with covariates, including space and time. For the peaks-over-threshold (POT) method, where only data exceeding a threshold are used for analysis [@Balkema1974; @Pickands1975], various models have been proposed to capture non-stationarity, including those of @davison1990models and @leadbetter91. In the parametric framework, non-stationarity can be incorporated within the appropriate distribution function for threshold exceedances by allowing the distribution’s (shape and scale) parameters to vary with covariates [see e.g. @coles2001introduction; @chavez2005gamExtremes and references therein]. Important assumptions underpinning this approach are that the data generating process is locally stationary and that observations from the data generating process can be considered approximately independent given covariates. The need for non-stationary extreme value threshold is well-recognised for environmental applications. @RbsTwn97 base their approach on a non-constant threshold to characterise the evolution of extreme sea currents. @northrop2011threshold propose a covariate-dependent threshold estimated using quantile regression, and @northrop2016thre propose a cross-validation procedure for threshold selection. We introduce a method for selection of a non-stationary threshold amenable to both parametric and semi-parametric approaches to inference. The basic assumption is that the shape parameter $\xi$ (the key parameter for quantifying tail-heaviness) does not depend on the covariate. This assumption is the starting point for conciliation between parametric and semi-parametric approaches. Applications of non-stationary extreme value analysis are more numerous within the remit of parametric inference than in the semi-parametric setting. For example, in an ocean engineering context, @forristall2004use performs extreme value analysis of significant wave height for directional octants. This approach (i) accommodates directional non-stationarity and (ii) allows extreme quantiles for specific directional sectors to be estimated. Choice of number and widths of directional sectors is an open problem (see e.g. @RssEA17b [@folgueras2019selection]). These choices constitute a difficult problem as the environmental extremes usually change smoothly with respect to direction, motivating use of various basis representations for parameters of the conditional distribution of threshold exceedances, as a function of covariate (see e.g. @jones2016statistics [@ZnnEA19a]). The goal in this paper is to combine parametric and semi-parametric modelling approaches to obtain a new method for inference on circular extreme data. Firstly, we augment the scope of the semi-parametric approach so that semi-parametric inference for quantities of interest in ocean engineering is possible and meaningful. This will be achieved via an adaptive method for threshold selection. We then present a comparative survey showing how the parametric approach can be complemented with semi-parametric methodology aiming at improved inference for directional extremes. In particular, we propose a unified procedure for inference that borrows insight from both frameworks and harmonising between them in terms of (i) model-fit, and (ii) estimated extreme value indices such as the shape parameter or extreme value index, $T$-year value or an extreme quantile including the right endpoint of the support of the underlying distribution. Finally, the main application for illustration of our approach involves directional extreme value analysis of hindcast storm peak significant wave height (henceforth referred to as [$H_S^{sp}$]{}) recorded at a northern North Sea location offshore Norway. We hope to demonstrate that the proposed approach to non-stationary extreme value analysis may be of practical benefit to practitioners in coastal and ocean engineering and environmental sciences. The remainder of the paper is organized as follows. The motivating application to directional modelling of [$H_S^{sp}$]{} is introduced in Section \[Sct:Dat\]. Section \[Sct:ThrStt\] provides key definitions and theoretical results underpinning the combined methodology for threshold selection in the presence of non-stationarity that will be developed in Section \[Sct:ThrEst\]. Sections \[Sct:SplML\] and \[Sct:ExtQnt\] detail the adapted estimators in this paper. Our main application of the parametric and semi-parametric combined methodology is presented in Section \[Sec:App\]. Finally, Section \[Sec:Summary\] lists the main contributions of the work. Motivating application {#Sct:Dat} ====================== The sample of data for the motivating application is described by [@RndEA15a]. The data corresponds to observations of storm severity and storm direction in the northern North Sea. Significant wave height (H$_{\mbox{s}}$) measures the roughness of the ocean surface, and can be defined as four times the standard deviation of the ocean surface elevation at a spatial location for a specified period of observation. The application sample is taken from the WAM hindcast of [@RstEA11], which provides time-series of significant wave height, (dominant) wave direction and season (defined as day of the year, for a standardised year consisting of 360 days) for three hour sea-states for the period September 1957 to December 2012 at a northern North Sea location in the vicinity of the black disk in the upper panel of Figure \[Fig:HsSPdata\]. A hindcast is a physical model of the ocean environment, incorporating pressure field, wind field and wind-wave generation models in particular; the hindcast model is calibrated to observations of the environment from instrumented offshore facilities, moored buoys and satellite altimeters in the neighborhood of the location for a period of time, typically decades. Extreme seas in the North Sea are dominated by winter storms originating in the Atlantic Ocean and propagating eastwards across the northern part of the North Sea. Due to their proximity to the storms, sea states at northern North Sea locations are usually more intense than in the southern North Sea. Occasionally, the storms travel south-eastward and intrude into the southern North Sea producing large sea states. Directions of propagation of extreme seas vary considerably with location, depending on land shadows of the British Isles, Scandinavia, and the coast of mainland Europe, and fetches associated with the Atlantic Ocean, Norwegian Sea, and the North Sea itself. In the northern North Sea the main fetches are the Norwegian Sea to the North, the Atlantic Ocean to the west, and the North Sea to the south. Extreme sea states from the directions of Scandinavia to the east and the British Isles to the south-west are not possible. The shielding by these land masses is more effective for southern North Sea locations, resulting in a similar directional distribution but reduced wave heights by comparison with northern North Sea locations. ![ Map showing the relevant location at North Sea (top); scatter-plot displaying the $H_S^{sp}$ data against 360 directions from which wave propagate, measured clockwise from north, expressed in the angular component $\theta= 0,1, \ldots, 359^\circ$(bottom).[]{data-label="Fig:HsSPdata"}](figs/StationMap3.jpeg "fig:")\ ![ Map showing the relevant location at North Sea (top); scatter-plot displaying the $H_S^{sp}$ data against 360 directions from which wave propagate, measured clockwise from north, expressed in the angular component $\theta= 0,1, \ldots, 359^\circ$(bottom).[]{data-label="Fig:HsSPdata"}](figs/HsSPdata "fig:") At the location of interest, observations of storm peak significant wave height ([$H_S^{sp}$]{}) are isolated from the hindcast time-series using the procedure described in [@EwnJnt08] as follows. Briefly, contiguous intervals of significant wave height above a low peak-picking threshold are identified, each interval assumed to correspond to a storm event. The peak-picking threshold corresponds to a directional-seasonal quantile of H$_{\mbox{s}}$ with specified non-exceedance probability, estimated using quantile regression. The maximum of significant wave height during the storm interval is taken as the storm peak significant wave height [$H_S^{sp}$]{}. The values of directional and seasonal covariates at the time of storm peak significant wave height are referred to as storm peak values of those variables. The resulting storm peak sample consists of 2941 values. With direction from which a storm travels expressed in degrees clockwise with respect to north, the lower panel of Figure \[Fig:HsSPdata\] shows a polar plot of observations of [$H_S^{sp}$]{} (in metres) versus direction. The land shadow of Norway (approximately the directional interval $(45^\circ,210^\circ)$) has a considerable effect on the rate and size of occurrences with direction. In particular, there is a dramatic increase in both rate and size of occurrences with increasing direction at around $210^\circ$, corresponding to Atlantic storm events from the south-west able to pass the Norwegian headland. We therefore should expect considerable directional variability in model parameter estimates for the sample. In contrast, the magnitude of the rate of change of both rate and size of occurrences with respect to season (not shown, but see @RndEA15a) is lower. Winter storms (approximately from October to March) are more intense and frequent. Only winter storms with storm peak events occurring in October to March including have been considered further in this work which corresponds to 1521 data points. Theoretical motivation for the stationary case {#Sct:ThrStt} ============================================== Here we summarise basic theoretical results and assumptions underpinning extreme value modelling for the stationary case. Further details are given in the Appendix. Extension to non-stationary will be present in Section \[Sct:SplML\]. Suppose the available sample consists of realizations of a sequence of independent and identically distributed (i.i.d) random variables $X_1, X_2, \dots, X_n$. The sequence can also be weakly dependent. We assume that all the random variables follow the same (unknown) distribution function, and for brevity use the symbol $X$ to refer to any of the random variables when there is no need to be more specific. The common distribution function is $F(x) = P(X \leq x)$, for every $x \in \mathbb{R}$. We denote by $x^F$ the right endpoint of the support of $F$, namely the ultimate value which bounds all possible observations from above, $$\label{EP} x^F= \sup\{x:\, F(x)<1\}.$$ We note that $x^F$ may be less than or equal to infinity. The extreme value (or extreme types) theorem (@FisherTippett:28 [@Gnedenko:43; @deHaan:70]) establishes that the limit distribution of linearly normalised partial maxima $M_n= \max(X_1, \dots, X_n)$, with real constants $a_n>0$ and $b_n\in \mathbb{R}$, must be one of only possible three extreme value distributions: Fréchet, Gumbel or Weibull. These three types can be nested in the one-parameter Generalized Extreme Value (GEV) distribution. Specifically, if there exist $a_n>0$ and $b_n\in \mathbb{R}$ such that, $$\label{DOA} \lim_{n\rightarrow \infty}P\big\{a_n^{-1}(M_n-b_n)\leq x\big\}= \lim_{n\rightarrow \infty} F^n(a_n x+b_n) = G(x),$$ for every continuity point of (non-degenerate) $G$, then $G$ must be a GEV distribution with distribution function given by: $$\label{DistNormMax} G_{\xi}(x)= \exp \bigl( -(1+\xi\, x)^{-1/\xi} \bigr),$$ for all $x$ such that $1+\xi x >0$. We then say that $F$ belongs to the max-domain of attraction of the GEV, for some $\xi\in {\ensuremath{{{\mathbb{R}}}}}$, and write $F\in \mathcal{D}(G_\xi)$. Parameter $\xi$ is conventionally referred to as the shape parameter in parametric literature, and the extreme value index (EVI) in semi-parametric literature of extremes. The Fréchet max-domain of attraction, corresponding to $\xi>0$, contains distributions exhibiting polynomial decay, such as the Pareto, Cauchy, Student’s $t$ and Fréchet itself. These distributions have infinite right endpoint $x^F$. All distribution functions belonging to $\mathcal{D}(G_\xi)$ with $\xi<0$, referred to as the Weibull max-domain of attraction, are short-tailed with finite $x^F$; examples include the uniform and beta distribution. For $\xi=0$, a continuity argument gives $G_{\xi=0}(x)=\exp\bigl(-e^{-x}\bigr)$, $x\in \mathbb{R}$. This gives rise to the Gumbel max-domain of attraction ($\mathcal{D}(G_\xi)$ with $\xi=0$, or $\mathcal{D}(G_0)$), a domain of particular interest in many applied fields, due to both simplicity of inference and the great variety of distributions possessing an exponential tail (with either $x^F < \infty$ or $x^F = \infty$). The normal, gamma and log-normal distributions are only a few members of $\mathcal{D}(G_0)$. Extreme value inference using block maxima has been practised for many years, from the time of [@Gumbel58] in application to hydrology. However, in this paper, inference is based on analysis of threshold exceedances (or POT), motivated by characterization of the max-domains of attraction above in terms of exceedances of high threshold. @Balkema1974 and @Pickands1975 established that the max-domain of attraction condition is equivalent to the assertion that the (conditional) distribution of $X$ given $X>u$, with $u$ near the right endpoint $x^F$, converges to the Generalized Pareto distribution (GPD) with distribution function $1+\log G_\xi$. Analysis of threshold exceedances (or POT) is potentially statistically more efficient than analysis of block maxima: in the former, all large values of threshold exceedance in the sample are admitted, including multiple occurrences of large values belonging to same block, which might be excluded in a block maximum analysis. The parametric and semi-parametric approaches to extreme value analysis considered in this paper have strong conceptual similarities. Both approaches involve estimation of three quantities. In the parametric setting, we estimate GPD shape and scale parameters, the latter being threshold-dependent; for estimation of T-year levels, we also need to estimate the rate of threshold exceedance. In the semi-parametric analysis, we focus on estimation of a single parameter, the so-called extreme value index (i.e. the semi-parametric equivalent of the GPD shape parameter); associated scale and location normalising functions (akin to $a>0$ and $b$ in on the real line) are estimated separately. Sensitivity to the extreme value threshold choice is a common critical feature of the approaches. Confirming the relative stability of estimated extreme value index (or shape parameter) and high quantiles (such as the $T$-year level), with near-zero exceedance probability $p$ with respect to threshold is a key diagnostic test for the analysis. This will be explained at length in Section \[Sct:ExtQnt\]. Non-stationary threshold selection {#Sct:ThrEst} ================================== We extend the univariate setting of Section \[Sct:ThrStt\] to the non-identically distributed case as follows. We assume that the covariate domain $\mathcal{S}$ is partitioned into $m$ intervals, with centroids $\theta_j$, and write $\Theta$ for the set $\{\theta_j\}_{j=1}^m$ of centroids. Specifically for directional analysis considered here, $\mathcal{S}=[0^\circ,360^\circ)$, and $\theta_j=j-1$, $j=1,...,360$. Suppose, for each $j$, that $X_1(\theta_j),\, \dots,\, X_{n}(\theta_j)$ consists of a sample of $n$ (unknown) independent identically-distributed (positive) random variables. We assume the extreme value theorem holds at each $\theta_j$, for sufficiently large $n$. The extreme value condition shows that the shape parameter (or extreme value index) $\xi(\theta_j)$ governs the tail behaviour of the underlying distribution [$F_{\theta_j}(x)=P\{X(\theta_j) \leq x\}$, $x\in {\ensuremath{{{\mathbb{R}}}}}$]{}. In the non-stationary case, our objective is estimation of [$\xi(\theta_j)$]{} (and associated parameters discussed in Section \[Sct:ThrStt\]) from a sample of threshold exceedances. We achieve this parametrically using maximum likelihood estimation for the GPD [@coles2001introduction; @davison1990models], and semi-parametrically using the moment estimator [@dekkers1989moment]. Enhanced versions of these estimators, designed to accommodate non-stationarity, will be introduced later on in this section. Extreme value statistics characterises the largest values in the sample and therefore choosing an appropriate threshold above which inference will take place is as essential as choosing between inference approaches. As noted in Section \[Sct:Int\], numerous authors have considered the estimation of non-stationary thresholds for extreme value analysis based on POT-GPD. Indeed, it appears sensible to consider a covariate-dependent threshold for inference using data showing strong covariate dependence. Motivated by the desire to admit the same proportion of the original sample for subsequent extreme value analysis given covariate, threshold estimation using a covariate-conditional quantile regression would seem an obvious choice [@northrop2011threshold; @northrop2016thre]. Using this approach, a constant threshold exceedance probability of $\tau$ given covariate might be sought. However, our studies have demonstrated that this approach might not strike the right bias-variance balance for efficient parameter estimation. Section \[Sct:ThrEst:Hrs\] below introduces our heuristic criterion devised for threshold selection, which relies on local estimates for $\xi$ on the covariate domain. These can be inferred using semi-parametric (Section \[Sct:ThrEst:SP\]) or maximum likelihood estimation (Section \[Sct:ThrEst:ML\]). Section \[Sct:ThrEst:Alg\] provides a simple algorithm for the threshold selection procedure. Heuristic criterion for threshold selection {#Sct:ThrEst:Hrs} ------------------------------------------- Let $\mathcal{N}(\theta_j, h)$ be the directional neighborhood with center at $\theta_j$ and with fixed radius $h>0$ defined by: $$\label{NeigbDirect} \mathcal{N}(\theta_j, h) = \{{\theta^* \in \mathcal{S}}:\, 0\leq d(\theta^, \theta_j) \leq h \},$$ for every $\theta_j \in \Theta$, equipped with the wrapped-Euclidean distance on $\mathcal{S}$: $$\label{circDist} {d(\theta^, \theta_j) := \min \big\{ \|\theta^* - \theta_j\|, 360 - \|\theta^* - \theta_j\|\big\}.}$$ We propose a heuristic threshold selection procedure that hinges on the propensity for extreme values to concentrate in the neighborhood $\mathcal{N}(\theta_j, h)$ of the centroid $\theta_j \in \Theta$. For each $\theta_{j}= j-1$ defined above, a threshold $u(\theta_j)$ is set automatically by drawing on the realisations of $X_i (\theta_{j'})$, $i=1, \ldots, n$, $\theta_{j'} \in \Theta \subset \mathcal{S}$, within lag $h$ of centroid $\theta_j$ resulting in a tally of: $$N(\theta_j)= {\ensuremath{\sum\limits_{i=1}^{n}}}{\ensuremath{\sum\limits_{j'=1}^{360}}} {\mathds{1}}_{\{\theta_{j'} \in \mathcal{N}(\theta_j, h)\}}\bigl( X_i(\theta_{j'}) \bigr)$$ observations. The indicator function ${\mathds{1}}_A(x)$ returns the value $1$ if $A$ holds true for $x$ and the value $0$ otherwise. A judicious choice of $h$ ensures a large enough number $N(\theta_j)$ is present, so that the extreme value theorem holds on $\mathcal{N}(\theta_j, h)$, for every $\theta_j \in \Theta$. The largest $k_j$ observations in $\mathcal{N}(\theta_j)$ are then taken as threshold exceedances for the direction-specific estimation of $\xi(\theta_j)$ and the optimal number of threshold exceedances is the number $k^_j$ satisfying $S_\phi(k^_j)={\underset{k}{\mathrm{min} \text{ }}} S_\phi(k_j)$ with: $$\label{autoChoice_k} S_\phi (k)= \frac{1}{k}\sum_{i \leq k} i^\phi \bigl\| \hat{\xi}_{i}(\theta_j) - \textrm{median}\bigl(\hat{\xi}_{1}(\theta_j), \hat{\xi}_{2}(\theta_j), \dots, \hat{\xi}_{k}(\theta_j)\bigr) \bigr\|,$$ where $0 \leq \phi < 0.5$, and $\hat{\xi}_{k}(\theta_j)$ stands for the designated estimator of $\xi(\theta_j)$ restricted to the $k$ upper observations amongst the $N(\theta_j)$ neighboring observations defined by $\mathcal{N}(\theta_j, h)$. The heuristic procedure was introduced by Reiss and Thomas [@reiss2007statistical cf.] and then studied in detail in @NFA2004. It facilitates an automatic choice of the threshold which can be understood intuitively as follows. For small $k$, the weighted deviations in tend to be large due to the inherently large variance of $\hat{\xi}_k(\theta_j)$. As $k$ increases, the summands in are expected to decrease until bias sets in and overrides the variance from which point $S_\phi$ is expected to increase again. Minimizing the weighted empirical distance is equivalent to optimizing the bias-variance trade-off by exploiting the settled behaviour of estimates $\{\hat{\xi}_k:\, k < N\}$ for appropriate $k$. A plethora of estimators for $\xi$ has been proposed in the literature, most notably, the moment estimator and parametric maximum likelihood estimator. These two estimators are extended here to directional estimators that not only rely on the magnitude of excesses above a threshold but also, and perhaps more critically, take into account the directional spread of exceedances relative to their centroid $\theta_j$. The motivation for the automatic selection of a moving threshold across $\theta_j \in \Theta$ is to couple threshold exceedances originating at every location $\theta_{j'} \in \mathcal{N}(\theta_j,h) \cap \Theta$ with their propensity $\omega(\theta_{j'})$ for spreading around $\theta_j \in \Theta$. Since we are dealing with circular data, the von Mises kernel is a natural choice for measuring this spread [cf. @PewseyNR14]. Precisely, we define weights: $$\label{NWweight} \omega(\theta_{j'}) := \frac{ \mathcal{K}_\eta(\theta_{j'})}{{\ensuremath{\sum\limits_{\theta_{j'} \in \mathcal{N}(\theta_j, h)}^{}}} \mathcal{K}_\eta( \theta_{j'})},$$ with the von Mises kernel implicitly defined on the centroid $\theta_j \in \Theta$ as: $$\mathcal{K}_\eta(s) := \frac{1}{2 \pi B_0(\eta)}\,\exp \big\{\eta \cos (s- \theta_j) \big\},$$ for $s\in \mathcal{S}$. The modified Bessel function of the first kind of order zero, $B_0(\eta)= \pi^{-1}\int_0^\pi e^{\eta \cos s}\, ds$, gives the required normalization in order to ensure $\mathcal{K}_\eta$ is in fact a density function. The concentration parameter $\eta > 0$ in the von Mises kernel controls the spread of the kernel in the sense that the greater the value of $\eta$, the greater the concentration and the lower the spread of the exceedances about the centroid $\theta_j\in \Theta$. Hence, this parameter plays a role similar to bandwidth $h$ intervening in \[NeigbDirect\], with both quantities playing out as important contributors to the degree of smoothness in the adaptive threshold estimation through the Moment (M) and maximum likelihood (ML) estimators given below in Sections \[Sct:ThrEst:SP\] and \[Sct:ThrEst:ML\]. Local semi-parametric estimation {#Sct:ThrEst:SP} -------------------------------- For each $\theta_j \in \Theta$, the extended version of the Moment estimator for $\xi(\theta_j)$ embedding directional weights is given by $$\label{MomMovThrKer} \hat{\xi}^{M}_k(\theta_j) \coloneqq M^{(1)}(\theta_j) + 1 - \frac{1}{2}\Big(1 - \frac{\bigl(M^{(1)}(\theta_j) \bigr)^2}{M^{(2)}(\theta_j)}\Big)^{-1},$$ with $$M^{(l)}(\theta_j) \coloneqq \sum_{ \substack{\theta_{j'}\in \mathcal{N}(\theta_j,h)\\ i =1, \ldots, n} } \omega(\theta_{j'}) \big(\log X_{i}(\theta_{j'})- \log X_{N(\theta_j)\,-k, N(\theta_j)} \big)^l\,{\mathds{1}}_{\{X_{i}(\theta_{j'}) > X_{N(\theta_j)\,-k, N(\theta_j)} \}} , \qquad l=1,2,$$ where $X_{N(\theta_j)\,-k, N(\theta_j)} $ denotes the $(k+1)^\text{th}$ largest value in the observed sample, whose directional covariate $\theta_{j'}\in \Theta$ belongs to $\mathcal{N}(\theta_j,h)$. This framework is key to the semi-parametric approach. The operative assumption relates to the asymptotic behavior of the $k$-th upper order statistics associated with the sample of i.i.d. positive random variables $X_i(\theta_j')-X_{N(\theta_j)\,-k, N(\theta_j)}$ established in the theory of extremes for threshold excesses. Conditionally on $X_{N(\theta_j)\,-k, N(\theta_j)}=u$, the common distribution function for these random variables is $F^{[u]}(t)= P\bigl(X(\theta_j)-u >t \|\,X(\theta_j)>t \bigr)$, for $t>0$ [cf. @deHaan2006extreme page 90]. See Appendix \[Sct:Append\] for a precise definition of $F^{[u]}$, and how it approaches the GPD function. Local maximum likelihood estimation {#Sct:ThrEst:ML} ----------------------------------- Along similar lines, the local directionally-weighted ML estimator $\hat{\xi}_k(\theta_j)$, for every $\theta_j \in \Theta$, is the result of maximizing, with respect to the parameter-vector $ \bigl( \xi(\theta_j), \sigma_u(\theta_j)\bigr)\in (-1, \infty) \times {\ensuremath{{{\mathbb{R}}}}}^+$, the weighted log-likelihood: $$\label{KernelLogLikNonStat} L\bigl(\xi(\theta_j), \sigma_u(\theta_j)\bigr) \coloneqq \sum_{ \substack{\theta_{j'}\in \mathcal{N}(\theta_j,h)\\ i =1, \ldots, n} } \omega(\theta_{j'}) \, \ell \big(\xi(\theta_j), \sigma_u(\theta_j) \|\,X_i(\theta_{j'})-X_{N(\theta_j)\,-k, N(\theta_j)} \big) {\mathds{1}}_{\{X_i(\theta_{j'})-X_{N(\theta_j)\,-k, N(\theta_j)} >0 \}},$$ with weights $\omega(\theta_{j'})$ as in , and: $$\ell \big(\xi(\theta_j), \sigma_u(\theta_j)\|\,y\big) = - \log \sigma_u(\theta_j) - \big(1+1/\xi(\theta_j)\big) \log \big( 1+\xi(\theta_j)\, y/ \sigma_u(\theta_j)\big)$$ when $\xi(\theta_j) \neq 0$. For $\xi(\theta_j) = 0$, a continuity argument yields $$\ell \big(\sigma_u(\theta_j) \| \,y \big) = -\log \sigma_u(\theta_j) -y/\sigma(\theta_j).$$ This ML formulation has been tailored for heuristic threshold selection, which itself has the semi-parametric method at its core. The link with parametric ML estimation is seen since, conditioned on the random number of exceedances $K=k$, random exceedances of threshold $X_{N(\theta_j)\,-k, N(\theta_j)}$ can be viewed as i.i.d. random variables with distribution function $F^{[u]}$ [details are deferred to part (i) of Appendix \[Sct:Append\]; cf. e.g. @reiss2007statistical page 234]. Algorithm for non-stationary threshold selection {#Sct:ThrEst:Alg} ------------------------------------------------ For each $\theta_j$, the local estimates $\hat{\xi}_k(\theta_j)$, $k=1, \ldots, N(\theta_j)-1$ used in and resulting $k_j^$, will reflect the extent of stationarity on $\mathcal{N}(\theta_j,h)$. The procedure described above for optimal selection of non-stationary threshold for all $\theta_j \in \Theta$ is outlined in Algorithm \[NPalg\]. We do not claim that this heuristic approach to extreme value threshold specification is optimal for every estimator one might devise for the extreme value index $\xi$, but we have found it useful in the applications context of Section \[Sec:App\]. Specify lag $h$, concentration $\eta$ and parameter $\phi$; Estimate weights $\omega(\theta_{j'})$ for $\theta_{j'}$ $\in \Theta$; Use the $k$ largest values in $\mathcal{N}(\theta_j,h)$ to estimate $\hat{\xi}_{k}(\theta_j)$ using or ; Calculate $S_\phi(k)$; Set $k_j^={\ensuremath{\displaystyle {\arg\min_{k}}}}\,S_\phi(k)$; Identify threshold $u(\theta_j)$ with the $(k_j^+1)^{\text{th}}$ largest value with direction in $\mathcal{N}(\theta_j,h)$; \[NPalg\] Spline-based maximum likelihood estimation {#Sct:SplML} ========================================== Given a non-stationary threshold $u(\theta_j)$, $\theta_j \in \Theta$, such as that obtained in Section \[Sct:ThrEst\], we proceed with parametric peaks over threshold analysis. We take the sample of threshold exceedances identified above, using the local maximum likelihood approach in Section \[Sct:ThrEst:ML\], and perform further parametric extreme value analysis. The purpose of this extra inference step is to mimic a conventional analysis of non-stationary threshold exceedances that might be undertaken in ocean engineering (see e.g. [@northrop2016thre]). Specifically, we assume a B-spline representation for the variation of GPD shape and scale parameters with covariate. We estimate spline coefficients using maximum penalized likelihood estimation, regulating the roughness of shape and scale with covariate to optimize predictive performance assessed using cross-validation. In Section \[Sct:ExtQnt\] we use the fitted model to infer $T$-year levels and (if appropriate) right endpoint. In the interest of physically meaningful inference, we assume that the shape and scale parameters vary smoothly with respect to covariate $\theta \in \mathcal{S}$, adopting smooth functions for $\xi(\theta)$ and $\log \sigma(\theta)$ using periodic cubic B-spline basis functions on $\mathcal{S}$ (see e.g. Chapter 5 of @wood2017generalized, @ZnnEA19a). On the index set $\Theta$ of covariate values $\theta_j, \ j=1,2,\dots,m$, we relate the values $\xi(\theta_j), \sigma(\theta_j)$ of GPD shape and scale to the periodic B-spline basis via basis matrix ${\boldsymbol{B}}$ with elements $B_{jb}$ such that: $$\label{bsplines} \xi(\theta_j) = \sum_{b=1}^{n_b} B_{jb} \beta_b^{(1)} \quad \text{ and } \quad \log \sigma(\theta_j) = \sum_{b=1}^{n_b} B_{jb} \beta_b^{(2)} , \qquad \theta_j \in \Theta,$$ where $n_b$ is the number of basis functions, and the $\beta$s are basis coefficients. The sample log-likelihood is: $$\label{LogLikNonStat} \ell({\boldsymbol{\beta}}) = - \sum_{j=1}^m \sum_{i=1}^{n_j} \Big( \log \sigma(\theta_j) + \big(1+\frac{1}{\xi(\theta_j)}\big) \log \big(1 + \frac{\xi(\theta_j)}{\sigma(\theta_j)} (X_i(\theta_j)-\widehat{u}(\theta_j) \ \big)\Big) {\mathds{1}}_{\{X_{i}(\theta_j) > \widehat{u}(\theta_j)\}},$$ with $n_j$ denoting the number of observations in the sample at covariate $\theta_j$, ${\boldsymbol{\beta}}^{(a)}=(\beta_1^{(a)},\beta_2^{(a)},\dots,\beta_{n_b}^{(a)})^\top$ for $a=1,2$, and ${\boldsymbol{\beta}}=({\boldsymbol{\beta}}^{(1)\top},{\boldsymbol{\beta}}^{(2)\top})^\top$. We set the number of spline knots on $\mathcal{S}$ to be more than sufficient to capture the anticipated parameter variability with covariate, and then penalize parameter roughness globally to obtain a model with good predictive performance. penalization in performed using first-order difference penalties for the coefficients in , $$P^{(a)} = \sum_{b=1}^{n_b-1} \Big(\beta_{b+1}^{(a)} - \beta_{b}^{(a)}\Big)^2 + \Big(\beta_{1}^{(a)} - \beta_{n}^{(a)}\Big)^2 = \boldsymbol{\beta}^{(a)\top} \boldsymbol{D}^\top \boldsymbol{D} \boldsymbol{\beta}^{(a)}, \qquad a =1,2,$$ with difference matrix ${\boldsymbol{D}}$ given by: $$\boldsymbol{D} = \begin{bmatrix} -1 & 1 & 0 & 0 & \cdot & \cdot \\ 0 & -1 & 1 & 0 & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & \cdot & 0 & -1 & 1 \\ 1 & 0 & \cdot & 0 & 0 & -1 \end{bmatrix}.$$ The penalized log-likelihood is then: $$\label{PenLogLikNonStat} \ell_{pen}(\boldsymbol{\beta}) = \ell(\boldsymbol{\beta}) - \lambda P^{(1)} - \kappa P^{(2)},$$ where $\lambda$ and $\kappa$ are smoothing parameters chosen maximise cross-validated predictive likelihood. In the application illustrated in Section \[Sec:App\], cross-validation is applied as follows. Each iteration of the cross-validation consists in using a bootstrap resample (of the original sample of threshold exceedances) as training set, and observations omitted from the bootstrap resample as test set. Note that the test sets corresponding to different bootstrap resamples may therefore overlap. The training set is used to estimate the model parameters, and the test set to assess prediction performance using mean squared prediction error (MSPE). The procedure is outlined in Algorithm \[Palg\]. Evaluate B-spline basis functions on index set $\Theta$; Specify sets of values of smoothing penalty coefficients $\lambda$ and $\kappa$ to consider; Estimate model parameters using bootstrap resample; Use estimated model to predict test observations (not occurring in bootstrap resample); Calculate the mean squared prediction error; Accumulate (average) mean squared prediction error; Select pair of values of $\lambda$ and $\kappa$ with best predictive performance, and evaluate spline coefficients for these choices of roughness coefficients; \[Palg\] Estimation of a non-stationary extreme quantile {#Sct:ExtQnt} ================================================ This section is devoted to the estimation of an extreme quantile, including that of the right endpoint in the case of a short-tailed distribution ($\xi<0$). We first present a class of maximum likelihood estimators for high quantiles in the parametric setting, and then introduce adapted versions of widely-used semi-parametric estimators amenable to the moving threshold $u(\theta)$, $\theta \in \mathcal{S}$, introduced in Section \[Sct:ThrEst\]. An algorithmic guide to inference of extreme quantiles and finite right endpoint, using any of the estimators discussed here, is given in Appendix B. In this section we assume a known (fixed, given) deterministic threshold $u(\theta)$, $\theta \in \mathcal{S}$ is available, as estimated in Section \[Sct:ThrEst:Alg\]. Exceedances are taken above threshold $X_{N(\theta)\,-k(\theta), N(\theta)}=u(\theta)$, where $k(\theta)$ is the number of exceedances in neighborhood $\mathcal{N}(\theta, h) \subset \mathcal{S}$ defined in . Further, let $k(\theta)/N(\theta)$ be the top sample fraction which will be retained for inference on extreme values, and $N(\theta)$ the total the number of observations in neighborhood $\mathcal{N}(\theta, h)$ for direction $\theta$. Given $u(\theta)$, the resulting random exceedances are distributed as $K=k(\theta)$ i.i.d. random variables with the same distribution function $F^{[u(\theta)]}$. This setting entails that any information in $X_{N(\theta)\,-k(\theta), N(\theta)}$ is disregarded [cf. @deHaan2006extreme page 90], and fits ideally to the parametric POT-GPD framework. Let $F_\theta$ be the actual distribution function underlying data measurements taken at direction $\theta \in \mathcal{S}$ and let $Q_\theta$ be the corresponding quantile function, i.e. $Q_\theta=F_\theta^\leftarrow$ with the left arrow indicating a generalized inverse. The basic theory for extremes (see Appendix \[Sct:Append\]) establishes the GPD function as the limit for the distribution of the linearly normalized exceedances over a threshold. In fact, conditions and ensure that an extreme quantile $F^\leftarrow(1-p)$, with $1-p > F(u)$ for some high threshold $u$, only depends on the tail of the distribution function $F$. Consequently, it will be possible to estimate an extreme quantile associated with a very small probability $p$ via the corresponding linear functional $\hat{b} + \hat{a} \,Q^H(1-p)$, with $Q^H$ now pertaining to the GP distribution function $H_{\xi}(x)= 1-(1+\xi x)^{-1/\xi}$, for all $x>0$ and $1+\xi x >0$, $\xi \in {\ensuremath{{{\mathbb{R}}}}}$. This setting carries over to both parametric and semi-parametric approaches, although with slight and yet potentially impactful differences. The most obvious of these is that following a parametric approach, the normalising constants $a$ and $b$ are ascribed to scale and location of the limiting GPD, whereas in the semi-parametric approach these are estimated as functionals of the sample analogue to the distribution function $F_{\theta}$. In the parametric setting, the level $x_p$ with small exceedance probability $p$ corresponds to the quantile of the distribution of the POT-exceedances at direction $\theta$. This suggests defining the $1/p$ extreme quantile as the value $x_{p}= Q^H(1-p)$, with normalizing constants $a>0$ and $b$ set to scale and location parameters of the GPD function $H$ for large $1/p$. Hence the local ML and spline ML estimator of $x_{p}$: $$\label{RLparametric} \widehat{x}_{p}^{ML}(\theta) = u(\theta) + \hat{\sigma}(\theta)\, \frac{\bigl(\frac{\varphi(u,\theta)}{p} \bigr)^{\hat\xi(\theta)}-1}{\hat\xi(\theta)},$$ with $\varphi(u,\theta)=k(\theta)/N(\theta)$, the sample fraction of exceedances of $u(\theta)$ within $\mathcal{N}(\theta,h)$. The estimator of an extreme level in can be interpreted as a $T$-year level used in hydrology and ocean engineering: $x(T)$ is that value exceeded on average once a year, i.e. $P\bigl(X> x(T)\bigr)= 1/T$, whereby we put $x(T)=x_{1/T}= Q^H(1-1/T)$. Local ML estimates for parameters $\xi(\theta_j)$ and $\sigma(\theta_j)$ on the index set $\theta_j \in \Theta$ for are obtained using the likelihood criterion in Section \[Sct:ThrEst\], now with constant weights $\omega(\theta_{j'})$ in . The values of $k(\theta_j)$ and $N(\theta_j)$ are estimated in Section \[Sct:ThrEst\]. In the spline ML approach of Section \[Sct:SplML\], $\varphi(u,\theta_j)$ is estimated as the probability of threshold exceedance for $\theta_j \in \Theta$ using logistic regression, with log-likelihood: $$\label{EllBeta} \ell({\boldsymbol{\beta}}) = \sum_{j=1}^m \tau_j \log[\nu_j] + (1-\tau_j) \log[1-\nu_j],$$ where $\tau_j$ is the sample proportion of threshold exceedances of $u$ at $\theta_j$. $\nu_j=(1+\exp[-\eta_j])$ is the probability of threshold exceedance at $\theta_j$, with ${\boldsymbol{\eta}} = \{\eta_j\}_{j=1}^m = {\boldsymbol{B}}{\boldsymbol{\beta}}$ for B-spline basis matrix ${\boldsymbol{B}}$ and parameter vector ${\boldsymbol{\beta}}$ to be estimated. Roughness penalization of ${\boldsymbol{\eta}}$, with optimal roughness coefficient $\mu$ estimated by cross-validation, ensures good predictive performance. The penalized log-likelihood thus takes the form $\ell_{pen}({\boldsymbol{\beta}}) = \ell({\boldsymbol{\beta}}) - \mu {\boldsymbol{P}}$, with ${\boldsymbol{P}}={\boldsymbol{\beta}}^\top {\boldsymbol{D}}^\top {\boldsymbol{D}} {\boldsymbol{\beta}}$ (see Section \[Sct:SplML\]). Estimation of $\xi(\theta)$ and $\sigma(\theta)$ for the spline ML approach is explained in Section \[Sct:SplML\]. In case $\xi(\theta)<0$, the limiting GP distribution function has a finite right endpoint which we also seek to estimate. A consistent estimator for this right endpoint follows readily from by setting $p=0$: $$\label{EPparametric} \hat{x}_{0}(\theta)= {u}(\theta) - \frac{\hat{\sigma}(\theta)}{\hat{ \xi}(\theta)}.$$ In the semi-parametric setting, we assume that the intermediate number $k(\theta)$ of exceedances above threshold $u(\theta)$ is such that $k(\theta) \rightarrow \infty$ and $k(\theta)/N(\theta) \rightarrow 0$ as $N(\theta)\rightarrow \infty$. That is, the number of threshold exceedances $k(\theta)$ remains negligible compared to the total number $N(\theta)$ of observations in neighborhood $\mathcal{N}(\theta,h)$. The latter can be rephrased in terms of the direction-specific sample size $n$ for the theoretical underpinning to domains of attraction that requires $n\rightarrow \infty$ also implies $N(\theta) \rightarrow \infty$. The proposed M estimator for the extreme quantile with small probability $p=p_n \rightarrow 0$ and $N(\theta)p/k(\theta) \rightarrow 0$, as $n \rightarrow \infty$, conditioned on the threshold $u(\theta)$ now follows: $$\label{HQsemiparametric} \widehat{x}^{M}_{p}(\theta) = u(\theta) + \widehat{a}_{\theta}^{M}\bigl( \frac{N(\theta)}{k(\theta)}\bigr)\, \frac{\Big( \frac{k(\theta) }{N(\theta) p}\Big)^{\hat{\xi}^{M}_{k}(\theta)}-1}{\hat{\xi}^{M}_{k}(\theta)},$$ with $k(\theta)$ assumed fixed. We then proceed in a semi-parametric way with the plug-in M estimator for the EVI defined in the same way as before in : $$\label{MomentEstPar} \hat{\xi}^{M}_k(\theta_j) \coloneqq M^{(1)}(\theta_j) + 1 - \frac{1}{2}\Big(1 - \frac{\bigl(M^{(1)}(\theta_j) \bigr)^2}{M^{(2)}(\theta_j)}\Big)^{-1},$$ and associated scale estimator: $$\label{afctEst} \hat{a}^M_{\theta}\Bigl(\frac{n}{k(\theta)}\Bigr):= u(\theta)\, M^{(1)}(\theta) \frac{1}{2} \Big(1 - \frac{(M^{(1)}(\theta))^2}{M^{(2)}(\theta)}\Big)^{-1},$$ with $M^{(l)}(\theta)$, $l=1,2$, updated to: $$M^{(l)}(\theta) = \frac{1}{k(\theta)}{\ensuremath{\sum\limits_{ i=1}^{k(\theta)-1}}}{\ensuremath{\sum\limits_{\theta_{j} \in \mathcal{N}(\theta,h)}^{}}} \big(\log X_{i}(\theta_j)- \log {u}(\theta_j)\big)^l\,{\mathds{1}}_{\{X_{i}(\theta_j) > {u}(\theta_j)\}}.$$ Expressions and show obvious similarities, and also distinctive traits of the M, local ML and spline ML approaches. Scale $a>0$ in is a function of the top sample fraction $k(\theta)/N(\theta)$, compared with the sample-free parameter GPD $\sigma>0$ in . In view of theoretical development in Appendix \[Sct:Append\](ii), the estimator of the scale function $\hat{a}^M$ refers to a consistent estimator $\hat{a}_{\theta}$ for the scale function $a^_\theta\bigl(\nicefrac{n}{k \omega(\theta)} \bigr)$ therein. The class of estimators for the right endpoint $x^F$ has been much studied from parametric and semi-parametric perspectives [cf. Chapter 4 of @deHaan2006extreme and connected references]. We now propose a model-free, data-driven estimator of $x^{F}(\theta)<\infty$, $\theta \in \mathcal{S}$, motivated by the general endpoint estimator of @FAN2014, coupled with non-stationary threshold $u(\theta)$. Specifically, we formulate an extreme value condition aiming to induce a partition of the Gumbel max-domain of attraction into a class of distributions with finite upper bound and a class containing the remainder. An example of a distribution function in the former class is the Negative Fréchet, with distribution function $F_{\alpha,\beta}(x)= 1-\exp\{-(\alpha-x)^{-\beta}\}$, $x \leq \alpha$, $\alpha \in {\ensuremath{{{\mathbb{R}}}}}$, $\beta>0$. Simple calculations show that $F_{\alpha,\beta}$ belongs to the Gumbel max-domain of attraction (hence with $\xi=0$) despite having finite right endpoint $x^F=\alpha<\infty$. In this sense, a semi-parametric estimator is likely to fare better than the parametric alternative in connection with small negative values of the shape parameter $\xi$. The parametric approach will tend to bear on the archetypical exponential distribution for inference (i.e., drawing on the POT-GDP with $\xi=0$ and infinite upper bound) and hence distributions associated with $\xi=0$ albeit with finite right endpoint will escape its grasp. With $\bigl\{Y_{i}\bigr\}_{i=1}^{N(\theta)}$ representing the random variables $X_i$s in a particular neighborhood $\mathcal{N}(\theta,h)$, defined in , we denote by $Y_{1,N(\theta)}\leq \ldots \leq Y_{N(\theta)-k(\theta),N(\theta)}\leq \ldots\leq Y_{N(\theta),N(\theta)}$ the corresponding ascending order statistics, and define the general endpoint estimator of $x^{F(\theta)}$, assumed finite, for every $\theta \in \mathcal{S}$, $$\label{generalEndpoint} \hat{x}_0^G(\theta) := Y_{N(\theta),N(\theta)} + \frac{1}{\log 2}{\ensuremath{\sum\limits_{i=0}^{k(\theta)-1}}} \log \Bigl(\frac{k(\theta)+i+1}{k(\theta)+i} \Bigr) \big( Y_{N(\theta)-k(\theta),N(\theta)} - Y_{N(\theta)-k(\theta)-i,N(\theta)}\big).$$ Notably, this estimator is valid for any $\xi(\theta)\leq 0$ deeming any prior estimation of the EVI unnecessary [cf. @FANR17]. Application to the storm peak significant wave height {#Sec:App} ===================================================== The methodology above provides an approach to applied non-stationary extreme value analysis incorporating elements from both parametric and semi-parametric inference as described. A key feature of the methodology is the estimation of a non-stationary threshold capturing the covariate dependence of large values of response so as to balance the number and spread of large observations on the covariate domain as explained in Section \[Sct:ThrEst\]. In this section, we apply the methodology to estimation of $T$-year values from the sample of storm peak significant wave height $H_S^{sp}$ on storm direction $\theta$ introduced in Section \[Sct:Dat\]. The mechanics of inference for extreme quantiles and right endpoint, if appropriate, is given in Appendix \[Sct:AppendOutline\]. Exploratory analysis of the sample, supported by previous analysis by [@RndEA15a], suggests that the covariate domain can be partitioned into five directional sectors assumed approximately homogeneous in terms of the characteristics of $H_S^{sp}$. Referring to Figure \[Fig:HsSPdata\], directional sectors corresponding to the following intervals of $\theta$ were identified. Sector 1 corresponds to $\theta \in (0^\circ,50^\circ] \cup (321^\circ,360^\circ]$, for storms propagating from the Norwegian Sea to the North; Sector 2 for $\theta \in (50^\circ,140^\circ]$ corresponds to the “land shadow” of Norway, with fetch-limited storms propagating from the coast with a more northerly direction relative to the normal to the coast; Sector 3 is $\theta \in (140^\circ,210^\circ]$, again for the Norwegian land-shadow, but with storms propagating from a more southerly direction; Sector 4 is $\theta \in (210^\circ,270^\circ]$ corresponding to storms from the Atlantic potentially “funnelled” by the Norwegian coast; and Sector 5 with $\theta \in (270^\circ,320^\circ]$, for more northerly Atlantic storms. Further information about the underlying physics is given in Section \[Sct:Dat\]. The partitioned sample is summarised in Figure \[Fig:ViolinData\], using so-called “violin” plots which add kernel density estimates to a box-whisker representation. The long-tailed behaviour of storms from the Atlantic is clear in Sectors 4 and 5, compared to the fetch-limited characteristics in storms from Sectors 2 and 3. Although Sector 4 exhibits the largest values of threshold exceedances in Figure \[Fig:ViolinData\], there is evidence from the kernel density plots that Sector 5 has a relatively long tail. In this section we seek to quantify tail-heaviness by estimating EVI / shape parameter $\xi$ using both parametric and semi-parametric approaches, and hence estimate extreme quantiles. `` ![ Violin plots of [$H_S^{sp}$]{} observations to the total of $N=1521$ data points.[]{data-label="Fig:ViolinData"}](figs/boxplotsViolinBySector_n.pdf "fig:") Using the approach in Section \[Sct:ThrEst\], non-stationary thresholds were estimated using the (parametric) ML estimator, and the (semi-parametric) M estimator for lag $h=30^\circ$, concentration $\eta=10$ and parameter $\phi=0.35$. Estimates are shown together in Figure \[Fig:thresholds\]. The general trends shown by the two estimates are in good agreement across the covariate domain. Subsequent parametric and semi-parametric inference for exceedance characteristics therefore has a relatively common starting point. ![ Adaptive threshold selection on the basis of the parametric ML estimator (black, solid line) and the semi-parametric M estimator with lag $h=30^\circ$, concentration $\eta=10$ and parameter $\phi=0.35$.[]{data-label="Fig:thresholds"}](figs/smooth_u_NP_h60_conc10_2.pdf) Estimates $\hat{\xi}(\theta_j)$ of EVI or GPD shape parameter $\xi$, gauging tail heaviness, for every centroid $\theta_j \in \Theta$ from each of the M, local ML an spline ML approaches is displayed in Figure \[Fig:EVI\], in terms of bootstrap means and 95% confidence intervals. Overall, there is good qualitative agreement between the three estimates. The estimates are also qualitatively plausible given other analyses of these data (@RndEA15a) and physical considerations. Effects of land shadows (e.g. $\theta \in (80,150)$) resulting in low $\xi$ are clear. The M estimator exhibits wider confident intervals. The spline ML estimator is smoothest with respect to covariate. Local ML and spline ML estimates make an additional asymptotic GPD assumption which the M estimate does not. The local ML estimator is in some senses intermediate, and this is reflected in the figure. Confidence limits exceed zero for all estimates, but this is more pronounced with the M estimation. Indeed, both M and local ML estimates for EVI do not differ significantly from zero for a large part of the covariate domain, suggesting that the data generating distribution lies in the Gumbel max-domain of attraction there. ![ Bootstrap mean and corresponding 95% confidence intervals for $\xi(\theta_j)$, $\theta_j \in \Theta$, based on: M (left), local ML (middle) and spline ML (right) estimators.[]{data-label="Fig:EVI"}](figs/boot_xi_M_h60_conc10_4.pdf "fig:") ![ Bootstrap mean and corresponding 95% confidence intervals for $\xi(\theta_j)$, $\theta_j \in \Theta$, based on: M (left), local ML (middle) and spline ML (right) estimators.[]{data-label="Fig:EVI"}](figs/boot_xi_ML_h60_conc10_4.pdf "fig:") ![ Bootstrap mean and corresponding 95% confidence intervals for $\xi(\theta_j)$, $\theta_j \in \Theta$, based on: M (left), local ML (middle) and spline ML (right) estimators.[]{data-label="Fig:EVI"}](figs/boot_xi_bsplines_h60_conc10_lambda_exp06_numIntKnots10_4.pdf "fig:") Estimation of $T$-year levels is straightforward once relevant POT-GPD parameters are estimated. Using expressions and , assuming $N_E$ occurrences of storm in observation period $T_0$, the $T$-year directional level corresponds to the value $x_p(\theta)$ such that $P\{ X(\theta) > x_p(\theta) \} = (T_0/N_E) \times T^{-1}$. Figure \[Fig:Tyearevent\] displays in a matrix rose-plots for the estimated $100$-year and $10,000$-year levels with accompanying 95% bootstrap confidence bands. Again, there is general qualitative agreement between the three estimates for $100$-year level (top row) and $10,000$-year level (bottom row), in terms of bootstrap mean. Uncertainties from the M estimate are somewhat larger, as might be expected recalling the evidence of Figure \[Fig:EVI\]. Not surprisingly, estimated extreme levels for directions with short fetches ($\theta \in [70,140)^\circ$) are low, whereas those corresponding to long fetches from the Atlantic Ocean and Norwegian Sea ($\theta \in [225,360)\cup0,40)^\circ$) are high. Estimated $100$-year return levels for [$H_S^{sp}$]{} fall between 15m and 20m the most severe sectors, in terms of bootstrap mean and confidence bands. The same is true of bootstrap means at the $10,000$-year level except for M estimates which exceed 20m. This is not inconsistent with evidence from Figure \[Fig:EVI\] regarding generally negative $\xi$ estimates. ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL100_M_h60_conc10_4.jpeg "fig:") ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL100_ML_h60_conc10_4.jpeg "fig:") ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL100_bsplines.jpeg "fig:") ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL10000_M_h60_conc10_4.jpeg "fig:") ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL10000_ML_h60_conc10_4.jpeg "fig:") ![ Rose diagrams for the $100$-year (top) and $10,000$-year (bottom) levels based on M (left), local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:Tyearevent"}](figs/RL10000_bsplines.jpeg "fig:") Assuming that threshold exceedances are indeed drawn from a short-tailed distribution independent of direction, we can also consider estimation of the largest possible value of $H_S^{sp}$ given the sample. Inspection of Figure \[Fig:EVI\] suggests that $\xi$ exceeds zero for all estimators for some values of covariate, and therefore it may be that the right endpoint is infinite there. Figure \[Fig:EP\] shows estimates corresponding to the general estimator for the finite right endpoint on the covariate domain, which does not require any estimation of $\xi$. Note that the general endpoint estimation does not rest on any value of the shape parameter/EVI, but rather on the less stringent assumption is that the distribution underlying the data has a finite endpoint. Figure \[Fig:EP\] also shows estimates of the right endpoint using outputs of local ML and spline ML estimation in , applied only when the estimated value of $\xi$ is negative. Figure \[Fig:Prop\] shows the proportion of bootstrap samples excluded from the ML-based inference, since $\hat\xi\geq 0$. ML estimation becomes more challenging as the true shape parameter $\xi$ approaches zero from negative values, with numerical optimization routines more than often experiencing convergence issues [see e.g. @GomesNeves2008 for a comparison between M and ML within the univariate semi-parametric setting]. ![ Rose diagrams for the finite right endpoint of [$H_S^{sp}$]{}. Three estimators are used: the general estimator (left), and the local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:EP"}](figs/boot_xF_gen.jpeg "fig:") ![ Rose diagrams for the finite right endpoint of [$H_S^{sp}$]{}. Three estimators are used: the general estimator (left), and the local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:EP"}](figs/boot_xF_localML_xiNeg.jpeg "fig:") ![ Rose diagrams for the finite right endpoint of [$H_S^{sp}$]{}. Three estimators are used: the general estimator (left), and the local ML (middle) and spline ML (right) estimators. Bootstrap means and corresponding 95% confidence bands are displayed as a function of direction measured clockwise from north.[]{data-label="Fig:EP"}](figs/boot_xF_splineML_xiNeg.jpeg "fig:") ![ Fraction of bootstrap samples excluded from the finite right endpoint estimation in Figure \[Fig:EP\] for failing to return a negative estimate of $\xi$, for (left) local ML and (right) spline ML.[]{data-label="Fig:Prop"}](figs/propNegEVI_localML.jpeg "fig:") ![ Fraction of bootstrap samples excluded from the finite right endpoint estimation in Figure \[Fig:EP\] for failing to return a negative estimate of $\xi$, for (left) local ML and (right) spline ML.[]{data-label="Fig:Prop"}](figs/propNegEVI_splineML.jpeg "fig:") Agreement between the three estimators for the finite right endpoint is not as good as that observed for $T$-year level. The characteristics of estimates from the general estimator and spline ML are in some agreement, apart from for northerly directions, for which spline ML suggests that a longer tail is present (and see Figure \[Fig:Prop\] for $\hat\xi>0$ from the North). For the Atlantic sector, local ML estimates are also in relatively good agreement with the others; however, local ML provides large numbers of estimates for $\xi$ exceeding zero for northerly and southerly storms. This produces large estimates for the right endpoint with large uncertainties. Indeed, were we to have attempted to estimate the right endpoint using semi-parametric M estimates for $\xi$ directly in , resulting endpoint estimates would also have been large and highly variable since a large proportion of M estimates for $\xi$ are fairly close to zero (cf. Figure \[Fig:EVI\]). In summary, application of the parametric and semi-parametric methodologies developed in Sections \[Sct:ThrEst\]-\[Sct:ExtQnt\] above to the sample of directional storm peak significant wave height suggests that estimates for extreme value index/ GPD shape, and $100$-year and $10,000$-year levels are in good qualitative agreement. Where differences occur, they can be understood and explained in terms of specific modelling assumptions made, rather than in terms of fundamental differences in the underlying approaches to extreme value analysis. Summarizing remarks {#Sec:Summary} =================== This paper presents a framework for inference on non-stationary peaks over threshold, reconciling approaches from semi-parametric and parametric extreme value analysis, in application to directional ocean storm severity. The key components of the framework are (a) estimation of non-stationary extreme value threshold, and (b) estimation of tail characteristics from threshold exceedances, including extreme quantiles and right endpoint when appropriate (finite). Threshold estimation is performed using a non-stationary extension of a heuristic approach proposed by @NFA2004 for semi-parametric moment (M) and parametric maximum likelihood (ML) estimators. Tail characteristics and extreme quantiles are then estimated, based on semi-parametric M, local parametric ML and spline ML estimators. We also develop a non-stationary semi-parametric general endpoint estimator (based on @FAN2014 for $\xi \leq 0$), and apply it with the standard right endpoint estimator (applicable for $\xi<0$) where appropriate. Inferences regarding directional thresholds for storm peak significant wave height are in good agreement over the covariate domain. Estimates for $100$ and $10,000$-year levels are also in reasonable agreement. Estimates for the right endpoint are more different across approaches, and are influenced by the specifics of modelling assumptions made associated with the different estimation strategies. For the application considered, both parametric and semi-parametric inference provides similar characterisations of extreme non-stationary ocean environments. Indeed, we illustrate how ideas from the semi-parametric and parametric schools of thought can be used in tandem to exploit the desirable features of the approaches, whilst overcoming some obvious pitfalls. For example, threshold estimation (used for both semi-parametric and parametric analysis) is motivated by an inherently non-parametric heuristic in Section \[Sct:ThrEst\]. Parametric approaches to non-stationary extremes are relatively well-studied due in part to the wide range of flexible covariate representations for GPD parameters for threshold exceedances, and associated methods for regression and assessment of model fit. In contrast, from a semi-parametric perspective, a tangible GPD exact fit need not be assumed, but avoiding a particular model choice from the outset generally results in increased uncertainty of estimates for EVI and high quantiles. However, we show in this work that semi-parametric and parametric approaches perform rather similarly when set up reasonably. By exploiting recent developments of extreme value theory for non-identically distributed observations [cf. @deHaan2015tail; @deHaanZhou20], we show that under reasonable and mildly restrictive assumptions, a suitable number of parameters can be introduced with the aim of optimizing the for bias/variance trade-off in the estimation of the various extreme value characteristics and/or indices. This optimization method relies heavily on an adaptive choice for the non-stationary threshold which determines where tail-related observations begin to show up in the available sample. Given threshold, the two streams of development mirror each other regarding tail inference. In the spline ML approach, a cubic B-splines representation with compact support is used: each basis function is non-zero on a specific interval of the covariate domain. This feature plays a similar role to bandwidth of directional neighborhoods in the semi-parametric M and local ML approaches. We anticipate that the framework presented here can be extended to address multidimensional covariates often encountered in practice. ACKNOWLEDGEMENT {#acknowledgement .unnumbered} --------------- Cláudia Neves gratefully acknowledges support from EPSRC-UKRI Innovation Fellowship grant EP/S001263/1 and project FCT-UIDB/00006/2020. ORCID {#orcid .unnumbered} ----- Evandro Konzen <https://orcid.org/0000-0002-6275-1681>\ Cláudia Neves <https://orcid.org/0000-0003-1201-5720>\ Philip Jonathan <https://orcid.org/0000-0001-7651-9181> Basis for inference on directional extremes via the POT method {#Sct:Append} ============================================================== The contents of this appendix build on Appendix B of @deHaan2015tail. This paper adds flexibility to the latter because we give allowances for $\xi=\xi(s)$ to vary with $s \geq 0$, where $s$ might represent direction $\theta$, or time or some other covariate. As a consequence, the right endpoint does not need to be assumed constant in $s$. We will not delve into the theoretical details in terms of explicit smoothness and boundedness conditions needing to be in place particularly by assuming $h=h_n>0$. These are clearly beyond the scope of this paper, but we envisage that the probabilistic underpinning to this work will stem from Chapter 9 of @deHaan2006extreme. At direction (or time) $s \geq 0$, let $\bigl( X_1(s), X_2(s), \ldots, X_n(s)\bigr)$ be a vector of i.i.d. random variables with common distribution function $F_s(x)$, for all $x \in \mathbb{R}$, absolutely continuous with right endpoint $x^{F_s} \leq \infty$. Assume $F_s \in \mathcal{D}(G_{\xi(s)})$, for some $\xi(s) \in \mathbb{R}$ and for every $s\geq 0$, i.e., that condition holds locally for each $s$. In this setting, Theorem 1.1.6 of @deHaan2006extreme ascertains that it is possible to replace $n$ with $t$ running over the real line in such a way that becomes equivalent to the following extreme value condition: there exists a positive function $a^_s$ such that $$\label{POTdomain} \lim_{t \uparrow x^{F_s}}\, \frac{ 1-F_s\bigl( t + x a^_s(t) \bigr)}{1-F_s(t)} = \frac{\log G_{\xi(s)}(x)}{\log G_{\xi(s)}(0)},$$ for all $x$ with $1+ \xi(s) x >0$. The limit in is the tail distribution function (also known as survival function) of the GP distribution with shape parameter $\xi(s) \in {\ensuremath{{{\mathbb{R}}}}}$, given by $\bigl(1+ \xi(s) x\bigr)^{-1/\xi(s)}$. The extreme value condition is often key for describing rare events’ behavior in lieu of the dual max-domain of attraction characterization $F\in \mathcal{D}(G_\xi)$. (i) Parametric approach {#i-parametric-approach .unnumbered} ----------------------- Taking a parametric view, the POT-domain of attraction condition prescribes the GP distribution as the proper fit to the normalized exceedances given these are above a certain high threshold near the right endpoint $x^{F_s}$. With minimal notational changes around a fixed (deterministic) threshold $u$, it is straightforward to see that condition implies $$\label{POTparam} \lim_{u \uparrow x^{F_s}} \, \Bigl\|P\{X_1(s) \leq x+u \, \|\, X_1(s) > u\} - H_{\xi(s), u, \sigma(u)}(x) \Bigr\|=0,$$ locally uniformly in $x> u$, for each $s\geq 0$, with $\sigma_s(u)>0$ (henceforth we omit the subscript $s$ for simplicity of notation) and $ H_{\xi, \mu, \sigma}(x):= 1- \bigl(1+\xi (x-\mu)/\sigma\bigr)^{-1/\xi}$, for all $x$ such that $0<H_{\xi, \mu, \sigma}(x)<1$, with location $\mu \in \mathbb{R}$ and scale $\sigma>0$. Informally, $F_s^{[u]}(x):= P\{X_1(s) \leq x+u \, \|\, X_1(s) > u\} \approx H_{\xi(s), u, \sigma(u)}(x)$, for all $x> 0$ and $u$ near the right endpoint $x^{F_s}$, with the scale parameter implicitly defined in terms of $s$ through the threshold $u(s)$. For each $s$, the limiting relation provides the probabilistic underpinning for fitting a GP distribution function to the unconditional tail distribution function $\overline{F_s}(x):= 1-F_s(x)$ with $x$ sufficiently large. This becomes more evident since $$\begin{aligned} & & F_s(x) = (1- F_s(u)) F_s^{[u]}(x) + F_s(u), \end{aligned}$$ whence, $$\label{POTapprox} \overline{F_s}(x)= \bigl(1-F_s^{[u]}(x)\bigr)(1- F_s(u)) \approx \bigl(1- H_{\xi(s), u, \sigma(u)}(x)\bigr)(1- F_s(u)),$$ for $ x > u$, as $u \rightarrow x^{F_s}$. Finally, we note that $$\label{Rep} \bigl(1- H_{\xi(s), u, \sigma(u)}(x)\bigr)(1- F_s(u))= H_{\xi(s), \mu^, \sigma^(u)}(x),$$ where $\mu^-u= \sigma(u) \,U_{H} \bigl(1- F_s(u)\bigr)$, $\sigma^(u) = \sigma(u) \bigl(1- F_s(u)\bigr)^{\xi(s)}$, and $U_H$ standing for the tail quantile function pertaining to the standard GPD, that is $$U_H(t):= \biggl(\frac{1}{1-H_{\xi(s), 0, 1}}\biggr)^{\leftarrow}(t)=\frac{t^{\xi(s)}-1}{\xi(s)},$$ for all $t \geq 1$ (the left arrow indicates the left-continuous inverse). The representation facilitates the view that, in practice, changes in the threshold (e.g. through covariates) will be reflected in the scale parameter. In turn, the approach for inference is reflected in the way we go choose to go about the term $1- F_s(u)$ for this to become statistically meaningful. In order to able to perform large sample inference drawing on the POT-GPD framework streamlined above, we now make the threshold dependent on the sample size $n$ and $u=u(n)$ will naturally become larger as $n$ goes to infinity. A parametric approach typically advocates for a large enough threshold to be fixed and inference to be conducted on the basis of the resulting POT framework, whereby the expected number of exceedances above the selected threshold is a random number $K_s$, say, satisfying $(n/K_s)(1- F_s(u_n)) \rightarrow 1$ in probability, as $n\rightarrow \infty$. This suggests estimation of $(1- F_s(u))$ via the analogous tail empirical distribution function (stepping up by $1/n$ at each observation) evaluated at $u$, adding up to $1-K_s/n$ in the above, associated with the random number $K_s$ of exceedances of $u$ at direction (or location) $s$. Hence, for a given (fixed) $k_s$, the location and scale parameters in become: $$\begin{aligned} \mu^(s)&=& u + \sigma(u) \,U_{H} \Bigl(1- \frac{k_s}{n}\Bigr),\\ \sigma^(u) &=& \sigma(u) \Bigl(1- \frac{k_s}{n}\Bigr)^{\xi(s)}.\end{aligned}$$ Therefore, the crux of parametric inference for extremes values lays in the estimation of the shape and scale parameters, respectively, $\xi(s)$ and $\sigma_s=\sigma^*(u)$. (ii) Semi-parametric approach {#ii-semi-parametric-approach .unnumbered} ----------------------------- It will be notationally cleaner to express the argument in terms of the pertaining tail quantile function $U_s:= \bigl(1/(1-F_s)\bigr)^\leftarrow$. Note that $U_s(t)$ is non-decreasing and provides a straightforward link to an extreme quantile, with the right endpoint representing the ultimate quantile: $\lim_{t\rightarrow \infty} U_s(t)=U_s(\infty)= x^{F_s}$. To this effect, we make $t$ in depend on the (possibly unknown) sample size $n$ at each location $s$ through replacing it by $U_s(n/k_s)$, where $k_s$ is an intermediate sequence of positive integers such that $k_s=k_s(n) \rightarrow \infty$ and $k_s/n \rightarrow 0$, as $n\rightarrow \infty$. This is possible because holds uniformly in $x$. Hence, we have for the left hand-side of : $$\frac{1-F_s\bigl(U_s(n/k_s)+ x \, a_s\bigl(U_s(n/k_s)\bigr) \bigr)}{1-F_s\bigl(U_s(n/k_s)\bigr)} = \frac{n}{k_s}\bigl(1-F_s\bigl(U_s(n/k_s)+ x \, a^\star_s(n/k_s) \bigr),$$ with $a_s^\star(n/k_s)=a_s\bigl(U_s(n/k_s)\bigr)$. For simplicity, we consider regularly spaced independent vectors $\bigl(X_{1}(s), X_{2}(s), \ldots, X_{n}(s)\bigr)$, $s=1,2, \ldots, m, \ldots$ with i.i.d. components, with partial tally of $N=n\times m \in \mathbb{N}$ observations across the whole system, and where $n$ is potentially unknown (without affecting inference on extremes), yet assumed large ($n\rightarrow \infty$). In this setting, the basic extreme value condition is: $$\label{DOAnonstat} \lim_{n \rightarrow \infty} \frac{N}{\omega(s) k}\Bigl[1-F_s\Bigl(\,U_s\bigl(\frac{n}{\omega(s) k}\bigr)+ x \, a^{\star}_s\bigl(\frac{n}{\omega(s) k}\bigr) \Bigr) \Bigr]= \bigl(1+\xi(s)x\bigr)^{-1/\xi(s)},$$ for all $x$ with $1+\xi(s)x>0$, uniformly in $s=1, 2, \ldots$, subject to $(1/m)\sum_{s= 1}^m \omega(s) \rightarrow 1$, as $m\rightarrow \infty$, i.e. the sequence of weights $\{\omega(m)\}_{m\in {\mathbb{N}}}$ is Cesàro summable. The latter is to maintain integrity, also ensuring that the stationary case is well-defined. In particular, the case of complete stationarity, corresponding to omni-directional data in the context of this paper, is recovered if $\omega(s)$ is uniformly distributed over the stipulated range for $s$. The interest lies in the estimation of the various extreme value indices $\xi(s)$, and the scale and location terms, respectively $a_s^\star(n/k_s)$ and $U_s(n/k_s)$, now with $k_s:= [\omega(s) \times k]$ and $[\bigcdot]$ standing for integer part. Since the $n$-th order statistic $X_{n-k_s:n}(s)$ is close to $U_s(n/k_s)$, if $k_s \rightarrow \infty$, $k_s/n \rightarrow 0$, as $n \rightarrow \infty$, we shall adopt it, as the usual estimator for the threshold $\widehat{U}_s(n/k_s)=X_{n-k_s:n}(s)$. The random adaptive threshold in this setting emulates the non-stationarity mirrored in the scale $\sigma_s>0$ which features the parametric setting (i). . Roadmap for application {#Sct:AppendOutline} ========================= The procedure for estimation of an extreme level and of the right endpoint, if appropriate, for any of the three methods is summarised in the following algorithm for clarity. The algorithm assumes that the non-stationary threshold $u(\theta)$, $\theta \in \mathcal{S}$ is already known. Specify data sample; non-stationary threshold $u(\theta)$; period of sample $T_0$, return period $T$; Isolate set of directional threshold exceedances; Specify window half-width $h$; Specify window half-width $h$; Specify values of smoothing coefficients $\lambda$ and $\kappa$ to consider; Specify details for B-spline basis function construction; Generate bootstrap resample from sample of threshold exceedances; Count number $k(\theta)$ of threshold exceedances in $\mathcal{N}(\theta,h)$; Count number $N(\theta)$ of observations in $\mathcal{N}(\theta,h)$ ; Estimate $\hat\xi(\theta)$ on $\mathcal{N}(\theta,h)$ using ; Estimate $\hat{a}_\theta$ on $\mathcal{N}(\theta,h)$ using ; Estimate high quantile with $p << 1/N(\theta)$ value using ; Estimate finite right endpoint ($x^{F_\theta}<\infty$) using ; Count number $k(\theta)$ of threshold exceedances in $\mathcal{N}(\theta,h)$; Count number $N(\theta)$ of observations in $\mathcal{N}(\theta,h)$ ; Estimate $\hat\xi(\theta)$, $\hat\sigma(\theta)$ on $\mathcal{N}(\theta,h)$ (using (4.6) with weights $\omega=1$); Estimate $T$-year return value using ; Estimate right endpoint (when $\hat\xi(\theta)<0$) using ; Estimate optimal smoothing parameters $\lambda$, $\kappa$ and hence estimate $\hat\xi(\theta)$, $\hat\sigma(\theta)$ (Algorithm 2); Estimate optimal smoothing parameter $\mu$ and fraction $\hat\tau(\theta)$ of threshold exceedances by logistic regression using ; Estimate $T$-year return value using , with $\hat\tau(\theta)$ in place of $k(\theta)/N(\theta)$; Estimate right endpoint (when $\hat\xi(\theta)<0$) using ; Accumulate bootstrap estimates for parameters, extreme levels or quantiles and for the right endpoint; Calculate bootstrap means and confidence intervals for parameters, return values and endpoint;
### startsection[subsubsection]{}[3]{}[10pt]{}[-1.25ex plus -1ex minus -.1ex]{}[0ex plus 0ex]{}[***]{} #### startsection[paragraph]{}[4]{}[10pt]{}[-1.25ex plus -1ex minus -.1ex]{}[0ex plus 0ex]{} biblabel\[1\][\#1]{} makefntext\[1\][\#1]{} Introduction ============ Active fluids are energized locally by motorized microscopic active particles such as kinesin-driven microtubules [@Sanchez2012] and myosin-actin complexes [@Mizuno370]. Therefore, their dynamics occur out of thermal equilibrium [@ramaswamy2017; @srimrmp]. Hydrodynamic instabilities of both polar and nematic active fluids have been studied using hydrodynamic theories and simulations for bulk [@sriram2002; @SaintillanShelley2008; @PahlavanSaintillan2011] as well as for confined fluids [@sriramThinfilm2009; @Maitra2018; @WoodhouseGoldstein2012; @Giomi_etal2012; @Norton2018; @Theillard2017; @Edwards2009]. In this paper, we consider the instabilities of active nematic fluids in the isotropic phase confined by an interface. The damping of a capillary wave on a flat interface between two passive viscous fluids is well-understood [@Lamb1994]. Likewise, theoretical studies of interfacial instabilities like the Rayleigh-Plateau capillary instability and Rayleigh-Taylor interface instability have been carried out for passive fluids [@Tomotika1935] including complex fluids such as polymer solutions [@Stanley1965; @goldin1969] and liquid crystals [@Cheong2001; @Cheong2002]. Less work has been done on interfacial instabilities in active fluids. It is natural to expect that the instabilities that occur in bulk active fluids can destabilize an otherwise stable interface, or make an already unstable interface more unstable. Work to date includes a study by Yang and Wang [@softmatter2014] of the Rayleigh-Plateau capillary instability of a thread of active polar fluid in the ordered state surrounded by a passive Newtonian fluid, a study by Whitfield and Hawkins [@2016njpactivedrop] of the instability of an spherical droplet of active polar fluid in the ordered state, and an analysis by Gao and Li [@activedroplet2017prl] of a self-driven droplet of an active nematic fluid. Also, Patteson et al.* [@Patteson2018] studied the propagation of active-passive interfaces in bacterial swarms. Recently, Maitra et al. [@sriramMembrane2014] explored the dynamics of an active membrane in an active polar medium, Mietke et al. studied the instabilities of an active membrane in a passive fluid [@MietkeJulicherSbalzarini2019], and V. Soni et al. studied the surface dynamics of an active colloidal chiral fluid [@VSoni_etal2018]. Here, we focus on linear stability analyses of active nematic fluids in the isotropic phase in with flat, cylindrical, or spherical interfaces. Our focus on the isotropic phase is motivated by recent experiments on active matter that show large regions in which the nematic order is small [@Wu_etal2017]. Our work is distinct from the other theoretical work just mentioned on interfacial instabilities in active fluids because we consider the active nematic to initially be in the isotropic state instead of the ordered state. We model the isotropic phase of an active nematic fluid by adding activity to de Gennes’ hydrodynamic model [@swimming2018; @DeGennes1969; @degennesbook; @deGennes1971] for the isotropic phase of a passive nematic. This model is appropriate for ‘shakers’ rather than ‘movers’ suspended in a liquid. The model shows that in the linear regime the isotropic active nematic fluid behaves like a viscoelastic fluid, with the viscosity and viscoelasticity growing large as the isotropic-nematic transition is approached [@Hatwalne2004]. However, our results for the stability of interfaces are qualitatively different from the passive viscoelastic fluid case due to the orientational degrees of freedom. We work in the limit of low Reynolds number, where viscous effects dominate inertial effects. For a passive fluid, deformations of a surface or spherical surface always relax, whereas a cylindrical thread is unstable to peristaltic deformations of sufficiently long wavelength. When the fluid is active, deformations of the surfaces in all three cases can be unstable. The instability of a bulk active isotropic fluid drives the instability of the flat and spherical surface, and enhances the Rayleigh-Plateau capillary instability of a cylinder. Our key results are as follows. In the all cases we consider, the coupled dynamics of the interface of the fluid and the nematic order parameter leads to two modes, with damped or growing propagating waves found for a sufficiently large dimensionless activity. Likewise, surface tension makes it harder for activity to destabilize an active film or an active fluid confined by a spherical interface, as compared to the unconfined case. For the cylindrical thread of radius $R$, harmonic perturbations of wavenumber $k<1/R$ are always unstable, just as in the passive case. Perturbations with $k>1/R$ become unstable above a critical activity increasing with $k$ and the surface tension of the interface. The remainder of the paper is organized as follows. In section \[model\], we introduce a hydrodynamic model for an active nematic fluid in the isotropic state. In section \[membrane\], we use this model to study the linear stability of a film bound by an interface. Next , in section \[thread\], we consider the stability of a thread of active fluid bound by either an interface. Finally, in section \[droplet\] analyze the stability of a spherical drop of active fluid. We offer concluding remarks in Section \[conclusion\]. Section \[appen\], the Appendix, contains additional details relevant to Section \[membrane\]. Model ===== The total free energy of an active isotropic nematic fluid with an interface is $\mathcal{F}= \mathcal{F}_n+\mathcal{F}_i$, where $\mathcal{F}_n$ is the free energy of the nematic fluid, and $\mathcal{F}_i$ is the energy of the interface. Denoting the nematic order parameter field by $Q_{\alpha\beta}$, the nematic free energy is $$\mathcal{F}_n=\int d^3x \left[ \frac{A}{2} Q_{\alpha\beta}Q_{\alpha\beta}+\frac{B}{3} Q_{\alpha\beta}Q_{\beta\gamma}Q_{\gamma\alpha}+\frac{C}{4}(Q_{\alpha\beta}Q_{\alpha\beta})^2\right], \label{Fn}$$ where we sum over repeated indices $\alpha, \beta,\dots$ which run over the three spatial coordinates. We consider the isotropic phase, for which $A>0$. In this case, Frank elasticity can be neglected as long as we are not too near the nematic transition. The interface energy is given by $$\mathcal{F}_i=\int\gamma \mathrm{d}S.\label{Fi}$$ where $\gamma$ is the interfacial tension and $\mathrm{d}S$ is the element of area. We use de Gennes’ hydrodynamic model [@DeGennes1969; @degennesbook; @deGennes1971] of the isotropic phase of a passive nematic fluid of uniform concentration, suitably modified [@swimming2018] to account for activity. In terms of the fluid velocity field $v_\alpha$, the strain rate and the vorticity tensors are given by $E_{\alpha\beta}=(\partial_\alpha v_\beta+\partial_\beta v_\alpha)/2$ and $\Omega_{\alpha\beta}=(\partial_\alpha v_\beta-\partial_\beta v_\alpha)/2$, respectively, where $\alpha,\beta=x,y,z$. The rate of change $R_{\alpha\beta}$ of the nematic order parameter $Q_{\alpha\beta}$ relative to the local background fluid is defined as $$R_{\alpha\beta}=\partial_t Q_{\alpha\beta}+\bm{v}\cdot\bm{\nabla}Q_{\alpha\beta}+\Omega_{\alpha\gamma}Q_{\gamma\beta}-Q_{\alpha\gamma}\Omega_{\gamma\beta}.$$ Then, the viscous stress $\sigma^v_{\alpha\beta}$ and equation of motion for the nematic order parameter $Q_{\alpha\beta}$ are given by [@swimming2018] $$\begin{aligned} &&\sigma^v_{\alpha\beta}=2\eta E_{\alpha\beta}+2(\mu+\mu_1) R_{\alpha\beta}+a'Q_{\alpha\beta}\label{stressa},\label{sigmaeq}\\ &&\Phi_{\alpha\beta}=2\mu E_{\alpha\beta}+\nu R_{\alpha\beta},\label{Qeq}\end{aligned}$$ where $\eta$ and $\nu$ are the shear and rotational viscosities, respectively, $\mu$ couples shear and nematic alignment, $a'$ and $\mu_1$ are activity parameters, and $\Phi_{\alpha\beta}$ is the molecular field defined as $\Phi_{\alpha\beta}\equiv-\delta \mathcal{F}/\delta Q_{\alpha\beta}$. From Eq. (\[Qeq\]) we can see that for the case of small shear rate and steady state, the principal axes of the order parameter align with the principal axes of the strain, with the case $\mu<0$ corresponding to the way that prolate particles align in shear, and the case $\mu>0$ corresponding to the way oblate particles align in shear. In passive fluids, $a'=0$ and $\mu_1=0$. In that case, the Onsager reciprocal relations [@onsager] are obeyed and the positive entropy production rate leads to the relation $\eta\nu-2\mu^2>0$. The active term $aQ_{\alpha\beta}$ appearing in Eq. accounts for the stress due to the force dipoles associated with the active particles [@sriram2002; @swimming2018] with $a'>0$ for contractile particles and $a'<0$ for extensile particles. Since time reversal symmetry and the Onsager relations are violated in active fluids when we do not keep track of the chemical reactions in the theory, the active term $\mu_1 R_{\alpha\beta}$ is allowed in Eq. . Various other approaches to the active matter equations also violate the Onsager reciprocal relation [@sriram2002; @WoodhouseGoldstein2012; @Norton2018]. In our entire analysis, we assume that $\mu_1$ is sufficiently small such that $\eta\nu-2\mu(\mu+\mu_1)>0$. Since we only study linear stability of the state with no order and no flow, we are justified in disregarding terms of higher order than quadratic in the order parameter. Thus, $\Phi_{\alpha\beta}\approx-AQ_{\alpha\beta}$ (with $A>0$ in the isotropic phase) and Eq. (\[Qeq\]) takes the form $$-AQ_{\alpha\beta} =2\mu E_{\alpha\beta}+\nu R_{\alpha\beta}.\label{Qeqn1}$$ Likewise, we ignore the higher order terms in $R_{\alpha\beta}$; thus, $R_{\alpha\beta} \approx \dot{Q}_{\alpha\beta}$. Our linearized equations are equivalent to the apolar case of the linearized equations of active matter that have appeared previously [@Hatwalne2004; @Kruse_etal2005; @MarenduzzoOrlandiniCatesYeomans2007; @LiverpoolMarchetti2006] when we set $\mu_1=0$; also we absorb a possible active term proportional to $Q_{\alpha\beta}$ in $\Phi_{\alpha\beta}$ in Eq. (\[Qeq\]). Assuming that $v_\alpha$ and $Q_{\alpha\beta}$ are proportional to $\exp( -\mathrm{i}\omega t)$, where the real part of $-\mathrm{i}\omega$ is the growth rate of the perturbations, we find using Eq. that $$Q_{\alpha\beta} =-\dfrac{2 \mu}{A-\mathrm{i}\omega\nu} E_{\alpha\beta}.\label{Qeq2}$$ Using Eq. , the viscous stress $\sigma^v_{\alpha\beta}$ is given by $$\sigma^v_{\alpha\beta}=2\eta_\mathrm{eff} E_{\alpha\beta},$$ where $$\begin{aligned} \eta_\mathrm{eff}&=&\dfrac{\eta A}{A-\mathrm{i}\omega\nu}\left[1-a -\dfrac{\mathrm{i}\omega\nu}{A}\left(1-\dfrac{2 \mu (\mu+\mu_1)}{\nu\eta} \right) \right]\label{etaprime}\\ &=&\eta\frac{1-a-\mathrm{i}\omega\tau_\mathrm{lc}'}{1-\mathrm{i}\omega\tau_\mathrm{lc}}.\end{aligned}$$ Here, we have defined the dimensionless activity by $a=a'\mu/\eta A$, and the relaxation times by $\tau_\mathrm{lc}=\nu/A$ and $\tau_\mathrm{lc}'=\tau_\mathrm{lc}[1-2\mu(\mu+\mu_1)/(\nu\eta)]$. Since we are assuming that $\eta\nu-2\mu(\mu+\mu_1)>0$, the isotropic phase of an infinite active nematic fluid is unstable against shear flow and local ordering [@swimming2018] when $a>1$. We will see in our instability analyses for the various geometries that the critical values of the dimensionless activity correspond to a negative effective shear viscosity, i.e. $a\ge1$. Note that in the oblate particle case of $\mu>0$, the critical value of the activity $a'$ is positive, meaning that significantly active contractile (puller) particles lead to instablity. For the prolate particle case of $\mu<0$, the critical activity is negative, meaning that sufficiently active extensile (pusher) particles lead to instability. It is apparent from the above equation that the effective viscosity of the fluid $\eta_\mathrm{eff}$ depends on the growth rate $-\mathrm{i}\omega$; in other words, the fluid behaves like a viscoelastic fluid due to the presence of the nematic molecules. At the special value of dimensionless activity $a=2 \mu (\mu+\mu_1)/\nu\eta=1-\tau_\mathrm{lc}'/\tau_\mathrm{lc}$, the effective viscosity $\eta_\mathrm{eff}$ is independent of $\omega$, and the fluid behaves like a Newtonian fluid with shear viscosity $\eta_\mathrm{eff}=\eta-2\mu(\mu+\mu_1)/\nu$. We will see below that at this special value of the activity the growth rate is that of a passive fluid. Since we ignore the inertia of the fluid, the force balance equation is given by $$\partial_\beta\sigma_{\alpha\beta}=0,\label{fb}$$ with $\sigma_\mathrm{\alpha\beta}=-p\delta_{\alpha\beta}+\sigma^v_{\alpha\beta}$. The pressure $p$ is the pressure arising from the incompressibility condition $\bm{\nabla}\cdot \textbf{v}=0$. We have disregarded the Ericksen stress $\sigma^\mathrm{e}_{\alpha\beta}=\mathcal{F}\delta_{\alpha\beta}-\partial{\mathcal F}/\partial(\partial_\beta Q_{\mu\nu})\partial_\alpha Q_{\mu\nu}$ [@DeGennes1969; @JulicherGrillSalbreaux2018] since it is at least quadratic order in $Q_{\alpha\beta}$. Then, using the incompressibility condition $\bm{\nabla}\cdot \textbf{v}=0$, the linearized Eq. can be simplified to $$\eta_\mathrm{eff}\nabla^2\textbf{v}-\bm{\nabla}p=0.\label{Stokes}$$ The incompressibility condition is imposed by representing $\textbf{v}$ as the curl of a stream function $\bm{\psi}$ i.e. $\textbf{v}=\bm{\nabla}\times\bm{\psi}$. For simplicity, we choose the form of $\bm{\psi}$ such that $\bm{\nabla}\cdot\bm{\psi}=0$. Taking the curl of Eq. yields $$\nabla^4\bm{\psi}=0,\label{fb3}$$ where $\nabla^4$ is the square of the Laplacian operator in three dimensions. We solve the above equation with the boundary conditions appropriate to the geometry at hand and calculate the forces on the interface due to the fluid. To describe the force per unit area acting on at the surface, we need to parametrize the surface as $\mathbf{X}(u^1,u^2)$, with coordinates $u^1$ and $u^2$. Due to the free energy associated with the surface \[see Eq. \], the force per unit area acting on the surface is given by [@Powers2010] $${\mathbf f}_m=2\gamma H\mathbf{n},$$ where $\mathbf{n}$ is the outward normal. Note that our convention is that $H$ is negative for a sphere or a cylinder. Since we disregard the inertia of the surface, the force balance equation at the surface reads $$\begin{aligned} (\sigma^+_{nn}-\sigma^-_{nn}) +2\gamma H&=&0\\\label{forbala2} (\sigma^+_{n\alpha}-\sigma^-_{n\alpha})+\partial_jX^\alpha&=&0, \end{aligned}$$ where $\sigma^\pm_{nn}=n^\alpha\sigma^\pm_{\alpha\beta}n^\beta$ and $\sigma_{n\alpha}=n^\beta\sigma_{\beta\alpha}$, with the plus and minus denoting the stress exerted on the interface from the $\mathbf{n}$ and $-\mathbf{n}$ sides, respectively. We close this section with estimates of the magnitudes of the liquid crystal relaxation time $\tau_\mathrm{lc}$ and the characteristic time scales for a film with interfacial tension or bending stiffness. A crude dimensional analysis estimate for $\tau_\mathrm{lc}=\nu/A$ is to suppose $\nu\approx\eta$, and to take $A=k_\mathrm{B}T/\ell^3$, where $k_\mathrm{B}T$ is thermal energy and $\ell$ is the length of the active particles. Using the viscosity of water, $\eta\approx10^{-3}\,$N-s/m$^2$, and $\ell\approx10\,\mu$m leads to $\tau_\mathrm{lc}\approx300\,$s. If the rods are $1\,\mu$m in length, then $\tau_\mathrm{lc}\approx0.3\,$s. However, since we are considering an active system, it is reasonable to suppose that $A$ is not determined by thermal energy, and that $A$, and the liquid crystal relaxation rate may be much bigger. For a film of thickness $d\approx1\,$mm and for the air-water surface tension $\gamma\approx70\times10^{-3}\,$N/m, the characteristic surface-tension driven relaxation time is $\tau_\mathrm{s}=\eta d/\gamma\approx0.1\,$ms. Thus, we expect the film relaxation time to be much shorter than the liquid crystal relaxation time, and we will focus on this limit. However, due to our uncertainty about the value of $A$, and also to show some of the range of possible phenomena, we also consider the case of $\tau_\mathrm{lc}\approx\tau_\mathrm{s}$. Instability of an active fluid film {#membrane} =================================== In this section, we study the instability of a flat interface of an active nematic fluid in its isotropic phase. The fluid is a film of thickness $d$ atop a solid substrate, with air above the film (Fig. \[schemf\]). We consider an air-fluid interface with constant uniform surface tension $\gamma$, and no bending stiffness. A film of passive fluid is always stable to sinusoidal perturbation, since the perturbation increases the surface area. Thus, the instability we study in this section arises from the activity of the fluid. The surface, which lies in the $zx$ plane in its unperturbed state, is subject to a transverse perturbation which is the real part of $h=\epsilon(t)\exp(\mathrm{i} kx)$, as shown in Fig. \[schemf\]. We assume that $\epsilon\propto\exp(-\mathrm{i}\omega t)$. ![(Color online.) Stability diagram showing when an interface of an active film is unstable as a function of dimensionless activity $a$ and dimensionless wavenumber $kd$ for the case of $\tau_\mathrm{s}\ll\tau_\mathrm{lc}$. The horizontal axis of the plot is scaled by $(\tau_\mathrm{lc}/\tau_\mathrm{s})^{1/4}$ since the longest wavelengths are unstable in this limit. The system is stable in the yellow-shaded region, and unstable in the unshaded region. Both the growing modes and the decaying modes propagate in the region between the two dashed lines.[]{data-label="critact"}](StabilityDiagramWtauslltauLC.eps){width="45.00000%"} The stream function is given by $\bm\psi=\psi\hat{\mathbf z}$, with $\psi$ a biharmonic function. For small deflections $kh\ll1$, the kinematic condition takes the form $$v_{y}(y=0)=\partial_t h,$$ where $v_y=-\partial_x\psi$. This condition, along with the conditions of zero tangential stress at the interface, $$\sigma_{xy}(y=0)=0,$$ and vanishing flow at $y=-d$, leads to $$\begin{aligned} \psi&=&-\frac{\mathrm{i}\omega\epsilon}{k}\mathrm{e}^{\mathrm{i}kx}\left\{\left[\cosh ky+\frac{\sinh ky }{F}\right]\right.\nonumber\\ &&+\left.\left[(1-2k^2d^2F)\frac{ky \cosh ky}{F}- kx \sinh ky\right]\right\},\end{aligned}$$ where $$F=\frac{\sinh 2kd-2kd}{\cosh 2kd+2k^2d^2+1}.\label{Fofkdeqn}$$ For small deflections the mean curvature is $H\approx-\partial^2_xh/2$, and the force balance equation on the interface becomes $$-\sigma_{yy}\|_{y=0} +\gamma \partial^2_x h=0.\label{forbal}$$ The stress component $\sigma_{yy}$ can be found by calculating the pressure from the $x$-component of the force balance equation . Once $\sigma_{yy}$ is calculated, we use normal stress balance at $y=0$ to obtain the characteristic equation $$-\mathrm{i}\omega=-\frac{\gamma k}{2\eta_\mathrm{eff}(\omega)}F(kd). \label{rootm}$$ In the passive Newtonian case with $a=0$ and with no coupling between the fluid and the liquid crystalline degrees of freedom, i.e. $\mu=\mu_1=0$, the growth rate has two branches that cross, one corresponding to the negative growth rate of a Newtonian film [@HenleLevine2007], with characteristic time scale $\tau_\mathrm{s}=\eta d/\gamma$, $$\begin{aligned} -\mathrm{i}\omega&\sim&-\frac{\gamma k}{2\eta},\quad kd\gg1\\ -\mathrm{i}\omega&\sim&-\frac{\gamma d^3k^4}{3\eta},\quad kd\ll1\label{longwavebranch} \end{aligned}$$ and one corresponding to the liquid crystalline relaxation rate, $-\mathrm{i}\omega=-1/\tau_\mathrm{lc}=-A/\nu$. When $\mu$ (or $\mu_1$) is nonzero and $a=0$, the growth rate curves repel each other instead of crossing, as in Fig. \[rootsplotm\], upper left panel. The active case is like the case of a passive viscoelastic fluid [@HenleLevine2007], for which the effective shear viscosity depends on $\omega$, and we must solve Eq. for $\omega$ as a function of $k$, which yields $$-\mathrm{i}\omega\tau_\mathrm{lc}'=\frac{a-1}{2}-\frac{kd F\tau_\mathrm{lc}}{4\tau_\mathrm{s}}\pm\sqrt{\left(\frac{a-1}{2}-\frac{kdF\tau_\mathrm{lc}}{4\tau_\mathrm{s}}\right)^2-\frac{kd F\tau_\mathrm{lc}'}{2\tau_\mathrm{s}}}, \label{growthflat}$$ where $F$ is given by Eq. (\[Fofkdeqn\]). As the activity increases, the splitting between the two growth rate curves decreases, until the value of $a= 2\mu \left(\mu +\mu _1\right)/\eta\nu$ is reached. At this special value of activity, $\eta_\mathrm{eff}$ is independent of $\omega$, and the branches of the growth rates cross as they do in the case of $a=0$ and $\mu=0$ (Fig. \[rootsplotm\], upper right panel). As the activity increases further, the real branches collapse into one branch for a range of wavevector, and the imaginary parts of the growth rate become nonzero in this same range (Fig. \[rootsplotm\], lower left panel). The critical activity $a=1$ corresponds to the point at which the effective shear viscosity vanishes. When $a>1$, one of the branches of the real part of the growth rate becomes positive, and the system is unstable for sufficiently long wavelengths (Fig. \[rootsplotm\], lower right panel). The critical activity $a_\mathrm{c}(k)$ at which the mode $k$ is marginally stable is found by demanding that $\mathrm{Re}(-\mathrm{i}\omega)=0$: $$a_\mathrm{c}(k)=1+\frac{1}{2}kd F(kd)\tau_\mathrm{lc}/\tau_\mathrm{s}.$$ Since $\tau_\mathrm{lc}/\tau_\mathrm{s}=(\nu\gamma)/(\eta A d)$, interfacial tension tends to suppress the instability for nonzero $k$. But even if $\tau_\mathrm{s}\ll\tau_\mathrm{lc}$, the longest wavelengths are always unstable. In this limit, the two branches of the uncoupled passive case cross when $kd\sim(\tau_\mathrm{s}/\tau_\mathrm{lc})^{1/4}$, which is why we plot the growth rates vs. $kd(\tau_\mathrm{lc}/\tau_\mathrm{s})^{1/4}$ in Fig. \[rootsplotm\]. The shapes of the real and imaginary parts of the growth rate curves for $\tau_\mathrm{lc}\approx\tau_\mathrm{s}$ and for $\tau_\mathrm{lc}\ll\tau_\mathrm{s}$ are qualitatively similar to the case of $\tau_\mathrm{s}\ll\tau_\mathrm{lc}$, with the main difference being that the band of unstable modes reaches further into the regime of short wavelength as $\tau_\mathrm{s}$ increases relative to $\tau_\mathrm{lc}$ (See Figs. \[rootsplotm2\] and \[rootsplotm3\] in the appendix). Note that the growth rate $-\mathrm{i}\omega$ always has an imaginary part when $a$ is sufficiently near $a_\mathrm{c}(k)$; when a mode is unstable with a sufficiently small growth rate, it also propagates. Propagating modes are found when $a_-\le a\le a_+$, where $$a_\pm=1+\frac{1}{2}kdF(kd)\tau_\mathrm{lc}/\tau_\mathrm{s}\pm\sqrt{\frac{1}{2}kdF(kd)\tau_\mathrm{lc}'/\tau_\mathrm{s}}. \label{aflatpm}$$ Also, there are no propagating modes without the interface, since $a_+-a_-\propto\sqrt{\tau'_\mathrm{lc}/\tau_\mathrm{s}}\propto\sqrt\gamma$. Figure \[critact\] shows when the interface is stable as a function of scaled dimensionless wavenumber and dimensionless activity for the case of $\tau_\mathrm{lc}/\tau_\mathrm{s}\gg1$. The system is always stable for $a<1$; as $a$ is increased beyond $a=1$, an increasingly large band of very long wavelength modes are unstable. Growing and decaying modes with a sufficiently small growth rate \[between the dashed lines in Fig. \[critact\], which are given by Eq. (\[aflatpm\])\] are also propagating. As $\tau_\mathrm{lc}/\tau_\mathrm{s}$ decreases, the band of unstable modes is limited to shorter and shorter wavelengths. To sum up, the time scale $\tau_\mathrm{lc}$ controls the rate of growth or decay of the modes, and the time scale $\tau_\mathrm{s}$ determines which modes become unstable. Since $a_\pm$ depends on $\tau_\mathrm{s}$, the velocity of propagation $\mathrm{Re}(\omega)/k$ is determined by $d/\tau_\mathrm{s}$. Rayleigh-Plateau capillary instability {#thread} ====================================== A fluid thread breaks into drops because perturbations of sufficiently long wavelength lower the area of the surface, and thus the energy. This instability is known as the Rayleigh-Plateau capillary instability [@Plateau1873; @Rayleigh1892]. In this section, we study how the presence of active nematic molecules in the liquid affects the Rayleigh-Plateau capillary instability. For simplicity, we disregard the outer fluid. While this approximation was natural in our study of the stability of a flat interface between air and an active fluid, it seems less natural for a thread of active fluid, since the thread must be supported by some surrounding fluid if it is not a jet. However, unlike the passive case of a stationary cylindrical interface [@Tomotika1935], accounting for the viscosity contrast leads to a complicated characteristic equation for the growth rate of the interface of an active thread. To avoid this complication and illustrate the essential physics, we assume the outer fluid is of sufficiently small viscosity that we may disregard it. We consider a cylindrical fluid thread of initial radius $R$, subject to an axisymmetric harmonic perturbation of wavenumber $k$ along the $x$ direction (see Fig. \[cylfig\]). The cylindrical coordinates are $(\rho,\theta,x)$. Initially the fluid is at rest, with a uniform pressure $p=\gamma/R$. The radius of the perturbed thread is given by the real part of $h(x,t)=R+\epsilon(t) \exp(\mathrm{i} k x)$, with $\epsilon k\ll1$. For an axisymmetric flow, we follow Happel and Brenner [@HappelBrenner1983] and define the stream function via $\bm{\psi}=-(\psi/\rho)\hat{\bm{\theta}}$. The stream function $\psi$ is related to velocity by $v_\rho=(1/\rho)\partial_x\psi$ and $v_x=-(1/\rho)/\partial_\rho\psi$. If we choose $\bm{\psi}=\Psi(\rho)\exp(\mathrm{i} k x) \hat{\bm{\theta}}$, then Eq. in cylindrical coordinates reduces to $$D^2\Psi=0,\label{eqpsic}$$ where [@Tomotika1935] $D\equiv\partial_\rho^2-(1/\rho) \partial_\rho-k^2$. The linearized kinematic condition at the interface, $\bm{\nabla}\times\bm{\psi}=\partial_t{h} \hat{\bm{\rho}}$, leads to $$\frac{k}{R}\Psi(\rho=R) =-\omega \epsilon.\\$$ ![(Color online.) Real (blue solid line) and imaginary (red dashed line) parts of the dimensionless growth rate $-\mathrm{i}\omega\tau_\mathrm{s}$ vs. dimensionless wavenumber $kR$ for $\tau_\mathrm{lc}/\tau_\mathrm{s}\gg1$. On this scale, the line corresponding the the branch $\mathrm{Re}(-\mathrm{i}\omega_-)\approx-1/\tau_\mathrm{lc}$ is along the horizontal axis. []{data-label="raycaptauslltaulc"}](RayCapTausllTaulc.eps){width="45.00000%"} The kinematic boundary condition and the condition of zero tangential stress, $\sigma_{x\rho}\|_{\rho=R}=0$, along with the condition of regularity at $\rho=0$, leads to the solution $$\psi=\epsilon\omega\mathrm{e}^{\mathrm{i}kx}\left[\frac{\rho^2I_0(k\rho)}{I_1(kR)}-\frac{kRI_0(kR)+I_1(kR)}{kI_1^2(kR)}\rho I_1(k\rho)\right].$$ where $I_0$ and $I_1$ are the Bessel functions of first kind. The growth rate is determined by the normal force balance equation, $$-\sigma_{\rho\rho}\|_{\rho=R}+2\gamma H=0.\label{forbscyl}$$ The pressure may be found from the $x$-component of the Stokes equation, Eq. (\[Stokes\]); with this pressure and the velocity field we may calculate $\sigma_{\rho\rho}=-p+2\eta_\mathrm{eff}\partial_\rho v_\rho$ and use the mean curvature expanded [@Ou-YangHelfrich] to linear order in $\epsilon$, $$H= -\dfrac{1}{2}\left[\frac{1}{R}+\epsilon \left(k^2 -\frac{1}{R^2}\right)\mathrm{e}^{\mathrm{i} k x}\right],\label{surftc}$$ in Eq. to find $$-\mathrm{i}\omega=\frac{\gamma}{2\eta_\mathrm{eff}(\omega) R}G,\label{RayPlatgrowth1}$$ where $$G=\frac{1-k^2R^2}{k^2 R^2 I_0^2(kR)/I^2_1(kR)-(1+k^2R^2)}.$$ When $\eta_\mathrm{eff}(\omega)=\eta$, the growth rate of Eq. is precisely that of a thread of a passive Newtonian viscous fluid thread [@Chandrasekhar1981]. Since the characteristic equation (\[RayPlatgrowth1\]) for the cylinder is of a similar form as the characteristic equation (\[rootm\]) for the planar surface, the growth rate is given by Eq. with $F$ replaced by $-G/k$ and $d$ replaced by $R$. (Note that in this section $\tau_\mathrm{s}=\eta/(\gamma R)$.) Figure \[raycaptauslltaulc\] shows the growth rate vs. dimensionless wavevector $kR$ for the case of $\tau_\mathrm{lc}\gg\tau_\mathrm{s}$. In this case, the growth rate is almost exactly the same as the classical result for a passive Newtonian fluid. The only dependence on activity or liquid crystalline parameters arises in the region near $kR=1$ where the real part of the growth rate vanishes. This fact can be seen by expanding the growth rate for small $\tau_\mathrm{s}/\tau_\mathrm{lc}$; away from the region where $G\ll1$, we have $$\begin{aligned} -\mathrm{i}\omega_-&\sim&\frac{1}{\tau_\mathrm{lc}}\\ -\mathrm{i}\omega_+&\sim&\frac{\tau_\mathrm{lc}}{\tau_\mathrm{lc}'}\frac{\gamma G}{2\eta R}.\end{aligned}$$ The effects of activity become apparent when the liquid crystal relaxation time is comparable to the film relaxation time, $\tau_\mathrm{lc}\sim\tau_\mathrm{s}$. The growth rate for several different dimensionless activities is shown in Fig. \[cylinder-a\]. In this case, the behavior of the growth rate with respect to activity is similar to behavior of the growth rate for a flat interface (compare with Fig. \[rootsplotm\]). The passive cylindrical thread is always unstable for modes with $kR<1$. Likewise, in the active case, modes with $kR<1$ are always unstable. Once $a>1$, modes with a wavenumber greater than $1/R$ can also be unstable; in particular, $\mathrm{Re}(-\mathrm{i}\omega)=0$ when $$a_\mathrm{c}(k)=1-\frac{G\tau_\mathrm{lc}}{2\tau_\mathrm{s}}.$$ Propagating modes are found when $a_-<a<a_+$, where $$a_\pm=1-\frac{G\tau_\mathrm{lc}}{2\tau_\mathrm{s}}\pm\sqrt{-\frac{2G\tau_\mathrm{lc}'}{\tau_\mathrm{s}}}.$$ Note that propagation only occurs when $kR>1$, i.e. $G(k)<0$. Figure \[raycaptausIStaulcPhase\] is the stablity diagram for the case of $\tau_\mathrm{lc}=\tau_\mathrm{s}$. ![(Color online.) Stability diagram showing when a cylindrical thread of active fluid is unstable as a function of dimensionless activity $a$ and dimensionless wavenumber $kR$ for the case of $\tau_\mathrm{s}=\tau_\mathrm{lc}$ and $\tau'_\mathrm{lc}/\tau_\mathrm{lc}=0.8$. The system is stable in the yellow-shaded region, and unstable in the unshaded region. Both the growing modes and the decaying modes propagate in the region between the two dashed lines.[]{data-label="raycaptausIStaulcPhase"}](RayCapTausISTaulcPhase.eps){width="45.00000%"} Instability of a spherical active droplet {#droplet} ========================================= A cylinder of active fluid is unstable, and breaks up into spherical droplets. A spherical droplet of a Newtonian fluid is always stable against surface tension since the spherical shape minimizes the surface energy. However, a spherical droplet of active fluid might go unstable due to activity. Here we carry out a linear stability analysis for a droplet of active nematic fluid in the isotropic phase (see Fig. \[schems\]). We assume that the spherical droplet of radius $R$ is subject to spherical harmonic perturbations such that the surface of the perturbed drop can be represented by $\mathbf{X}(\theta, \phi)=(R+\epsilon(t) Y^m_l(\theta, \phi))\hat{\bm{ r }}$ with $\epsilon\ll R$. We choose the following form of the stream function $\bm{\psi}$ to enforce the condition $\bm{\nabla}\cdot\bm{\psi}=0$: $$\bm{\psi}=-v( r )\dfrac{1}{\sin\theta}\dfrac{dY^m_l(\theta, \phi)}{d\phi} \hat{\bm{\theta}}+v( r )\dfrac{dY^m_l(\theta, \phi)}{d\theta}\hat{\bm{\phi}},$$ where $Y^m_l(\theta, \phi)$ is a spherical harmonic. Inserting this stream function in the Stokes equations, we find that the function $v( r )$ obeys $$D^2v( r )=0,\label{eqpsis}$$ where $$D\equiv \dfrac{1}{ r ^2}\left[ \dfrac{d}{d r }\left( r ^2 \dfrac{d}{d r }\right) -l(l+1)\right].$$ The boundary conditions on the interface are the linearized kinematic condition, $\bm{\nabla}\times\bm{\psi}=\partial_t\textbf{X}$, and the linearized zero shear stress condition: $$\begin{aligned} -l(l+1) \dfrac{v( r =R)}{R} =-\mathrm{i}\omega\epsilon,\\ \sigma_{\phi r }( r =R)=0,\\ \sigma_{\theta r }( r =R)=0.\end{aligned}$$ The solution of Eq. with the above boundary conditions is given by $$v(r)=\mathrm{i}\omega \epsilon\frac{ r ^l R^{-l-1} \left[l (l+2) R^2-\left(l^2-1\right) r ^2\right]}{l (l+1) (2 l+1)}$$ With this solution, we get the following expression for $\sigma_{ r r }$ after integrating the $ r $-component of Eq. with respect to $ r $:\ $$\sigma_{ r r }( r ,\theta, \phi)=-2\mathrm{i} \omega\eta_\mathrm{eff} \epsilon \mathcal{G}[l]Y^m_l(\theta, \phi)+C,\label{strrs}$$ where $$\mathcal{G}[l]=\frac{ (l-1) r ^{l-2} R^{-l-1} \left[\left(-l^3+4 l+3\right) r ^2+l^2 (l+2) R^2\right]}{l (2 l+1)}.$$ In the unperturbed state, the surface tension leads to a constant pressure $C$ via the Young-Laplace law. Since we suppose that there is no fluid outside the drop, the force balance equation at the surface of the drop (in the limit $\epsilon\ll R$) is given by (see Eq. ) $$\sigma_{ r r }(R,\theta, \phi)-2\gamma H=0.\label{surffs}$$ The mean curvature $H$ is given to first order in $\epsilon$ by [@Ou-YangHelfrich], $$H=-\left[\frac{1}{R}+\epsilon\frac{ (l-1)(l+2) }{2R^2}Y^m_l(\theta, \phi)\right].\label{surftss}$$ We see from Eqs. and that, for the $l=1$ mode, there are no changes in $\sigma_{ r r }$ or the Laplace pressure $2\gamma H$ due to the perturbation, because to leading order, the $l=1$ mode is equivalent to the displacement of the droplet along the $z$ direction (see Fig. \[schems\]). Therefore, we consider modes with $l>1$. From Eq. , and , we find that $C=-2\gamma/R$ and $$-\mathrm{i}\omega=-\frac{\gamma}{2\eta(\omega) R}\frac{l(l+2)(2l+1)}{2l^2+4l+3}.\label{roots}$$ When $\eta$ is independent of $\omega$, this result is precisely the relaxation rate for perturbations of a sphere with surface tension in the limit that viscosity dominates inertia [@Chandrasekhar1959; @Reid1960]. Equation (\[roots\]) is quadratic in $\omega$, and the real parts of its two roots represent growth rates of the perturbation. The critical dimensionless activity $a_c(l)$ for the $l$th harmonic perturbation calculated is given by $$a_{c}(l)= 1+\dfrac{\tau_\mathrm{lc}}{2 \tau_\mathrm{s}}\frac{l (l+2) (2 l+1)}{2 l ^2+4l+3}.$$ Since the smallest value of $l$ is 2, the critical value of the dimensionless activity above which droplet becomes unstable is given by $$a_c(l=2)\simeq 1+\dfrac{\tau_\mathrm{lc}}{ \tau_\mathrm{s}}.$$ Therefore, critical dimensionless activity for a spherical droplet $a_c(l=2)$ is larger than its value for the unconfined fluid. Also, $a_c(l=2)$ decreases with $R$: smaller active droplets are more stable. Fig. \[spherel\] shows that $a_{c}(l)$ increases almost linearly with $l$. Discussion and Conclusion {#conclusion} ========================= In this paper we have studied the effect of activity on the stability of flat, cylindrical, and spherical interfaces. In all cases, the bulk instability of the active fluid, which is characterized by a vanishing effective shear viscosity, leads to spontaneous shear flows that can destabilize an interface that would be stable in the case of a passive fluid. Furthermore, all three geometries showed oscillatory behavior at suitably large activity, corresponding to propagating damped or growing modes. The presence of propagating modes (damped or growing) at zero Reynolds number is qualitatively different from the passive fluid case, where no propagation is seen at zero Reynolds number. The propagating modes in our linear stability analysis may be the seed for propagating modes at large amplitude, as seen in numerical calculations of active membranes [@MietkeJulicherSbalzarini2019]. We made several approximations in this paper to make our calculation tractable. We neglected the Frank elasticity, which meant that the base state that we expanded about is uniform, $Q_{\alpha\beta}=0$. If we had included Frank elasticity, we would have to specify anchoring conditions for $Q_{\alpha\beta}$. For the case of planar or homeotropic anchoring, the base state would be nonuniform, and its stability would be more difficult to analyze by the technique we employ. The case of a zero-torque anchoring condition would lead to a uniform base state, but it would still make our calculation more complicated since we would not be able to eliminate $Q_{\alpha\beta}$ by simply solving an algebraic equation, and we would not be able to lump all the liquid-crystalline and active effects into the effective frequency-dependent viscosity $\eta_\mathrm{eff}(\omega)$. It would be interesting to generalize our calculations to include Frank elasticity, since it has been shown that Frank elasticity (or equivalently rotational diffusion in the work of Woodhouse and Goldstein) leads to spontaneous flow even for undeformed confining surfaces [@WoodhouseGoldstein2012]. A second major simplification is our neglect of the outer fluid. Because we neglected the viscosity of the outer fluid, we only had to solve a quadratic equation to find the branches of the growth rate. Including the outer fluid is more realistic, and it will lead to a more complicated characteristic equation, and more branches. Also, if we use the thermal energy scale to estimates the material parameters (questionable in a active system), we are led to $\tau_\mathrm{lc}\gg\tau_\mathrm{s}$, which makes the interesting activity-driven phenomena such as instability and oscillation occur at long wavelength in the case of the flat film, but only in a narrow regime near $kR\approx1$ in the case of the cylindrical thread. When the viscosity of the outer fluid is accounted for, the growth rate of the passive cylindrical thread vanishes [@Tomotika1935] at $k=0$, which will also lead to interesting activity-driven behavior at long wavelength in the cylinder. Finally, all of the calculations we did for interfaces could be modified to apply to the case of an active fluid bound by a membrane, which could be more relevant for biological phenomena. Appendix {#appen} ======== \[plotappendix\] In this appendix we display more plots of the growth rate and the stability diagram for the case of the film of thickness $d$ (Section \[membrane\]). Fig. \[rootsplotm2\] shows the real and imaginary parts of the growth rate for $\tau_\mathrm{lc}=\tau_\mathrm{s}$, whereas Fig. \[rootsplotm3\] shows the same quantities for the case of $\tau_\mathrm{s}/\tau_\mathrm{lc}\gg1$. In all case, the shape of the curves is qualitatively similar, but the scale of wavevectors where the instability and oscillations changes, with the instability and oscillations occurring when $kd\sim(\tau_\mathrm{s}/\tau_\mathrm{lc})^{1/4}$ when $\tau_\mathrm{s}/\tau_\mathrm{lc}\ll1$, when $kd\sim1$ when $\tau_\mathrm{s}/\tau_\mathrm{lc}\sim1$, and when $kd\sim\tau_\mathrm{s}/\tau_\mathrm{lc}$ when $\tau_\mathrm{s}/\tau_\mathrm{lc}\gg1$. Figure \[critact2\] shows the stability diagram for $\tau_\mathrm{s}=\tau_\mathrm{lc}$ (upper panel) and $\tau_\mathrm{s}\gg\tau_\mathrm{lc}$ (lower panel). ![(Color online.) Stability diagrams showing when an interface of an active film is unstable as a function of dimensionless activity $a$ and dimensionless wavenumber $kd$. The top panel shows the case of $\tau_\mathrm{s}=\tau_\mathrm{lc}$. The system is stable in the yellow-shaded region, and unstable in the unshaded region. Both the growing modes and the decaying modes propagate in the region between the two dashed lines. The bottom panel shows the case of $\tau_\mathrm{s}\gg\tau_\mathrm{lc}$, with $kd$ scaled by $\tau_\mathrm{lc}/\tau_\mathrm{s}$ since the instability occurs over a wide band of wavenumbers.[]{data-label="critact2"}](StabilityDiagram.eps){width="3.5in"} Conflicts of interest ===================== There are no conflicts to declare. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported in part by National Science Foundation Grant Nos. CBET-1437195 (TRP) and National Science Foundation Grant No. MRSEC-1420382 (RAP and TRP). We are grateful to Dan Blair, Kenny Breuer, and Ian Wong for helpful discussions.
--- abstract: \| The random grid search (RGS) is a simple, but efficient, stochastic algorithm to find optimal cuts that was developed in the context of the search for the top quark at Fermilab in the mid-1990s. The algorithm, and associated code, have been enhanced recently with the introduction of two new cut types, one of which has been successfully used in searches for supersymmetry at the Large Hadron Collider. The RGS optimization algorithm is described along with the recent developments, which are illustrated with two examples from particle physics. One explores the optimization of the selection of vector boson fusion events in the four-lepton decay mode of the Higgs boson and the other optimizes SUSY searches using boosted objects and the razor variables.\ author: - 'P.C. Bhat$^{\,a}$[^1], H.B. Prosper$^{\,b}$[^2], S. Sekmen$^{\,c}$[^3], C. Stewart$^{\,d}$[^4]' bibliography: - 'bibliography.bib' title: Optimizing Event Selection with the Random Grid Search --- Acknowledgements {#acknowledgements .unnumbered} ================ The work of PCB and HBP is supported in part by the U.S. Department of Energy, under contract number DE-AC02-07CH11359 with Fermilab, and under grant number DE-SC0010102, respectively. The work of SS is supported by the financial support of the National Research Foundation of Korea (NRF), funded by the Ministry of Science & ICT under contract NRF-2008-00460 and by the U.S. Department of Energy through the Distinguished Researcher Program from the Fermilab LHC Physics Center. [^1]: pushpa@fnal.gov [^2]: harry@hep.fsu.edu [^3]: ssekmen@cern.ch [^4]: stewart@broadinstitute.org
--- abstract: \| In traditional models for word-of-mouth recommendations and viral marketing, the objective function has generally been based on reaching as many people as possible. However, a number of studies have shown that the indiscriminate spread of a product by word-of-mouth can result in [overexposure]{}, reaching people who evaluate it negatively. This can lead to an effect in which the over-promotion of a product can produce negative reputational effects, by reaching a part of the audience that is not receptive to it. How should one make use of social influence when there is a risk of overexposure? In this paper, we develop and analyze a theoretical model for this process; we show how it captures a number of the qualitative phenomena associated with overexposure, and for the main formulation of our model, we provide a polynomial-time algorithm to find the optimal marketing strategy. We also present simulations of the model on real network topologies, quantifying the extent to which our optimal strategies outperform natural baselines. author: - Rediet Abebe - 'Lada A. Adamic' - Jon Kleinberg bibliography: - 'sigproc.bib' title: Mitigating Overexposure in Viral Marketing --- Introduction {#intro} ============ A rich line of research has studied the effectiveness of marketing strategies based on person-to-person recommendation within a social network — a process often termed [viral marketing]{} [@jurvetson-viral] and closely connected to the broader sociological literature on the [diffusion of innovations]{} in social networks [@rogers-diffusion]. A key genre of theoretical question that emerged early in this literature is the problem of optimally “seeding” a product in a social network through the selection of a set of initial adopters [@dr; @kkt; @rd]. In this class of questions, we consider a firm that has a product they would like to market to a group of agents on a social network; it is often the case that the firm cannot target all the participants in the network, and so they seek to target the most influential ones so as to maximize exposure and create a cascade of adoptions. Approaches to this question have generally been based on objective functions in which the goal is to maximize the number of people who are reached by the network cascade — or more generally, in which the objective function monotonically increases in the number of people reached. [[[The dangers of overexposure.]{}]{}]{} Separately from this, lines of research in both marketing and in the dynamics of on-line information have provided diverse evidence that the benefits of a marketing campaign are not in fact purely increasing in the number of people reached. An influential example of such a finding is the [Groupon effect]{}, in which viral marketing via Groupon coupons leads to lower Yelp ratings. In [@group], they note the negative effect Groupon has on average Yelp ratings and provide arguments for the underlying mechanism; one of their central hypotheses is that by using Groupon as a matchmaker, businesses may be attracting customers from a portion of the population that is less inclined to like the product. In another example, Kovcs and Sharkey [@good] discuss a setting on Goodreads where books that win prestigious awards (or are short-listed for them) attract more readers following the announcement, which again leads to a drop in the average rating of the book on the platform. Aizen et al. [@aizen-batting] show a similar effect for on-line videos and other media; they receive a discontinuous drop in their ratings when a popular blog links to them, driving users to the item who may not be interested in it. Research in marketing has shown that exposure to different groups and influence between such groups can help or hurt adoption [@berger2007consumers; @hb; @turf]. For example, Hu and Van den Bulte [@hb] argue that agents adopt products to boost their status; and so as word-of-mouth effects for a product become stronger among middle-income individuals, there might be a negative impact on adoption among higher-income individuals. Similar behavior is observed in health campaigns. In [@health], Wakefield et al. discuss the importance of segmenting populations and exposing groups to anti-smoking campaigns whose themes the group is most susceptible to in order to maximize the impact of future campaigns. We think of these effects collectively as different forms of [overexposure]{}; while reaching many potential customers is not a concern in and of itself, the empirical research above suggests that there may exist particular subsets of the population — potentially large subsets — who will react negatively to the product. When a marketing cascade reaches members of this negatively inclined subset, the marketing campaign can suffer negative payoff that may offset the benefits it has received from other parts of the population. This negative payoff can come in the form of harm to the firm’s reputation, either through latent consumer impressions and effects on brand loyalty [@reputation2; @reputation1] or through explicitly visible negative reviews on rating sites. Despite the importance of these considerations in marketing, they have not been incorporated into models of influence-based marketing in social networks. What types of algorithmic issues arise when we seek to spread a word-of-mouth cascade through a network, but must simultaneously ensure that it reaches the “right” part of the audience — the potential customers who will like the product, rather than those who will react negatively to it? [[[The present work: A model of cascades with the risk of overexposure.]{}]{}]{} In this paper, we propose a basic theoretical model for the problem of seeding a cascade when there are benefits from reaching positively inclined customers and costs from reaching negatively inclined customers. There are many potential factors that play a role in the distinction between positively and negatively inclined customers, and for our model we focus on a stylized framework in which each product has a known parameter $\phi$ in the interval $[0,1]$ that serves as some measure for the breadth of its appeal. At this level of generality, this parameter could serve as a proxy for a number of things, including quality; or a one-dimensional combination of price and quality; or — in the case where the social network represents a population defined by a specific interest — compatibility with the core interests of network’s members. Each node in the network is an [agent]{} who will evaluate the product when they first learn of it; agents differ in how [critical]{} they are of new products, with agents of low criticality tending to like a wider range of products and agents of high criticality tending to reject more products. Thus, each agent $i$ has a criticality parameter $\theta_i$ in the interval $[0, 1]$; since we assume that the firm has a history of marketing products to this network over a period of time, it knows this parameter $\theta_i$. When exposed to a product, an agent accepts the product if $\phi \geq \theta_i$ and advertises the product to their neighbors, leading to the potential for a cascade. However, if $\phi < \theta_i$, then the agent rejects the product, which results in a negative payoff to the firm; the cascade stops at such agents $i$, since they do not advertise it to their neighbors. The firm’s goal is to advertise the product to a subset of the nodes in the network — the [seed set]{} — resulting in a potential cascade of further nodes who learn about the product, so as to maximize its overall payoff. This payoff includes a positive term for each agent $i$ who sees the product and has $\phi \geq \theta_i$, and a negative term for each agent $i$ who sees the product and has $\phi < \theta_i$; agents who are never reached by the cascade never find out about the product, and the firm gets zero payoff from them. [[[Overview of Results.]{}]{}]{} We obtain theoretical results for two main settings of this problem: the [unbudgeted case]{}, in which the firm can initially advertise the product to an arbitrary seed set of nodes, and the [budgeted case]{}, in which the firm can advertise the project to at most $k$ nodes, for a given parameter $k$. We note that typically in influence maximization problems, the unbudgeted case is not interesting: if the payoff is monotonically increasing in the number of nodes who are exposed to the product, then the optimal unbudgeted strategy is simply to show the product to everyone. In a world with negative payoffs from overexposure, however, the unbudgeted optimzation problem becomes non-trivial: we must tradeoff the benefits of showing the product to customers who will like it against the negatives that arise when these customers in turn share it with others who do not. For the unbudgeted problem, we give a polynomial-time algorithm for finding the optimal seed set. The algorithm uses network flow techniques on a graph derived from the underlying social network with the given set of parameters $\theta_i$. In contrast, we provide an NP-hardness result for the budgeted problem. We then provide a natural generalization of the model: rather than each agent exhibiting only two possible behaviors (rejecting the product, or accepting it and promoting it), we allow for a wider range of agent behaviors. In particular, we will assume each agent has three parameters which control whether the agent ignores the product, views but rejects the product, accepts the product but does not broadcast it to its neighbors, and accepts the product and advertises it to neighbors. We show how to extend our results to this more general case, obtaining a polynomial-time algorithm for the unbudgeted case and an NP-hardness result for the budgeted case. Finally, we perform computational simulations of our algorithm for the unbudgeted case on sample network topologies derived from moderately-sized social networks. We find an interesting effect in which the performance of the optimal algorithm transitions between two behaviors as $\phi$ varies. For small $\phi$ the payoff grows slowly while a baseline that promotes the product to every agent $i$ with $\theta_i < \phi$ achieves negative payoff (reflecting the consequences of overexposure). Then, for large $\phi$, the payoff grows quickly, approaching a simple upper bound consisting of all $i$ for which $\theta_i < \phi$. Preliminaries {#prelim} ============= There is a product with a parameter $\phi \in [0, 1]$, measuring the breadth of its appeal. $G$ is an unweighted, undirected graph with $n$ agents as its nodes. For each agent $i$, the agent’s criticality parameter $\theta_i \in [0, 1]$ measures the minimum threshold for $\phi$ the agent demands before adoption. Thus, higher values of $\theta_i$ correspond to more critical agents. We assume that these values are fixed and known to the firm. The firm chooses an initial set of agents $S \subseteq V$ to “seed” with the product. If an agent $i$ sees the product, it accepts it if $\theta_i \leq \phi$ and rejects it if $\theta_i > \phi$. We say that an agent $i$ is [accepting]{} in the former case and [rejecting]{} in the latter case. Each accepting agent who is exposed to the product advertises it to their neighbors, who then, recursively, are also exposed to the product. We will assume throughout that the firm chooses a seed set consisting entirely of accepting nodes (noting, of course, that rejecting nodes might subsequently be exposed to the product after nodes in the seed set advertise it to their neighbors). We write $V(S)$ for the set of agents exposed to the product if the seed set is $S$. Formally, $V(S)$ is the set of all agents $i$ who have a path to some node in $j \in S$ such that all of the internal nodes on the $i$-$j$ path are accepting agents; this is the “chain of recommendations" by which the product reached $i$. Among the nodes in $V(S)$, we define $V^+(S)$ to be the set of agents who accept the product and $V^-(S)$ to be the set of agents who reject the product. The payoff function associated with seed set $S$ is: $$\begin{aligned} \label{payoff1} \pi (S) = \|V^+(S)\|- \|V^-(S)\|.\end{aligned}$$ We can, more generally, assume that there is a payoff of $p$ to accepting the product and a negative payoff of $q$ to rejecting the product, and we set the payoff function to be: $$\begin{aligned} \label{payoff2} \pi(S) = p \|V^+(S)\| - q \|V^-(S)\|.\end{aligned}$$ We will call this the generalized payoff function, and simply refer to Equation \[payoff1\] as the payoff function. The overarching question then is: \[qq1\] Given a set of agents $V$ with criticality parameters $\theta_i$ ($i \in V$) on a social network $G = (V, E)$, and given a product of quality $\phi$, what is the optimal seed set $S \subseteq V$ that the firm should target in order to maximize the payoff given by Equation (\[payoff2\])? In contrast to much of the influence maximization literature, we assume that the agents’ likelihood of adoption, once exposed to this product, is not affected by which of their neighbors have accepted or rejected the product. This differs from, for instance, models in which each agent requires a certain fraction (or number) of its neighbors to have accepted the product before it does; or models where probabilistic contagion takes place across the edges. These all form interesting directions for further work; here, however, we focus on questions in which the intrinsic appeal of the product, via $\phi$, determines adoption decisions, and the social network provides communication pathways for other agents to hear about the product. Before proceeding to the main result, we develop some further terminology that will be helpful in reasoning about the seed sets. Let $i$ be an accepting node, and let $S = \{i\}$. Then we say that $V(S)$ is the [cluster]{} of $i$, denoted by $C_i$; we call $V^+(S)$ the [interior]{} of $C_i$ and denote it by $C_i^o$, and we call $V^-(S)$ the [boundary]{} of $C_i$ and denote it by $C_i^b$. We denote the payoff corresponding to the seed set $S= \{i\}$ by $\pi_i$. Note that, $$\pi_i = p\|C_i^o\| - q\|C_i^b\|.$$ Given an accepting node $i \in V$, and a node $j \in C_i^o$, we have $C_i = C_j$. \[lm:cluster-equiv\] If $j$ is in the interior of $C_i$, then there exists a path $(k_1, k_2, \cdots, k_\ell)$ in $G$, where $i = k_1$ and $j = k_\ell$ such that each node along the path has $\theta \leq \phi$. (That is, each node $k_i$ is exposed to and accepts the product as a result of $k_{i-1}$’s advertisement.) We would like to prove that if $S = \{j\}$, then $i$ would be exposed to the product. Equivalently, we want to show there exists a path from $j$ to $i$ of nodes with $\theta \leq \phi$; but this is precisely the path $(k_\ell, k_{\ell - 1}, \cdots, k_1)$. For an arbitrary seed set $S$, the set $V(S)$ may consist of multiple interior-disjoint clusters, which we label by $\{C_1, C_2, \cdots, C_k\}$, where $k \leq \|S\|$. Note that each of these clusters might be associated with more than one agent in the seed set and that $\cup_{i = 1}^k C_i = V(S)$. (Likewise, $\cup_{i = 1}^k C_i^o = V^+(S)$ and $\cup_{i = 1}^k C_i^b = V^-(S)$.) Given a seed-set $S$ and corresponding clusters, a direct consequence of Lemma \[lm:cluster-equiv\] is that adding more nodes already contained in these clusters to the seed-set does not change the payoff. Given a set $S'$ of accepting nodes such that $S \subseteq S' \subseteq V(S)$, we have $\pi(S) = \pi(S')$. It therefore suffices to seed a single agent within a cluster. Given a cluster $C_i$, we will simply pick an arbitrary node in the interior of the cluster to be the canonical node $i$ and use that to refer to the cluster even if $C_i$ is formed as a result of seeding another node $j \in C_i^o$. Main Model {#unbudgeted} ========== Given that all $\theta_i$ are known to the firm, a naive approach would suggest to seed all $i$ where $\pi_i \geq 0$. While this is guaranteed to give a nonnegative payoff, $S$ need not be optimal. \[counter\] Consider the graph below, where nodes in blue accept the product and those in red reject the product. \[fig:countereg\] (n1) at (0, 0.5) [1]{}; (n2) at (0, 1.5) [2]{}; (n3) at (1.5, 0) [3]{}; (n4) at (1.5, 1) [4]{}; (n5) at (1.5, 2) [5]{}; (n6) at (3, 0.5) [6]{}; (n7) at (3, 1.5) [7]{}; /in [n1/n4, n1/n2, n1/n5, n1/n3, n2/n4, n2/n5, n2/n3, n3/n6, n3/n7, n4/n6, n4/n7, n5/n6, n5/n7, n6/n7]{} () – (); Suppose $ p = q = 1$. Then, a naive approach would set $S = \emptyset$, since each of the resulting clusters to the blue nodes has negative payoff. However, setting $S = \{1, 2, 6, 7\}$ has payoff $1$. This phenomenon is a result of the fact that the clusters $C_i$ might [have]{} boundaries that intersect non-trivially. Thus, there could be agents whose $\pi_i < 0$, but $C_i^b$ is in a sense “paid for” by seeding other agents; and hence we could have a net-positive payoff from including $i$ subject to seeding other agents whose cluster boundaries intersect with $C_i^b$. Using this observation, we will give a polynomial-time algorithm for finding the optimal seed set under the generalized payoff function using a network flow argument. We first begin by constructing a flow network. Given an instance defined by $G$ and $\phi$, we let $\{C_1, C_2, \cdots, C_k\}$ be the set of all distinct clusters in $G$, with disjoint interiors. We form a flow network as follows: set $A = \{1, 2, \cdots, k\}$ corresponding to the canonical nodes of the clusters above and $R$ be the set of agents in the boundaries of all clusters. We add an edge from the source node $s$ to each node $i \in A$ with capacity $p \cdot \|C_i^o\|$ and label this value by $\operatorname{cap}_i$, and an edge from each node $j \in R$ to $t$ with capacity $q$. We add an edge between $i$ and $j$ if and only if $j \in C_i^b$, and set these edges to have infinite capacity. We denote this corresponding flow network by $G_N$. For the above example, $G_N$ is: \[scale=1,auto=left,every node/.style=[circle,fill=gray!20]{}\] (n1) at (0, 0) ; (n2) at (0, 2) ; (n3) at (2, 0) [3]{}; (n4) at (2, 1) [4]{}; (n5) at (2, 2) [5]{}; (n7) at (-2, 1) [s]{}; (n8) at (4, 1) [t]{}; (n7) to (n1); (n7) to (n2); (n1) to (n3); (n1) to (n4); (n1) to (n5); (n2) to (n3); (n2) to (n4); (n2) to (n5); (n3) to (n8); (n4) to (n8); (n5) to (n8); In this example, the edges out of $s$ have capacities $2p$ and edges into $t$ have capacities $q$. Edges between blue and red notes have infinite capacity. Assuming $4p \geq 3q$, the min-cut $(X,Y)$ has $Y = \{t\}$ and all other nodes in $X$. Given a min-cut $(X,Y)$ in $G_N$, the optimal seed set in $G$ is $A \cap X$. The min-cut $(X,Y)$ must have value at most $q \|R\|$ since we can trivially obtain that by setting the cut to be $(V(G_N) \backslash \{t\}, \{t\})$. Given a node $i \in X$, which we recall corresponds to the canonical node of a cluster, if a node $j \in R$ is exposed to the product as a result of seeding any node in the cluster, then $j \in X$. Otherwise, we would have edge $(i, j)$ included in the cut, which has infinite capacity contradicting the minimality of the cut $(X,Y)$. Therefore, the min-cut will include all nodes in the seed set as well as all nodes that are exposed to the product as a result of the corresponding seed set in $S$. Note that the edges across the cut are of two forms: $(s, i)$ or $(j, t)$, where $i \in A$ and $j \in R$. The first set of edges contribute $\sum_{i \in A \cap Y} \operatorname{cap}_i$ (recall $\operatorname{cap}_i = p \cdot \| C_i^o\|$) and the latter contributes $\| R \cap X\| q$. Therefore, the objective for finding a min-cut can be equally stated as minimizing, $${\sum_{i \in A \cap Y} \operatorname{cap}_i + \|R \cap X\| q}.$$ over cuts $(X,Y)$. Note that $$\sum_{i \in A } \operatorname{cap}_i = \sum_{i \in A \cap X} \operatorname{cap}_i + \sum_{i \in A \cap Y} \operatorname{cap}_i.$$ Therefore, we have: $$\begin{aligned} &{\left(}{\sum_{i \in A \cap Y} \operatorname{cap}_i + \|R \cap X\| q}{\right)}\\ & = {\left(}{ \sum_{i \in A} \operatorname{cap}_i - \sum_{i \in A \cap X} \operatorname{cap}_i + \|R \cap X\| q}{\right)}\\ & = \sum_{i \in A} \operatorname{cap}_i - {\left(}{ \sum_{i \in A \cap X} \operatorname{cap}_i } - \|R \cap X\| q{\right)}\end{aligned}$$ Note the term ${\left(}{ \sum_{i \in A \cap X} \operatorname{cap}_i } - \|R \cap X\| q{\right)}$ is precisely what the payoff objective function is maximizing, giving a correspondence between the min-cut and optimal seed set. We therefore have the main result of this section: There is a polynomial-time algorithm for computing the optimal seed set for Problem \[qq1\] when there are no budgets for the size of the seed set. An interesting phenomenon is that the payoff is not monotone in $\phi$ even when considering optimal seed sets. Take the following example: Suppose we are given the network below with the numbers specifying the $\theta_i$ for each corresponding node: \[scale=0.9,auto=left,every node/.style=[circle,fill=gray!20]{}\] (n1) at (2, -1) [0.2]{}; (n2) at (2, 1) [0.2]{}; (n3) at (4, 0) [0.5]{}; (n4) at (6, 1) [1]{}; (n5) at (6, -1) [1]{}; (n6) at (8, 1) [1]{}; (n7) at (8, -1) [1]{}; /in [n1/n2, n1/n3,n2/n3, n3/n4, n3/n5, n3/n6, n3/n7,n4/n5,n4/n6,n4/n7,n5/n6,n5/n7,n6/n7]{} () – (); If $\phi \in [0, 0.2)$, we cannot do better than the empty-set. If $\phi \in [0.2, 0.5)$, the seed set that includes either of the two left-most nodes gives a payoff of $1$, which is optimal. For the case where $\phi \in (0.5, 1)$, the empty-set is again optimal. This example gives a concrete way to think about overexposure phenomena such as the Groupon effect [@group] discussed in the introduction. Viewed in the current terms, we could say that by using Groupon, one could increase the broad-appeal measure of the product (e.g., cheaper, signaling higher quality, etc), which therefore exposes the product to portions of the market that would have previously not been exposed to it, and this could lead to a worse payoff. Generalized Model ================= We now consider the generalized model where there are three parameters corresponding to each agent $i$, $\tau_i \leq \theta_i \leq \sigma_i$. An agent considers a product if $\tau_i \leq \phi$, adopts a product if $\theta_i \leq \phi$, and advertises it to their friends if $\sigma_i \leq \phi$. If an agent is exposed to a product but $\phi < \tau_i$, then the payoff associated with the agent is 0. If $\phi \in [\tau_i, \theta_i)$, then the agent rejects the product, for a payoff of $-q < 0$. As before, there is a payoff of $p > 0$ if the agent accepts the product; however, the agent only advertises the product to its neighbors after accepting if $\phi \geq \sigma_i$. We therefore have four types of agents: - Type I: Agents for which $\phi < \tau_i$, - Type II: Agents for which $\phi \in [\tau_i, \theta_i)$, - Type III: Agents for which $\phi \in [\theta_i, \sigma_i)$, and - Type IV: Agents for which $\phi \geq \sigma_i$. We denote the set of all agents of Type I by $T_1$ (and likewise for the other types). The basic model above is the special case where $\tau_i = 0$ and $\theta_i = \sigma_i$ for all $i \in V$. In this instance, we only have agents of Types II and IV. Given any seed set $S$, we note: 1. $\pi(S) = \pi(S \cup T_1)$, 2. $\pi(S) \leq \pi(S \cup T_3)$. \[lm:types\] These follow from the observation that: 1. Agents of Type I are those that do not look at the product since $\phi$ is below their threshold $\tau_i$, and thus do not affect the payoff function when added to any seed set. 2. Agents of Type III are those for which $\phi \geq \theta_i$, and therefore they accept the product, but do not advertise it to their friends. Therefore, adding such an agent to the seed set increases the payoff by exactly $p > 0$ per agent added. In the simplest case, we have $ \tau_i = \theta_i$ and $\sigma_i = 1$, such that agents will be either of Type I or III, and the optimal seed set is precisely $S = T_3$. That is, agents only view a product if they are going to accept it and they never advertise it to their neighbors, and there is no cascade triggered as a result of seeding agents. When this is not the case, we will note that the results in the previous section can be adapted to this setting to find an optimal seed set efficiently. Given a network $G$ in this generalized setting, consider a corresponding network $G' = (V', E')$, which is the subgraph of $G$ consisting of agents of only Types II and IV. We can then apply the algorithm in the previous section to this subgraph $G'$ to find an optimal seed set. We claim that the union of this with agents of Type III yields an optimal seed set in $G$. Given a product with a value $\phi$ and a network $G$ with agents of parameters $\tau_i, \theta_i,$ and $\sigma_i$, there is a polynomial-time algorithm for finding an optimal seed set to maximize the payoff function. Given such a graph $G$, and a corresponding subgraph $G'$ with an optimal seed set $S'$, we argue that $S' \cup T_3$ is an optimal seed set in $G$. For the sake of contradiction, suppose $S$ is an optimal seed set in $G$, such that $\pi(S) > \pi(S' \cup T_3)$. By Lemma \[lm:types\], we can assume that $S$ does not include any agent of Type I and includes all agents of Type III. This assumption implies $\pi( S \backslash T_3) > \pi(S')$. But since $S \backslash T_3 \subseteq G'$, this contradicts the optimality of $S'$. Returning to the implications for the Groupon effect, we note that a firm can efficiently maximize its payoff over the choice of [both]{} the seed set and $\phi$. Suppose the firm knows the parameters $\tau_i, \theta_i, \sigma_i$ for each of the agents $i$. Given $n$ agents, these values divide up the unit interval into at most $4n$ subintervals $I_j$. It is easy to see that the payoff depends only on which subinterval $\phi$ is contained in, but does not vary within a subinterval. Earlier, we saw the payoff need not be monotone in $\phi$; but by trying values of $\phi$ in each of the $4n$ subintervals, the firm can determine a value of $\phi$ and a seed set that maximize its payoff. Budgeted Seeding is Hard {#budgeted} ======================== In this section, we show that the seeding problem, even for the initial model, is NP-hard if we consider the case where there is a budget $k$ for the size of the seed set and we want to find $S$ subject to the constraint that $\|S\| \leq k$. In the traditional influence maximization literature, we leverage properties of the payoff function such as its submodularity or supermodularity to give algorithms that find optimal or near-optimal seed sets. The payoff function here, however, is neither submodular nor supermodular. \[counter2\] Take the network shown in figure below. Set $p = q = 1$: \[fig:countereg2\] \[scale=1.25,auto=left,every node/.style=[circle,fill=gray!20]{}\] (n1) at (1, 0) [1]{}; (n2) at (0.25, 1) [2]{}; (n3) at (1, 2) [3]{}; (n4) at (2, 0.5) [4]{}; (n5) at (2, 1.5) [5]{}; (n6) at (3, 1.5) [6]{}; (n7) at (3, 0.5) [7]{}; /in [n1/n2, n2/n3, n1/n4, n2/n4,n1/n5,n2/n5,n3/n4,n3/n5,n5/n6,n6/n4,n5/n6,n1/n3,n4/n7,n5/n7,n6/n7]{} () – (); Supermodularity states that given $S' \subseteq S$ and $x \notin S$, $$f(S' \cup \{x\}) - f(S') \leq f(S \cup \{x\}) - f(S).$$ Set $S' = \{1 \}$, $S = \{1, 6\}$, and let $x$ be node $7$. Then, supermodularity would give: $$\begin{aligned} \pi(\{1, 7\}) - \pi(1) & \leq \pi \{1, 6, 7\} - \pi\{1, 6\}\\ 2& \leq 0\end{aligned}$$ Submodularity states that, $$f(S' \cup \{x\}) - f(S') \geq f(S \cup \{x\}) - f(S).$$ A counterexample to this is obtained by setting $S' = \emptyset$, $S = \{1\}$ and $ x = \{7\}$. Here, we show an even stronger hardness result: it is NP-hard to decide if there is a (budgeted) set yielding positive payoff. Since it is NP-hard to tell whether the optimum in any instance is positive or negative, it is therefore also NP-hard to provide an approximation algorithm with any multiplicative guarantee — a sharp contrast with the multiplicative approximation guarantees available for budgeted problems in more traditional influence maximization settings. The decision problem of whether there exists a seed set $S$ with $\|S\| \leq k$ and $\payoffvar(S) > 0$ is NP-complete. We will prove this using a reduction from the NP-complete Clique problem on $d$-regular graphs: given a $d$-regular graph $G$ and a number $k$, the question is to determine whether there exists a [$k$-clique]{} — a set of $k$ nodes that are all mutually adjacent. (We also require $d \geq k$.) We will reduce an instance of $k$-clique on $d$-regular graphs to an instance of the decision version of our budgeted seed set problem as follows. Given such an instance of Clique specified by a $d$-regular graph $G$ and a number $k$, we construct an instance of the budgeted seed set problem on a new graph $G'$ obtained from $G$ as follows: we replace each $(i, k) \in E$, with two new edges $(i, j), (j, k)$, where $j$ is a new node introduced by [subdividing]{} $(i,k)$. Let $V$ be the set of nodes originally in $G$, and $V'$ the set of nodes introduced by subdividing. In the seed set instance on $G'$, we define $\theta_i$ and $\phi$ such that $\theta_i < \phi$ for all $i \in V$ and $\theta_i > \phi$ for all $i \in V'$. We define the payoff coefficients $p, q$ by $q = 1$ and $p = d - (k-1)/2 + \epsilon$ for some $0 < \epsilon < 1/n^2$. We will show that $G$ has a $k$-clique if and only if there is a seed set of size at most $k$ in $G'$ with positive payoff. First, suppose that $S$ is a set of $k$ nodes in $G$ that are all mutually adjacent, and consider the corresponding set of nodes $S$ in $G'$. As a seed set, $S$ has $k$ accepting nodes and $kd - {k \choose 2}$ rejecting neighbors, since $G$ is $d$-regular but the nodes on the ${k \choose 2}$ subdivided edges are double-counted. Thus the payoff from $S$ is $kp - (kd - {k \choose 2}) = k(p - d + (k-1)/2)$, which is positive by our choice of $p$. For the converse, suppose $S$ has size $k' \leq k$ and has positive payoff in $G'$. Since the seed set consists entirely of accepting neighbors (any others can be omitted without decreasing the payoff), $S \subseteq V$, and hence so the neighbors of $S$ reject the product. If $S$ induces $\ell$ edges in $G$, then the payoff from $S$ includes a negative term from each neighbor, with the nodes on the $\ell$ subdivided edges double-counted, so the payoff is $k'p - (k'd - \ell)$. If $\|S\| = k' < k$, then since $\ell \leq {k' \choose 2}$, the payoff is at most $k'p - (k'd - {k' \choose 2}) = k'(p - d + (k'-1)/2)$, which is negative by our choice of $p$. If $\|S\| = k$ and $\ell \leq {k \choose 2} - 1$, then the payoff is at most $kp - (kd - ({k \choose 2} - 1)) = k(p - d + (k-1)/2 - 1/k)$, which again is negative by our choice of $p$. Thus it must be that $\|S\| = k$ and $\ell = {k \choose 2}$, so $S$ induces a $k$-clique as required. Experimental Results ==================== In this section, we present some computational results using datasets obtained from SNAP (Stanford Network Analysis Project). In particular, we consider an email network from a European research institution [@email2; @email1] and a text message network from a social-networking platform at UC-Irvine [@text]. The former is a directed network of emails sent between employees over an 803-day period, with $986$ nodes and $24929$ directed edges. The latter is a directed network of text messages sent between students through an online social network at UC Irvine over a 193-day period, with $1899$ nodes and $20296$ directed edges. In both networks, we use the edge $(i,j)$ to indicate that $i$ sent at least one email or text to node $j$ over the time period considered. For both of these networks, we present results corresponding to the general model. We consider 100 evenly-spaced values of $\phi$ in $[0, 1]$ and compare the seed set obtained by our algorithm with some natural baselines. The parameters $\tau_i \leq \theta_i \leq \sigma_i$ for each agent are chosen as follows: we draw three numbers independently from an underlying distribution (we analyze both the uniform distribution on $[0,1]$ and the Gaussian distribution with mean $0.5$ and standard deviation $0.1$); we then sort these three numbers in non-decreasing order and set them to be $\tau_i, \theta_i,$ and $\sigma_i$ respectively. For each $\phi$ we run 100 trials and present the average payoff. The average time to run one simulation is $0.915$ seconds for the text network and $0.454$ seconds for the email network. This includes the time to read the data and assemble the network; the average time spent only on computing the min-cut for the corresponding network is $0.052$ and $0.054$ seconds respectively. For each of these figures, we give a natural upper-bound which is the number of agents such that $\theta_i < \phi$. This includes agents of Type III and IV. We note that the seed set obtained by our algorithm often gives a payoff close to this upper bound. We compare this to two natural baselines: the first sets agents of Type III to be the seed set and the second sets agents of both Types III and IV to be the seed set. We show that the first baseline performs well for lower values of $\phi$, where the second baseline underperforms significantly; and, the second picks up performance significantly for higher values of $\phi$ while the first baseline suffers. The seed set obtained by our algorithm, on the other hand, outperforms both baselines by a notable margin for moderate values of $\phi$. This gap in performance corresponds to the overexposure effect in our models. \[fig:unifgen\] \[fig:gaussgen\] Each of the figures show that the seed set chosen by our algorithm outperforms these natural baselines. In Figure 2, we note for $\phi < 2/3$ the optimal seed set obtained through our algorithm is close to picking only agents of Type III. Adding on agents of Type IV performs worse than both seeding just agents of Type III or the optimal seed set. This changes for $\phi$ values over $2/3$. Here, the number of agents of Type III drops, and thus the payoff obtained by seeding agents of Type III drops with it. On the other hand, the seed set consisting of all agents of Types III and IV picks up performance, coming close to the optimal seed set for $\phi \approx 0.7$. This behavior appears in both networks. Further Related Work ==================== As noted in the introduction, our work — through its focus on selecting a [seed set]{} of nodes with which to start a cascade — follows the motivation underlying the line of theoretical work on influence maximization [@dr; @kkt; @rd]. There has been some theoretical work showing the counter-intuitive outcome where increased effort results in a less successful spread. An example is [@sela], where they show that due to the separation of the infection and viral stage, there are cases where an increased effort can result in a lower rate of spread. A related line of work has made use of rich datasets on digital friend-to-friend recommendations on e-commerce sites to analyze the flow of product recommendations through an underlying social network [@leskovec-ec06j]. Further work has experimentally explored influence strategies, with individuals either immediately broadcasting their product adoption to their social network, or selecting individuals to recommend the product to [@aral2011creating]. The consequences of negative consumer reactions have been analyzed in a range of different domains. In the introduction we noted examples involving Groupon [@group], book prizes [@good], and on-line media collections [@aizen-batting]. Although experimentally introduced negative ratings tend to be compensated for in later reviews, positive reviews can lead to herding effects [@muchnik2013social]. There has also been research seeking to quantify the economic impact of negative ratings, in contexts ranging from seller reputations in on-line auctions [@bajari-auction-reputation; @resnick-value-reputation] to on-line product reviews [@pang-lee-sentiment-book]. This work has been consistent in ascribing non-trivial economic consequences to negative consumer impressions and their articulation through on-line ratings and reviews. Recent work has also considered the rate at which social-media content receives “likes” as a fraction of its total impressions, for quantifying a social media audience’s response to cascading content [@rotabi-www17-audience]. The literature on pricing goods with network effects is another domain that has developed models in which consumers are heterogeneous in their response to diffusing content. The underlying models are different from what we pursue here; a canonical structure in the literature on pricing with network effects is a set of consumers with different levels of willingness to pay for a product [@katz-shapiro-net-ext]. This willingness to pay can change as the product becomes more popular; a line of work has thus considered how a product with network effects can be priced adaptively over time as it diffuses through the network [@arthur-pricing-social; @hartline-pricing-social]. The variation in willingness to pay can be viewed as a type of “criticality,” with some consumers evaluating products more strictly and others less strictly. But a key contrast with our work is that highly critical individuals in these pricing models do not generally confer a negative payoff when they refuse to purchase an item. Conclusion ========== Theoretical models of viral marketing in social networks have generally used the assumption that all exposures to a product are beneficial to the firm conducting the marketing. A separate line of empirical research in marketing, however, provides a more complex picture, in which different potential customers may have either positive or negative reactions to a product, and it can be a mistake to pursue a strategy that elicits too many negative reactions from potential customers. In this work, we have proposed a new set of theoretical models for viral marketing, by taking into account these types of overexposure effects. Our models make it possible to consider the optimization trade-offs that arise from trying to reach a large set of positively inclined potential customers while reducing the number of negatively inclined potential customers who are reached in the process. Even in the case where the marketer has no budget on the number of people it can expose to the product, this tension between positive and negative reactions leads to a non-trivial optimization problem. We provide a polynomial-time algorithm for this problem, using techniques from network flow, and we prove hardness for the case in which a budget constraint is added to the problem formulation. Computational experiments show how our polynomial-time algorithm yields strong results on network data. Our framework suggests many directions for future work. It would be interesting to integrate the role of negative payoffs in our model here with other technical components that are familiar from the literature on influence maximization, particularly the use of richer (and potentially probabilistic) functions governing the spread from one participant in the network to another. For example, when nodes have non-trivial thresholds for adoption — requiring both that they evaluate the product positively and also that they have heard about it from at least $k$ other people, for some $k > 1$ — how significantly do the structures of optimal solutions change? It will also be interesting to develop richer formalisms for the process by which positive and negative reactions arise when potential customers are exposed to good or bad products. With such extended formalisms we can more fully bring together considerations of overexposure and reputational costs into the literature on network-based marketing. Acknowledgements {#acknowledgements .unnumbered} ================ The first author was supported in part by a Google scholarship, a Facebook scholarship, and a Simons Investigator Award and the third author was supported in part by a Simons Investigator Award, an ARO MURI grant, a Google Research Grant, and a Facebook Faculty Research Grant.
--- abstract: 'We investigate the phase structure of pure $SU(2)$ LGT at finite temperature in the mixed fundamental and adjoint representation modified with a $\mathbb{Z}_2$ monopole chemical potential. The decoupling of the finite temperature phase transition from unphysical zero temperature bulk phase transitions is analyzed with special emphasis on the continuum limit. The possible relation of the adjoint Polyakov loop to an order parameter for the finite temperature phase transition and to the topological structure of the theory is discussed.' author: - \| Andrea Barresi, Giuseppe Burgio[^1], Michael Müller-Preussker\ $\;$\ Humboldt-Universität zu Berlin, Institut für Physik, 10115, Germany title: 'Finite temperature phase transition, adjoint Polyakov loop and topology in $SU(2)$ LGT[^2]' --- INTRODUCTION ============ Pure $SU(N)$ lattice gauge theories within the fundamental representation of the gauge group show a finite temperature deconfinement phase transition together with the breaking of a global $\mathbb{Z}_N$ center symmetry. But if confinement is a feature of the Yang-Mills continuum degrees of freedom it should be independent of the group representation for the lattice action. As Polyakov’s center symmetry breaking mechanism is available only to half-integer representations of the group, a finite temperature investigation of Wilson’s action for $SU(2)$ in the adjoint representation, i.e. $SO(3)$, might offer interesting insight to the present understanding of confinement. The $SU(2)$ mixed fundamental-adjoint action was originally studied by Bhanot and Creutz [@1BC81]: $$\label{eq1} S\!=\!\sum_{P}\Bigg[\!\beta_{A}\Bigg(1-\frac{\mathrm{Tr}_{A}U_{P}}{3}\Bigg)+\beta_{F}\Bigg(1-\frac{\mathrm{Tr}_{F}U_{P}}{2}\Bigg)\!\Bigg]$$ They found the well known non-trivial phase diagram characterized by first order $T=0$ bulk phase transition lines. A similar phase diagram is shared by $SU(N)$ theories with $N\ge 3$ [@2BC81]. Halliday and Schwimmer [@1HS81] found a similar phase diagram using a Villain discretization for the center blind part of action (\[eq1\]) $$S=\!\!\sum_{P}\!\!\Bigg[\!\beta_{V}\Bigg(\!1-\frac{\sigma_{P}\mathrm{Tr}_{F}U_{P}}{2}\!\Bigg)\!+\beta_{F}\Bigg(\!1-\frac{\mathrm{Tr}_{F}U_{P}}{2}\!\Bigg)\!\Bigg]$$ $\sigma_{P}$ being an auxiliary $\mathbb{Z}_2$ plaquette variable. By defining $\mathbb{Z}_2$ magnetic monopole and electric vortex densities $M=1-\langle\frac{1}{N_{c}}\sum_{c}\sigma_{c}\rangle$, $E=1-\langle\frac{1}{N_{l}}\sum_{l}\sigma_{l}\rangle$ with $\sigma_{c}=\prod_{P\epsilon\partial c}\sigma_{P}$ and $\sigma_{l}=\prod_{P\epsilon\hat{\partial} l}\sigma_{P}$ they argued that the bulk phase transitions were caused by condensation of these lattice artifacts. They also suggested [@2HS81] a possible suppression mechanism via the introduction of chemical potentials of the form $\lambda\sum_{c}(1-\sigma_{c})$ and $\gamma\sum_{l}(1-\sigma_{l})$. Recently Gavai and Datta [@1G99] explicitely realized this suggestion, studying the $\beta_{V}-\beta_{F}$ phase diagram as a function of $\lambda$ and $\gamma$. They found lines of second order finite temperature phase transitions crossing the $\beta_V$ and $\beta_F$ axes for $\lambda\ge 1$ and $\gamma\ge 5 $. In the limiting case $\beta_{F}=0$ and $\gamma=0$, i.e. $SO(3)$ theory with a $\mathbb{Z}_2$ monopole chemical potential, a quantitative study is difficult because of the lack of an order parameter. The $\mathbb{Z}_2$ global symmetry remains trivially unbroken. A thermodynamical approach [@2G99] shows a steep rise in the energy density for asymmetric lattices with $N_{\tau}=2,4$ and a peak in the specific heat at least for $N_{\tau}=2$, supporting the idea of a second order deconfinement phase transition. The authors have seen the adjoint Polyakov loop to fluctuate around zero below the phase transition and to take the values $1$ and $-\frac{1}{3}$ above the phase transition as $\beta_V\to \infty$. ADJOINT ACTION WITH CHEMICAL POTENTIAL ====================================== We study an adjoint representation Wilson action modified by a chemical potential suppressing the $\mathbb{Z}_2$ magnetic monopoles $$S=\frac{4}{3}\beta_{A}\sum_{P}\Bigg(1-\frac{\mathrm{Tr}_{F}^{2}U_{P}}{4}\Bigg)+\lambda\sum_{c}(1-\sigma_{c})$$ The link variables are taken in the fundamental representation only to improve the speed of our simulations, after checking that with links represented by $SO(3)$ matrices nothing changes. A standard Metropolis algorithm is used to update the links. The term $\sigma_{c}=\prod_{P\epsilon\partial c}\mathrm{sign}(\mathrm{Tr}_{F}U_{P})$ is completely center blind, i.e. $U_{\mu}(x)\rightarrow -U_{\mu}(x)\Rightarrow\sigma_{c}\rightarrow\sigma_{c}\;\;\;\forall \mu,x,c$. \[fig:btlam\] ![The phase diagram in the $\beta_A-\lambda$ plane for various $N_\tau$.](7bis.eps "fig:"){width="45.00000%"} Fig. 1 shows the phase diagram in the $\beta_A-\lambda$ plane at finite temperature. The two phases (I-II) are separated by a bulk first order line at which $\mathbb{Z}_2$ monopoles condense, phase I being continously connected with the physical $SU(2)$ phase as $\beta_F$ is turned on. Finite temperature lines, at which $\langle L_A \rangle$ shows a jump, cross the plane more or less horizontally. Putting aside the order parameter problem, the scaling behaviour at the critical temperature $T_c\equiv\frac{1}{aN_\tau}$ as a function of $\beta_A$ and $\lambda$ turns out difficult in phase I, whereas in phase II it shows a nice scaling behaviour in $\beta_A$ at fixed $\lambda \gtrsim 1$. SYMMETRY AND ORDER PARAMETER ============================ A quantitative study of the observed finite temperature transition is viable either relying on pure thermodynamical quantities [@2G99] or defining a reasonable order parameter, i.e. by understanding the underlying symmetry breaking mechanism, if any. The only hints we have are the change in the distribution of the adjoint Polyakov line operator $\frac{1}{3}\mathrm{Tr} L_A(\vec{x})$ and the values it takes in the continuum limit. After maximal abelian gauge (MAG) [@MAG] and abelian projection it is indeed possible to establish an exact global symmetry which can be broken at the phase transition and a related order parameter. Taking $$O_{\mu}(x)=I+ \sin 2 \theta_{\mu}(x) T_3 + (1-\cos 2 \theta_{\mu}(x))T_3^2$$ as the projected link in the adjoint theory, with $\vec{T}$ the adjoint representation generators of the Lie algebra, it is easy to see that the “parity” operator $P=I+2 T_3^2$ acting on all links living at fixed time as $$P O_{\mu}(x)=I- \sin 2 \theta_{\mu}(x)T_3 + (1+\cos 2 \theta_{\mu}(x))T_3^2$$ leaves all the plaquettes (and thus the action) invariant, while changing the Polyakov line. If $\Theta_L (\vec{x}) =\sum_{n=0}^{N_{\tau}-1} \theta_4(\vec{x}+\,n\,a\,\hat{4})$ is the Polyakov line global abelian phase, then for the spatial average $\langle \mathrm{Tr}L_A\rangle=1+2\langle \mathrm{cos} 2 \Theta_L (\vec{x})\rangle$ and $\langle \mathrm{Tr}PL_A\rangle=1-2\langle \mathrm{cos} 2 \Theta_L (\vec{x})\rangle$. If this symmetry is broken at the phase transition, then $\langle \mathrm{Tr}L_A\rangle=1$ below and $\langle \mathrm{Tr}L_A\rangle=1\pm2\Delta$ above, with $\Delta=\langle \mathrm{cos} 2 \Theta_L\rangle$. Thus, a reasonable order parameter should be $\|\Delta\|=\frac{1}{2}\|\langle\mathrm{Tr}L_A\rangle-1\|$. \[fig:dist\] ![$\Theta_L(\vec{x})$ volume distribution below ($\Delta=0$) and above the transition ($\Delta=\pm 1$) for typical configurations.](hist2.ps) ![$\Theta_L(\vec{x})$ volume distribution below ($\Delta=0$) and above the transition ($\Delta=\pm 1$) for typical configurations.](hist3.ps "fig:") ![$\Theta_L(\vec{x})$ volume distribution below ($\Delta=0$) and above the transition ($\Delta=\pm 1$) for typical configurations.](hist4.ps "fig:") Fig. 2 shows the volume distribution of the Polyakov line angle at the phase transition for some typical configurations. Although such a sharp change can be observed also for the full ${\rm Tr} \mathrm{L}_A(\vec{x})$ distribution, in the latter case a quantitative analysis is made difficult by the asymmetry of the values at which it peaks. In the abelian projected case, after MAG, $\Theta_L(\vec{x})$ is clearly flat below the phase transition, peaking around $0 (\pi)$ and $\frac{\pi}{2}$ above. In Fig. 3 the proposed order parameter is plotted as a function of $\beta_A$ for $\lambda=1$ and $N_\tau=4$. A singular behaviour around $\beta_A\simeq 1$ is starting to show at $V=16^3$. At $N_\tau=6$ the critical $\beta_A$ increases by roughly $35\%$. \[fig:magn\] ![Ensemble average of $\|\Delta\|$ vs. $\beta_A$.](magn_new.eps "fig:") The results show that the proposed symmetry breaking mechanism is plausible and that the order parameter behaves as one expects for a $2^{\rm nd}$ order transition, although more data at higher volumes and a study of the susceptibility would be necessary to asses such statements. The analysis of Binder cumulants is also feasible with our definitions. A study of the critical exponents and of the cluster properties of $\langle \mathrm{Tr}L_A \rangle$ would be as well interesting in order to establish whether the features of such a system are similar to those of the usual $SU(2)$ phase transition. All these questions will be addressed in a forthcoming paper. CONCLUSIONS =========== We have studied the phase diagram of the adjoint Wilson action with a chemical potential $\lambda$ for the $\mathbb{Z}_2$ magnetic monopoles. The finite temperature phase transition can be decoupled from the bulk phase transition both for positive and negative $\lambda$. The scaling behaviour of $\beta_A$ with $N_\tau$ is established in both cases, although the type I phase presents some difficulties in taking the continuum limit. In the context of abelian dominance we propose a symmetry breaking mechanism and an order parameter for the phase transition, giving promising results for numerical simulations. A deeper numerical analysis and possible extensions of the definitions will be the subject of a forthcoming paper. This work was funded by a EU-TMR network under the contract FMRX-CT97-0122 and by the DFG-GK 271. [9]{} G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. M. Creutz and K.J.M. Moriarty, Nucl. Phys. B210 (1982) 59. I.G. Halliday and A. Schwimmer, Phys. Lett. B101 (1981) 327. I.G. Halliday and A. Schwimmer, Phys. Lett. B102 (1981) 337. S. Datta and R.V. Gavai, Nucl. Phys. Proc. Suppl. 83 (2000) 366-368. S. Datta and R.V. Gavai, Phys. Rev. D60 (1999) 34505. A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.J. Wiese, Phys. Lett. B198 (1987) 516. [^1]: Address from Nov. $1^{\rm st}$ 2001: School of Mathematics, Trinity College, Dublin 2, Ireland. [^2]: Talk given by A. Barresi at Lattice2001, Berlin. HU-EP-01/41
--- abstract: 'Following many theories that predict the existence of the multiverse and by conjecture that our space-time may have a generalized geometrical structure at the fundamental level, we are interested in using a non-commutative geometry (NCG) formalism to study a suggested two-layer space that contains our 4-dimensional (4-D) universe and a re-derived photon propagator. It can be shown that the photon propagator and a cosmic microwave background (CMB) angular correlation function are comparable, and if there exists such a multiverse system, the distance between the two layers can be estimated to be within the observable universe’s radius. Furthermore, this study revealed that our results are not limited to CMB but can be applied to many other types of radiation, such as X-rays.' author: - \| Sahar Arabzadeh $^{1}$ [^1] and Kamran Kaviani $^{2}$[^2]\ \ title: Multiverse effects on the CMB angular correlation function in the framework of NCG --- [Keywords]{}: Non-commutative geometry, Multiverse, CMB, Gauge theory Introduction ============ There are many theories about the early universe that predict the existence of the multiverse. The concept of such a system was first proposed in the many-world interpretation of quantum mechanics. Since then, it has been studied within the framework of string theory[@Everett], so the idea of an inflationary multiverse is derived from the idea of eternal inflation [@Linde; @Garica]. In the standard picture, there exist infinite number of pocket universes (bubbles) such that each universe corresponds to a vacuum. Each pocket universe contains an infinite number of universes; in that case, each pocket universe has a positive cosmological constant [@Bousso]. However, the multiverse comprises a larger physical structure which contains our 4-D universe as well. The other universes may lie beyond our observable universe: by that definition, they are unobservable. Therefore, it is imperative that researchers find observable evidence of the existence of the multiverse. There have been various researches conducted on a cosmic microwave background (CMB) that may provide that observable evidence [@Salem; @Kaleban]. In this manuscript, we examine the effect of the multiverse on the two-point function of quantum electrodynamics (QED) in a two-layer space within a framework of non-commutative geometry (NCG). We chose this framework following Connes’ analogue, according to the conjecture that our space-time may have a generalized geometrical structure at a fundamental level. This manuscript is organized into several sections. In section \[ms\], we start with the mathematical set-up which is sufficient for our present job. In the next section, a two-layer space is examined as a multiverse, and we investigate the effect of such a system on a photon propagator of our 4-D universe. The resulting potential can be compared with a CMB two-point angular correlation function, as the observable evidence of the existence of a multiverse. The results showed that the Coulomb potential in each layer of this two-layer system and the CMB two-point angular correlation are comparable. Furthermore, if there exists such a multiverse system, a distance of two layers can be estimated to be within the observable universe’s radius. In addition, the corresponding Coulomb potential eventuates a tiny redshift in hydrogen atom levels, so that if the distance between two layers is considered to be within the observable universe’s radius, the mentioned redshift will be on the order of $10^{-25} eV$. We will also show that this result is not limited to CMB but can be applied to all monochromatic waves, such as X-rays. Mathematical setup {#ms} ================== In NCG, a spectral triple ($A, H, D$) includes all geometrical information of a non-commutative space in which $A$ is an involutive algebra of bounded operators on a Hilbert space $H$, and the generalized Dirac operator $D$ is a self-adjoint operator on $H$ but does not belong to $A$ [@Connes]. As an example for ordinary manifolds, the algebra $A$ can be the algebra of smooth functions on manifolds. The Hilbert space is the vector space of Square-integrable functions. This method can be generalized to the other $C^$ algebras which are not commutative and so there is no corresponding manifold for them. In this approach, The Dirac operator plays a metric role and determines the distance between two points (states) of the geometrical space, using the Connes’ formula of distance [@Connes; @Landi; @Connes; @2]: $$d(1,2)=\sup_{a\in A} \lbrace \vert a_1 - a_2 \vert : \Vert [D,a] \Vert \leq 1\rbrace .$$ In the above formula $a_1$ and $a_2$ are elements of the algebra. One can easily obtain and re-derive usual distance in a commutative space by using the above formula. In many cases the term $[D,a]$ denotes by $d a$ means differential of $a$. We remember that the commutator satisfies Leibniz rule. In addition, a Connes’ p-form, $\omega_p$ can be derived from the following relation [@Landi]: $$\label{omega} \omega_p=\sum\limits_j a_0^j [D,a_1^j] [D,a_2^j]\;...\;[D,a_p^j]\;\;\; , a_i^j\in A,$$ In NCG fiber bundles can be presented as modules over algebra. By using Serre-Swan theorem [@Connes; @Landi], one may generalize the Yang-Mills field which is the connection on fiber bundle by introducing a self-adjoint algebraic one-form as the gauge field. Suppose $\mathbb{A}$ exists as a self-adjoint one-form that represents a gauge potential. The corresponding field strength or curvature, $\theta$, is defined by $\theta = \mathbb{A}^2 + d\mathbb{A}$, which is a two-form, and the Yang-Mills action function is of the form: $$\label{yms} \mathcal{S}= YM(\mathbb{A})\equiv tr_\omega ((\mathbb{A}^2 + d\mathbb{A})^2),$$ where $tr_\omega$ is the Dixmier trace [@Landi] and plays the role of ordinary trace and the role of integration for discrete part of space and over the continues structures respectively. The following equation shows the relation between the two-layer space action functional and corresponding Lagrangian. $$\mathcal{S}= \int tr ((\mathbb{A}^2 + d\mathbb{A})^2) d^4 x$$ Note that the discrete dimension of the two-layer space is zero so we just drop the $\int d^4 x$ to write the Lagrangian [@Landi]. $$\label{lagra} \mathcal{L}= tr ((\mathbb{A}^2 + d\mathbb{A})^2)$$ Two-layer space {#tls} =============== The underlying assumption is that, within a multiverse, separated universes have only gravitational interactions. Suppose a set of points. A constraint on some subset of these points to have no Yang-Mills interaction with others creates a partition of the set. One may perform this by defining an equivalence relation as: point $i\sim$ point$j$, if only there exists a Yang-Mills interaction between them. Let us call each equivalence class a universe (see FIG. 1). These universes have only gravitational interactions with each other. In terms of NCG, they have only geometrical interactions. It is expected after these considerations that the distance between each two universes should be so large as the long-range interactions can be ignored. ![We call each subset of points that have Yang-Mills interaction with each other,a universe.](uni){width="5.5cm"} We assume that there exists a universe like ours at a specific distance. That universe is studied in the framework of NCG by consideration of a two-layer space, each layer containing a 4-D compact spin manifold denoted by $Y$. To introduce the desired spectral triple for this space, the algebra $A$ is the algebra of block diagonal matrices in which their entries are smooth real functions, $A= C_\mathbb{R}^\infty (Y) \oplus C_\mathbb{R}^\infty (Y)$. The Hilbert space $H$ is the direct sum of two Hilbert spaces on $Y$, $H= H(Y) \oplus H(Y)$. The entries of the Dirac operators are first-order differential operators that act on the spinors of $Y$. Due to the existence of $\mathbb{Z}_2$ equivalence, the block diagonal entries of $D$ and the off-diagonal elements are equal, respectively. In addition, because $[D, a]$ for $a \in A$ is a multiplication operator, off-diagonal blocks should be a multiplication operator [@Chams], which results in a self-adjoint operator. Therefore, the most general form of the Dirac operator is as follows: $$\label{dirac} D= \begin{pmatrix} i \gamma_\mu (\partial^\mu+ ie\mathbb{A}^\mu+\cdots) & \psi+ \gamma^5 m\\ \psi+ \gamma^5 m & i \gamma_\nu (\partial^\nu+ ie\mathbb{A}^\nu+\cdots)\\ \end{pmatrix},$$ where $\psi$ and $m$ are real functions. $\mathbb{A}$ is the gauge potential. Suppose that $\psi$ is equal to unit, and we limit our calculations to QED for simplicity. By using equation (2), the generalized one-form is as follows: $$\label{phi} \omega=\begin{pmatrix} \gamma^\mu \omega_{1 \mu} & \bar{\gamma} \phi\\ -\bar{\gamma} \phi^{\dagger} & \gamma^\nu \omega_{2 \nu}\\ \end{pmatrix}.$$ where $\bar{\gamma}=\psi+\gamma^5 m$. Using the above $\omega$ as a Yang-Mills field, one may write the field strength, $\theta$ as: $$\label{scale} {\scalebox{0.78}{$ \theta=\begin{pmatrix} \gamma^{\mu\nu}(\partial_\mu \omega_{1 \nu}-\partial_\nu \omega_{1 \mu})+2m\gamma^5(\phi-\phi^{\dagger})-\bar{\gamma}^2(\phi^{\dagger}\phi) & (m+\phi) \gamma^\mu \gamma^5 (\omega_{1 \mu} - \omega_{2 \mu})- m \gamma^{5} \gamma^{\mu} D_{\mu} \phi^\dagger \\ -(m+\phi) \gamma^\mu \gamma^5 (\omega_{1 \mu} - \omega_{2 \mu})+ m \gamma^{5} \gamma^{\mu} D_{\mu} \phi &\gamma^{\mu\nu}(\partial_\mu \omega_{2 \nu}-\partial_\nu \omega_{2 \mu})+2 m \gamma^5 (\phi-\phi^{\dagger})-\bar{\gamma}^2(\phi^{\dagger}\phi) \\ \end{pmatrix},$}}$$ where $\gamma^{\mu\nu} =\frac{1}{2}(\gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu) $. See Appendix A for more calculations. The one-form, $\omega$, and curvature, $\theta$, are consistent with Chamseiddine et al.’s results for the two-layer space [@Chams]. By substituting $e\mathbb{A}_{i\mu}$ as the $\omega_{i \mu}$ and by using equation (\[lagra\]), one can easily write the Yang-Mills Lagrangian as follows: $$\label{Lagr} \begin{array}{c} \mathcal{L}=-8e(( \partial^\mu \mathbb{A}^\nu_1-\partial^\nu \mathbb{A}_1^\mu)(\partial_\mu \mathbb{A}_{1 \nu}-\partial_\nu \mathbb{A}_{1\mu})+\\ \\ ( \partial^\rho \mathbb{A}_2^\sigma-\partial^\sigma \mathbb{A}_2^\rho)(\partial_\rho \mathbb{A}_{2\sigma}-\partial_\sigma \mathbb{A}_{2\rho})-(m+\phi)^2(\mathbb{A}_1^\mu -\mathbb{A}_2^\mu)(\mathbb{A}_{1 \mu}-\mathbb{A}_{2 \mu}))\\ \\ - 4 m^2 D_\mu \phi^\dagger D^\mu \phi + 2(1+m^2)^2 (\phi^\dagger \phi)^2 -8m^2 (\phi^\dagger \phi)\\ \end{array},$$ Let us return to equation (\[phi\]) and shield the Yang-Mills field. Then, having two almost non-Yang-Mills interacting 4-D universes by tuning $\phi$, $\phi$ connects the two layers through Yang-Mills interactions. To retain the gauge invariance, we keep the field $\phi$. However, we take it as tiny as possible to be able to continue shielding the Yang-Mills field. The dominant terms of the Lagrangian function become the following form: $$\label{Lag} \begin{array}{c} \mathcal{L}=-8e(( \partial^\mu \mathbb{A}^\nu_1-\partial^\nu \mathbb{A}_1^\mu)(\partial_\mu \mathbb{A}_{1 \nu}-\partial_\nu \mathbb{A}_{1\mu})+\\ \\ ( \partial^\rho \mathbb{A}_2^\sigma-\partial^\sigma \mathbb{A}_2^\rho)(\partial_\rho \mathbb{A}_{2\sigma}-\partial_\sigma \mathbb{A}_{2\rho})-\\ \\ m^2(\mathbb{A}_1^\mu -\mathbb{A}_2^\mu)(\mathbb{A}_{1 \mu}-\mathbb{A}_{2 \mu}))\\ \end{array}.$$ By using the above Lagrangian and choosing a Feynman gauge, one may obtain a photon propagator as follows (see Appendix B for more details): $$\label{prop} \tilde{D}_F^{\mu\nu}(k) = \frac{i m^2g^{\mu\nu}}{k^2(k^2+2m^2) - i \varepsilon}.$$ This propagator has two poles, which suggests a mass-less photon and a tachyon-type photon with a mass order of $m$. Since we do not see evidence of such photons (for example, in the thermodynamic properties), we expect $m$ to be small enough. We will study two consequences of choosing this propagator in the subsections \[scf\] and \[cp\]. The spatial correlation function {#scf} -------------------------------- We are now ready to compare the resultant propagator with the two-point angular correlation function from the Wilkinson Microwave Anisotropy Probe (WMAP) nine-year results [@Copi; @CMB]. The spatial correlation function can be calculated from the Fourier transform of equation (\[prop\]) and then integrated on the time because within the CMB, we detect photons from all time ranges in each specific direction. $$\label{new} \begin{array}{c} \tilde{D}_F^{\mu\nu}(r)\equiv \int_0^T \mathrm{d} t \tilde{D}_F^{\mu\nu}(x) = \int_0^T \mathrm{d} t \int \frac{\mathrm{d}^4 k}{(2\pi)^4}\;\frac{i m^2g^{\mu\nu} \mathrm{e}^{-ikx}}{k^2(k^2+2m^2) - i \varepsilon} \\ \\ = \int_0^T \mathrm{d} t \int_{-\infty}^{+\infty} \mathrm{d}k_0 \int \frac{ \mathrm{d}^3 k}{(2\pi)^4}\;\frac{i m^2g^{\mu\nu}}{k^2(k^2+2m^2) - i \varepsilon} \mathrm{e}^{-ik_0 t}\mathrm{e}^{i\vec{k}.\vec{r}}=\\ \\ \int_{-T}^T \mathrm{d} t \int_{0}^{+\infty} \mathrm{d}k_0 \int \frac{ \mathrm{d}^3 k}{(2\pi)^4}\;\frac{i m^2g^{\mu\nu}}{k^2(k^2+2m^2) - i \varepsilon} \mathrm{e}^{-ik_0 t}\mathrm{e}^{i\vec{k}.\vec{r}}= \\ \\ \int_{0}^{+\infty} \mathrm{d}k_0 \int \frac{ \mathrm{d}^3 k}{(2\pi)^4}\;\frac{2 i m^2g^{\mu\nu} \mathrm{e}^{i\vec{k}.\vec{r}}}{(k_0^2 -\vert k \vert^2)(k_0^2 -\vert k \vert^2+2m^2)} \frac{sin(k_0 T)}{k_0}=\\ \\ \int \frac{\mathrm{d}^3 k}{(2\pi)^3}\;\frac{i m^2g^{\mu\nu}\mathrm{e}^{i\vec{k}.\vec{r}}}{\vert k \vert ^2(\vert k \vert^2- 2m^2) - i \varepsilon}+\int \frac{\mathrm{d}^3 k}{(2\pi)^3}\;\frac{i m^2g^{\mu\nu} cos(\vert k\vert T)}{\vert k \vert ^2(2m^2) - i \varepsilon}\\ \end{array}$$ where $T$ is the age of the universe. The first term is a multiple of the Coulomb potential, as we will see in the next section. To have a better sense about the other term, consider a sinusoidal wave, which travels in limited-time intervals. Obviously, it is not a monochrome wave because its Fourier transform contains all frequency contributions. However, if it travels in the time from $-\infty$ to $+\infty$, it will be monochrome. Notice that in the case of the CMB correlation function, photons can be estimated to be monochrome (with a frequency of about 160.2 GHz). Now in the limit $T \to \infty$ and using the third line in equation (\[new\]), one can easily show the following relation: $$\begin{array}{c} \tilde{D}_F^{\mu\nu}(r)=\int_{0}^{+\infty} \int \frac{\mathrm{d}k_0 \mathrm{d}^3 k}{(2\pi)^3}\;\frac{i m^2g^{\mu\nu}}{k^2(k^2+2m^2) - i \varepsilon} \mathrm{e}^{i\vec{k}.\vec{r}} \delta(k_0)\\ \\ =\int \frac{\mathrm{d}^3 k}{(2\pi)^3}\;\frac{i m^2g^{\mu\nu}}{\vert k \vert ^2(\vert k \vert^2- 2m^2) - i \varepsilon}\mathrm{e}^{i\vec{k}.\vec{r}}dt\\ \\ = \frac{i g^{\mu\nu}}{4 \pi}\left(\frac{cos(\sqrt{2}m r)}{r} \right)\\ \end{array}$$ The $\delta(k_0)$ indicates that only low-frequency photons contribute to the propagator. This condition occurs when we study the correlation function of monochrome waves. Assuming that the radiation energy density at two separated points are almost the same, we expect that the energy of the propagating photon to be low. ![Photon spatial correlation divided to $i\;g^{\mu\nu}$ vs. $\theta$.(for $m=0.03634 (G ly)^{-1}$)](6p){width="7cm"} ![The two-point angular correlation function from the WMAP 9 year results [@Copi].](observed1){width="7cm"} For better comparison with the CMB two-point correlation function, consider our universe effectively as a 1-dimensional ring with the radius of the observable universe $R$ (about 46 billion light years), where we are standing at the center of the so-called ring. We can substitute $r$ in the equation(\[vr\]) with $R\theta$, where $\theta$ is a viewing angle. Since $m$ is dimensionally proportional to the inverse of the length, it can be substituted with $\frac{k}{R}$, where $k$ is some real factor. The FIG. 2 is plotted using $m=0.03634 (G ly)^{-1}$. This value is obtained by fitting the zeroes of two curves in FIG. 2 and the “KQ75y9 mask” in FIG .3. by using Least Mean Square method. The effect of increasing $m$ is increasing the oscillation of $D_F$. X-ray is a monochromatic wave too and one may see that the X-ray two-point angular correlation function has a similar behaviour [@Xray]. This is another observable effect of $m$ in our universe Let us consider $C(\theta)$ as the angular two-point correlation function, which is defined as the average product between the temperature of two points’ angles $\theta$ apart [@Sophie]. $$C(\theta)=\overline{T(\hat{\Omega}_1),T(\hat{\Omega}_2)}\vert_{\hat{\Omega}_1.\hat{\Omega}_2=cos(\theta)}$$ Here $T(\hat{\Omega})$ is the fluctuation around the mean value of the temperature, in direction $\hat{\Omega}$ in the sky. This temperature is microwave radiation, or electromagnetic radiation, which is spatially averaged in all directions. Therefore, we can compare it with a QED spatial propagator. FIG. 2 is plotted for $m=\frac{3.38}{R}$ in which a row estimation means that by considering $m$ as the inverse of distance between two layers, the distance between these two universes exists in the order of the radius of an observable universe. This is the observable evidence, as the signature of the multiverse within our 4-D universe. The effect on Coulomb potential and Hydrogen energy levels {#cp} ---------------------------------------------------------- In the case of two distinguishable fermions’ scattering, the leading order contribution to the Feynman amplitude is an equation (\[lead\]), where $p$, $p'$, $k$ and $k'$ are incoming and outgoing momentums respectively: $$\label{lead} \begin{array}{c} i \mathcal{M}= - i e^2 \overline{u}(p') \gamma_\mu u(p) \frac{ g^{\mu\nu} m^2}{q^2\left( q^2 +2m^2\right)-i \varepsilon}\overline{u}(k')\gamma_\nu u(k),\\ \\ q=p'-p=k'-k.\\ \end{array}$$ In the non-relativistic limit: $$i \mathcal{M}= \frac{- i e^2 m^2}{\vert q\vert^2(\vert q\vert^2-2m^2)} (2m \eta '^\dagger \eta) (2m \eta '^\dagger \eta),$$ where $\eta$ is a two-component constant spinor. We compare this with the Born approximation for the scattering amplitude within non-relativistic quantum mechanics: $$\langle p'\vert iT\vert p\rangle = -i\tilde{V}(q) (2\pi)\delta (E_{p'}- E_p).$$ For the QED interaction in this kind of multiverse system, the Coulomb potential in the momentum and coordinate space are in the following forms, respectively (See Appendix C for more details.): $$\label{vr} \begin{array}{c} \tilde{V}(q) =\frac{e^2m^2}{\vert q\vert^2(\vert q\vert^2-2m^2)}\\ \\ V(r)= \frac{e^2}{4 \pi}\left(\frac{cos(\sqrt{2}m r)}{r} \right)\\ \end{array}$$ The Coulomb potential of the form of equation (\[vr\]) forces a change in the hydrogen atoms’ energy levels. There is a perturbation in the hydrogen atoms’ Hamiltonian as follows: $$\begin{array}{c} V_0 = \frac{e^2}{4 \pi r},\\ \\ V=V_0+\Delta V= \frac{e^2}{4 \pi}\left(\frac{cos(\sqrt{2}m r)}{r} \right),\\ \\ \Delta H=\Delta V= \frac{e^2}{4 \pi} \frac{\left( cos(\sqrt{2}m r)-1\right)}{r}.\\ \end{array}$$ where $V_0$ is the Coulomb potential, and $\Delta V$ is the perturbation term. The first-order perturbation for some energy levels are listed below (for $m= \frac{3.38}{R}, R= 46 (G ly)^{-1}$): $$\Delta E= \langle\psi\vert \Delta V \vert \psi \rangle$$ Where $\psi = R_{nl}(r) Y_{lm}(\theta , \phi)$. For example: $$\begin{array}{c} R_{nl}=R_{10}\;\;\Rightarrow\; \Delta E=- O(10^{-25}) eV\\ \\ R_{nl}=R_{20}\;\;\Rightarrow\; \Delta E=- O(10^{-26}) eV\\ \\ R_{nl}=R_{21}\;\;\Rightarrow\; \Delta E=0. eV\\ \end{array}$$ One can easily check that all energy levels have a miniscule non-measurable redshift. Conclusion {#Co} ========== The goal of this manuscript was to investigate the effects of a multiverse system on the Green function of the Yang-Mills theory within the framework of non-commutative geometry. To research an observable evidence of the multiverse in our 4-D universe, we constructed a two-layer space. Here, each space corresponded to a universe and obtained the photon propagator in every single layer. Based on our calculations, the resultant photon spatial correlation function was comparable with the CMB and the X-ray two-point angular correlation. Our model predicts that all monochromatic wave two-point angular correlation has the same behaviour with the CMB. Based on our calculations and by fitting zeroes of our model and observed correlation curves, if there exists such a multiverse, the distance between the two layers lies within the order of the observable universe. To investigate more effects of the multiverse, the Coulomb potential and its effect on Hydrogen atom energy levels was re-derived by considering new photon propagator. It was observed that the perturbation of Hydrogen atom Hamiltonian and changes in energy levels are in the order of $10^{-25} eV$ and are so tiny that can be observed. One-form and Curvature {#A} ====================== The most general form of Dirac operator is according to equation (\[dirac\]): $$D= \begin{pmatrix} i \gamma_\mu (\partial^\mu+ \cdots) & \psi+ \gamma^5 m\\ \psi+ \gamma^5 m & i \gamma_\nu (\partial^\nu+ \cdots)\\ \end{pmatrix}.$$ All of the zero order components of $D$ are denoted by dots. They do not contribute to the one-form presentation. (Algebra elements and zero order components are commutative.) By using equation (2), one-form is shown to be in the following form: $$\omega= \sum_j a_0^j [D,a_1^j]= \begin{pmatrix} \gamma^\mu \omega_{1\mu} & (\psi + \gamma^5 m) \phi\\ -(\psi + \gamma^5 m) \tilde{\phi} & \gamma^\mu \omega_{2\mu}\\ \end{pmatrix},$$ where we use variables $a_{0}^j=a_{01}^j\oplus a_{02}^j$ and $a_{1}^j=a_{11}^j\oplus a_{12}^j$ then we have the following notations: $$\begin{array}{cc} \omega_{1\mu} = \sum_j a_{j1}^0 \partial_\mu a_{j1}^1 ; & \phi= \sum_j a_{j1}^0 (a_{j2}^1-a_{j1}^1) ;\\ \\ \omega_{2\mu}= \sum_j a_{j2}^0 \partial_\mu a_{j2}^1 ; & \tilde{\phi}= \sum_j a_{j2}^0 (a_{j2}^1-a_{j1}^1) .\\ \end{array}$$ In order to have a self-adjoint one-form which can play the role of connection and vector potential the variables are chosen such that $\tilde{\phi} = \phi ^\dagger$. Now the curvature $\theta$ can be derived. $$\theta = \omega^2 + d \omega=\omega^2+[D, \omega]$$ $$\omega^2 =\begin{pmatrix} \gamma ^\mu \omega_{1 \mu} \gamma ^\rho \omega_{1 \rho}-\bar{\gamma}^2(\phi^{\dagger}\phi) & (\phi) \gamma^\mu \gamma^5 (\omega_{1 \mu}- \omega_{2 \mu}) \\ -(\phi) \gamma^\mu \gamma^5 (\omega_{1 \mu}- \omega_{2 \mu}) &\gamma ^\nu \omega_{2 \nu} \gamma ^\rho \omega_{2 \rho}-\bar{\gamma}^2(\phi^{\dagger}\phi) \\ \end{pmatrix}$$ $${\scalebox{0.9}{$ d \omega =\begin{pmatrix} \gamma^{\mu\nu}(\partial_\mu \omega_{1 \nu}-\partial_\nu \omega_{1 \mu})+2m\gamma^5(\phi-\phi^{\dagger}) & (m) \gamma^\mu \gamma^5 (\omega_{1 \mu} - \omega_{2 \mu})- m \gamma^{5} \gamma^{\mu} D_{\mu} \phi^\dagger \\ -(m) \gamma^\mu \gamma^5 (\omega_{1 \mu} - \omega_{2 \mu})+ m \gamma^{5} \gamma^{\mu} D_{\mu} \phi &\gamma^{\mu\nu}(\partial_\mu \omega_{2 \nu}-\partial_\nu \omega_{2 \mu})+2 m \gamma^5 (\phi-\phi^{\dagger}) \\ \end{pmatrix}$}}$$ Notice that the first term of $\omega^2$ diagonal components are traceless so it does not contribute to the Lagrangian (\[Lagr\]) and we omitted it in equation (\[scale\]). Photon Propagator {#B} ================= The Lagrangian of QED in the two-layer multiverse system is in the form of equation (\[Lag\]). One may find an equation of motion by using the Euler-Lagrange equation, which results: $$\begin{array}{c} (\partial^2 g^{\mu\nu} -\partial ^\mu \partial^\nu) \mathbb{A}_{1\nu}=-m^2 (\mathbb{A}_1^\mu-\mathbb{A}_2^\mu)\\ \\ (\partial^2 g^{\rho\sigma} -\partial ^\rho \partial^\sigma) \mathbb{A}_{2\rho}=-m^2 (\mathbb{A}_2^\sigma-\mathbb{A}_1^\sigma)\\ \end{array}$$ It is now possible to solve these two coupled equations: $$\mathbb{A}_{2\mu}=\frac{1}{m^2}(\partial^2 g^{\nu \mu}-\partial ^\mu \partial ^\nu -m^2 g^{\nu \mu})\mathbb{A}_{1\nu}\\$$ By substituting $\mu$ index with $\rho$ in the second line, we have the following relations: $$\begin{array}{c} \label{f} \frac{\left( (\partial^2-m^2)g^{\mu\sigma}-\partial^\mu \partial^\sigma \right) \left( (\partial^2-m^2)g^\nu_\mu-\partial_\mu \partial^ \nu\right)\mathbb{A}_{1\nu}}{m^2}=m^2\mathbb{A}^{1\sigma}\\ \\ \frac{\left(\left((\partial^2-m^2)g^{\mu\sigma}-\partial^\mu \partial^\sigma \right) \left( (\partial^2-m^2)g^\nu_\mu-\partial_\mu \partial^ \nu\right)-m^4 g^{\sigma\nu} \right) \mathbb{A}_{1\nu}}{m^2}=0\\ \end{array}$$ The Fourier transform of the equation (\[f\]) leads to the following relation: $$\left(\frac{k^2+2m^2}{m^2}\right) \left( k^2 g^{\sigma\nu}-k^\sigma k^\nu \right) \tilde{D}_{F\;\nu\rho}=i\delta^\sigma_\rho$$ One cannot derive the Feynman propagator from above equation because the coefficient of $\tilde{D}_{F\;\nu\rho}$ is a singular $4\times 4$ matrix. This problem occurs due to gauge symmetry. By using the Faddeev-Popov method [@Peskin], the Lagrangian and equation of motions change, respectively, as shown in the following form: $$\mathcal{L} \longrightarrow \mathcal{L}-\frac{1}{2\xi_1}(\partial^\mu \mathbb{A}_{1 \mu})^2-\frac{1}{2\xi_2}(\partial^\nu \mathbb{A}_{2 \nu})^2$$ $$\begin{array}{c} (\partial^2 g^{\mu\nu} -(1-\frac{1}{\xi_1})\partial ^\mu \partial^\nu) \mathbb{A}_{1\nu}=-m^2 (\mathbb{A}_1^\mu-\mathbb{A}_2^\mu)\\ \\ (\partial^2 g^{\rho\sigma} -(1-\frac{1}{\xi_2})\partial ^\rho \partial^\sigma) \mathbb{A}_{2\rho}=-m^2 (\mathbb{A}_2^\sigma-\mathbb{A}_1^\sigma)\\ \end{array}$$ Specific values $\xi_1=1$ and $\xi_2=1$ are chosen to make the computation. Then, the result is a propagator in the form of an equation (\[prop\]): $$\tilde{D}_F^{\mu\nu}(k) = \frac{i m^2g^{\mu\nu}}{k^2(k^2+2m^2)- i \varepsilon}$$ Coulomb Potential {#C} ================= Suppose scattering of two distinguishable* fermions. The leading order of contribution is then an equation (\[lead\]): $$\begin{array}{c} i \mathcal{M}= -2 i e^2 \overline{u}(p') \gamma_\mu u(p) \frac{ g^{\mu\nu} m^2}{q^2\left( q^2 +2m^2\right)-i \varepsilon}\overline{u}(k')\gamma_\nu u(k)\\ \\ q=p'-p=k'-k\\ \end{array}$$ To obtain the potential, in a non-relativistic limit: $$\overline{u}(p') \gamma^0 u(p)= u^\dagger (p') u(p) \approx 2m \xi '^\dagger \xi$$ The other terms,$\overline{u}(p') \gamma^i u(p)$ for $i=1,2,3$, can be neglected compare to $\overline{u}(p') \gamma^0 u(p)$ in non relativistic limit [@Peskin]. Thus we have: $$i \mathcal{M}= \frac{-2 i e^2 m^2}{\vert q\vert^2(\vert q\vert^2-2m^2)-i\varepsilon} (2m \xi '^\dagger \xi) (2m \xi '^\dagger \xi)$$ We compare this with the Born approximation in which the scattering amplitude in non-relativistic quantum mechanics appears: $$\langle p'\vert iT\vert p\rangle = -i\tilde{V}(q) (2\pi)\delta (E_{p'}- E_p)$$ $$\tilde{V}(q) =\frac{2e^2m^2}{\vert q\vert^2(\vert q\vert^2-2m^2)-i\varepsilon}$$ The repulsive potential can be derived from inversing the Fourier transform. $$\begin{array}{c} V(x)=\int \frac{\mathrm{d}^3 q}{(2\pi)^3} \frac{2e^2m^2}{\vert q\vert^2(\vert q\vert^2-2m^2)-i\varepsilon} \mathrm{e}^{iq.x}\\ \\ =\frac{2 e^2 m^2}{4 \pi ^2} \int_0^\infty \mathrm{d}\vert q \vert \;\left(\frac{\mathrm{e}^{i\vert q \vert r}-\mathrm{e}^{-i\vert q \vert r}}{i \vert q \vert r}\right) \frac{1}{\vert q \vert^2 -2m^2 -i \varepsilon}\\ \\ =\frac{2 e^2 m^2}{4 \pi ^2 r} \int_{-\infty}^\infty \frac{\mathrm{e}^{i \vert q \vert r}}{\vert q \vert (\vert q \vert^2 -2m^2-i \varepsilon)}\\ \end{array}$$ The contour of this integral can be closed above in the complex plane, and we pick up the residue of $0$ and $\sqrt{2 m^2+i \varepsilon }$. Therefore, we find the potential as follows: $$V(r)= \frac{e^2}{4 \pi}\left( -\frac{1}{r}+\frac{2 cos(\sqrt{2}m r)}{r} \right)$$ References {#references .unnumbered} ========== [1]{} H. Everett, Rev. Mod. Phys. 29, 454 (1957). A. D. Linde, D. A. Linde and A. Mezhlumian, From the Big Bang theory to the theory of a stationary universe Phys. Rev. D 49, 1783 (1994). Garriga, J., Schwartz-Perlov, D., Vilenkin, A., ans Winitzki, S. (2006). Probabilities in the inflationary multiverse. Journal of Cosmology and Astroparticle Physics, 2006(01), 017. Bousso, R., and Freivogel, B. (2007). A paradox in the global description of the multiverse. Journal of High Energy Physics 2007(06), 018. Salem, M. P. (2012). The CMB and the measure of the multiverse. Journal of High Energy Physics, 2012(6), 1-28. Kleban, Matthew, Thomas S. Levi, and Kris Sigurdson. “Observing the multiverse with cosmic wakes.” Physical Review D 87.4 (2013): 041301. A. H. Chamseddine, G. Felder, and J.Fr ̈ohlich,Gravity in noncommutative geometry, Commun. Math. Phys. 155, 205 (1993). A. Connes, Non commutative geometry, Academic Press, Cambridge, Massachusetts (1994). A. H. Chamseddine, A. Connes, The Spectral Action Principle, Commun. Math. Phys. 186, 731-750 (1997). Hale, M. (2002). Path integral quantisation of finite noncommutative geometries. Journal of Geometry and Physics, 44(2), 115-128. G. Landi, An Introduction to Noncommutative Spaces and Their Geometries, Springer, New York, (1997). Connes, A. (2000). A short survey of noncommutative geometry. Journal of Mathematical Physics, 41(6), 3832-3866. Craig J. Copi, Dragan Huterer, Dominik J. Schwarz, and Glenn D. Starkman, Large-Angle Anomalies in the CMB, Advances in Astronomy, 2010 (2010), Article ID 847541. S. Zhang, Isotropy in the two-point angular correlation function of the CMB arXiv:Astro-ph 1109.6936v2 M. E. Peskin, D. V. Schroeder, An Introduction To Quantum Field Theory, Boulder, Colorado (October 2, 1995). BB.C.J. Copi, D. Huterer, D.J. Schwarz and G.D. Starkman ,Lack of large-angle TT correlations persists in WMAP and Planck, Monthly Notices of the Royal Astronomical Society, 451(3), 2978-2985, 2015. V. Springel, C.S. Frenk, S.D.M. White, The large-scale structure of the Universe, Nature 440.7088 (2006) 1137-1144. [^1]: sr.arabzadeh@gmail.com [^2]: kamran.kaviani@gmail.com
--- abstract: 'We propose a definition of nonclassicality for a single-mode quantum-optical process based on its action on coherent states. If a quantum process transforms a coherent state to a nonclassical state, it is verified to be nonclassical. To identify nonclassical processes, we introduce a representation for quantum processes, called the process-nonclassicality quasiprobability distribution, whose negativities indicate nonclassicality of the process. Using this distribution, we derive a relation for predicting nonclassicality of the output states for a given input state. We experimentally demonstrate our method by considering the single-photon addition as a nonclassical process, and predicting nonclassicality of the output state for an input thermal state.' author: - 'Saleh Rahimi-Keshari' - Thomas Kiesel and Werner Vogel - Samuele Grandi - Alessandro Zavatta and Marco Bellini title: Quantum Process Nonclassicality --- #### Introduction. The ability of detecting any nonclassicality generated by any quantum device enables us to manipulate and control the evolution of a quantum state. In particular, this plays a central role in implementation of quantum information processing and communication [@Nielsen-Chuang]. In general, the problem of the characterization of an unknown quantum device is addressed by means of quantum process tomography [@PCZ97; @DL01; @MRL08]. A general method for quantum process tomography was recently proposed that is based on probing a quantum process (described by a completely positive and linear map $\mathcal{E}$) using coherent states to characterize the process tensor in the Fock basis, with a fixed maximum number of photons [@lobino2008; @rahimi-keshari2011; @LobinoQPT2; @lvovsky2012]. However, any photon-number cutoff will transform a classical state into a nonclassical one, as a finite sum of nonclassical states is always nonclassical. Therefore, the previously known methods are not able to distinguish quantum processes whose outputs are classical for any classical input state from those that may convert classical input states into nonclassical output states. For this purpose a universal nonclassicality test of the output states is indispensable. Nonclassicality of quantum states is characterized by the Glauber-Sudarshan representation [@Glauber; @Sudarshan] of the density operator $\hat \rho$, $$\hat \rho=\int \text{d}^2 \alpha P(\alpha) {\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}. \label{Glauber-Sudarshan}$$ If the $P$ function has the form of a classical probability density, the corresponding quantum state is said to have classical analogue [@Titulaer], otherwise the state is referred to as nonclassical [@Mandel]. However, in practice the $P$ function is highly singular for many quantum states, so that it cannot be used to experimentally check the nonclassicality in general. A recently proposed method for verifying nonclassicality of quantum states is to use a regularized version of the $P$ function, referred to as the nonclassicality quasiprobability distribution (NQD), $P_\Omega(\beta)$. Its negativities indicate the nonclassicality of any quantum state [@Kiesel], and it can be directly sampled by homodyne detection [@Kiesel-POmega-Spats; @Kiesel-POmega-Squeeze]. The relation between the NQD and the $P$ function is easily formulated by their Fourier transforms, i.e., the characteristic functions $\Phi_\Omega(\xi)$ and $\Phi(\xi)$, respectively. The function $\Phi_\Omega$ is obtained by multiplying $\Phi$ with a proper filter function, for a detailed discussion of the requirements we refer to [@Kiesel]. An example of such a filter function is $$\Omega_w(\xi)=\frac{1}{\mathcal N} \int \text{d}^2\eta e^{-\|\eta\|^4} e^{-\|\frac{\xi}{w} + \eta\|^4} \ , \label{nonclas-filter}$$ where $\mathcal N$ ensures the normalization $\Omega_w(0){=}1$. In this Letter, we propose a definition of nonclassicality for single-mode quantum-optical processes. We introduce a method for detecting nonclassical processes by testing the nonclassicality of the output states for coherent states at the input. If there exists an input coherent state leading to a nonclassical output state, the quantum process is nonclassical. This method enables us to identify nonclassical quantum processes that may transfer classical input states into nonclassical output states. We derive a relation for predicting the NQD of the output state for given input states. Moreover, we experimentally demonstrate our method by verifying the single-photon addition to be a nonclassical process and predicting nonclassicality of the output state for an input thermal state. #### Nonclassical processes. For a general input quantum state $\hat \rho$, the output of a quantum process described by the map $\mathcal{E}(\hat \rho)$ is obtained by using Eq. (\[Glauber-Sudarshan\]) and the linearity of the map, $$\mathcal{E}(\hat \rho)=\int \text{d}^2 \alpha P_{\text{in}}(\alpha) \mathcal{E}({\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}) \label{proc-out}$$ where $P_{\text{in}}(\alpha)$ is the $P$ function of the input state. As the map may not be trace-preserving in general, the output state, $\hat{\rho}_{\mbox{\tiny $\mathcal{E}$}} \propto \mathcal{E}(\hat{\rho}) $, is obtained from this expression simply by normalization. The $P$ function of the output state is given by $$P_{\text{out}}(\beta) = \int \text{d}^2 \alpha P_{\text{in}}(\alpha) P_{\mathcal{E}}(\beta \|\alpha), \label{P-in-out}$$ where $P_{\mathcal{E}}(\beta \| \alpha)$ is the $P$ function of the output state of the process conditioned on the input state being the coherent state $\|\alpha\rangle$. From it follows that if the output states of a quantum process for input coherent states are classical states, i.e., having $P$ functions that are positive semidefinite, then the output of the process for any classical input quantum states will be always a classical state. This motivates us to define nonclassicality of quantum processes as follows. A quantum process is nonclassical if it transforms an input coherent state to a nonclassical state. Therefore, based on this definition, a [classical process]{} transforms all coherent states to classical states, and the output state is classical for any classical input state. Also, nonclassicality of the output state for only one coherent state is sufficient evidence that the process is nonclassical. As the regularized version of the $P$ function, $P_\Omega(\alpha)$, is an appropriate representation of quantum states for experimentally verifying nonclassicality, we introduce a related characterization of quantum processes. The regularized version of $P_{\mathcal E}(\beta \| \alpha)$, denoted as $P_{\Omega,\mathcal{E}}(\beta \| \alpha)$, a regular function of two complex variables, is a representation of the process that we use to verify its nonclassicality. This conditioned quasiprobability distribution is denoted as the process-nonclassicality quasiprobability distribution (PNQD), which unambiguously identifies the nonclassicality of a given quantum process. For a nonclassical process there exists an input coherent state ${\left\| \alpha_0\right\rangle}$ and $\beta_0$ such that $P_{\Omega,\mathcal{E}}(\beta_0 \|\alpha_0){<}0$. We note that the nonclassicality of a quantum process does not imply that the output state is nonclassical for any classical input states, as we shall see in the following example of the cat-generation process. Having knowledge of the PNQD, $P_{\Omega,\mathcal{E}}(\beta \| \alpha)$, and using , one can find the NQD of the output state for an input state described by $P_{\text{in}}(\alpha)$ via $$P_{\Omega}(\beta)= \int \text{d}^2\alpha P_{\text{in}}(\alpha) P_{\Omega,\mathcal{E}}(\beta \| \alpha) \ . \label{nonclas-quas-in-out}$$ In case the $P$ function of the input state is highly singular, by using the Parseval identity the NQD of the output state can be obtained as $$P_{\Omega}(\beta)= \int \text{d}^2\nu \Phi_{\text{in}}(\nu) \tilde{P}_{\Omega,\mathcal{E}}(\beta \| \nu) \ ,$$ where $\Phi_{\text{in}}(\nu)$ is the characteristic function of the $P$ function and $\tilde{P}_{\Omega,\mathcal{E}}(\beta \| \nu)$ is the Fourier transform of $P_{\Omega,\mathcal{E}}(\beta \|\alpha)$. For unknown quantum processes the PNQD can be estimated by sampling the NQD of the output states [@Kiesel-POmega-Spats; @Kiesel-POmega-Squeeze] for a sufficiently large number of input coherent states. In principle, for any unknown quantum process, $P_{\Omega,\mathcal{E}}(\beta\|\alpha)$ can even be uniquely determined by knowing the action of the process on only an arbitrary small compact set of input coherent states ${\left\| \alpha\right\rangle}$ [@Supplement]. #### Examples of classical and nonclassical processes. Examples of classical processes include the photon subtraction and the interaction of a state with a thermal bath; for further details on classical maps, cf. Ref. [@Gehrke]. For the photon-subtraction process, the map $$\mathcal{E}({\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}) =\hat{a} {\left\| \alpha\right\rangle}{\left\langle \alpha\right\|} \hat{a}^{\dagger} = \|\alpha \|^2 {\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}$$ yields a classical state. Consequently, the output will be classical for any classical input state [@zavatta2008]. However, a nonclassical input state may be transformed to an output state with modified nonclassical properties [@grangier2006], see Fig. \[fig:POmega1\]. Hence the output state of a classical process for a nonclassical input state may not be classical. ![(Color online) The NQD for the output of the single-photon subtraction process, acting on a squeezed vacuum state with variances $V_x {=} 0.5$ and $V_p {=} 3.0$. We used the filter function with the filter width $w {=} 1.5$. The negativities indicate the nonclassicality of the state.[]{data-label="fig:POmega1"}](PhotonSubSqueezedState.jpg){width="0.8\columnwidth"} Let us consider a model for the process of decoherence caused by a thermal bath of mean occupation number $\bar{n}$ [@Richter-Vogel], the characteristic function of the $P$ function of the state at time $t$ is given by [@Marian] $$\Phi(\xi,t)= \exp\left[-\|\xi\|^2 (\bar{n}-(\bar{n}+1)e^{-2\gamma t})\right] \Phi_Q(\xi e^{-\gamma t},0) \ ,$$ where $\Phi_Q(\xi,0)$ ($\Phi_Q(\xi,0){=}\exp[-\|\xi\|^2]\Phi(\xi,0)$) is the characteristic function of the $Q$ function of the state at time $t{=}0$, $t$ is the interaction time, and $\gamma$ is the damping rate. The $Q$ function of any quantum state is a positive semidefinite function [@perelomov]. Hence, for $\frac{\bar{n}}{\bar{n}+1}{>}e^{-2\gamma t}$ the $P$ function is positive semidefinite, as it is given by the convolution of two positive semidefinite functions, and the output state for any initial nonclassical state is always classical. In this case, this process is classical. An example of a nonclassical process is the cat-generation process. The unitary evolution associated with the Hamiltonian $\hat{H}_{\text{Kerr}}{=}\chi(\hat{a}^{\dagger}\hat{a})^2$, generates the Schrödinger cat state at time $t{=}\frac{\pi}{2\chi}$ ($\hbar{=}1$) [@Milburn; @YS86] $$\begin{aligned} \mathcal{E}({\left\| \alpha\right\rangle}{\left\langle \alpha\right\|})&=e^{-i\frac{\pi}{2}(\hat{a}^{\dagger}\hat{a})^2}{\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}e^{i\frac{\pi}{2}(\hat{a}^{\dagger}\hat{a})^2}\nonumber \\ &=\frac{1}{2}({\left\| \alpha\right\rangle}+i{\left\| -\alpha\right\rangle})({\left\langle \alpha\right\|}-i{\left\langle -\alpha\right\|}) \ .\end{aligned}$$ This is a nonclassical process, as the PNQD takes on negative values; see Fig. \[fig:POmega1b\]. As would be expected, this nonclassical process converts a classical state to a nonclassical one. However, for certain classical input states the output state can be classical. As the corresponding unitary operator is a function of the photon number operator $\hat{n}{=}\hat{a}^\dagger\hat {a}$, it leaves the photon number states $\|n\rangle \langle n\|$ unchanged. As a consequence, any statistical mixture of photon number states remains unchanged. Therefore, the output state of this process for an input thermal state is the same thermal state, which yields a classical output from a nonclassical process with classical input. ![(Color online) The PNQD of the cat-generation process, for an input coherent state with $\alpha{=}2.0$. The filter function is the same as in Fig. \[fig:POmega1\].[]{data-label="fig:POmega1b"}](SchrodingerCat){width="0.8\columnwidth"} Last but not least, even a nonclassical process with nonclassical input can have a classical output state. A simple example is the application of the squeezing operation on a squeezed input state. For example, when a squeezed vacuum input state is squeezed again with the same amount of squeezing but in the quadrature orthogonal to the original squeezing, the squeezed state transforms into the vacuum state. #### Experimental demonstration of a nonclassical process. Based on our definition, the single-photon addition process is a nonclassical process, as it transforms the vacuum state to the single-photon state, i.e., a coherent state with zero amplitude to a nonclassical state. In the following, we experimentally demonstrate our method by applying it to the single-photon addition process. Stimulated parametric down-conversion is used to generate the single-photon-added coherent states [@zavatta04:science; @zavatta05]. The core of the experimental setup is a $\chi^{(2)}$-nonlinear crystal \[$\beta$-barium borate (BBO), type I\] pumped by a 90-mW UV beam obtained by frequency doubling 1.5-ps pulses at 785 nm from a mode-locked Ti:sapphire laser. The spontaneous parametric down-conversion from the crystal consists in pairs of entangled photons emitted in two well-defined directions called signal and idler. When a seed coherent state is injected in the crystal along the signal direction, stimulated emission also takes place. A single-photon avalanche silicon detector is placed along the idler beam after spatial and spectral filtering. A click from this detector heralds the generation of a single-photon-added coherent state in a well-defined spatiotemporal signal mode, which is then characterized by time-domain homodyne detection [@zavatta02:josab]. In the experiment, we analyzed the photon-added states with 13 different input coherent-state amplitudes. For each acquisition the homodyne phase was varied between zero and $\pi$ and actively locked to 10 different values by monitoring the DC level from the homodyne receiver. ![image]({POmegaSPACS-0.00-1.2.jpg}){width="32.00000%"} ![image]({POmegaSPACS-0.46-1.2.jpg}){width="32.00000%"} ![image]({POmegaSPACS-1.12-1.2.jpg}){width="32.00000%"} Now we prove experimentally that this process is a nonclassical one. To estimate the PNQD of this process from experimentally recorded quadrature distributions, we use the sampling approach which has already been applied to determine the NQD in [@Kiesel-POmega-Squeeze]. The PNQD is reconstructed by using the filter function with the filter width $w{=}1.2$. The effect of the quantum efficiency is removed as described in [@Supplement]. The obtained results are shown for three different input coherent states in Fig. \[fig:nqp:spacs\]. We observe negativities for different amplitudes of the coherent input state ${\left\| \alpha\right\rangle}$, with decreasing negativity for increasing $\alpha$. Obviously, the negativity appears close to the origin of phase space, i.e., at $\beta{=}0$. Therefore, we examine the dependence of the PNQD on the input amplitude $\alpha$ at $\beta{=}0$ more closely; see Fig. \[fig:nqp:spacs:cross\]. It is clearly seen that the negativity is statistically significant for low input amplitudes $\alpha$, which eventually yields the sought experimental proof of the nonclassicality of the process. For larger amplitudes, the negativity vanishes at $\beta{=}0$. However, this does not mean that the output state for an input coherent state with large amplitude is classical. As the single-photon-added coherent states are nonclassical for any input amplitude [@zavatta05], one will find negativities of the PNQD at values of $\beta$ different from zero. ![(Color online) PNQD with $w{=}1.2$ for different input amplitudes $\alpha$ for the single-photon-addition process, evaluated at $\beta {=} 0$. The error bars correspond to one standard deviation. The green solid line represents the theoretical expectation.[]{data-label="fig:nqp:spacs:cross"}]({POmegaAt0_1.2_fit.jpg}){width="\columnwidth"} By using the experimentally estimated PNQD for the single-photon-addition process, we are able to estimate the NQD of the output state via Eq. (5) for a thermal input state with low mean photon number. The fact that photon addition is probabilistic is properly taken into account; for details see [@Supplement]. In Fig. \[fig:POmega:SPATS\] we show the predicted NQD of the output state for a thermal input state, displaying strongly significant negativities, which prove nonclassicality of the output. This estimate of the NQD of the output state is in good agreement with the directly measured NQD of single-photon-added thermal states [@Kiesel-POmega-Spats]. ![(Color online) Estimated NQD of the output state of the single-photon addition process for a thermal input state with mean thermal photon number $\bar n {=} 0.5$. The standard deviation, shown by the blue shaded area, is mostly hidden by the linewidth, the systematic error is displayed by the red shaded area, see [@Supplement].[]{data-label="fig:POmega:SPATS"}]({POmegaSPATS_0.50_1.2_fit.jpg}){width="\columnwidth"} #### Conclusions. We have proposed a definition of nonclassicality of a quantum process through its action on coherent states. Based on this definition, any quantum process that transforms a coherent state to a nonclassical one is identified as a nonclassical process, which may transfer a classical state to a nonclassical one. For classical processes the output state is guaranteed to be classical for any input classical states. A classical process can also be useful to transform the nonclassical properties of the input state into another form, which is desired for some applications. Nonclassical processes are necessary for the generation of nonclassical states, and subsequently they can be used to create entanglement by overlapping them on a beam splitter [@kim2002; @wang2002]. Conversely, interference of classical states will not generate entanglement. The presented method enables us to check whether an unknown quantum device can generate nonclassical states and to predict nonclassicality of the output state. 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Saleh Rahimi-Keshari$^$, Thomas Kiesel, Werner Vogel, Samuele Grandi, Alessandro Zavatta, and Marco Bellini\ Mathematical properties of the PNQD =================================== By definition, the process-nonclassicality quasiprobability distribution (PNQD) is given by $$\begin{aligned} P_{\Omega,\mathcal{E}}(\beta\|\alpha) &=\frac{1}{\pi^2} \int d^2\xi \Omega_{w}(\xi) e^{\beta\xi^{}-\xi\beta^{}}\nonumber\\ &\qquad \times\text{Tr}[e^{\hat{a}^{\dagger}\xi}e^{-\hat{a}\xi^{}} \mathcal{E}({\left\| \alpha\right\rangle}{\left\langle \alpha\right\|})].\end{aligned}$$ By using the Kraus decomposition of the quantum process, $\mathcal{E}(\rho) = \sum_{i=1}^{L}\hat{K}_{i}^{} \, \rho \,\hat{K}_{i}^{\dagger}$, where $L\leq \text{dim}(\mathcal{H})^2$, $\sum_{i=1}^{L}\hat{K}_{i}^{} \,\hat{K}_{i}^{\dagger}\leq\mathcal{I}$ and $\hat{K}_{i}$ are some Kraus operators on $\mathcal{H}$, the above equation becomes $$\begin{aligned} P_{\Omega,\mathcal{E}}(\beta\|\alpha)&= \frac{1}{\pi^2} \sum_{i=1}^{L}\int d^2\xi \Omega_{w}(\xi) e^{\beta\xi^{}-\xi\beta^{}}\nonumber\\ &\qquad \times{\left\langle \alpha\right\|}\hat{K}_{i}^{\dagger}e^{\hat{a}^{\dagger}\xi}e^{-\hat{a}\xi^{}} \hat{K}_{i}^{} {\left\| \alpha\right\rangle}.\end{aligned}$$ Consequently, we can write the PNQD as an expectation value $$P_{\Omega,\mathcal{E}}(\beta\|\alpha) = {\left\langle \alpha\right\|}\mathcal{E}_(\hat O(\beta)){\left\| \alpha\right\rangle}$$ with respect to the input coherent state, where we have defined $$\mathcal{E}_(\hat O_w(\beta)) = \sum_{i=1}^L \hat K_i^\dagger \hat O_w(\beta) \hat K_i$$ and $$\hat O_w(\beta)\equiv \frac{1}{\pi^2} \int d^2\xi \Omega_{w}(\xi) e^{\hat a^{\dagger}\xi}e^{-\hat a\xi^{}} e^{\beta\xi^{}-\xi\beta^{}}.$$ It has been shown in [@UniNonclWitness] that $\hat O_w(\beta)$ is bounded, and ${\rm Tr}(\hat O_w(\beta)) = \pi^{-1}$. Therefore, also the operator $\mathcal{E}_(\hat O_w(\beta))$ is bounded [@Kraus]. According to [@Cahill; @MW], this implies that this function can be expressed as an everywhere convergent power series in terms of $\alpha$ and $\alpha^\ast$. Hence, for any unknown quantum process $\mathcal E$ and any complex number $\beta$, $P_{\Omega,\mathcal{E}}(\beta\|\alpha)$ is uniquely determined by any small compact set of input coherent states ${\left\| \alpha\right\rangle}$. Determination of nonclassicality quasiprobability distributions =============================================================== Phase-sensitive nonclassicality quasiprobability distribution ------------------------------------------------------------- We reconstruct the nonclassicality quasiprobability distribution (NQD) $P_\Omega(\beta)$ from the quadrature distributions with the help of the relation $$P_\Omega(\beta) = \int_{-\infty}^\infty dx \int_0^\pi\frac{d\varphi}{\pi} p(x;\varphi) f_\Omega(x, \varphi; \beta, w).$$ The pattern function $f_\Omega(x,\varphi; \beta, w)$ is given by $$\begin{aligned} f_\Omega(x,\varphi; \beta, w) =& \int_{-\infty}^\infty db\,\frac{\|b\|}{\pi} e^{i b x} e^{2i\|\beta\| b \sin(\arg(\beta)-\varphi-\tfrac{\pi}{2})} \nonumber\\ &\times e^{b^2/2}\Omega_w(b)\nonumber , \end{aligned}$$ where $\Omega_w(b)$ is the nonclassicality filter. In practice, the nonclassicality quasiprobability can be estimated from quadrature-phase value pairs $(x_i, \varphi_i)_{i=1}^N$ as $$P_\Omega(\beta) = \frac{1}{N} \sum_{i=1}^N f_\Omega(x_i,\varphi_i; \beta, w).$$ The data consists of $266000$ points taken at ten different phase values. To avoid certain computational artifacts, the evaluation is performed as described in the Supplementary Material of [@NonclQuasiProbSqueeze]. As the quantum efficiency slightly differs for each state, we decided to remove its effect and show the results for the states with quantum efficiency $\eta = 1$. This can be achieved by some simple rescaling: $$P_\Omega(\beta; \eta = 1, w) = \eta P_{\Omega} (\sqrt{\eta}\beta;\eta, w/\sqrt{\eta}).\label{eq:remove:eta}$$ On the left side of the equation, we have the nonclassicality quasiprobability of the ideal state. Therefore, it is sufficient to sample the nonclassicality quasiprobability of the lossy state with rescaled width and $\beta$. For details, see also [@NonclQuasiProbSPATS; @NonclWitnessSimulation]. Phase-randomized nonclassicality quasiprobability distribution -------------------------------------------------------------- If we want to predict the outcome of a phase-insensitive quantum process for a phase-insensitive input quantum state (like a thermal state), it is sufficient to examine only the phase-randomized output states of the process. In general, however, even a classical process can lead to a phase-sensitive output state from a phase-insensitive input state, for example by coherent displacement. For the phase-randomized nonclassicality quasiprobability, $$\bar P_\Omega(a) = \frac{1}{2\pi}\int_0^{2\pi}d\phi P_\Omega(a e^{i\varphi}),$$ we can use the phase-randomized pattern function $$\begin{aligned} \bar f_\Omega(x; a, w) &=& \frac{1}{2\pi} \int_0^{2\pi}d\phi f_\Omega(x, \varphi; a e^{i\phi}, w)\\ &=& \int_{-\infty}^\infty db\,\frac{\|b\|}{\pi} e^{i b x} J_0(2a b) e^{b^2/2}\Omega_w(b).\end{aligned}$$ As it can be seen, the phase argument of the pattern function disappears. Prediction of output state of the single-photon addition process for an input thermal state =========================================================================================== There is one critical point in the consideration of probabilistic quantum processes, such as the single-photon addition process, see [@TomoCreation]: The output of the quantum process formalism is given by $$\hat\rho_{\rm out} \propto \mathcal E_{\text{add}}(\hat \rho_{\rm in}) = \hat a^\dagger \hat\rho_{\rm in}\hat a.$$ However, for probabilistic quantum processes the term on the right side is not a valid quantum state, since the density matrix is not correctly normalized. Therefore, we do not observe the right side directly, but its quantum state $$\hat\rho_{\rm out} = \frac{\hat a^\dagger \hat\rho_{\rm in}\hat a}{{\rm Tr}(\hat a^\dagger \hat\rho_{\rm in}\hat a)} = \frac{\hat a^\dagger \hat\rho_{\rm in}\hat a}{1 + \langle\hat a^\dagger\hat a\rangle_{\rm in}},$$ Therefore, if we insert the $P$ representation of the quantum state $$\hat \rho_{\rm in} = \int d^2\alpha P_{\rm in}(\alpha) \|\alpha\rangle\langle\alpha\|,$$ and the output state for input coherent states $$\hat\rho_{\rm out}(\alpha) = \frac{\hat a^\dagger{\left\| \alpha\right\rangle}{\left\langle \alpha\right\|}\hat a}{1+\|\alpha\|^2},$$ we find that $$\begin{aligned} \hat \rho_{\rm out} &=& \frac{1}{1 + {\rm Tr}(\hat\rho_{\rm in} \hat a^\dagger\hat a)} \int d^2\alpha P_{\rm in}(\alpha) \mathcal E_{\text{add}} (\|\alpha\rangle\langle\alpha\|) \nonumber\\ &=& \frac{1}{1 + \langle\hat a^\dagger\hat a\rangle_{\rm in}} \int d^2\alpha P_{\rm in}(\alpha) (1+\|\alpha\|^2)\hat\rho_{\rm out}(\alpha),\end{aligned}$$ where $\langle\hat a^\dagger\hat a\rangle_{\rm in} $ is the mean photon number of the input state for which we want to predict the output. So far, we have sampled the nonclassicality quasiprobabilities of the output states $\hat\rho_{\rm out}(\alpha)$ for different coherent input states, $P_{\Omega,\mathcal{E}}(\beta\|\alpha)$. From these, we can predict the NQD of the output state for an input state as $$P_\Omega(\beta) = \frac{1}{1 + \langle\hat a^\dagger\hat a\rangle_{\rm in}}\int d^2\alpha P_{\Omega,\mathcal{E}}(\beta\|\alpha) P_{\rm in}(\alpha)(1+\|\alpha\|^2),$$ where $P_{\rm in}(\alpha)$ is the $P$ function of the input state. If the latter is independent of the phase, the integral simplifies to $$P_\Omega(\beta) = \frac{2\pi}{1 + \langle\hat a^\dagger\hat a\rangle_{\rm in}} \int_0^{\infty} d a\ a \bar P_{\Omega,\mathcal{E}}(\beta\|a) P_{\rm in}(a)(1+a^2),$$ where $ \bar P_{\Omega,\mathcal{E}}(\beta\|a)$ is the phase-randomized PNQD described above. In practice, we evaluate this integral from the final number of measured input coherent state with the trapezoidal rule. 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--- abstract: 'Recent progress on the nature of short duration Gamma Ray Bursts (GRBs) has shown that a fraction of them originate in the local universe. These systems may well be the result of giant flares from Soft Gamma Repeaters (SGRs) believed to be magnetars (neutron stars with extremely large magnetic fields $\ge10^{14}\rm{G}$). If these magnetars are formed via the core collapse of massive stars, then it would be expected that the bursts should originate from predominantly young stellar populations. However, correlating the positions of BATSE short bursts with structure in the local universe reveals a correlation with all galaxy types, including those with little or no ongoing star formation. This is a natural outcome if, in addition to magnetars forming via the core collapse of massive stars, they also form via Accretion Induced Collapse following the merger of two white dwarfs, one of which is magnetic. We investigate this possibility and find that the rate of magnetar production via WD-WD mergers in the Milky Way is comparable to the rate of production via core collapse. However, while the rate of magnetar production by core collapse is proportional to the star formation rate, the rate of production via WD-WD mergers (which have long lifetimes) is proportional to the stellar mass density, which is concentrated in early-type systems. Therefore magnetars produced via WD-WD mergers may produce SGR giant flares which can be identified with early type galaxies. We also comment on the possibility that this mechanism could produce a fraction of the observed short duration GRB population at low redshift.' author: - 'R. Chapman, A. J. Levan, R. S. Priddey, and N. R. Tanvir' - 'G. A. Wynn and A. R. King' - 'M. B. Davies' title: 'Soft Gamma Repeaters and Short Gamma Ray Bursts: making magnetars from WD-WD mergers' --- Recent observations of short GRBs have shown them to be associated with a variety of host galaxy types (e.g. Gehrels et al. 2005; Fox et al. 2005; Berger et al. 2005). Tanvir et al. (2005) have performed correlation analyses indicating that up to $25\%$ of short duration GRBs originate in the local universe (within 100 Mpc), and this correlation (seen with all galaxy types) was strongest when restricted to Sbc and earlier types. SGRs are thought to be formed in the core collapse of massive stars, and due to relatively short lifetimes ($\sim10^4$ years; e.g. Kouveliotou 1999) would therefore be expected predominantly in star forming galaxies, while essentially none should be seen in ellipticals. Here we consider an alternative model for the creation of SGRs, and thus potentially GRBs: namely SGRs which are created via the Accretion Induced Collapse (AIC) of white dwarfs to neutron stars (e.g. Nomoto and Kondo 1991). Usov (1992) and King, Pringle, & Wickramasinghe (2001) have suggested that the merger of two white dwarfs (one or more of which was highly magnetic) may result in the production of a magnetar via AIC. However, the required field strengths for the white dwarfs are very large. For typical white dwarf and neutron star parameters, white dwarf B-fields of several hundred MG are necessary for magnetar creation. Such fields are relatively rare, but do exist within the magnetic white dwarf population. Figure \[f1\] shows the distribution of magnetic fields in isolated white dwarfs and in magnetic CVs. To estimate the formation rate of SGRs via the WD-WD channel within the Milky Way, we construct a mass distribution containing both magnetic and non-magnetic CO white dwarfs (Figure \[f2\]). The fraction of magnetic WD ($\rm{B}~>2\rm{MG}$) is $\sim9\%$. For magnetic WDs we calculate the B-field formed upon collapse to a neutron star of radius $10^6$cm. Picking a WD at random from the entire mass distribution, and a second from a gaussian centred on the mass of the first produces binaries with mass ratios close to unity, in agreement with observations. The fraction of double degenerate systems formed above the Chandrasekhar mass ($\rm{M_c}$) is $\sim25\%$, and $\sim40\%$ of these contain at least one magnetic WD. In $\sim10\%$ of the double degenerate population, the B-fields are strong enough to form a magnetar upon AIC after merger. Population syntheses suggest a merger rate of $3\rm{x}10^{-3}\rm{yr}^{-1}$ for WD binaries with masses $>\rm{M_c}$, and thus we expect a galactic rate of magnetar production $3\rm{x}10^{-4}\rm{yr}^{-1}$ via WD-WD mergers, comparable to the rate via core collapse. Note that this is a conservative estimate based on the magnetic field of the newly formed magnetar arising solely from flux conservation of the progenitor field during collapse. It is eminently plausible, and probably inevitable, that any seed field will be significantly amplified by an $\alpha-\omega$ dynamo mechanism within the newly formed neutron star given sufficiently rapid rotation (Thompson & Duncan 1993). Taking a 30Mpc radius sample from the Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et al. 1991), we extrapolate these results via galaxy type (T-type), mass and SFR to predict the rate of magnetar formation via both routes (Figure \[f3\]) where it can be seen that the the rates of each channel within the local universe are comparable and thus we may expect to see a correlation between the locations of short bursts and all galaxy types. The rate of formation in $\rm{T}<4$ galaxies accounts for $\sim70\%$ of the total rate. Therefore SGR flares appearing as short GRBs may be found in all galaxy types. AJL & NRT are grateful to PPARC for postdoctoral and senior research fellowship awards. AJL also thanks the Swedish Institute for support while visiting Lund. Astrophysics research at Leicester and Hertfordshire is funded by a PPARC rolling grant. RC is grateful to the University of Hertfordshire for a studentship. MBD is a Royal Swedish Academy Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. ARK gratefully acknowledges a Royal Society–Wolfson Research Merit Award. Berger, E., et al.2005, Nature, 438, 988 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., et al.1991, Third Reference Cat. of Bright Galaxies, (Berlin: Springer-Verlag) (VizieR Online Data Cat. VII/155) Fox, D. B., et al. 2005, Nature, 437, 845 Gehrels, N., et al.2005, Nature, 437, 851 King, A. R., Pringle, J. E., & Wickramasinghe, D. T. 2001, MNRAS, 320, L45 Kouveliotou, C. 1999, Proceedings of the National Academy of Science, 96, 5351 Liebert, J., Bergeron, P., & Holberg, J. B. 2005, ApJS, 156, 47 Nomoto, K., & Kondo, Y. 1991, ApJL, 367, L19 Norton, A. J., Wynn, G. A., & Somerscales, R. V. 2004, ApJ, 614, 349 Schmidt, G. D., et al.2003, ApJ, 595, 1101 Tanvir, N., Chapman, R., Levan, A., & Priddey, R. 2005, Nature, 2005, 438, 991 (T05) Thompson, C., & Duncan, R. C. 1993, , 408, 194 Usov, V. V. 1992, Nature, 357, 472 Vanlandingham, K. M., et al. 2005, AJ, 130, 734 Wickramasinghe, D. T., & Ferrario, L. 2000, PASP, 112, 873
--- abstract: 'Let $\CZhat$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\CC$ to $\CZhat$ is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order.' address: - 'School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332' - 'Department of Mathematics, Duke University, Durham, NC 27708' - 'Department of Mathematics, North Carolina State University, Raleigh, NC 27607' author: - Jennifer Hom - Adam Simon Levine - Tye Lidman bibliography: - 'bib.bib' title: - - Knot concordance in homology cobordisms ---
--- abstract: 'In this paper, we discuss the effects of electromagnetic field on the dynamical instability of a spherically symmetric expansionfree gravitational collapse. Darmois junction conditions are formulated by matching interior spherically symmetric spacetime to exterior Reissner-Nordstr$\ddot{o}$m spacetime. We investigate the role of different terms in the dynamical equation at Newtonian and post Newtonian regimes by using perturbation scheme. It is concluded that instability range depends upon pressure anisotropy, radial profile of energy density and electromagnetic field, but not on the adiabatic index $\Gamma$. In particular, the electromagnetic field reduces the unstable region.' author: - \| M. Sharif$^1$ [^1] and M. Azam$^{1,2}$ [^2]\ $^1$ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan.\ $^2$ Division of Science and Technology, University of Education,\ Township Campus, Lahore-54590, Pakistan. title: 'Effects of Electromagnetic Field on the Dynamical Instability of Expansionfree Gravitational Collapse' --- [Keywords:]{} Local anisotropy of pressure; Instability; Electromagnetic field.\ [PACS:]{} 04.20.-q; 04.40.-b; 04.40.Dg; 04.40.Nr. Introduction ============ The stability/instability of self-gravitating objects has great importance in general relativity. It is well-known that different ranges of stability would imply different kinds of evolution in the collapse as well as structure formation of self-gravitating objects. The adiabatic index $\Gamma$ defines the range of instability which is less than $\frac{4}{3}$ for a spherically symmetric distribution of isotropic perfect fluid [@1]. Also, it is obvious that a stellar model can exist only if it is stable against fluctuations. A stable stationary black hole solution under perturbations tells the final state of dynamical evolution of a gravitating system. The expansion scalar, $\Theta$, measures the rate at which small volumes of the fluid may change. In the expanding sphere, the increase in volume due to increasing area of external surface must be reimbursed with the increasing area of internal boundary surface. A similar behavior of surface area can be observed in the case of contraction. Thus we have to keep $\Theta$ vanishing in each case. Skripkin [@2] explored the central explosion of a spherically symmetric fluid distribution with expansionfree scalar. This leads to the formation of Minkowskian cavity at the center of the fluid. Eardley and Smarr [@3] investigated that the collapse of self-gravitating fluids would lead to formation of naked singularity for inhomogeneous energy density but to black hole for homogenous case. It was found that expansionfree model requires locally anisotropic fluid and inhomogeneous energy density [@4]-[@6]. Herrera et al. [@7] found that inhomogeneous expansionfree dust models with negative energy density has no physical significance. The same authors [@8] discussed cavity evolution in relativistic self-gravitating fluid. Rosseland [@9] was the first to study self-gravitating spherically symmetric charged fluid distribution. Since then many people have considered the effect of electromagnetic charge on the structure and evolution of self-gravitating systems [@10]-[@14]. Di Prisco et al. [@15] explored the effect of charge on the relation between the Weyl tensor and the inhomogeneity of energy density and concluded that Coulomb repulsion might prevent the gravitational collapse of the sphere. Thirukkanesh and Maharaj [@16] investigated that gravitational attraction is compensated by the Coulomb’s repulsive force along with gradient pressure in a gravitational collapse. Sharif and Abbas [@17] discussed the effect of electromagnetic field on spherically symmetric gravitational collapse with cosmological constant. Sharif and Sundas [@18] used Misner-Sharp formalism to discuss charged cylindrical collapse of anisotropic fluid and found that electric charge increases the active gravitational mass. It is evident that anisotropy, free streaming radiation, thermal conduction and shearing viscosity affect the evolution of self-gravitating systems. In literature [@19; @20], it is shown that the thermal effects reduce the range of instability. Chan et al. [@21] explored that the instability range depends upon the local anisotropy of the unperturbed fluid. The same authors [@22] found the effects of shearing viscous fluid on the instability range. Chan [@23] studied collapsing radiating star with shearing viscosity and concluded that it would increase anisotropy of pressure as well as the value of effective adiabatic index. Horvat et al. [@24] explored that instability of anisotropic star occurs at higher surface compactness when the anisotropy of the pressure is present. Herrera et al. [@25] discussed the dynamical instability of expansionfree fluid at Newtonian and post Newtonian order and found that the range of instability is determined by the anisotropic pressure and radial profile of the energy density. In a recent paper [@26], this problem has been explored in $f(R)$ gravity. In this paper, we take spherically symmetric distribution of collapsing fluid along with electromagnetic field and investigate how electromagnetic field would affect the range of instability. Darmois Junction conditions [@27] are used to match the interior spherically symmetric spacetime to exterior Reissner-Nordstr$\ddot{o}$m (RN) spacetime on the external hypersurface and on the internal hypersurface Minkowski spacetime within the cavity to the fluid distribution. We find that electromagnetic field, energy density and anisotropic pressure affect the stability of the system. The paper has the following format. In section 2, we discuss Einstein-Maxwell equations and some basic properties of anisotropic fluid. Section 3 provides the formulation of junction conditions. In section 4, the perturbation scheme is applied on the field as well as dynamical equations. We discuss the Newtonian and post Newtonian regimes and obtain the dynamical equation in section 5. Results are summarized in the last section. Fluid Distribution and the Field Equations ========================================== Consider a spherically symmetric distribution of charged collapsing fluid bounded by a spherical surface $\Sigma$. The line element for the interior region is the most general spherically symmetric metric given by $$\label{1} ds^2_-=-A^2(t,r)dt^{2}+B^2(t,r)dr^{2}+R^2(t,r)(d\theta^{2} +\sin^2\theta{d\phi^2}),$$ where we assume comoving coordinates inside the hypersurface $\Sigma$. The interior coordinates are taken as $x^{-0}=t,~x^{-1}=r,~x^{-2}=\theta,~x^{-3}=\phi$. It is assumed that the fluid is locally anisotropic and the energy-momentum tensor for such a fluid is given by $$\label{2} T^-_{\alpha\beta}=(\mu+p_{\perp})u_{\alpha}u_{\beta}+p_{\perp}g_{\alpha\beta}+ (p_r-p_{\perp})\chi_{\alpha} \chi_{\beta},$$ where $\mu$ is the energy density, $p_{\perp}$ the tangential pressure, $p_r$ the radial pressure, $u_{\alpha}$ the four-velocity of the fluid and $\chi_{\alpha}$ is the unit four-vector along the radial direction. Using the following definitions in comoving coordinates $$\label{3} u^{\alpha}=A^{-1}\delta^{\alpha}_{0},\quad \chi^{\alpha}=B^{-1}\delta^{\alpha}_{1},$$ we can write $$u^{\alpha}u_{\alpha}=-1,\quad\chi^{\alpha}\chi_{\alpha}=1,\quad \chi^{\alpha}u_{\alpha}=0.$$ The expansion scalar is defined as $$\label{4} \Theta=u^{\alpha}_{;\alpha}=\frac{1}{A}\left(\frac{\dot{B}}{B} +2\frac{\dot{R}}{R}\right).$$ Here dot and prime represent derivatives with respect to $t$ and $r$ respectively. The Maxwell equations can be written as $$\begin{aligned} \label{5} F_{\alpha\beta}=\phi_{\beta,\alpha}-\phi_{\alpha,\beta}, \quad F^{\alpha\beta}_{;\beta}=4{\pi}J^{\alpha},\end{aligned}$$ where $\phi_\alpha$ is the four potential and $J^{\alpha}$ is the four current. The electromagnetic energy-momentum tensor is given by $$\begin{aligned} \label{7} E_{\alpha\beta}=\frac{1}{4\pi}\left(F^\gamma_{\alpha}F_{\beta\gamma} -\frac{1}{4}F^{\gamma\delta}F_{\gamma\delta}g_{\alpha\beta}\right),\end{aligned}$$ where $F_{\alpha\beta}$ is the Maxwell field tensor. Since the charge is at rest with respect to comoving coordinates, the magnetic field will be zero. Thus we can write $$\begin{aligned} \label{8} \phi_{\alpha}=\left({\phi}(t,r),0,0,0\right),\quad J^{\alpha}={\xi}u^{\alpha},\end{aligned}$$ where $\xi$ is the charge density. The conservation of charge requires $$\label{9} q(r)=4\pi\int^r_{0}{\xi}B{R^2}dr$$ which is the electric charge interior to radius $R$. Using Eq.(\[1\]), the Maxwell equations (\[5\]) yield $$\begin{aligned} \label{10} {\phi''}-\left(\frac{A'}{A}+\frac{B'}{B}-2\frac{R'}{R}\right){\phi'} &=&4\pi\xi{AB^{2}},\\\label{11} \dot{\phi'}-\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B} -2\frac{\dot{R}}{R}\right){\phi'}&=&0.\end{aligned}$$ Integration of Eq.(\[10\]) implies $$\begin{aligned} \label{12} {\phi'}=\frac{qAB}{R^{2}}.\end{aligned}$$ The electric field intensity is defined as $$\begin{aligned} \label{13} E(t,r)=\frac{q}{4{\pi}R^{2}}.\end{aligned}$$ The Einstein field equations $$\label{14} G^-_{\alpha\beta}=8\pi\left(T^{-}_{\alpha\beta} +E^-_{\alpha\beta}\right),$$ for the interior metric gives the following set of equations $$\begin{aligned} \nonumber &&8{\pi}A^{2}({\mu}+2{\pi}E^2)=\left(\frac{2\dot{B}}{B} +\frac{\dot{R}}{R}\right)\frac{\dot{R}}{R}\\\label{15}&-&\left(\frac{A}{B}\right)^2 \left[\frac{2R''}{R}+\left(\frac{R'}{R}\right)^2-\frac{2B'R'}{BR} -\left(\frac{B}{R}\right)^2\right],\\\label{16} &&0=-2\left(\frac{\dot{R'}}{R}-\frac{\dot{R}A'}{RA} -\frac{\dot{B}R'}{BR}\right), \\\nonumber &&8{\pi}B^{2}(p_{r}-2{\pi}E^2)=-\left(\frac{B}{A}\right)^2\left [\frac{2\ddot{R}}{R}-\left(\frac{2\dot{A}}{A} -\frac{\dot{R}}{R}\right) \frac{\dot{R}}{R}\right]\\\label{17} &+&\left(\frac{2A'}{A}+\frac{R'}{R}\right)\frac{R'}{R} -\left(\frac{B}{R}\right)^2, \\\nonumber &&8{\pi}R^{2}(p_{\perp}+2{\pi}E^2)=8{\pi}R^{2}(p_{\perp} +2{\pi}E^2)\sin^{-2}\theta\\\nonumber &=&-\left(\frac{R}{A}\right)^2\left[\frac{\ddot{B}}{B} +\frac{\ddot{R}}{R}-\frac{\dot{A}}{A} \left(\frac{\dot{B}}{B}+\frac{\dot{R}}{R}\right) +\frac{\dot{B}\dot{R}}{BR}\right]\\\label{18} &+&\left(\frac{R}{B}\right)^2\left[\frac{A''}{A} +\frac{R''}{R}-\frac{A'B'}{AB}\right. \left.+\left(\frac{A'}{A}-\frac{B'}{B}\right)\frac{R'}{R}\right].\end{aligned}$$ The mass function is defined as follows [@28] $$\label{19} m(t,r)=\frac{R}{2}(1-g^{\alpha\beta}R_{,\alpha}R_{,\beta}) =\frac{R}{2}\left(1+\frac{\dot{R}^2}{A^2} -\frac{R'^2}{B^2}\right)+\frac{q^2}{2R}.$$ Differentiating this equation with respect to $r$ and using Eq.(\[15\]), we get $$\label{20} m'=4\pi{\mu}R'R^2+16\pi^2R^2E(RE'+2R'E).$$ The proper time and radial derivatives are given by $$\label{21} D_{T}=\frac{1}{A}\frac{\partial}{\partial t},\quad D_{R}=\frac{1}{R'}\frac{\partial}{\partial r},$$ where $R$ is the areal radius of the spherical surface. The velocity of the collapsing fluid is defined by the proper time derivative of $R$, i.e., $$\label{22} U=D_{T}R=\frac{\dot{R}}{A}$$ which is always negative in case of collapse. Using this expression, Eq.(\[19\]) can be written as $$\label{23} \tilde{E}\equiv\frac{R'}{B}=\left[1+U^{2}-\frac{2m}{R} +\left(\frac{q}{R}\right)^2\right]^{1/2}.$$ The conservation of energy-momentum tensor yields $$\label{24} (T^{-\alpha\beta}+E^{-\alpha\beta})_{;\beta}u_\alpha=-\frac{1}{A} \left[\dot{\mu}+(\mu+p_r)\frac{\dot{B}}{B} +2(\mu+p_\perp)\frac{\dot{R}}{R}\right]=0$$ which can be rewritten as $$\label{25} \dot{\mu}+(\mu+p_r)A\Theta+2(p_\perp-p_r)\frac{\dot{R}}{R}=0,$$ and $$\begin{aligned} \nonumber (T^{-\alpha\beta}+E^{-\alpha\beta})_{;\beta}\chi_{\alpha}&=&\frac{1}{B} \left[p'_{r}+(\mu+p_r)\frac{A'}{A}+2(p_{r}-p_\perp)\frac{R'}{R} \right.\\\label{26}&-&\left.\frac{E}{R}(4{\pi}RE'+8{\pi}R'E)\right]=0.\end{aligned}$$ Junction Conditions =================== In this section, we formulate the Darmois junction conditions for the general spherically symmetric spacetime in the interior region and RN spacetime in the exterior region. The line element for RN spacetime in Eddington-Finkelstein coordinates is given as $$\label{27} ds^2_+=-\left(1-\frac{2M}{\rho}+\frac{Q^2}{\rho^2}\right)d\nu^2 -2d{\rho}d{\nu}+\rho^2(d\theta^2+\sin^2{\theta}d\phi^2),$$ where $M$, $Q$ and $\nu$ are the total mass, charge and retarded time respectively. For smooth matching of the interior and exterior regions, Darmois conditions [@27] can be stated as follows:\ \ 1. The continuity of the line elements over $\Sigma$ $$\label{28} \left(ds^{2}_{-}\right)_{\Sigma}=\left(ds^{2}_{+}\right)_{\Sigma} =\left(ds^{2}\right)_{\Sigma}.$$ 2. The continuity of the extrinsic curvature over $\Sigma$ $$\label{29} \left[K_{ij}\right]=K^{+}_{ij}-K^{-}_{ij}=0,\quad (i,j=0,2,3).$$ The boundary surface $\Sigma$ in terms of interior and exterior coordinates can be defined as $$\begin{aligned} \label{30} f_{-}(t,r)&=&r-r_{\Sigma}=0,\\\label{31} f_{+}(\nu,\rho)&=&\rho-\rho(\nu_{\Sigma})=0,\end{aligned}$$ where $r_{\Sigma}$ is a constant. Using Eqs.(\[30\]) and (\[31\]), the interior and exterior metrics take the following form over $\Sigma$ $$\begin{aligned} \label{32} (ds^2_{-})_{\Sigma}&=&-A^2(t,r_{\Sigma})dt^{2}+R^2(t,r_{\Sigma})(d\theta^{2} +\sin^2\theta{d\phi^{2}}),\\\label{33} (ds^2_{+})_{\Sigma}&=&-\left(1-\frac{2M}{\rho_{\Sigma}} +\frac{Q^2}{\rho^2_{\Sigma}}+2\frac{d\rho_{\Sigma}}{d\nu}\right)d\nu^2 +\rho^2_{\Sigma}(d\theta^2+\sin^2\theta{d\phi^2}).\end{aligned}$$ The continuity of the first fundamental form implies $$\begin{aligned} \label{34} \frac{dt}{d\tau}&=&A(t, r_{\Sigma})^{-1},\quad R(t,r_{\Sigma})=\rho_{\Sigma}(\nu),\\\label{35} \left(\frac{d\nu}{d\tau}\right)^{-2}&=&\left(1-\frac{2M}{\rho_{\Sigma}} +\frac{Q^2}{\rho^2_{\Sigma}}+2\frac{d\rho_{\Sigma}}{d\nu}\right).\end{aligned}$$ For the second fundamental form, we evaluate outward unit normals to $\Sigma$ by using Eqs.(\[30\]) and (\[31\]) as follows $$\begin{aligned} \label{36} n^{-}_{\alpha}&=&\left(0,B(t,r_{\Sigma}),0,0\right),\\\label{37} n^{+}_{\alpha}&=&\left(1-\frac{2M}{\rho_{\Sigma}} +\frac{Q^2}{\rho^2_{\Sigma}}+2\frac{d\rho_{\Sigma}}{d\nu}\right)^{-\frac{1}{2}} \left(-\frac{d\rho_{\Sigma}}{d\nu},1,0,0\right).\end{aligned}$$ The non-vanishing components of the extrinsic curvature in terms of interior and exterior coordinates are $$\begin{aligned} \label{38} &&K^{-}_{00}=-\left[\frac{A'}{AB}\right]_{\Sigma},\quad K^{-}_{22}=\left[\frac{RR'}{B}\right]_{\Sigma},\quad K^{-}_{33}=K^{-}_{22}\sin^{2}\theta, \\\label{39} &&K^{+}_{00}=\left[\left(\frac{d^{2}\nu}{d\tau^{2}}\right) \left(\frac{d\nu}{d\tau}\right)^{-1} -\left(\frac{d\nu}{d\tau}\right)\left(\frac{M}{\rho^{2}} -\frac{Q^{2}}{\rho^{3}}\right)\right]_{\Sigma},\\\label{40} &&K^{+}_{22}=\left[\left(\frac{d\nu}{d\tau}\right) \left(1-\frac{2M}{r}-\frac{Q^2}{r^2}\right)r +\left(\frac{dr}{d\tau}\right)r\right]_{\Sigma},\\\label{41} &&K^{+}_{33}=K^{+}_{22}\sin^{2}\theta.\end{aligned}$$ Making use of Eqs.(\[29\]), (\[34\]) and (\[35\]), we get $$\begin{aligned} \label{42} M\overset{\Sigma}=m(t,r)\quad \Longleftrightarrow\quad q(r)\overset{\Sigma}=Q\end{aligned}$$ and $$\begin{aligned} \nonumber &&2\left(\frac{\dot{R'}}{R}-\frac{\dot{R}A'}{RA}-\frac{\dot{B}R'}{BR}\right) \overset{\Sigma}=-\frac{B}{A}\left[\frac{2\ddot{R}}{R}-\left(\frac{2\dot{A}}{A}- \frac{\dot{R}}{R}\right) \frac{\dot{R}}{R}\right]\\\label{43} &&+\frac{A}{B}\left[\left(\frac{2A'}{A}+\frac{R'}{R}\right)\frac{R'}{R} -\left(\frac{B}{R}\right)^2\right],\end{aligned}$$ where $q(r)=Q$ has been used. Comparing Eq.(\[43\]) with Eqs.(\[16\]) and (\[17\]), we obtain $$\label{44} p_{r}\overset{\Sigma}=0.$$ The expansionfree models require the existence of internal vacuum cavity within the fluid distribution. The matching of Minkowski spacetime within cavity to the fluid distribution on $\Sigma^{(i)}$ (boundary surface between cavity and fluid) gives $$\label{45} m(t,r)\overset{\Sigma^{(i)}}{=}0,\quad p_{r}\overset{\Sigma^{(i)}}{=}0.$$ The Perturbation Scheme ======================= This section is devoted to perturb the field equations, Bianchi identities and all the material quantities by using the perturbation scheme [@19; @20] upto first order. Initially, all the quantities have only radial dependence, i.e., fluid is in static equilibrium. After that, all the quantities and the metric functions have time dependence as well in their perturbation. These are given by $$\begin{aligned} \label{46} A(t,r)&=&A_0(r)+\lambda T(t)a(r),\\\label{47} B(t,r)&=&B_0(r)+\lambda T(t)b(r),\\\label{48} R(t,r)&=&R_0(r)+\lambda T(t)c(r),\\\label{49} E(t,r)&=&E_0(r)+\lambda T(t)e(r),\\\label{50} \mu(t,r)&=&\mu_0(r)+\lambda {\bar{\mu}}(t,r),\\\label{51} p_r(t,r)&=&p_{r0}(r)+\lambda {\bar{p_r}}(t,r),\\\label{52} p_{\perp}(t,r)&=&p_{\perp0}(r)+\lambda{\bar{p_{\perp}}}(t,r),\\\label{53} m(t,r)&=&m_0(r)+\lambda{\bar{m}}(t,r),\\\label{54} \Theta(t,r)&=&\lambda {\bar{\Theta}}(t,r),\end{aligned}$$ where $0<\lambda\ll1$. By the freedom allowed in radial coordinates, we choose $R_0(r)=r$. The static configuration (unperturbed) of Eqs.(\[15\])-(\[18\]) is obtained by using Eqs.(\[46\])-(\[52\]) as follows $$\begin{aligned} \label{55} 8{\pi}\left(\mu_{0}+2{\pi}E^{2}_{0}\right) =\frac{1}{(B_0r)^2}\left(2r\frac{B_0'}{B_0}+B_0^2-1\right),\\\label{56} 8{\pi}\left(p_{r0}-2{\pi}E^{2}_{0}\right) =\frac{1}{(B_0r)^2}\left(2r\frac{A_0'}{A_0}-B_0^2+1\right),\\\label{57} 8{\pi}\left(p_{\perp0}+2{\pi}E^{2}_{0}\right) =\frac{1}{B_0^2}\left[\frac{A_0''}{A_0}-\frac{A_0'}{A_0}\frac{B_0'}{B_0} +\frac{1}{r}\left(\frac{A_0'}{A_0}-\frac{B_0'}{B_0}\right)\right].\end{aligned}$$ The corresponding perturbed field equations become $$\begin{aligned} \nonumber 8{\pi}{\bar\mu}+32{\pi}^2{E_0}Te&=&-\frac{2T}{B_0^2} \left[\left(\frac{c}{r}\right)''-\frac{1}{r} \left(\frac{b}{B_0}\right)' -\left(\frac{B_0'}{B_0}-\frac{3}{r}\right) \left(\frac{c}{r}\right)'\right.\\\label{58} &-&\left.\left(\frac{b}{B_0}-\frac{c}{r}\right) \left(\frac{B_0}{r}\right)^2\right]-16{\pi}\frac{Tb}{B_{0}} \left({\mu_{0}}+2{\pi}E^{2}_{0}\right),\\\label{59} 0&=&2\frac{\dot{T}}{A_0B_0}\left[\left(\frac{c}{r}\right)' -\frac{b}{rB_0}-\left(\frac{A'_0}{A_0}-\frac{1}{r}\right)\frac{c}{r}\right],\end{aligned}$$ $$\begin{aligned} \nonumber 8{\pi}{\bar{p_{r}}}-32{\pi}^2{E_0}Te&=&-\frac{2\ddot{T}}{A_0^2}\frac{c}{r}+ \frac{2T}{rB_0^2}\left[\left(\frac{a}{A_0}\right)'+ \left(r\frac{A_0'}{A_0}+1\right)\left(\frac{c}{r}\right)'\right.\\\label{60} &-&\left.\frac{B_0^2}{r}\left(\frac{b}{B_0}-\frac{c}{r}\right)\right] -16{\pi}\frac{Tb}{B_0}\left({p_{r0}}-2{\pi}E^{2}_{0}\right),\\\nonumber 8{\pi}{\bar{p_{\perp}}}+32{\pi}^2{E_0}Te&=&-\frac{\ddot{T}}{A_0^2} \left[\frac{b}{B_0}+\frac{c}{r}\right]+\frac{T}{B_0^2} \left[\left(\frac{a}{A_0}\right)'' +\left(\frac{c}{r}\right)''\right.\\\nonumber &+&\left.\left(\frac{2A_0'}{A_0}-\frac{B_0'}{B_0}+\frac{1}{r}\right) \left(\frac{a}{A_0}\right)'- \left(\frac{A_0'}{A_0}+\frac{1}{r}\right) \right.\\\nonumber&\times&\left.\left(\frac{b}{B_0}\right)' +\left(\frac{A_0'}{A_0} -\frac{B_0'}{B_0}+\frac{2}{r}\right) \left(\frac{c}{r}\right)'\right]\\\label{61} &-&16{\pi}\frac{Tb}{B_0}\left({p_{\perp0}}+2{\pi}E^{2}_{0}\right).\end{aligned}$$ The Bianchi identities (\[24\]) and (\[26\]) for the static configuration yields $$\begin{aligned} \label{62} \frac{1}{B_0}\left[p_{r0}'+(\mu_0+p_{r0})\frac{A_0'}{A_0} +\frac{2}{r}(p_{r0}-p_{\perp0})\right] -\frac{4\pi{E_0}}{B_0{r}}\left[2E_0+rE'_{0}\right]=0,\end{aligned}$$ which can be rewritten as $$\begin{aligned} \label{63} \frac{A'_0}{A_0}&=&-\frac{1}{\mu_0+p_{r0}} \left[p'_{r0}+\frac{2}{r}(p_{r0}-p_{\perp0})- \frac{4{\pi}E_0}{r}(2E_0+rE'_0)\right].\end{aligned}$$ The perturbed configurations imply $$\begin{aligned} \label{64} &&\frac{1}{A_0}\left[\dot{\bar{\mu}}+(\mu_0+p_{r0})\dot{T}\frac{b}{B_0} +2(\mu_0+p_{\perp0})\dot{T}\frac{c}{r}\right]=0, \\\nonumber &&\frac{1}{B_0}\left[\bar{p'_r}+(\mu_0+p_{r0}){T}\left(\frac{a}{A_0}\right)' +(\bar{\mu}+\bar{p_r})\frac{A'_0}{A_0} \right.\\\nonumber &+&\left.2(p_{r0}-p_{\perp0}){T}\left(\frac{c}{r}\right)'+2(\bar{p_r} -\bar{p_\perp})\frac{1}{r}\right]\\\label{65} &-&\frac{4\pi{E_0}T}{{B_0}r}\left(4e+2r{E_0}\left(\frac{c}{r}\right)' +re'+re\frac{E'_0}{E_0}\right)=0.\end{aligned}$$ Integration of Eq.(\[64\]) yields $$\begin{aligned} \label{66} \bar\mu=-\left[(\mu_0+p_{r0})\frac{b}{B_0}+2(\mu_0+p_{\perp0})\frac{c}{r}\right]T.\end{aligned}$$ The expansion scalar turns out to be $$\begin{aligned} \label{67} \bar\Theta&=&\frac{\dot{T}}{A_0}\left(\frac{b}{B_0}+\frac{2c}{r}\right).\end{aligned}$$ Using expansionfree condition, it follows $$\label{68} \frac{b}{B_0}=-2\frac{c}{r}.$$ Inserting this value in Eq.(\[59\]), we obtain $$\label{69} c=k\frac{A_0}{r^2},$$ where $k$ is an integration constant. Using Eq.(\[68\]) in (\[66\]), we get $$\label{70} \bar{\mu}=2(p_{r0}-p_{\perp0})T\frac{c}{r}.$$ This shows that perturbed energy density comes from the static configuration of pressure anisotropy. Similarly, the unperturbed and perturbed configuration for Eq.(\[19\]) lead to $$\begin{aligned} \label{71} m_0&=&\frac{r}{2}\left(1-\frac{1}{B_0^2}\right) +8\pi^2{E^2_0}r^3,\\\label{72} \bar{m}&=&-\frac{T}{B_0^2}\left[r\left(c'-\frac{b}{B_0}\right) +(1-B_0^2)\frac{c}{2}\right] +8\pi^2E_0{T}\left(2r^3+3r^2c{E_0}\right).\end{aligned}$$ Using the matching condition (\[44\]), Eq.(\[51\]) implies $$\label{73} p_{r0}\overset{\Sigma}=0,\quad \bar{p}_{r}\overset{\Sigma}=0.$$ Inserting these values in Eq.(\[60\]), we obtain $$\label{74} \ddot{T}(t)-\alpha(r){T}(t)\overset{\Sigma}=0,$$ where $$\begin{aligned} \nonumber \alpha(r)&\overset{\Sigma}=&\left(\frac{A_{0}}{B_{0}}\right)^2 \left[\left(\frac{a}{A_0}\right)'+\left(r\frac{A_0'}{A_0}+1\right) \left(\frac{c}{r}\right)'\right.\\\label{75} &-&\left.\frac{B_0^2}{r}\left(\frac{b}{B_0}-\frac{c}{r}\right) +16{\pi}^2rE_{0}B_{0}\left(eB_0+bE_{0}\right)\right]\frac{1}{c}.\end{aligned}$$ In order to explore instability region, all the functions involved in the above equation are taken such that $\alpha_{\Sigma}$ is positive. The corresponding solution of Eq.(\[74\]) is given by $$\label{76} T(t)=-\exp(\sqrt{\alpha_{\Sigma}}t).$$ This shows that the system starts collapsing at $t=-\infty$ with $T(-\infty)=0$ keeping it in static position. It goes on collapsing with the increase of $t$. Newtonian and Post Newtonian Terms and Dynamical Instability ============================================================ This section investigates the terms corresponding to Newtonian (N), post Newtonian (pN) and post post Newtonian (ppN) regimes. This is done by converting relativistic units into c.g.s. units and expanding upto order $c^{-4}$ in the dynamical equation. For the N approximation, it is assumed that $$\mu_0\gg p_{r0},\quad\mu_0\gg p_{\perp0}.$$ For the metric coefficients expanded upto pN approximation, we take $$\label{77} A_0=1-\frac{Gm_0}{c^2r},\quad B_0=1+\frac{Gm_0}{c^2r},$$ where $G$ is the gravitational constant and $c$ is the speed of light. Using Eqs.(\[56\]) and (\[71\]), it follows that $$\label{78} \frac{A_0'}{A_0}=\frac{8\pi{p_{r0}}r^3+2m_0-32{\pi^2}{E^2_0}r^3} {2r(r-2{m_0}+16{\pi^2}{E^2_0}r^3)},$$ which together with Eq.(\[62\]) leads to $$\begin{aligned} \label{79} p_{r0}'&=&-\left[\frac{8\pi{p_{r0}}r^3+2m_0-32{\pi^2} {E^2_0}r^3}{2r(r-2{m_0}+16{\pi^2}{E^2_0}r^3)}\right](\mu_0+p_{r0}) \\\nonumber&+&\frac{2}{r} (p_{\perp0}-p_{r0})+\frac{4{\pi}E_0}{r}(2E_0+rE'_0).\end{aligned}$$ In view of dimensional analysis, this equation can be written in c.g.s. units as follows $$\begin{aligned} \label{80} p_{r0}'&=&-G\left[\frac{c^{-2}8\pi{p_{r0}}r^3+2m_0-32c^{-2}{\pi^2} {E^2_0}r^3}{2r(r-2Gc^{-2}{m_0}+16Gc^{-4}{\pi^2}{E^2_0}r^3)}\right] (\mu_0+c^{-2}p_{r0})\\\nonumber &+&\frac{2}{r} (p_{\perp0}-p_{r0})+\frac{4{\pi}E_0}{r}(2E_0+rE'_0).\end{aligned}$$ When we expand this equation upto $c^{-4}$ order and rearrange lengthy calculations, we have $$\begin{aligned} \label{81} p_{r0}'&=&-G\frac{\mu_0m_0}{r^2}+\frac{2}{r}(p_{\perp0}-p_{r0})+ \frac{4\pi}{r}\left(2E^2_0+r{E_0}E'_0\right)\\\nonumber&-& \frac{G}{c^{2}r^3}\left(2G{\mu_0}m^{2}_0+p_{r0}m_{0}r+4{\pi} \mu_{0}p_{r0}r^4-16\pi^2{E^2_0}{\mu_0}r^4\right)\\\nonumber &-&\frac{G}{c^{4}r^4}\left(4G^2{\mu_0}m^{2}_0 +2Gp_{r0}m^{2}_{0}r+4{\pi}\mu_{0}p_{r0}r^4\right.\\\nonumber &-&\left.32\pi^2G{E^2_0}m_0{\mu_0}r^4 -16\pi^2{E^2_0}{p_{r0}}r^5\right).\end{aligned}$$ Here the terms with coefficient $c^0$ are called N order terms, coefficient with $c^{-2}$ of pN order and with $c^{-4}$ are of ppN order terms. The relationship between $\bar{\mu}$ and $\bar{p}_r$ is given by [@19; @20] $$\label{82} \bar{p}_r=\Gamma\frac{p_{r0}}{\mu_0+p_{r0}}\bar{\mu}.$$ It is noted that the fluid under the expansionfree condition evolves without being compressed [@29]. Thus the adiabatic index $\Gamma$ (which measures the variation of pressure for a given variation of density) is irrelevant here for the case of expansionfree evolution as the perturbed energy density depends on the static configuration. Using Eq.(\[70\]) in the above equation, it follows that $$\label{83} \bar{p_r}=2\Gamma\frac{p_{r0}}{\mu_0+p_{r0}}(p_{r0}-p_{\perp0})T\frac{c}{r}.$$ From Eqs.(\[55\]) and (\[71\]), we get $$\begin{aligned} \label{84} \frac{B_0'}{B_0}&=&\frac{8\pi{\mu_{0}}r^3-2m_0-32{\pi^2}{E^2_0}r^3} {2r(r-2{m_0}+16{\pi^2}{E^2_0}r^3)}.\end{aligned}$$ Next, we develop dynamical equation by substituting Eq.(\[61\]) along with Eqs.(\[68\]) and (\[76\]) in Eq.(\[65\]) and neglecting the ppN order terms $\bar{p_r},~\bar{\mu} \frac{A'_0}{A_0}$, it follows that $$\begin{aligned} \nonumber &&8\pi(\mu_0+p_{r0})r\left(\frac{a}{A_0}\right)' +16\pi(p_{r0}-p_{\perp0})r\left(\frac{c}{r}\right)'\\\nonumber &-&64\pi({p_{\perp0}+2\pi{E^2_0}})\frac{c}{r} -32\pi^{2}{E_0}\left(2e+2r{E_0}\left(\frac{c}{r}\right)' +re'+re\frac{E'_0}{E_0}\right)\\\nonumber &-&\frac{2}{B^2_{0}}\left[\left(\frac{a}{A_0}\right)'' +\left(\frac{c}{r}\right)''+\left(2\frac{A'_0}{A_0}-\frac{B_0'}{B_0} +\frac{1}{r}\right)\left(\frac{a}{A_0}\right)'\right.\\\label{85} &+&\left.\left(3\frac{A'_0}{A_0}-\frac{B_0'}{B_0} +\frac{4}{r}\right)\left(\frac{c}{r}\right)'\right] -2\frac{\alpha_{\Sigma}}{A^2_0}\frac{c}{r}=0.\end{aligned}$$ In order to discuss instability conditions of this equation upto pN order, we evaluate the following terms of dynamical equation. Under expansionfree condition, Eq.(\[60\]) can be written as $$\begin{aligned} \nonumber \left(\frac{a}{A_0}\right)'&=&-\frac{k{A_0}}{r^2}\left[16\pi(p_{r0} -2\pi{E^2_0})B^2_0-\alpha_{\Sigma}\left(\frac{B_0}{A_0}\right)^2 \right.\\\label{86}&+&\left.\left(\frac{A'_0}{A_0}\right)^2 -\frac{2}{r}\frac{A'_0}{A_0}+\frac{3}{r^2}(B^2_0-1)\right] -16{\pi}^2er{B^2_0}{E_0},\end{aligned}$$ where Eqs.(\[69\]) and (\[74\]) has been used. We can write two more equations by using Eqs.(\[86\]) and (\[69\]) as follows $$\begin{aligned} \nonumber &&\left(\frac{a}{A_0}\right)''+\left(2\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{1}{r}\right)\left(\frac{a}{A_0}\right)'\\\nonumber &=&k\frac{A_0}{r^2} \left[16\pi{p_{r0}}{B^2_0}\left(\frac{1}{r}-\frac{B'_0}{B_0}\right) -16\pi{p_{r0}}'{B^2_0}+\frac{2}{r}\left(\frac{A'_0}{A_0}\right)' \right.\\\nonumber &-&\left.\frac{2}{r}\frac{A'_0}{A_0}\frac{B'_0}{B_0} +\frac{1}{r^2}\frac{A'_0}{A_0}(5-9B^2_0)-\frac{3}{r^2}\frac{B'_0}{B_0}(B^2_0+1) +\frac{9}{r^3}(B^2_0-1)\right]\\\nonumber&+&\alpha_{\Sigma}k \frac{A_0}{r^2}\left(\frac{B_0}{A_0}\right)^2\left(\frac{A'_0}{A_0} -\frac{B'_0}{B_0}+\frac{1}{r}\right)-16\frac{kA_0{\pi}^2}{r^2} \left[12E^2_{0}\frac{A'_0}{A_0}\right.\\\nonumber&+&\left.2{E^2_0} \left(2\frac{B'_0}{B_0}-\frac{1}{r}\right)+8E_0E'_0\right]+32{\pi^2}r \left(3eE_0\frac{B'_0}{B_0}+eE'_0+e'E_0\right)\\\label{87} &-&64{\pi^2}erE_0\frac{A'_0}{A_0}. \\\nonumber &&\left(\frac{c}{r}\right)''+\left(3\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{4}{r}\right)\left(\frac{c}{r}\right)'\\\label{88}&=&k\frac{A_0}{r^3} \left[\left(\frac{A'_0}{A_0}\right)'-\frac{A'_0}{A_0}\frac{B'_0}{B_0} -\frac{11}{r}\frac{A'_0}{A_0} +\frac{3}{r}\frac{B'_0}{B_0}\right].\end{aligned}$$ Combining Eqs.(\[87\]) and (\[88\]), it follows that $$\begin{aligned} \nonumber &-&\frac{2}{B^2_0}\left[\left(\frac{a}{A_0}\right)''+\left(\frac{c}{r}\right)'' +\left(2\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{1}{r}\right)\left(\frac{a}{A_0}\right)'\right.\\\nonumber&-&\left. \left(3\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{4}{r}\right)\left(\frac{c}{r}\right)'\right]- 2\frac{\alpha_{\Sigma}}{A^2_0}\frac{c}{r}\end{aligned}$$ $$\begin{aligned} \nonumber &=&32{\pi}k\frac{A_0}{r^2} \left[p'_{r0}+p_{r0}\left(\frac{B'_{0}}{B_{0}}-\frac{1}{r}\right)\right] -6k\frac{A_0}{B^2_{0}r^3}\left[\left(\frac{A'_0}{A_0}\right)' \right.\\\nonumber&-&\left.\frac{A'_0}{A_0}\frac{B'_0}{B_0}-(3B^2_{0}+2) \frac{1}{r}\frac{A'_0}{A_0}-\frac{1}{r}B_{0}B'_{0}+\frac{3}{r^2}(B^2_{0}-1)\right] \\\nonumber&-&2\frac{\alpha_{\Sigma}k}{A_{0}r^2} \left(\frac{A'_0}{A_0}+\frac{B'_0}{B_0}\right)-16\frac{kA_0{\pi}^2}{r^2} \left[12E^2_{0}\frac{A'_0}{A_0}+2{E^2_0}\left(2\frac{B'_0}{B_0}-\frac{1}{r}\right) \right.\\\label{89}&+&\left.8E_0E'_0\right]+32{\pi^2}r \left(3eE_0\frac{B'_0}{B_0}+eE'_0+e'E_0\right) -16{\pi^2}erE_0\frac{A'_0}{A_0}.\end{aligned}$$ Inserting Eqs.(\[86\]) and (\[89\]) in Eq.(\[85\]) and making use of Eqs.(\[63\]), (\[77\]) and (\[84\]), we obtain dynamical equation at pN order (with $c = G = 1$) $$\begin{aligned} \nonumber &&-\frac{8\pi}{r^2}\left\{\left(1-\frac{m_0}{r}\right)\left[2p'_{ro} +\frac{2}{r}(5p_{r0}-p_{\perp0})-8\pi{E_0E'_0} -\frac{24\pi{E^2_0}}{r}\right]\right.\\\nonumber&+&\left.4\pi{\mu_0}r(4p_{r0}-{E^2_0})+ (\mu_0+p_{r0})r\left[\frac{3}{r^2}(B^2_0-1)-{\alpha_\Sigma} \left(\frac{B_0}{A_0}\right)^2\right]\right\}\\\nonumber &+&(\mu_0+p_{0})r\left[6\frac{m_0}{r^3}-\alpha_{\Sigma} \left(1+3\frac{m_0}{r}\right)\right]-{\mu_0}r\left(\frac{3}{r^2} +4\alpha_{\Sigma}\right)\left(\frac{m_0}{r}\right)^2\\\nonumber &+&\frac{32{\pi}k}{r^2}\left[p'_{r0}-\frac{p_{r0}}{r} +4{\pi}r{\mu_{0}}p_{r0}-\frac{m_0}{r}{p'_{r0}}-16{\pi}^2E^2_0r^2p_{r0}+ \frac{m_0}{r}p_{r0}\right.\\\nonumber&\times&\left. (64{\pi}^3E^2r^3{\mu_0}+16{\pi}^2E^2_0r^2)\right] -6\frac{k}{r^3}\left\{\frac{\mu'_{0}}{\mu^2_0}\left[p'_{r0} +\frac{2}{r}(p_{r0}-p_{\perp0})\right.\right.\\\nonumber &-&\left.\left.\frac{4{\pi}E_0}{r}(2E_0+rE'_0)\right]\right. -\left(1-3\frac{m_0}{r}\right) \frac{1}{\mu_0}\left[p''_{r0}-\frac{2}{r^2}(p_{r0}-p_{\perp0}) \right.\\\nonumber&+&\left.\frac{2}{r}(p'_{r0}-p'_{\perp0})- \frac{4\pi}{r}(3E_0E'_0+rE_0E''_0+E^{'2}_0) +\frac{4{\pi}E_0}{r^2}(2E_0+rE'_0)\right]\\\nonumber &+&\frac{8\pi{r^3}\mu_0-2{m_0}-32{\pi}^2E^2_0r^3}{2r^2{\mu_0}}\left[p'_{r0} +\frac{2}{r}(p_{r0}-p_{\perp0})-\frac{4{\pi}E_0}{r}(2E_0\right. \\\nonumber&+&\left.rE'_0)\right]+\left(5-9\frac{m_0}{r}\right) \frac{1}{r\mu_0}\left[p'_{r0}+\frac{2}{r^2}(p_{r0}-p_{\perp0}) -\frac{4{\pi}E_0}{r}(2E_0+rE'_0)\right]\\\nonumber &-&{4\pi}{\mu_0}\left[1+\frac{m_0}{r}+2 \left(\frac{m_0}{r}\right)^2\right]+\frac{1}{r^2}\left[7\frac{m_0}{r}-5 \left(\frac{m_0}{r}\right)^2\right]+16{\pi^2}E^2_0\\\nonumber &+&\left.64{\pi^3}\mu_0{E^2_0}r^2+48{\pi^2}E^2_0\frac{m_0}{r}\right\} -2\frac{\alpha_{\Sigma}k}{r^2} \left\{-\left(1+\frac{m_0}{r}\right) \frac{1}{r\mu_0}\left[p'_{r0}\right.\right.\end{aligned}$$ $$\begin{aligned} \nonumber &+&\left.\left.\frac{2}{r}(p_{r0}-p_{\perp0})- \frac{4{\pi}E_0}{r}(2E_0+rE'_0)\right]+{4\pi}\left[1+3\frac{m_0}{r}+ \left(\frac{m_0}{r}\right)^2\right]r\mu_{0}\right.\\\nonumber &-&\left.\frac{1}{r}\left[\frac{m_0}{r}+3 \left(\frac{m_0}{r}\right)^2\right]-\left(1+\frac{m_0}{r}\right) 64{\pi}^3E^2_0{\mu_0}r^3-32{\pi}^2m_0E^2_0\right\}\\\label{90} &-&16\frac{kA_0{\pi}^2}{r^2} \left[16\pi{\mu_0}E^2_0r^2-\frac{2E^2_0}{r}\left(1-\frac{m_0}{r}\right) +8E_0E'_0\right]=0.\end{aligned}$$ Using the fact that $\mu_0\gg p_{r0}$, we discard the terms belonging to pN and ppN order like $\frac{p_{r0}}{\mu_0}$ in the above equation to obtain dynamical equation at N approximation as follows $$\begin{aligned} \nonumber &&24\pi{\mu_0}+8\pi{\mid{p'_{r0}}\mid}r+2\left(\alpha_{\Sigma} -\frac{21}{r^2}\right)\frac{m_0}{r}+\frac{32{\pi}^2r^3{E_0}}{k} \left(2e+re'+re\frac{E'_0}{E_0}\right)\\\label{91} &&=32\pi(5p_{r0}-2p_{\perp0})-416{\pi}^2{E^2_0} -32{\pi^2}rE_0E'_0.\end{aligned}$$ Here $p'_{r0}<0$ shows that pressure is decreasing during collapse of expansionfree fluid. Using Eq.(\[20\]) in the above equation, we get $$\begin{aligned} \nonumber &&\frac{4\pi}{9}{\mid{p'_{r0}}\mid}+\alpha_{\Sigma}\frac{{m_0}r^2}{9} +\frac{16{\pi}^2r^6{E_0}}{9k} \left(2e+re'+re\frac{E'_0}{E_0}\right)\\\nonumber &=&\frac{16\pi}{9}(5p_{r0}-2p_{\perp0})r^3 +\frac{4\pi}{3}\left(7\int^{r}_{r_{{\Sigma}^{(i)}}}{\mu_0}r^2dr-{\mu_0}r^3\right)\\\label{92} &+&\frac{16\pi^2}{3}\left[7\int^{r}_{r_{{\Sigma}^{(i)}}}({E_0E'_0}r^3+2E^2_0r^2)dr -\frac{1}{3}(13{E^2_0}r^3-r^4E_0E'_0)\right].\end{aligned}$$ For the instability of expansionfree fluid, we require that each term in Eq.(\[92\]) must be positive. For this purpose, the positivity of the first term of Eq.(\[92\]) leads to $p_{r0}>(\frac{2}{5})p_{\perp0}$ and the positivity of the last two terms is determined by considering the radial profile of the energy density and electromagnetic field in the form $\mu_0={\gamma}r^m$ and $E_0={\delta}r^n$ respectively. Here $\gamma,~\delta$ are the positive constants and $m,~n$ are constants defined in the interval $(-\infty,\infty)$. Using these solutions, the last two terms of Eq.(\[92\]) will be positive for $m\neq-3$ and $n\neq-2,2$ if $$\begin{aligned} \label{93} r>r_{{\Sigma}^{(i)}}\left(\frac{7}{4-m}\right)^{\frac{1}{m+3}},\end{aligned}$$ and $$\begin{aligned} \label{94} r>r_{{\Sigma}^{(i)}}\left(\frac{14+7n}{2-n}\right)^{\frac{1}{2n+3}}.\end{aligned}$$ These two equations define the range of instability. Thus instability of the system is subject to the consistency of Eqs.(\[93\]) and (\[94\]). For $m=-3$, we obtain from Eq.(\[93\]) $$\begin{aligned} \label{95} \frac{8\pi{\gamma}}{6}\left[7\log \left(\frac{r}{r_{{\Sigma}^{(i)}}}\right)-1\right],\end{aligned}$$ which defines the instability region for $r>r_{{\Sigma}^{(i)}}1.15$. For $n=-2,2$, the range of instability is not defined. Now we find the instability range of Eqs.(\[93\]) and (\[94\]) for the remaining values of $m$ and $n$. For this purpose, we consider the following two cases.\ \ Case (i) Here we take $m\leq0$ and $n\leq0$. For $m=0$, Eq.(\[93\]) gives $r>{r_{{\Sigma}^{(i)}}}1.20$ which shows that the region of instability decreases from $1.20$ to $1.15$ as $m$ varies from $0$ to $-3$. When $m$ varies from $-3$ to $-\infty$, the unstable region is swept out by the whole fluid, i.e., $r>{r_{{\Sigma}^{(i)}}}$. For $n=0$, Eq.(\[94\]) yields $r>{r_{{\Sigma}^{(i)}}}1.91$. This indicates that instability range varies from $1.91$ to $2.33$ as $n$ varies from $0$ to $-1$, i.e., decreases for this region and vanishes for $n\leq-2$.\ \ Case (ii) When $m\geq0$ and $n\geq0$, we see from Eq.(\[93\]) that the range of instability decreases as $m$ increases and vanishes for $m\geq4$, while for $n\geq0$, the range of instability in Eq.(\[94\]) varies from $1.91$ to $1.83$ as $n$ varies from $0$ to $1$, i.e., the range of instability increases as $n$ approaches to $1$ and vanishes for $n\geq2$. In other words, electromagnetic field reduces the instability region in the interval $(-2,2)$.\ It is mentioned here that, for the pN approximation, the physical behavior of the dynamical equation is essentially the same by considering the relativistic effects upto first order $$\begin{aligned} \nonumber &&24\pi{\mu_0}+8\pi{\mid{p'_{r0}}\mid}r+2\left(\alpha_{\Sigma} -\frac{21}{r^2}\right)\frac{m_0}{r}\\\nonumber&+&16\pi{\mid{p'_{r0}}\mid}m_0 +8\pi{\alpha_{\Sigma}}p_{r0}r^2+6\left(\alpha_{\Sigma} -\frac{5}{r^2}\right)\left(\frac{m_0}{r}\right)^2\\\nonumber&+&\frac{32{\pi}^2r^3{E_0}}{k} \left(2e+re'+re\frac{E'_0}{E_0}\right)-24\pi{\mu_0}\left(\frac{m_0}{r} -\alpha_{\Sigma}m^2_0\right)\\\nonumber&+&64{\pi^2}E^2_0\left(1-\frac{m_0}{r}\right) +\frac{64{\pi^2}E^2_0}{r}\left(5+\alpha_{\Sigma}r-7\frac{m_0}{r}\right)\\\nonumber &=&32\pi(5p_{r0}-2p_{\perp0})+16{\pi}(2p_{r0}-p_{\perp0})\frac{m_0}{r} +128{\pi^3}{\mu_0}E^2_0r(1-\alpha_{\Sigma}r^2)\\\label{98} &+&\frac{16\pi^2}{k}r^4E_0{\mu_0} +32{\pi^2}E_0E'_0\left(9-\frac{2m_0}{r}\right).\end{aligned}$$ Concluding Remarks ================== This paper investigates the effects of electromagnetic field on the instability range of expansionfree fluid at Newtonian and post Newtonian regimes. In general, the instability range is defined by the adiabatic index $\Gamma$ which measures the compressibility of the fluid. On the other hand, in our case, the instability range depends upon the radial profile of the energy density, electromagnetic field and the local anisotropy of pressure at N approximation, but independent of the adiabatic index $\Gamma$. This means that the stiffness of the fluid at Newtonian and post Newtonian regimes does not play any role at the instability range. It is interesting to note that independence of $\Gamma$ requires the expansionfree collapse (without compression of the fluid). This shows the importance of local anisotropy, inhomogeneity energy density and electromagnetic field in the structure formation as well as evolution of self-gravitating objects. We see from Eqs.(\[93\]) and (\[94\]) that in the absence of electromagnetic field the region of instability is taken to the whole fluid. However, with the inclusion of electromagnetic field, the region of instability decreases. Thus the system is unstable in the interval $(-2,2)$ and stable for the remaining values of $n$. Also, Eqs.(\[93\]) and (\[94\]) define the instability range of the cavity associated with the expansionfree fluid. We would like to mention here that the unstable range will be customized differently for different parts of the sphere as the energy density and electromagnetic field are defined by the radial profile. [Acknowledgments]{} We would like to thank the Higher Education Commission, Islamabad, Pakistan, for its financial support through the [Indigenous Ph.D. 5000 Fellowship Program Batch-VII]{}. One of us (MA) would like to thank University of Education, Lahore for the study leave. [40]{} Chandrasekhar, S.: Astrophys. J. 140(1964)417. Skripkin, V.A.: Soviet Physics-Doklady. 135(1960)1183. Eardley, D.M. and Smarr, L.: Phys. Rev. D19(1979)2239. Herrera, L. and Santos, N.O.: Phys. Rep. 286(1997)53. Herrera, L., Di Prisco, A., Martin, J., Ospino, J., Santos, N.O. and Troconis, O.: Phys. Rev. D69(2004)084026. Ivanov, B.: Int. J. Theor. Phys. 49(2010)1236. Herrera, L., Le Denmat, G. and Santos, N.O.: Phys. Rev. D79(2009)087505. Herrera, L., Le Denmat, G. and Santos, N.O.: Class. Quantum Grav. 27(2010)135017. Rosseland, S.: Mon. Not. R. Astron. Soc. 84(1924)720. de la Cruz, V. and Israel, W.: Nuovo Cimento A51 (1967)744. Bekenstein, J.: Phys. Rev. D4(1971)2185. Olson, E. and Bailyn, M.: Phys. Rev. D13(1976)2204. Mashhoon, B. and Partovi, M.: Phys. Rev. D20(1979)2455. Ghezzi, C.: Phys. Rev. D72(2005)104017. Di Prisco, A., Herrera, L., Denmat, G.Le., MacCallum, M.A.H. and Santos, N.O.: Phys. Rev. D76(2007)064017. Thirukkanesh, S. and Maharaj, S.D.: Math. Meth. Appl. Sci. 32(2009)684. Sharif, M. and Abbas, G.: Mod. Phys. Lett. A24(2009)2551. Sharif, M. and Sundas, F.: Gen. Relativ. Gravit. 43(2011)127. Herrera, L., Santos, N.O. and Le Denmat, G.: Mon. Not. R. Astron. Soc. 237(1989)257. Chan, R., Kichenassamy, S., Le Denmat, G. and Santos, N.O.: Mon. Not. R. Astron. Soc. 239(1989)91. Chan, R., Herrera, L. and Santos, N.O.: Mon. Not. R. Astron. Soc. 265(1993)533. Chan, R., Herrera, L. and Santos, N.O.: Mon. Not. R. Astron. Soc. 267(1994)637. Chan, R.: Mon. Not. R. Astron. Soc. 316(2000)588. Horvat, D., Ilijic, S. and Marunovic, A.: Class. Quantum Grav. 28(2011)25009. Herrera, L., Santos, N.O. and Le Denmat, G.: Gen. Relativ. Gravit. 44(2012)1143. Sharif, M. and Kausar, H.R.: JCAP 07(2011)022. Darmois, G.: Memorial des Sciences Mathematiques (Gautheir-Villars, 1927) Fasc. 25. Misner, C.W. and Sharp, D.: Phys. Rev. 136(1964)B571. Herrera, L., Santos, N.O., and Wang, A.: Phys. Rev. D78(2008)084026. [^1]: msharif.math@pu.edu.pk [^2]: azammath@gmail.com
[APPENDIX]{} Outer Warp + Flat Extension =========================== Adding $\alpha$ Dependence to Outer Warp ---------------------------------------- TB96 describe how to calculate the SED for an outer warp seen at various inclination angles. Their general method for calculating the SED also includes a dependence on azimuthal viewing angle, although their detailed treatment of various occultation effects (the star blocking the far side of the disk, the disk blocking the star, etc.) does not include this dependence. Since these disks are non-axissymetric the SED can depend substantially on the azimuthal viewing angle, $\alpha$, of the observer. In this section we describe how we have added this dependence into the equations of TB96. We do not include a detailed description of the derivation of the equations, but merely state most of them and some of the geometric logic behind their modification. The first modification is to equation (6) of TB96, which describes the calculation of the flux from the disk based on the temperature of the disk. This equation assumes that the disk is viewed along $\alpha=0$ so that the upper disk from 0 to $\pi/2$ looks the same as the upper disk from $3\pi/2$ to $2\pi$. When $\alpha\neq0$ this symmetry is broken and the individual components of the disk must be considered. While the temperature of a concave, or convex, piece of the disk does not change with viewing angle, the orientation of each of the concave, or convex pieces, changes and must be treated separately. For the outer warp this becomes: $$\begin{aligned} \textstyle F_{\nu,{\bf u}} = \int^{R_{disk}}_{r_{min}}\left[\int_0^{\pi/2}B(T_{concave})f_{up}+\int_0^{\pi/2}B(T_{convex})f_{low}+\int_{\pi/2}^{\pi}B(T_{concave})f_{low}+\int_{\pi/2}^{\pi}B(T_{convex})f_{up}\right]\nonumber\\ \textstyle +\int^{R_{disk}}_{r_{min}}\left[\int_{\pi}^{3\pi/2}B(T_{convex})f_{up}+\int_{\pi}^{3\pi/2}B(T_{concave})f_{low}+\int_{3\pi/2}^{2\pi}B(T_{concave})f_{up}+\int_{3\pi/2}^{2\pi}B(T_{convex})f_{low}\right]\end{aligned}$$ The next change comes in the appendix to the functional form of the parameter $C$. The function $C$ is used to define the points that are along the line of sight with the star. If the line intersects the star then we need to worry about whether the disk blocks the star or the star blocks the disk. If this line does not intersect the star then the disk cannot block the star and the star cannot block the disk. The definition of $C$ changes from $C=r\sin\theta$ to $C=r\sin (\theta-\alpha )$. Also the radial part of the deformation used in equation (A6) of TB96 is taken to be $H(r)=gR_{disk}\left(\frac{r}{R_{disk}}\right)^n\cos\alpha$. For $\alpha=\pi/2$ the disk along the line of sight is flat and the radial part of the height will remain at zero, while along $\alpha=\pi$ the disk curves below the midplane as expected. The final change comes when calculating the stellar flux. In equation (A12) of TB96 we take $h(r,\theta)$ to be $h(r,\alpha)$ since this represents the part of the disk that will block the star. As the azimuthal angle increases the disk blocks less of the star because the height of the disk is smaller. We make more changes to how the stellar flux is calculated, which are described below, but when it comes to the occultation of the star by the disk this is the only change. In the end we are able to run our models from $0<i<\pi$ and $-\pi/2<\alpha<\pi/2$. Due to the symmetry of the disk this covers all possible viewing angles allowing us to accurately model the precession of the warp, as well as observe the warp from an arbitrary angle. Flat extension of Outer Warp ---------------------------- We have taken the outer warp model and added a flat extension beyond it in order to treat disks where the warp is not at the outer edge of the disk. The warp will shadow the outer disk, changing its temperature structure. For simplicity we assume that the outer disk is a flat blackbody. The temperature can be derived using the same formula as with the warped disk (Equation 6), but with different definitions for the integration boundaries. Half of the flat extension will be shadowed while half will not be shadowed. For the side of the flat extension that is beyond the part of the warp the goes below the midplane there is no shadowing of the disk and the integration ranges over: $$\begin{aligned} \varepsilon_{min}=0\nonumber\\ \varepsilon_{max}=\pi/2\nonumber\\ \\ \delta_{min}=0\nonumber\\ \delta_{max}=\arcsin(R_/d)\nonumber\\ d^2=r^2+h^2\nonumber\\\end{aligned}$$ For the part of the flat extension that lies behind the warp that stretches above the midplane the definition of $\delta_{min}$ changes. In this case $\delta_{min}$ is set by the angle between the warp and the point $P(r,\theta)$ in the disk. $$\delta_{min}=\arctan\left(\frac{gR_{warp}\cos(\theta)}{(r-R_{warp})}\right)$$ This takes into account shadowing of the flat extension to the disk due to the warp. Once the temperature structure has been determined the flux can be derived using the equation for the flux from a disk (Equation 5). Inner Warp ========== Temperature Profile of the Inner Warp ------------------------------------- In this section we describe the method for calculating the SED of a disk with an inner warp. In the text we laid out the basic equations from TB96 that are needed to calculate the temperature structure. As mentioned in the text the essential difference between the inner warp and outer warp comes in calculating $\delta_{max},\delta_{min},\varepsilon_{min},\varepsilon_{max}$ for each point $P(r,\theta)$, which are used in equation 6. From here the disk is split into two sides that are treated separately. The convex side is the side of the disk that faces the star on the inner edge and receives the most direct heating from the star. For this side, the integration ranges over: $$\begin{aligned} \varepsilon_{min}=0\nonumber\\ \varepsilon_{max}=\pi/2\nonumber\\ \\ \delta_{min}=-\arctan(\partial h/\partial r)\nonumber\\ \delta_{max}=\arcsin(R_/d)\nonumber\\ d^2=r^2+h^2\nonumber\\\end{aligned}$$ Figure \[del\_vex\] demonstrates the limits on $\delta$ for the convex side of the disk. The definition of $\delta_{min}$, demonstrated in figure \[delblock\_vex\], comes from the inner disk blocking light from the top of the star. The inner disk will limit the field of view of the point $P(r,\theta)$ as it looks toward the star. Traveling out from the star, less of the star will be seen by the disk because the shallower slope of the disk will cause more of the star to be blocked. In the limit of a flat disk far from the star $\delta_{min}$ approaches zero and the disk can only see half of the star. If the point $P(r,\theta)$ on the disk is close enough to the star then the disk can see all of the star and $\delta_{min}=-\delta_{max}$. The limits on $\varepsilon$ assume that the scale height of the disk does not change across the face of the disk, which will be an accurate approximation far from the star. The concave side of the disk is the side that does not directly face the star. Since it does not face the star much of the inner parts of the disk will be blocked by the warp and will only be heated by viscous dissipation. The condition for the point $P(r,\theta)$ on the concave part of the disk to see any of the star is: $$\frac{h(r_{min},\theta)}{r-r_{min}}<\frac{R_}{r}$$ If the point $P(r,\theta)$ meets this condition then this point can see some of the star and $\delta_{min}$ becomes (fig \[delmin\_cave\]) $$\delta_{min}=\arctan(\frac{h(r_{min},\theta)}{r-r_{min}})$$ The rest of the limits stay the same. In the limit of a perfectly flat disk $\delta_{min}$ approaches 0 and the point $P(r,\theta)$ is irradiated by only half of the star. For a large warp the only heating by this side of the disk will be from viscous dissipation because the warp will block the star over most of the disk. Calculating the SED for an Inner Warp ------------------------------------- In this section we describe the procedure for converting the temperature structure into a spectral energy distribution (SED). From TB96, the flux emitted by the disk is $$F_{\nu,{\bf u}}= \int\int_{disk surface} B_{\nu}(T(r,\theta))dS{\bf n_d}\cdot{\bf u}$$ In this case [u]{} is the vector along the line of sight to the observer from the center of the star and ${\bf n_d}$ is the normal to the disk at the point $P(r,\theta)$. The vector ${\bf u}$ can be defined in terms of the azimuthal and polar angles to the line of sight, $\alpha$ and $i$ respectively. The area of the disk along the line of sight is given by $$\textstyle dS {\bf n_d}\cdot{\bf u} = r\left[\left(\frac1{r}\frac{\partial h}{\partial \theta}\sin\theta-\frac{\partial h}{\partial r}\cos\theta\right)\cos\alpha\sin i-\left(\frac1{r}\frac{\partial h}{\partial\theta}\cos\theta+\frac{\partial h}{\partial r}\sin\theta\right)\sin\alpha\sin i+\cos i\right]drd\theta$$ The angle $\alpha$ ranges from $-\pi$/2 to $\pi$/2 while the inclination $i$ ranges from 0 to $\pi$. This covers all possible viewing angles of the disk, since the symmetry of the disk makes some viewing angles redundant. Splitting up the equation for the flux from the disk helps to make the problem simpler to understand and more tractable. It also fits with the fact that we do not need to calculate the temperature structure of the entire disk. The symmetry of the disk allows us the calculate the temperature of the convex and concave side from $0<\theta<\pi/2$ and then apply this temperature profile to the rest of the disk. The integral is split into eight parts: $$\begin{aligned} \textstyle F_{\nu,{\bf u}} = \int^{R_{disk}}_{r_{min}}\left[\int_0^{\pi/2}B(T_{concave})f_{up}+\int_0^{\pi/2}B(T_{convex})f_{low}+\int_{\pi/2}^{\pi}B(T_{concave})f_{low}+\int_{\pi/2}^{\pi}B(T_{convex})f_{up}\right]\nonumber\\ \textstyle +\int^{R_{disk}}_{r_{min}}\left[\int_{\pi}^{3\pi/2}B(T_{convex})f_{up}+\int_{\pi}^{3\pi/2}B(T_{concave})f_{low}+\int_{3\pi/2}^{2\pi}B(T_{concave})f_{up}+\int_{3\pi/2}^{2\pi}B(T_{convex})f_{low}\right]\label{eqn_flux}\end{aligned}$$ where $f(r,\theta)=dS {\bf n_d}\cdot{\bf u}p(r,\theta)$, and $p(r,\theta)$ is a binary function used to determine if the point $P(r,\theta)$ is visible to the observer. The integration is done over both the upper and lower sides of the disk in order to account for inclination angles greater than $90^{\circ}$ where the lower half of the disk is visible. If the inclination is 0 then the observer is face on to the upper half of the disk, which has both a concave and convex side. If the inclination is $180^{\circ}$ then the observer is face on to the lower half of the disk, which includes both a concave and a convex side. Treating each quarter of the disk separately allows us to use the symmetry of the temperature profile but still treat general azimuthal viewing angles. ### Calculating the value of p The above description sets out the basics for how to calculate the temperature structure and SED for a warped inner disk. Most of this is derived from TB96, which treated these situations generally enough to apply to any type of warp. The main differences between this inner warp and the outer warp from TB96 comes from the calculation of $p(r,\theta)$. This section describes the conditions used to calculate $p(r,\theta)$ for the particular warp used here. The first condition is that the observer is facing the point $P(r,\theta)$. For inclinations less than $90^{\circ}$ the observer will see mostly the upper half of the disk, while at inclinations greater than $90^{\circ}$ the observer will see mostly the lower half of the disk. There are select inclinations close to edge on where at inclinations less than $90^{\circ}$ some of the lower disk can be seen. For example, if figure \[delblock\_vex\] had an observer in the upper left viewing the disk close to edge on they would be able to see some of the lower convex side that is illustrated in the figure. In general it can be determined if the observer is facing the point $P(r,\theta)$ based on the dot product ${\bf n_d}\cdot{\bf u}$. The normal is defined as extending on the upper side of the disk and the dot product will be greater than zero if ${\bf n_d}$ and ${\bf u}$ lie along the same direction. Therefore, the upper part of the disk can be seen if the dot product is greater than 0 while the lower part of the disk can be seen when the dot product is negative. Now we determine if the point $P(r,\theta)$ is blocked by either the star or the disk. First we consider whether the star blocks the far side of the disk. This applies for inclinations less than $90^{\circ}$ where the star may block part of the upper convex side, as is demonstrated in figure \[inclim\]. The limit at which this condition applies is given by $$\tan i_{lim}=\frac{r_{min}-R_\cos i_{lim}}{gr_{min}\cos\alpha+R_\sin i_{lim}}$$ [cccc]{} 0.005& 59.7 & 78.2 & 83.9\ 0.01& 59.4 & 77.9 & 83.7\ 0.03& 59.3 & 76.7 & 82.5\ 0.05& 57.2 & 75.6 & 81.4\ 0.07& 56.1 & 74.5 & 80.3\ 0.1& 54.5 & 72.8 & 78.6\ 0.3& 44.7 & 62.3 & 67.8\ 0.5& 36.7 & 53.1 & 58.3\ If the inclination is greater than $i_{lim}$ then part of the disk is blocked by the star. In this case we can use the discussion of TB96 to determine if the point $P(r,\theta)$ is blocked by the star (figure \[disk\_block\] and table \[ilim\]). Defining $C=r\sin(\theta-\alpha)$ the only time the star can block the disk is when $C<R_$ otherwise $p(r,\theta)=1$. If $C<R_$ then $p(r,\theta)=1$ only if $r\cos\theta<r_D\cos\theta_D$, where $\theta_D=\pi-\arcsin(C/r_D)$ and $r_D$ is the positive root of the following equation: $$\left(-H(r_D)\sqrt{1-\left(\frac{C}{r_D}\right)^2}-z_M\right)\sin i+\left(r_D\sqrt{1-\left(\frac{C}{r_D}\right)^2}+x_M\right)\cos i=0$$ where $H(r)=gr_{min}\left(\frac r{r_{min}}\right)^{-n}\cos\alpha$ and M is the point on the northern hemisphere of the star in the plane (P,[u]{},z) such that [u]{} is tangent to the star at this point: $$\begin{aligned} x_M=-\sqrt{R_^2-C^2}\cos i\\ y_M=C\\ z_M=\sqrt{R_^2-C^2}\sin i\end{aligned}$$ Now consider inclinations greater than $90^{\circ}$. In this case the disk can still be blocked by the star if the warp is small enough and the inclination angle is close enough to 90 degrees (figure \[disk\_block2\]). This affects both the upper and lower part of the disk, but only from $\theta\in[\alpha+\pi/2,\alpha+3\pi/2]$. The condition for the point P to be hidden by the star is: $$\|h(r,\theta)\|<\frac{R_\cos(i-\pi/2)\tan(\pi-i)-r}{tan(\pi-i)}$$ As above this only applies when $C=R_\sin(\theta-\alpha)<R_$. Another possibility to consider when the inclination is greater than $90^{\circ}$ is that the warp on the lower concave side is steep enough that is blocks part of this side of the disk (figure \[disk\_block3\]). This condition only applies to parts of the lower concave side, from $\theta\in[\alpha+\pi/2,\alpha+3\pi/2]$. The condition is that $\gamma_1<\gamma_2$ where $\gamma_1=i-\pi/2$ and $$\gamma_2=\arcsin\left(\frac{h_m-h(r,\theta)}{PM}\right)$$ In this case M is the highest point on the disk on the line of sight to point P. For $r\sin(\pi-\theta)>r_{min}$ the disk does block itself, but for $r\sin(\pi-\theta)<r_{min}$ we have $r_m=r_{min}$ and $\theta_m=\arcsin(r\sin(\theta-\alpha)/r_m)$. The quantity PM is the distance between the point P and the point M ($\sqrt{(h_m-h)^2+(r_m-r)^2}$). Once all of the conditions have been considered the flux from the disk can be calculated using equation \[eqn\_flux\]. These conditions would apply to any type of warp whose maximum height above the midplane occurs at the inner edge of the disk, as opposed to at the outer edge of the disk, regardless of the exact functional form of the warp (ie. power law vs. exponential). Spiral Wave =========== Temperature Profile of Spiral Wave ---------------------------------- The third type of disk that we attempt to model contains a spiral wave. As with the warped disks we follow the derivation of TB96 to derive the temperature structure and SED for this disk. The derivation for the temperature structure is very similar to that of the inner warp, only with slightly different definitions of the boundaries. For the part of the disk inside the wave, the disk is not blocked by the wave but the amount of the star seen can change. For points far from the wave, the disk is like a flat disk and $\delta_{min}=0$. For points on the wave, as it rises above the midplane, more of the lower half of the star will become visible. How much of the lower half of the star is visible depends on the location and height of the point on the wave. In this case the lower limit on $\delta$ is: $$\delta_{min}=-\arctan(h/(r-r_{min}))\\$$ This limit will continue to increase until the point on the wave can see the entire star and then $\delta_{min}=-\delta_{max}$. The other limits stay the same as in the previous models: $$\begin{aligned} \varepsilon_{min}=0\nonumber\\ \varepsilon_{max}=\pi/2\nonumber\\ \\ \delta_{max}=\arcsin(R_/d)\nonumber\\ d^2=r^2+h^2\nonumber\\\end{aligned}$$ For the parts of the disk behind the wave, some of the star may be obscured. In this case: $$\begin{aligned} \delta_{min}=\arctan(h_{sw}/(r-r_{sw}))\\ h_{sw}=gr_{min-sw}(1-m\theta/2\pi)\nonumber\\ r_{sw}=r_{min-sw}(1+n\theta)\nonumber\\\end{aligned}$$ This is similar to the concave side of the inner warp, where the warp can obscure part of the star. The only difference is that the maximum height of the disk does not occur at the inner edge of the disk, but instead occurs at the location of the spiral wave. When $\delta_{min}>\delta_{max}$ then the entire star is blocked and that point on the disk is only heated by viscous dissipation. With these definitions and equation (6) we can calculate the temperature of the disk. Calculating SED of Spiral Wave ------------------------------ The flux from the disk is given by: $$F_{\nu,{\bf u}}=\int^{R_{disk}}_{r_{min}}\int^{2\pi}_{0}B(T_{disk})f_{up}$$ There is no symmetry in the disk that allows us to split the disk into different parts, as with the concave and convex pieces of the inner warp. We also only consider inclinations less than $90^{\circ}$, where we only see the upper disk, since the lower disk will look the same as the upper disk. The one occultation effect we include is the blocking of the disk by the wave. For $\alpha-\pi/2<\theta<\alpha+\pi/2$, the near side of the disk, the wave can block the part of the disk that is at smaller radius than the wave. For $\alpha+\pi/2<\theta<\alpha+3\pi/2$, the side of the disk on the other side of the star from the observer, the outer disk can be blocked by the wave. This effect can become important for modest inclinations, given the typical wave heights we consider here. To determine if a point on the disk is blocked by the wave we first need to determine where the line of sight intersects the wave. This is illustrated in figure \[wave\_block1\] and is given by: $$x = r\sin\theta = r_{min-sw}(1+n\theta_M)\sin(\theta_M)$$ Here $\theta_m$ is the azimuthal coordinate of the point where the wave intersects the line of sight. We assume that the point M lies at the peak of the spiral wave. This is only an approximation, although the narrowness of the wave make it an accurate one. The angle between the line connecting the points P and M and the midplane is $\gamma$ (Fig. \[wave\_block\]). When $\gamma>\pi/2-i$ then point $P(r,\theta)$ is blocked. $$\tan\gamma=\left(\frac{h_M-h_P}{r_M-r_P}\right)$$ We ignore occultation effects due to the star blocking the disk, which we did consider in the inner and outer warp model. Based on our experience with the warped disks and the typical dimensions of the disk, these are negligible effects that will only play a role very close to edge on. We also do not consider situations where the wave on the near side of the disk can block the far side of the disk. The exclusion of these two effects prevents us from considering the spiral wave at inclinations very close to $90^{\circ}$. Stellar Flux ============ Next we consider the flux coming from the star. We follow a similar procedure as above where the equation for the stellar flux is modulated by a binary function ($\varepsilon(\phi,\psi)$) which equals 1 when that part of the star is not blocked by the disk and it equals zero when the star is blocked by the disk. The angles $\phi,\psi$, shown in figure \[star\], are the azimuthal and polar angles of a point on the surface of the star relative to the center of the star and the z axis (the same z axis as for the disk). The x-axis of this coordinate system is in the same direction as the line of sight, and will differ from the x-axis of the disk by the angle $\alpha$. The flux from the star is: $$F_=B_{\nu}\int\int_{surface}\varepsilon(\phi,\psi)d{\bf A}$$ To determine the surface over which we integrate, we need to know the points of the star that are seen by the observer (ie. which side of the star is facing the observer). These points will be those that have ${\bf u}\cdot d{\bf A}\geq0$ where $$\begin{aligned} {\bf u}=\sin i\hat{x}+\cos i\hat{z}\\ d{\bf A}=R_^2\sin\psi d\psi d\phi(\cos\phi\sin\psi\hat{x}+\sin\phi\cos\psi\hat{y}+\cos\psi\hat{z})\end{aligned}$$ The evaluation of $\varepsilon(\phi,\psi)$ will depend on the type of warp/wave and the orientation of the observer. First consider inclinations less than $90^{\circ}$. In this case the warp/wave streching above the midplane may block some of the star. The entire star will be blocked if the following condition is met: $$h(r,\alpha)-r\tan(\pi/2-i)>R_$$ where r is location of the peak of the vertical disturbance and $h$ is the maximum height of the warp or wave at the angle $\alpha$. The exact value of $r$ and $h$ will depend on whether we are considering the outer warp, inner warp, or spiral wave (ie. $r=r_{min}$, $h(r,\alpha)=gr_{min}\cos(\alpha)$ for the inner warp). This condition is illustrated in figure \[star\_block\] for the inner warp, which is the disk that is the most likely to occult the star. None of the star will be blocked if the inclination is less than $i_{lim}$ (discussed earlier). A generic version of the equation for $i_{lim}$ that can be applied to all of the disks is: $$\tan i_{lim}=\frac{r-R_\cos i_{lim}}{h(r,\alpha)+R_\sin i_{lim}}$$ When the inclination falls between these two limits only a fraction of the star is blocked. We can use the discussion of TB96 section A2.2 to determine if a point of the star’s surface is hidden by the star The point Q is a point on the surface of the star that intersects the line of sight and the upper edge of the disk. If the point $N(\phi,\psi)$ lies above Q then the observer can see this part of the star, otherwise it is hidden and $\varepsilon(\phi,\psi)=0$. The vertical coordinate of Q, $z_Q$, is the greatest root of the following equation: $$(1+\tan^2 i)z_Q^2-2\tan i(h\tan i-r\cos\alpha)z_Q+(h\tan i-r\cos\alpha)^2-R_^2+R_^2\sin^2\psi\cos^2\phi=0$$ If $z_N\geq z_Q$ then $\varepsilon(\phi,\psi)=1$ otherwise $\varepsilon(\phi,\psi)=0$ with $z_N$ being given by: $$z_N=-R_\sin\psi\sin\phi\sin i+R_\cos\psi\cos i$$ For the inner and outer warp we consider inclinations greater than $90^{\circ}$ where the disk can still block part of the star, although it is less likely because the disk curves away from the observer. This is illustrated in figure \[star\_block2\] for the inner warp, but can also apply to the outer warp in the limit that $h$ goes to zero. The point at which the line of sight is perpendicular to the normal of the disk sets a limit to the distance z above the midplane that an observer can see. If this distance is less than the radius of the star then some of the star is blocked by the disk. The point D at which the the line of sight is perpendicular to the disk occurs when ${\bf u}\cdot{\bf n}=0$ (figure \[star\_block2\]). If this condition is met at a radius $r_D$ then a point $N(\phi,\psi)$ on the stellar surface will be blocked if $$R_*\cos\psi>h(r_D,\alpha)+r_D\tan(i-\pi/2)$$ If there is no point at which ${\bf u}\cdot{\bf n}=0$ then $r_D=r_{min}$ and the same condition for being able to see the star is used. When this condition is met $\varepsilon(\phi,\psi)=0$, otherwise $\varepsilon(\phi,\psi)=1$. All of these different occultations are combined to determine $\varepsilon(\phi,\psi)$. The flux is determined by integrating over the entire surface and added to the flux from the disk to create the observed SED.
--- abstract: 'We propose $S$=+1 baryon interpolating operators, which are based on an exotic description of the antidecuplet baryon like diquark-diquark-antiquark. By using one of the new operators, the mass spectrum of the spin-1/2 pentaquark states is calculated in quenched lattice QCD at $\beta=6/g^2=6.2$ on a $32^3\times48$ lattice. It is found that the $J^P$ assignment of the lowest $\Theta(uudd \bar s)$ state is most likely $(1/2)^-$. We also calculate the mass of the charm analog of the $\Theta$ and find that the $\Theta_c(uudd \bar c)$ state lies much higher than the $DN$ threshold, in contrast to several model predictions.' author: - Shoichi Sasaki title: Lattice study of exotic $S=+1$ baryon --- Recently, LEPS collaboration at Spring-8 has observed a very narrow resonance $\Theta^{+}(1540)$ in the $K^{-}$ missing-mass spectrum of the $\gamma n \rightarrow nK^{+}K^{-}$ reaction on $^{12}C$ [@Nakano:bh]. A remarkable observation is its strangeness is $S$=+1, which means that the observed resonance must contain a strange antiquark. Thus, the $\Theta^{+}(1540)$ should be an exotic baryon state with the minimal quark content $uudd \bar s$. This discovery is subsequently confirmed in different reactions by several other collaborations [@Exp]. It should be noted, however, that the experimental evidence for the $\Theta^{+}(1540)$ is not very solid yet since there are a similar number of negative results to be reported [@Hicks:2004vd]. Theoretically the existence of such a state was predicted long time ago by the Skyrme model [@Skyrmion]. However, the prediction closest in mass and width with the experiments was made by Diakonov, Petrov and Polyakov using a chiral-soliton model [@Diakonov:1997mm]. They predicted that it should be a narrow resonance and stressed that it can be detected by experiment because of its narrow width. In a general group theoretical argument with flavor $SU(3)$, $S$=+1 pentaquark state should be a member of antidecuplet or higher dimensional representation such as 27-plet or 35-plet. Both the Skyrme model and the chiral-soliton model predict that the lowest $S$=+1 state appears in the antidecuplet, $I$=0, and its spin and parity should be $(1/2)^{+}$. Experimentally, spin, parity and isospin of the $\Theta^{+}(1540)$ are not determined yet. After the discovery of the $\Theta^{+}(1540)$, many model studies for the pentaquark state are made with different spin, parity and isospin. Lattice QCD in principle can determine these quantum numbers of the $\Theta^{+}(1540)$, independent of such arbitrary model assumptions or the experiments. We stress that there is substantial progress in lattice study of excited baryons recently [@Sasaki:2003xc]. Especially, the negative parity nucleon $N^(1535)$, which lies close to the $\Theta^{+}(1540)$, has become an established state in quenched lattice QCD . Here we report that quenched lattice QCD is capable of studying the $\Theta^{+}(1540)$ as well. Indeed, it is not so easy to deal with the $qqqq{\bar q}$ state rather than usual baryons ($qqq$) and mesons ($q \bar q$) in lattice QCD. The $qqqq{\bar q}$ state can be decomposed into a pair of color singlet states as $qqq$ and $q \bar q$, in other words, it can decay into two-hadron states even in the quenched approximation. For instance, one can start a study with a simple minded local operator for the $\Theta^{+}(1540)$, which is constructed from the product of a neutron operator and a $K^{+}$ operator such as $\Theta = \varepsilon_{abc}(d^{T}_{a}C\gamma_5 u_{b})d_{c}({\bar s}_{e}\gamma_5 u_{e})$. The two-point correlation function composed of this operator, in general, couples not only to the $\Theta$ state (single hadron) but also to the two-hadron states such as an interacting $KN$ system . Even worse, when the mass of the $qqqq{\bar q}$ state is higher than the threshold of the hadronic two-body system, the two-point function should be dominated by the two-hadron states. Thus, a specific operator with as little overlap with the hadronic two-body states as possible is desired in order to identify the signal of the pentaquark state in lattice QCD. ![image](EffMassK1506_Pos_v3.eps) ![image](EffMassK1506_Neg_v3.eps) For this purpose, we propose some local interpolating operators of antidecuplet baryons based on an exotic description as diquark-diquark-antiquark. There are basically two choices as ${\bar 3}_{c}\otimes{\bar 3}_{c}$ or ${\bar 3}_{c}\otimes 6_{c}$ to construct a color triplet diquark-diquark cluster . We adopt the former for a rather simple construction of diquark-diquark-antiquark. We first introduce the flavor antitriplet (${\bar 3}_{f}$) and color antitriplet (${\bar 3}_{c}$) diquark field $$\begin{aligned} \Phi_{\Gamma}^{i, a}(x)=\frac{1}{2}\varepsilon_{ijk} \varepsilon_{abc}q_{j,b}^{T}(x)C\Gamma q_{k,c}(x)\end{aligned}$$ where $C$ is the charge conjugation matrix, $abc$ the color indices, and $ijk$ the flavor indices. $\Gamma$ is any of the sixteen Dirac $\gamma$-matrices. Accounting for both color and flavor antisymmetries, possible $\Gamma$s are restricted within $1$, $\gamma_5$ and $\gamma_5\gamma_{\mu}$ which satisfy the relation $(C\Gamma)^T=-C\Gamma$. Otherwise, the above defined diquark operator is identically zero. Hence, three types of flavor ${\bar 3}_{f}$ and color ${\bar 3}_{c}$ diquark; scalar ($\gamma_5$), pseudoscalar ($1$) and vector ($\gamma_5\gamma_{\mu}$) diquarks are allowed [@Hadron2003]. The color singlet state can be constructed by the color antisymmetric parts of diquark-diquark $({\bar 3}_{c}\otimes{\bar 3}_{c})_{\rm antisym}={3}_{c}$ with an antiquark (${\bar 3}_{c}$). In terms of flavor, ${\bar 3}_{f}\otimes{\bar 3}_{f}\otimes {\bar 3}_{f}={1}_{f}\oplus 8_{f}\oplus 8_{f}\oplus \overline{10}_{f}$. Manifestly, in this description, the $S$=+1 state belongs to the flavor antidecuplet [@Hosaka:2003jv]. Automatically, the $S$=+1 state should have isospin zero. Then, the interpolating operator of the $\Theta(uudd{\bar s})$ is obtained as $$\begin{aligned} \Theta(x)=\varepsilon_{abc} \Phi_{\Gamma}^{s, a}(x) \Phi_{\Gamma '}^{s, b}(x)C{\bar s}^{T}_{c}(x)\end{aligned}$$ for $\Gamma \neq \Gamma '$. The form $C{\bar s}^{T}$ for the strange antiquark field is responsible for giving the proper transformation properties of the resulting pentaquark operator under parity and Lorentz transformations [@Sugiyama:2003zk]. Note that because of the color antisymmetry, the combination of the same types of diquark is not allowed. Consequently, we have three different types of exotic $S$=+1 baryon operators through the combination of two different types of diquarks, which have different spin-parity [@Hadron2003]: $$\begin{aligned} \label{eq:1stOP} \Theta^{1}_{+}&=& \varepsilon_{abc}\varepsilon_{aef}\varepsilon_{bgh} (u_{e}^{T}Cd_{f})(u_{g}^{T}C\gamma_{5}d_{h})C{\bar s}^{T}_{c} ,\\ \label{eq:2ndOP} \Theta^{2}_{+, \mu}&=& \varepsilon_{abc}\varepsilon_{aef}\varepsilon_{bgh} (u_{e}^{T}C\gamma_{5}d_{f})(u_{g}^{T}C\gamma_{5}\gamma_{\mu}d_{h})C{\bar s}^{T}_{c} ,\\ \label{eq:3rdOP} \Theta^{3}_{-, \mu}&=& \varepsilon_{abc}\varepsilon_{aef}\varepsilon_{bgh} (u_{e}^{T}Cd_{f})(u_{g}^{T}C\gamma_{5}\gamma_{\mu}d_{h})C{\bar s}^{T}_{c} \end{aligned}$$ where the subscript $``+(-)"$ refers to positive (negative) parity since these operators transform as ${\cal P}\Theta_{\pm}({\vec x},t){\cal P}^{\dag} =\pm\gamma_{4}\Theta_{\pm}(-{\vec x},t)$ (for $\mu = 1,2,3$) under parity. The first operator of Eq. (\[eq:1stOP\]) is proposed for QCD sum rules in a recent paper [@Sugiyama:2003zk] independently. In this description, the operator of exotic $\Xi_{3/2}$ ($ssdd {\bar u}$ or $uuss {\bar d}$) states, which are members of the antidecuplet, can be treated by interchanging $u$ and $s$ or $d$ and $s$ in the above operators. If a strange antiquark is simply replaced by a charm antiquark, the proposed pentaquark operators can be regarded as the anti-charmed analog of the isosinglet pentaquark state, $\Theta_c(uudd {\bar c}$). Recall that any of local type baryon operators can couple to both positive- and negative-parity states since the parity assignment of an operator is switched by multiplying the left hand side of the operator by $\gamma_5$. The desired parity state is obtained by choosing the appropriate projection operator, $1\pm \gamma_4$, on the two-point function $G(t)$ and direction of propagation in time. Details of the parity projection are described in Ref. [@Sasaki:2001nf]. We emphasize that the second and third operators, Eqs. (\[eq:2ndOP\]) and (\[eq:3rdOP\]), can couple to both spin-1/2 and spin-3/2 states. By using them, it is possible to study the spin-orbit partner of the spin-1/2 $\Theta$ state, whose presence contradicts the Skyrme picture of the $\Theta$ [@Glozman:2003sy]. However, we will not pursue this direction in this article. We utilize only the first operator of Eq. (\[eq:1stOP\]), which couples only to a spin-1/2 state. Under the assumption of the highly correlated diquarks, we simply omit a quark-exchange diagram between diquark pairs contributing to the full two-point function in the following numerical simulations. We generate quenched QCD configurations on a lattice $L^3\times T=32^3\times48$ with the standard single-plaquette Wilson action at $\beta=6/g^{2}=6.2$ ($a^{-1}=2.9$ GeV). The spatial lattice size corresponds to $La\approx 2.2 {\rm fm}$, which may be marginal for treating the ground state of baryons without large finite volume effect. Our results are analyzed on 240 configurations. The light-quark propagators are computed using the Wilson fermions at four values of the hopping parameter $\kappa=\{0.1520, 0.1506, 0.1497, 0.1489\}$, which cover the range $M_{\pi}/M_{\rho}=0.68-$0.90. $\kappa_{s}=0.1515$ and $\kappa_{c}=0.1360$ are reserved for the strange and charm masses, which are determined by approximately reproducing masses of $\phi(1020)$ and $J/\Psi(3097)$. We calculate a simple point-point quark propagator with a source location at $t_{\rm src}=6$. To perform precise parity projection, we construct forward propagating quarks by taking the appropriate linear combination of propagators with periodic and anti-periodic boundary conditions in the time direction. This procedure yields a forward in time propagation in the time slice range $0<t<T-t_{\rm src}$. ![ An example of the effective mass for the negative parity state of the charm analog $\Theta_c(uudd{\bar c})$ at $\kappa=0.1506$ for up and down quarks and the charm quark $\kappa_c=0.1360$. []{data-label="fig:ThetaC"}](EffMassK1506_CNeg_v3.eps) In this calculation, the strange (charm) quark mass is fixed at $\kappa_s$ ($\kappa_c$) and the up and down quark masses are varied from $M_{\pi}\approx 1.0$ GeV ($\kappa=0.1489$) to $M_{\pi}\approx 0.6$ GeV ($\kappa=0.1520$). Then, we perform the extrapolation to the chiral limit using five different $\kappa$ values. We first calculate the effective masses $M_{\rm eff}(t)=\ln\{G(t)/G(t+1)\}$ for both parity states of the spin-1/2 $\Theta(uudd{\bar s})$. For example, Figs. \[fig:Theta\] show effective masses for the positive parity channel and the negative parity channel at $\kappa=0.1506$ for up and down quarks with the fixed strange quark. Statistical uncertainties in both figures are estimated by a single elimination jack-knife method. In Fig. \[fig:Theta\] (a), the effective mass plot for the positive parity state shows a very short plateau albeit with large statistical errors. This plateau terminates at $t\approx 17$ and then the rather noisy signals appear after $t=18$ and become reduced around the $KN$ threshold. We remark that the positive parity $\Theta$ state can decay into the $KN$ state in a P-wave where the two hadrons should have a nonzero momentum. However, all momenta are quantized as ${\vec p}_{n}=2\pi{\vec n}/L$ on a system of finite volume. The $KN$ threshold is defined as the total energy of the non-interacting $KN$ state with the smallest nonzero momentum $\|{\vec p}_{\rm min}\|=2\pi/L$ in lattice units. Here, we stress the following two points. First, there is [no clear signal for the $KN$ state]{} to be observed in the effective mass plot. It means that our proposed interpolating operator couples weakly to the $KN$ scattering state. Secondly, our observed plateau in Fig. \[fig:Theta\] (a) is considerably higher than the $KN$ threshold. While the observed asymptotic state can be identified as a pentaquark (single hadron) state, our results seem to give no indication of the $\Theta^{+}(1540)$ state in the positive parity channel. In the negative parity channel, the gross feature is similar to the case of the positive parity. Fig. \[fig:Theta\] (b) shows that a clear plateau appears in the range $15\le t \le 20$. The relatively noisy signals appear around the S-wave $KN$ threshold after $t=21$ and continue toward the maximum time slice $t=T-t_{\rm src}$ for the forwarding propagation. The errors after $t=21$ are probably underestimated. The correlators for the heavier mass state in this euclidean time region usually have many orders of magnitude deviation and the distribution is non-Gaussian. Therefore, the signals after $t=21$ are inconclusive. We perform a covariant single exponential fit [^1] to the two-point function in the plateau region $15\le t \le 20$, where the respective $\chi^2$ is indeed most favorable. The estimated mass is clearly higher than the $KN$ threshold, which is evaluated as the total energy of the non-interacting $KN$ state with zero momentum. The excitation energy of the observed asymptotic state from the $KN$ threshold is roughly consistent with the experimental value. Although, without a finite volume analysis, it cannot be excluded that the observed plateau stems from only a mixture of the $KN$ scattering states; we may regard it as a pentaquark state with a mass close to the experimental value of the $\Theta^{+}(1540)$. As a strange antiquark is simply replaced by a charm antiquark, we can explore the anti-charmed pentaquark $\Theta_c$($uudd{\bar c}$) as well. A similar identification for the $\Theta_c$ state can be made in the negative parity channel. The effective mass plot (Fig. \[fig:ThetaC\]) shows that a plateau, which terminates at $t\approx21$, is much higher than the $DN$ threshold. The relatively noisy signals appear around this threshold after $t\approx21$. The observed asymptotic state is identified with a pentaquark (single hadron) state similarly. In Fig. \[fig:Theta5Q\] we show the mass spectrum of the $\Theta(uudd{\bar s})$ states with the positive parity (open squares) and the negative parity (open circles) as functions of the pion mass squared. Mass estimates are obtained from covariant single exponential fits in the appropriate fitting range. All fits have a confidence level larger than 0.3 and $\chi^{2}/N_{DF}<1.2$. It is evident that the lowest state of the isosinglet $S$=+1 baryons has [the negative parity]{}. We evaluate the mass of the $\Theta(uudd{\bar s})$ with both parities in the chiral limit. A simple linear fit for all five values in Fig. \[fig:Theta5Q\] yields $M_{\Theta(1/2^-)}$=0.62 (3) and $M_{\Theta(1/2^+)}$=1.00 (5) in lattice units. If we use the scale set by $r_0$ from Ref. [@Necco:2001xg], we obtain $M_{\Theta(1/2^-)}=1.84 (8)$ GeV and $M_{\Theta(1/2^+)}=2.94 (13)$ GeV. It is worth quoting other related hadron masses. The chiral extrapolated values for the kaon, the nucleon and the $N^$ state are $M_{K}=0.53 (1)$ GeV, $M_{N}=1.06 (2)$ GeV and $M_{N^}=1.76 (5)$ GeV in this calculation. Our obtained $\Theta(1/2^-)$ mass is slightly overestimated in comparison to the experimental value of the $\Theta^{+}(1540)$, but comparable to our observed $N^$ mass, which is also overestimated. Needless to say, the evaluated values should not be taken too seriously since they do not include any systematic errors. Such a precise quantitative prediction of hadron masses is not the purpose of the present paper. Rather, we emphasize that, our results strongly indicate the $J^P$ assignment of the $\Theta^{+}(1540)$ is most likely $(1/2)^-$. This conclusion is consistent with that of a recent lattice study [@Csikor:2003ng] (if one corrects the parity assignment of their operator [@Fodor]) and that of QCD sum rules approach [@Sugiyama:2003zk]. Results for the lowest-lying spin-1/2 $\Theta_c$ state, which has the negative parity, are also included in Fig. \[fig:Theta5Q\]. The $\Theta_c$ state lies much higher than the $DN$ threshold in contradiction with several model predictions . The chiral extrapolated value of the $\Theta_c$ mass is 3.45(7) GeV, which is about 500 MeV above the $DN$ threshold ($M_{D}$=1.89(1) GeV) in our calculation. This indicates that the anti-charmed pentaquark $\Theta_c$ is not to be expected as a bound state. ![ Masses of the spin-1/2 $\Theta(uudd{\bar s})$ states with both positive parity (open squares) and negative parity (open circles) as functions of pion mass squared in lattice units. The charm analog $\Theta_c(uudd{\bar c})$ state (open diamonds) is also plotted. Horizontal short bar represents the $KN$($DN$) threshold estimated by $M_{N}+M_{K}$($M_{N}+M_{D}$) in the chiral limit. []{data-label="fig:Theta5Q"}](FIG_ThetaSqMass_v3.eps) We have calculated the mass spectrum of the $S$=+1 exotic baryon, $\Theta(uudd{\bar s})$, and the charm analog $\Theta_c(uudd{\bar c})$ in quenched lattice QCD. To circumvent the contamination from hadronic two-body states, we formulated the antidecuplet baryon interpolating operators using an exotic description like diquark-diquark-antiquark. Our lattice simulations seem to give no indication of a pentaquark in the positive parity channel to be identified with the $\Theta^+(1540)$. In contrast the simulations in the negative parity channel can easily accommodate a pentaquark with a mass close to the experimental value. Although more detailed lattice study would be desirable to clarify the significance of this observation, the present lattice study favors spin-parity $(1/2)^-$ for the $\Theta^+(1540)$. We have also found that the lowest spin-1/2 $\Theta_c$ state, which has the negative parity, lies much higher than the $DN$ threshold, in contrast to several model predictions . To establish the parity of the $\Theta^+(1540)$, more extensive lattice study is required. Especially, a finite volume analysis is necessary to disentangle the pentaquark signal from a mixture of the $KN$ scattering states. It is also important to explore the chiral limit. This calculation was performed using relatively heavy quark mass so that one may worry about a level switching between both parity states toward the chiral limit as observed in the case of excited baryons . We remark that a study for the non-diagonal correlation between our pentaquark operators and a standard two-hadron operator should shed light on the structure of the very narrow resonance $\Theta^{+}(1540)$. The possible spin-orbit partner of the $\Theta$ state is also accessible by using two of our proposed operators. We plan to further develop the present calculation to involve more systematic analysis and more detailed discussion. It is a pleasure to acknowledge A. Hosaka, T. Nakano and L. Glozman for useful comments. I would also like to thank T. Doi, M. Oka and T. Hatsuda for fruitful discussions on the subject to determine the parity of the $\Theta$ state, and S. Ohta for helpful suggestions and his careful reading of the manuscript. 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However, nobody knows an appropriate analytic form for the two-point function of an unstable state [in finite volume]{} on the lattice.
--- abstract: \| Background modeling techniques are used for moving object detection in video. Many algorithms exist in the field of object detection with different purposes. In this paper, we propose an improvement of moving object detection based on codebook segmentation. We associate the original codebook algorithm with an edge detection algorithm. Our goal is to prove the efficiency of using an edge detection algorithm with a background modeling algorithm. Throughout our study, we compared the quality of the moving object detection when codebook segmentation algorithm is associated with some standard edge detectors. In each case, we use frame-based metrics for the evaluation of the detection. The different results are presented and analyzed. author: - title: \| Foreground-Background Segmentation Based on Codebook and Edge\ Detector --- codebook; edge detector; video segmentation; mixture of gaussian; Introduction ============ The detection of moving objects in video sequence is the first step in video surveillance system. The performance of the visual surveillance depends on the quality of the object detection. Many segmentation algorithms extract moving objects from image/video sequences. The goal of segmentation is to isolate moving objects from stationary and dynamic background. The variation of local or global light intensity, the object shadow, the regular or irregular background and foreground have an impact on the results of object detection. The object detection techniques are subdivided in three categories which are without background modeling, with background modeling and combined approach. The techniques based on background modeling are recommended in case of dynamic background observed by a static camera. These techniques generally model the background with respect to relevant image features. So Foreground pixels can be determined if the corresponding features from the input image significantly differ from those of the background model. Three methods are used: background modeling, background estimation, background substraction. Many research works have already been done [@kim; @mog; @mog1; @mog2]. Generally, background modeling techniques improve the foreground-background segmentation performance significantly in almost every challenging environment. They have better performance in both outdoor and indoor environments. The objects are integrated in background if they remain static over a specific delay. Sudden variation of light intensity make background model unstable. The method proposed in [@kim] has better performance in these situations. Kim et al. [@kim] propose a real time foreground background segmentation using codebook model. This algorithm works in two steps which are learning phase and update phase. The learning consists to determine a background model which is compared to the input image. The model is updated with new image. In this work, we are interested to combine the algorithm proposed in [@kim] with an edge detection algorithm. The main idea is to highlight the boundaries of objects in a scene. The using of an edge detector will verify if foreground pixels detected by the codebook algorithm belong to an object or not. In this paper we have explored three edge detectors : Sobel operator, Laplace of Gaussian operator and Canny edge detector. The paper consists of five sections. In Section \[reviewcodebook\] we made a review on moving object detection using codebook. In section \[proposed\] we presented the proposed algorithm for foreground-background segmentation. In Section \[result\] we presented the experimental results and we used some measures to evaluate the performance of the system. Finally in Section \[conclude\] we ended this work with further directions. Object Detection Based on Codebook {#reviewcodebook} ================================== The basic codebook background model is proposed in [@kim]. This method is widely used for moving object detection in case of stationary and dynamic background. In this method, each pixel is represented by a codebook $ \mathcal{C}=\{c_{1}, c_{2},......., c_{L}\}$. The length of codebook is different from one pixel to another. Each codeword $c_{i}$, $i=1,........,L$ is represented by a RGB vector $v_{i}$ $ (R_{i},G_{i},B_{i})$ and a 6-tuples $ aux_{i}=${[Ǐ]{}$_{i}$, [Î]{}$_{i}$, $f_{i}$, $p_{i}$, $\lambda_{i}$, $q_{i}$} where Ǐ and Î are the minimum and maximum brightness of all pixels assigned to this codeword $c_{i}$, $f_{i}$ is the frequency at which the codeword has occurred, $\lambda_{i}$ is the maximum negative run length defined as the longest interval during the training period that the codeword has not recurred, $p_{i}$ and $q_{i}$ are the first and last access times, respectively, that the codeword has occurred. The codebook is created or updated using two criteria. The first criteria is based on color distorsion $\delta$ whereas the second is based on brightness distorsion. We calcule the color distorsion $\delta$ using equation (\[eq\]). $$\label{eq} \delta=\sqrt{\|\|p_{t}\|\|^2-C_{p}^2}$$ In this equation, $C_{p}^2$ is the autocorrelation of R, G and B colors of input pixel $p_{t}$ and the codeword $c_{i}$, normalized by brightness. The autocorrelation value is given by equation (\[eq1\]). $$\label{eq1} C_{p}^2=\frac{(R_{i}R+G_{i}G+B_{i}B)^2}{R_{i}^2+G_{i}^2+B_{i}^2}$$ According to [@kim], the brightness $I$ has delimited by two bounds. The lower bound is $I_{low}=\alpha$[Î]{}$_{i}$ and the upper limit is $I_{hi}=min\{\beta$[Î]{}, $ \displaystyle \frac{\text{{\em \v{I}}}}{\alpha}\}$. For an input pixel which have R, G and B colors, the formula of the brightness is given by equation (\[eq2\]). $$\label{eq2} I=\sqrt{R^2+G^2+B^2}$$ For each input pixel, if we find a codeword $c_{i}$ which respect these two criteria (distorsion criteria and brightness criteria) then we update this codeword by setting $v_{i} $ to $ (\frac{f_{i}R_{i}+R}{f_{i}+1},\frac{f_{i}G_{i}+G}{f_{i}+1},\frac{f_{i}B_{i}+B}{f_{i}+1})$ and $ aux_{L} $ to {$min(I,$[Ǐ]{}$_{i}$), $max(I,$[Î]{}$_{i}$), $f_{i}+1$, $max(\lambda_{i},t-q_{i})$, $p_{i}$, $t$}. If we don’t find a matched codeword, we create a new codeword $c_{K}$. In this case, $v_{K}$ is equal to $(R,G,B)$ and $ aux_{K}$ is equal to {$I$, $I$, $1$, $t-1$, $t$, $t$}. After the training period, if an incoming pixel matches to a codeword in the codebook, then this codeword is updated. If the pixel doesn’t match, his information is put in cache word and this pixel is treated as a foreground pixel. If a cache word is matched more frequently so this cache word is put into codebook. Although the original codebook is a robust background modeling technique, there are some failure situations. Firstly, for example, in winter, people commonly use black coats. If foreground-background segmentation is done using the codebook method, it may adopt black colour as background for many pixels. That is why a lot of pixels are incorrectly segmented. Secondly, if an object in the scene stops its motion, then it is absorbed in the background. Kim et al. [@kim] indicate tuning parameter to overcome this problem, but these modifications reduce the global performance of the algorithm in another situation. Due to the performance of the proposed method by [@kim], several researchers continue by digging further. These improvements can be classified into four points. The first point is the improvement of the algorithm suggested by Kim et al. [@kim] by changing algorithm’s parameters. In this category, Ilyas et al. [@atif] proposed to use maximum negative run length $\lambda$ and frequency $f_{i}$ to decide whether to delete codewords or not. They also proposed to move cache codeword into the codebook when access frequency $f_{i}$ is large. In [@cheng], Cheng et al. suggested to convert pixels from RGB to YUV space. After this conversion they use the V component to build single gaussian model, making the whole codebook. Shah et al [@shah] used a statistical parameter estimation method to control adaptation procedure. Pal et al. [@amit] spreaded codewords along boundaries of the neighboring layers. According to this paper, pixels in dynamic region will have more than one codeword. The second point is about the improvement of the codebook algorithm by changing algorithm’s model. Some papers such as [@anup; @huo] are proposed in this category. Doshi et al. [@anup] proposed to use the V component in HSV representation of pixels to represent the brightness of these pixels. They suggested an hybrid cone-cylinder model to build the background model. Donghai et al. [@huo] proposed codebook background modeling algorithm based on principal component analysis (PCA). The model overcomes the mistake of gaussian mixture model sphere model and codebook cylinder model. The third point concerns the improvement of codebook algorithm by extension on pixels. Doshi et al. [@anup] proposed to convert pixel from RGB to HSV color space and Wu et al. [@wu] suggested to extend codebook in both temporal and spatial dimensions. Then the proposed algorithm in [@wu] is based on the context information. Fang et al. [@fang] proposed to convert pixels from RGB to HSL color space, and use L component as brightness value to reduce amount of calculation. The fourth point concerns the improvement of the algorithm proposed in [@kim] by combining it with other methods. In this category, some papers are proposed such as [@li; @yu; @liB]. Li et al. [@li] suggested to combine gaussian mixture model and codebook whereas Wu et al. [@yu] proposed to combine local binary pattern (LBP) with codebook to detect object. LBP texture information is used to establish the first layer of background. Li et al [@liB] proposed to use single gaussian to model codewords. It builds a texture-wise background model by LBP. This work proposes moving object detection based on the combination of codebook with edge detector. We use the gradient information of the pixel to improve the detection. Proposed Algorithm {#proposed} ================== Our proposed algorithm consists to combine the codebook with an edge detector algorithm. The goal of this combination is to improve the moving object detection in video. After running the codebook algorithm for foreground-background segmentation we proposed to find the convex hull of each contour which have been detected in the result. We computed an edge detection algorithm and applied it on the original frame which have been converted from color image to grayscale. We perform a two-level thresholding. We thresholded image by using the edge detector response and pixels are displayed only if the gradient is greater than a value $\varphi$. The value of $\varphi$ is given by formula (\[phi\]). $$\label{phi} \varphi=G(1-\theta)$$ In equation (\[phi\]), $G$ is the maximum gradient of input image and $\theta$ is a variable which value belongs to \[0 1\]. The value of $\theta$ depends on the characteristics of the input sequence. This double thresholding allows us to select only the major edges. After that we also find the convex hull of contours which have been detected in thresholded image. At this step the potential objects which are on the frame are detected. Finally a comparison between the pixels detected by codebook and the pixel detected once thresholding is done. The role of this comparison is to identify effective foreground pixels. An effective foreground pixel is a pixel which has been classified to foreground pixels by codebook and has been detected to be an object’s pixel by the edge detector. The detailed algorithm is given by Algorithm 1. In this algorithm, we assume that : - for each image of the sequence, the result of the segmentation is given by $r$; - the input pixel $p_{t}$ has R, G and B colors; - $N$ is the number of images that we use for the training; - $L$ is the length of codebook; - the size of the input image $F_{t}$ is $m\times n$; - $\varPsi$, $t_{1}$, $t_{2}$, $r$, $\varpi$ are the grayscale images which have same size with initial image $F_{t}$. - $Threshold$ $(x)$ is a procedure which thresholds the image $x$. The detailed procedure is given in Algorithm 2; - $BGS(F_{t})$ is a procedure which subtracts the current image $F_{t}$ from the background model. It’s described in [@kim]. For all pixels of the frame $F_{t}$, this procedure searches a matched codeword in the codebook. If a pixel doesn’t match to any codeword, this pixel is treated as a foreground pixel. [m[2cm]{} m[2cm]{} m[2cm]{}]{}\ ![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](in000872.jpg "fig:")&![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](cb000872.png "fig:")&![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](CodebookSeuiller.png "fig:")\ &&\ ![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](sobel.png "fig:")&![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](sobelSeuiller.png "fig:")&![(a) image $F_{t}$, (b) image $\varPsi$, (c) image $t_{1}$, (d) image $\varpi$ (using Sobel operator), (e) image $t_{2}$, (e) image $r$.[]{data-label="step"}](sb000872.png "fig:")\ &&\ $l \leftarrow 0, t \leftarrow 1$ $t \leftarrow$ detectedge(G) The figure \[step\] illustrates results obtained by the intermediate steps of the proposed algorithm. Experimental Results and Performance {#result} ==================================== In this section, we present the performance of the proposed approach by comparing with the codebook algorithm [@kim] and mixture of gaussian algorithm [@mog]. The section consists on two subsections. The first subsection presents the experimental results whereas the second presents and analyzes the performance of each algorithm. Experimental Results -------------------- For the validation of our algorithm, we have selected two benchmarking datasets from [@changelien] covered under the work done by [@changepaper]. They are “Canoe” and “fountain01” datasets. The experiment environment is IntelCore7@2.13Ghz processor with 4GB memory and the programming language is C++. The parameters settings for mixture of gaussian were $\alpha$ = 0.01, $\rho$ = 0.001, K = 5, T = 0.8 and $\lambda$ = 2.5$\sigma$. These parameters were suggested in [@mog]. According to [@kim], for codebook, parameter $\alpha$ is between 0.4 and 0.7, and parameter $\beta$ belongs \[1.1 1.5\]. In this work, we take $\alpha= 0.4$ and $\beta = 1.25$. The parameter $\theta$ of our proposed method depends on the dataset. For “Canoe” dataset we use $\theta=0.85$ whereas for “fountain01” dataset we use $\theta=0.80$. The results of segmentation are given in Figure \[fig\]. [m[1cm]{} m[3.3cm]{} m[3.3cm]{} m[3.5cm]{} m[3.5cm]{}]{}\ Original Frame&![image](in000872.jpg)&![image](in000888.jpg)&![image](in000728.jpg)&![image](in000740.jpg)\ Ground Truth&![image](gt000872.png)&![image](gt000888.png)&![image](gt000728.png)&![image](gt000740.png)\ CB results&![image](cb000872.png)&![image](cb000888.png)&![image](cb000728.png)&![image](cb000740.png)\ MoG results&![image](mog000872.png)&![image](mog000888.png)&![image](mog000728.png)&![image](mog000740.png)\ MCBSb results&![image](sb000872.png)&![image](sb000888.png)&![image](sb000728.png)&![image](sb000740.png)\ MCBLp results&![image](lap000872.png)&![image](lap000888.png)&![image](lap000728.png)&![image](lap000740.png)\ MCBCa results&![image](ca000872.png)&![image](ca000888.png)&![image](ca000728.png)&![image](ca000740.png)\ We assume that : - CB means codebook; - MoG means mixture of gaussian; - MCBSb means combination of codebook and Sobel; - MCBLp means combination of codebook and Laplacian of Gaussian operator; - MCBCa means combination of codebook and Canny edge detector. Performance Evaluation and Discussion ------------------------------------- In each case, we use an evaluation based on ground truth to show the performance of the segmentation algorithm. The ground truth has been obtained by labelling objects of interest in the original frame. The ground truth based metrics are : true negative (TN), true positive (TP), false negative (FN) and false positive (FP). A pixel is a true negative pixel when both ground truth and system result agree on the absence of object. A pixel is a true positive pixel when ground truth and system agree on the presence of objects. A pixel is a false negative (FN) when system result agree of absence of object whereas ground truth agree of the presence of object. A pixel is a false positive (FP) when the system result agrees with the presence of object whereas ground truth agree with the absence of object. With these metrics, we compute other parameters which are: - [False positive rate]{} (FPR) using formula (\[fpr\]); $$\label{fpr} FPR=1-\frac{TN}{TN+FP}$$ - [True positive rate]{} (TPR) using formula (\[tpr\]); $$\label{tpr} TPR=\frac{TP}{TP+FN}$$ - [Precision]{} (PR) using formula (\[pr\]); $$\label{pr} PR=\frac{TP}{TP+FP}$$ - [F-measure]{} (FM) using formula (\[fm\]); $$\label{fm} FM=\frac{2\times PR\times TPR}{PR+TPR}$$ We also compare the segmentation methods by using [percentage of correct classification]{} (PCC) and [Jaccard coefficient]{} (JC). PCC is calculate with formula (\[pcc\]) and JC is calculate with formula (\[jc\]). $$\label{pcc} PCC=\frac{TP+TN}{TP+FN+FP+TN}$$ $$\label{jc} JC=\frac{TP}{TP+FP+FN}$$ We present the results in Table \[rec\] and Table \[rec1\]. -- ------- --------------- ------- ------- -------------- 1.62 0.36 0.29 0.31 [0.24]{} 41.01 [89.82]{} 86.29 86.01 81.89 35.08 [88.17]{} 63.08 65.09 43.39 95.98 [99.18]{} 97.94 98.01 97.27 21.27 [78.85]{} 46.07 48.24 27.71 -- ------- --------------- ------- ------- -------------- : Comparison of different metrics according to experiments with dataset “canoe”[]{data-label="rec"} -- ------- ------- --------------- --------------- --------------- 1.09 1.59 [0.43]{} 0.45 [0.43]{} 2.24 4.01 6.82 [7.31]{} 6.79 4.17 7.63 11.58 [12.50]{} 11.53 98.86 98.40 [99.51]{} 99.49 [99.51]{} 2.13 3.97 6.14 [6.67]{} 6.11 -- ------- ------- --------------- --------------- --------------- : Comparison of different metrics according to experiments with dataset “fountain01”[]{data-label="rec1"} In these Tables, we assume that, value in bold are the optimal value of the row. We analyzied the results in Table \[rec\] and Table \[rec1\] throught two steps. At first, we make a comparison between codebook, mixture of gaussian and our method based on the combination of codebook and edge detector. At the second stage, we make a comparative study of the performance of the system obtained after combination of codebook with the three edge detectors. All experiments confirm that when codebook is combined with an edge detector, we get better result than original codebook. Experiments with “canoe” dataset proove that mixture of gaussian has good result than codebook. According to our results, the choice between mixture of gaussian and our method based on the combination of codebook and edge detector depends upon the application and the dataset’s characteristics. Experimentals Results with dataset “fountain01” show that our method is better than the mixture of gaussian approach. However, according to results of experiments with dataset “Canoe” we have two cases : 1. if we want to minimize the false alarms then FPR should be minimized. In this case, experiments show that method based on the combination of codebook and edge detector has the best result; 2. if we don’t want to miss any foreground pixel we need to maximize TPR and FM. In this case, Experiments with allow us to use mixture of gaussian for segmentation. 3. The experimental results also proove that the choice of edge detection algorithm depends upon the application. For example, for an application in which the real-time parameter is not important, the use of Canny operator is recommended if we want to minimize the false alarms. But if we need to improve TPR then we use a Laplacian of Gaussian operator. If we want to make a real-time application, we need to use Sobel operator, because the complexity of Sobel operator is less than Laplacian of Gaussian operator and Canny operator. The using of Sobel operator increases the codebook algorithm processing time by 19.55% (23.33% for laplacian of Gaussian and 28.15% for canny edge detector). Conclusion {#conclude} ========== In this paper, we present a novel algorithm to segment moving objects with an approach combining the codebook and edge detector. Firstly, we segment sequence using codebook algorithm. This segmentation help us to know background pixels and foreground pixels. After that, by using edge detector, we show the object boundaries in each sequence. Then we set all foreground pixels which are not object’s pixel to background pixel. The results can be summarized as follow : - our method outperforms the codebook algorithm [@kim] in accuracy; - in [@kim], authors claimed that codebook algorithm works better than the mixture of gaussian algorithm. This is not always true; - the choice between our algorithm and the mixture of gaussian algorithm [@mog] depends on the input dataset’s characteristics and the final application; - the choice of edge detection algorithm which combines with codedook algorithm depends also on the characteristics of the sequence and the final application . In the future, we will propose an extended version by adding a region based information in order to improve the compactness of the foreground object. Acknowledgment {#acknowledgment .unnumbered} ============== This work is partially financially supported by the Association AS2V and Fondation Jacques De Rette, France. We are also grateful to Professor Kokou Yetongnon for his fruitful comments. [1]{} K. Kim, T. H. Chalidabhonse, D. Harwood, L. Davis, “Real-time foreground-background segmentation using codebook model”, Real-Time Imaging, 2005, Vol. 11, pp. 167-256. C. Stauffer, W. Grimson, “Adaptive background mixture models for real-time tracking”, Conference on Computer Vision and Pattern Recognition, 1999, Vol. 2, pp. 246-252. J. Cheng, J. Yang, Y. Zhou, “A novel adaptive gaussian mixture model for background subtraction”. Proceeding of 2nd Iberian Conference on Pattern Recognition and Image Analysis, 2005, Shanghai, China, pp. 587-593. N. Friedman, S. Russel, “Image segmentation in a video sequences: A probabilistic approach”, Proceeding of IEEE Conference on Computer Vision and Pattern Recognition, 2008, pp. 1-6. A. Ilyas, M. Scuturici, S. Miguet, “Real time foreground-background segmentation using a modified codebook model”, International Conference on Advanced Video and Signal Based Surveillance, 2009, USA, pp. 454-459. X. Cheng, T. Zheng, L. Renfa, “A fast motion detection method based on improved codebook model”, Journal of Computer and Development, 2010, vol. 47, pp. 2149-2156. M. Shah, J. D. Deng, B. J. Woodford, “Enhanced codebook model for real-time background subtraction”, ICONIP(3) 2011, pp. 449-458. A. Pal, G. Schaefer, M. E. Celebi. “Robust codebook-based video background subtraction”, ICASSP 2010,pp. 1146-1149. A Doshi, M. M. Trivedi, “Hybrid cone-cylinder codebook model for foreground detection with shadow and highlight suppression”, AVSS, 2006, vol. 19, pp. 121-133. H. Donghai, Y. Dan, Z. Xiaohong, H. Mingjian, “Principal Component Analysis Based Codebook Background Modeling Algorithm”, Acta Automatica Sinica, 2012, vol. 38, pp. 591-600. M. Wu, X. Peng, “Spatio-temporal context for codebook- based dynamic background subtraction”, International Journal of Electronics and Communications, 2010, Vol. 64, pp. 739-747. X. Fang, C. Liu, S. Gong, Y. Ji, “Object Detection in Dynamic Scenes Based on Codebook with Superpixels”, Asian Conference on Pattern Recognition, 2013, pp. 430-434. Y. Li, F. Chen, W. Xu, Y. Du, “Gaussian-based codebook model for video background subtraction”, Lecture Notes in Computer Science, 2006, Vol. 4222, pp. 762-765. W. Yu, D. Zeng, H. Li, “Layered video objects detection based on LBP and codebook”, ETCS, 2009, pp. 207-213. B. Li, Z. Tang, B. Yuan, Z. Miao, “Segmentation of moving foreground objects using codebook and local binary patterns”, CISP, 2008, 239-243. ChangeDetection. ChangeDetection Video Database, 2012. \[Online\]. Available from: http://www.changedetection.net/ N. Goyette, P. M. Jodoin, F. Porikli, J. Konrad, P. Ishwar, “changedetection.net: A new change detection benchmark dataset”, in Proc. IEEE Workshop on Change Detection (CDW’12), 2012. Providence, RI.
--- abstract: 'Mirror matter models have been suggested recently as an explanation of neutrino puzzles and microlensing anomalies. We show that mirror supernovae can be a copious source of energetic gamma rays if one assumes that the quantum gravity scale is in the TeV range. We show that under certain assumptions plausible in the mirror models, the gamma energies could be degraded to the 10 MeV range (and perhaps even further) so as to provide an explanation of observed gamma ray bursts. This mechanism for the origin of the gamma ray bursts has the advantage that it neatly avoids the “baryon load problem”.' address: \| $^1$Department of Physics, University of Maryland, College Park, MD-20742[^1]\ $^2$ Department of Physics, Tel Aviv University, Tel Aviv, Israel\ and Department of Physics, University of South Carolina, Columbia, SC-29208\ $^3$ Department of Physics, Southern Methodist University, Dallas, TX-75275. author: - 'Rabindra N. Mohapatra$^1$, Shmuel Nussinov$^2$ and Vigdor L. Teplitz$^3$' date: 'September, 1999' title: TeV Scale Quantum Gravity and Mirror Supernovae as Sources of Gamma Ray Bursts --- epsf.tex (\#1 width \#2)[=\#2 ]{} Introduction ============ The origin of the gamma ray bursts (GRBs) observed for over three decades still remains unclear[@piran]. The GRBs are short, intense photon bursts with photon energies in the keV and MeV range although bursts with energy spectra extending above a GeV have been observed. The isotropy and $\frac{dN}{dV}$(intensity) distributions and the high redshift galaxies associated with some GRBs indicate that the sources of GRBs are located at cosmological distances. The specific nature of the sources remains however unclear. If unbeamed, the sources must emit total $\gamma$-ray energies of $10^{51}$ to $10^{53}$ ergs[@piran][^2] This is very much reminiscent of typical supernova energies. However, most supernovae (e.g. type II supernovae) cannot be these sources, since $\gamma$-rays with typical radiation lengths of 100 gm/cm$^2$ cannot penetrate the large amount ($\sim 10~M_{\odot}$) of overlying ejecta. Many of the models for unbeamed (beamed) GRBs use massive compact sources to produce neutrinos which annihilate to form fireballs of $e^+e^-$’s and $\gamma$’s[@goodman; @eichler]. The fireballs expand and cool adiabatically, until the temperature (or the transverse energy) is low enough so that the $e^+e^-$ annihilate into the $\gamma$’s. To avoid the ‘baryon load’’ problem and the absorption of $\gamma$’s, fairly “bare collapses” are required[@dar]. Accretion induced collapses and binary neutron star mergers[@piran] were considered but it is not clear whether these are sufficiently “baryon clean”. One “baryon clean” source candidate based novel particle physics is a neutron star to strange quark star transition. Other recent suggestions[@kluz; @bli] invoked the existence of sterile neutrinos[@sterile]. If the emitted neutrinos undergo maximal oscillation to the sterile neutrinos[@kluz], the latter can penetrate the baryon barrier and subsequently normal neutrinos will appear via the $\nu_s-\nu$ oscillation. In this scenario, the last “back” conversion occur at relatively large distances [^3] and the $\nu\bar{\nu}\rightarrow e^+e^-$ which goes like $R^{-8}$[@goodman] is inefficient[^4]. Similar difficulties are encountered by models utilizing exact “mirror” symmetric theories[@foot] where the sterile (mirror) neutrinos emitted in a mirror star collapse oscillate into ordinary neutrinos. In this note, we propose another GRB scenario in the context of the asymmetric mirror models[@berezhiani]. It utilizes the conversion of $\nu'+\bar{\nu}',\gamma'+\gamma'\rightarrow e^+e^-, \gamma\gamma$ etc inside the mirror star, where primed symbols denote mirror particles. Since the familiar electrons and photons do not interact with mirror matter, the expanding fireball is not impeded and we have an ideal bare collapse. The resulting photons expected to have initial energies of $\approx$ GeV, can be processed in this expansion down to the MeV part of the GRB spectra observed. Furthermore, if the source is embedded in the disk of a galaxy, further degrading can take place due to the “minibaryon load” of the disk resulting in keV gamma rays as well as possibly structure in the gamma ray spectrum. The key requirement is that the conversion process be fast enough so that a finite fraction of the collapse energy is indeed converted into ordinary matter. As we will see this naturally obtains[@sila] if we can have a low scale (of order of a TeV) for quantum gravity[@nima][^5] In section 2 we give a brief review of the assumptions of mirror matter models within which we work. In section 3 we outline our scenario, computing the initial $\gamma$ energies, and a brief discussion of possible fireball mechanism for degradation of the photon energies. We also discuss the effect of a baryon cloud (“mini-baryon load”) which can lead to further degradation of gamma energies. We work within the framework of TeV scale gravity using the results of Silagadze[@sila] for the production of familiar matter from mirror matter. We conclude in section 4 with a brief discussion. Asymmetric mirror model and large scale structure in the mirror sector ====================================================================== Let us begin with a brief overview of the asymmetric mirror matter model and the the parameters describing fundamental forces in the mirror sector. In asymmetric mirror matter models[@berezhiani], one considers a duplicate version of the standard model with an exact mirror symmetry[@lew] which is broken[@berezhiani] in the process of gauge symmetry breaking. Denoting all particles and parameters of the mirror sector by a prime over the corresponding familiar sector symbol (e.g. mirror quarks are $u',d',s',$ etc and mirror Higgs field as $H'$, mirror QCD scale as $\Lambda'$) we assume that $<H'>/<H>=\Lambda'/\Lambda\equiv \zeta$[@tep2]. This is admittedly a strong assumption for which there is no particle physics proof, but it does provide a certain degree of economy. Of course, if one envisioned the weak interaction symmetry to be broken by a new strong interaction such as technicolor in both sectors, then it is possible to argue that such a relation emerges under certain assumptions. There also exists a cosmological motivation for assuming $<H'>/<H>=\Lambda'/\Lambda\simeq 15$. One can show that in this case the mirror baryons can play the role of cold dark matter of the universe[@tep2; @bere]. The argument goes as follows: one way to reconcile the mirror universe picture with the constraints of big bang nucleosynthesis (BBN) is to assume asymmetric inflation with the reheating temperature in the mirror sector being slightly lower than that in the normal one[@reheat]. Taking the allowed extra number of neutrinos at the BBN to be 1 implies $(T'_R/T_R)^3\leq 0.25$. One can then calculate the contribution of the mirror baryons to $\Omega$ to be $$\begin{aligned} \Omega_{B'}\simeq (T'_R/T_R)^3 \zeta \Omega_B\end{aligned}$$ Since one expects, under the above assumption, the masses of the proton and neutron to scale as the $\Lambda$ in both sectors, if we assume that $\Omega_B\simeq 0.07$, then this implies $\Omega_{B'}\simeq 0.26$ leading to a total matter content $\Omega_m\simeq 0.33$. Thus familiar and mirror baryons together could explain the total matter content of the universe without need for any other kind of new particles. An important implication of this class of mirror models is that the interaction strengths of weak as well as electromagnetic processes (such as Compton scattering cross sections etc) are much smaller than that in the familiar sector. This has implications for the formation of structure in the mirror sector. Structure formation in a similar asymmetric mirror model was studied in Ref.[@tep1] where it was shown that despite the weakness in the mirror particle processes, there are cooling mechanisms that allow mirror condensates to form as the universe evolves. The basic idea is that the mirror matter provides gravitational wells into which the familiar matter gets attracted to provides galaxies and their clusters. However due to weakness of the physical processes, the mirror matter is not as strongly dissipative as normal matter. So for instance in our galaxy, the familiar matter is in the form of a disk due to dissipative processes whereas mirror stars which form the halo are not in disk form. In contrast, in the symmetric mirror model[@foot], the mirror matter would also be in a disk form and therefore could not help in explaining observed spherical galactic halos. Furthermore, since mirror matter condensed first in view of the lower temperature, it is reasonable to expect that mirror star formation largely took place fairly early (say $z\geq 1$) and the subsequent rate is much lower. In what follows to understand the observed GRBs we would require a mirror star formation rate of about one per million year per galaxy (to be contrasted with about 10/year/galaxy for familiar stars). In the asymmetric mirror model, it has been shown that there are simple scaling laws (first reference in [@tep2]) for the parameters of the mirror stars: (i) the mass of the mirror stars scale as $\zeta^{-2}$; (ii) the radius of the mirror stars also scales like $\zeta^{-2}$ whereas (iii) the core temperature scales slightly faster than $\zeta$. Here $\zeta$ denotes the ratio of the mass scales in the mirror and familiar sectors and is expected to be of order 15-20 from considerations of neutrino physics[@berezhiani]. Due to the higher temperature of the mirror stars, they will “burn” much faster and will reach the final stage of the stellar evolution sooner. Because of the $\zeta^{-4}$ decrease of weak cross sections and the increase in particle masses we do not expect mirror star collapse to result in explosion. Rather there should be neutrino emission and black hole formation. Thus we would expect that there will be an abundant supply of mirror “supernovae.” We will show in the next section that these could be the sources of the GRBs. Low quantum gravity scale and production of familiar photons in mirror supernovae ================================================================================= In a mirror supernova, one would expect most of the gravitational binding energy to be released via the emission of mirror neutrinos as in the familiar case. However, in the asymmetric mirror matter model, we expect the temperature of the collapsing star to be higher. We have $NT=GM^2/R$ where $N$ is the number of mirror baryons in the star (about $M_{\odot}/\zeta m_p$). At $\zeta = 10$ the maximum mirror star mass is about $M_{\odot}$ so that $T$ is about a $GeV$ where we have taken the radius of the collapsed mirror star to be about a kilometer. Let us now estimate the production cross section for the familiar photons in the collision of the mirror photons in the core. The most favorable case occurs if we assume that the quantum gravity scale is in the TeV range[@sila]. In this case, assuming two extra dimensions[@nima] and following reference [@han], we estimate the cross section $\sigma_{{\gamma'}{\gamma'}\rightarrow \gamma\gamma}$ to be, $$\begin{aligned} \sigma_{{\gamma'}{\gamma'}\rightarrow \gamma\gamma} \simeq \frac{1}{10}\frac{s^3}{\Lambda^8}\end{aligned}$$ where $s$ is the square of the total center of mass energy. For $s= 1$ GeV$^2$ and $\Lambda \simeq 1$ TeV, we get, $\sigma_{{\gamma'}{\gamma'}\rightarrow \gamma\gamma}\simeq 10^{-52}$ cm$^2$. We estimate the rate of energy loss per unit volume to into familiar, not mirror, photons to be roughly $$\begin{aligned} \frac{dQ}{dtdV}\simeq cn^2_{\gamma'}2E_{{\gamma'}}\sigma_{{\gamma'}{\gamma'} \rightarrow\gamma\gamma}\end{aligned}$$ Multiplying by the volume of the one kilometer black hole gives about $10^{52}$ erg/s. This energy is of the right order of magnitude for the total energy release in the case of unbeamed or mildly beamed GRBs. However the initial energy of individual photons obtained via $\nu'\rightarrow \gamma$ conversion is essentially that of the mirror neutrinos i.e. $E_{\gamma}(t=0)\approx E_{\nu'}\approx 3 T_{mirror}$. The spectrum of the latter- just like that of ordinary neutrinos obtained in the core cooling of ordinary type II supernovae- is expected to be roughly thermal with $T_{mirror}\approx $ GeV, which is roughly 100 times higher than for familiar collapse. While in some GRBs, photons of energies in the range of GeVs to TeVs have been observed, the bulk of the spectrum is in the MeV/keV region. Reprocessing the initial photons leading to energy degradation is therefore important. Two distinct mechanisms contribute to reprocessing: (i) Fireball evolution and (ii) Overlying putative familiar material. Let us discuss both these mechanisms. [Mechanism (i):]{} At t=0, we have, because of universality of gravitational interactions an equal number of familiar $e^+e^-$ produced with the photons. The resulting dense $e^+e^-\gamma$ “fireball” constitutes a highly opaque plasma. There is an extensive literature dealing with the evolution of such fireballs[@goodman; @dar]. In the case where this evolution is free from the effects of overlaying matter (i.e. the effects of (ii) are negligible), the discussion becomes almost model independent and many features can be deduced from overall energetics and thermodynamic considerations. Thus at t=0 when a fraction $\epsilon$ of the mirror neutrinos convert to $\gamma$’s (and/or $e^+e^-$’s), the latter have a blackbody spectrum with temperature $T_{\nu'}$. However the overall normalization, i.e. the energy density $$\begin{aligned} U_{\gamma}= \epsilon U_{\nu'} = \epsilon a T^4_{\nu'}\end{aligned}$$ falls short by a factor $\epsilon$ of the universal black body energy density at such a temperature. Fast processes of the form $\gamma\gamma\rightarrow e^+e^-\rightarrow 3\gamma$ (allowed in the thermal environment) will then immediately reequilibrate the system at $$\begin{aligned} T_{\gamma} \sim \epsilon^{\frac{1}{4}} T_{\nu'} \approx \left(\frac{1}{3}-\frac{1}{30}\right) T_{\nu'}\end{aligned}$$ (corresponding to GRB energies between $10^{48}-10^{52}$ ergs and mirror supernova energies of $10^{52}-10^{53}$ ergs). Subsequent evolution can further increase $N_{\gamma}$ and correspondingly decrease $\overline{E}_{\gamma}$ down towards the MeV range. Independent of this, mere expansion reduces the transverse photon energy according to $E^{tr}_{\gamma}\approx (R/r)T_{\gamma}(t=0)$, where R is the size of the source and $r$ is the current $\gamma$ location. (The last expression which parallels that for adiabatic cooling simply reflects the geometrical convergence of trajectories of colliding $\gamma$’s which become more and more parallel with distance $r$.) Since $E_{tr}$ controls the center of mass energy of the $\gamma \nu$ collisions, the $\gamma\gamma\rightarrow e^+e^-$ processes become kinematically forbidden and the density of $e^+e^-$ pairs falls exponentially i.e. $n_{e^+e^-}\approx e^{-\frac{m_e}{T_{tr}(r)}}$ eventually leaving freely propagating $\gamma$’s. [Mechanism (ii):]{} A “mini-baryon load” of familiar material encountered by the outgoing $\gamma$’s could further reduce the photon energy. Also the presence such matter in conjunction with mild beaming could induce the very short time structure often observed. In order to have an effective degrading of the emitted photon energies, we will need an appropriate density of familiar matter which can be estimated as follows. Let us assume a density profile of the form: $$\begin{aligned} \rho(R)=\frac{\rho_0 R^2_0}{R^2+R^2_0}\end{aligned}$$ Then we demand the constraint that $\int \rho(R) dR \simeq 100$ gm/ cm$^2$ where 100 gm/cm$^2$ represents the radiation length of photons in matter. This implies $\rho_0 R_0 \simeq 100$ gm/cm$^2$. The kinematical requirement of having comoving baryonic plus fireball system requires $$\begin{aligned} \gamma_B\equiv \frac{f W_{GRB}}{M_{Baryo}}\approx \gamma_{Fireball}\approx \frac{E_{e^+e^-}}{2m_e}\end{aligned}$$ where $f$ is the fraction of energy imparted to baryons and $\gamma_B$ is the Lorentz factor. Using $M_{baryo}\approx \frac{4\pi}{3}(\rho_0R_0)R^2_0$, we find $$\begin{aligned} R_0=10^{12}~cm~\left[\frac{(W/(10^{50}~ergs))}{(E/100~MeV) (\rho_0R_0/100~gm~cm^{-3})}\right]^{1/2}\end{aligned}$$ so that for the nominal values of the total GRB energy, the fireball processed energy of individual $e^+, e^-,\gamma$ and the column density, we find $R_0=10^{12}$ cm so that $\rho_0= 10^{-10}$ gr/cm$^{3}$ and $M_{baryo}\approx 10^{25} ~gr \simeq 10^{-8} M_{\odot}$. It is interesting to note that in the present scenario, GRB’s originating from mirror supernovae in the galactic halos, which most likely would not face the “minibaryon load”, may have only the first stage i.e. energy degradation by fireball mechanism and hence will have a harder spectra and smoother time profile. (Clearly discerning such a component in the GRB population will be quite interesting.) On the other hand the GRBs originating from supernovae in the disk of galaxies will have degradation due to both mechanisms and therefore more structure in the spectra as well as a softer spectra. Beyond the immediate neighbourhood of the mirror star there would be further energy degradation from interaction with interstellar matter ranging from molecular clouds to interstellar comets. There is not however sufficient material in one kilopersec to overcome the small value of the Thompson cross section i.e. $n_e\sigma_T \ell\sim 10^{-2}$ as against a required value of one. Discussion ========== Section 3 shows, we believe, that mirror matter supernovae, within the asymmetric mirror matter model, can provide a plausible explanation for gamma ray bursts. The scenario requires some coupling between the mirror and familiar sectors. In Section 3, we have used the couplings provided by TeV range quantum gravity following the estimate of reference ([@sila]), but other coupling mechanisms (such as a small $\gamma-\gamma'$ mixing) might be possible as well. Given TeV scale gravity, it is noteworthy that the same value of $\zeta$ required by other “manifestations” of mirror matter gives both an appropriate upper limit to the energy of the familiar gammas produced and an appropriate cross section section for their production. A major advantage of this GRB explanation is that it solves the baryon load problem in a natural way. In this model, we would expect production of GeV neutrinos at nearly the same rate as $e^+e^-$ and $\gamma\gamma$ etc. For GRBs located in our galaxy, they should be observable in detectors such as Super-Kamiokande. If this model is correct, given the short lifetime of the mirror stars[@tep2], the GRB frequency of $10^{-6}$/year/galaxy must be a result of low mirror star formation rate, which as mentioned above is not an unreasonable assumption. Finally, it is tempting to speculate that, if the primary GRB mechanism is to produce a fireball in the many MeV temperature range, there should exist a GRB population with temperatures in that range. In view of the fact that most of the data on GRBs comes from BATSE detector which triggers mostly on $\gamma$’s below 300 keV, it appears that such a population is not necessarily excluded by current data. This possibility that mirror matter can explain GRBs adds to a growing list of arguments that asymmetric mirror matter should be taken seriously. These include: (1) the requirement in many string theories that mirror matter exist; (2) the fact that the same range for $\zeta$ that was required in Section 3 for GRBs gives a mirror neutrino at the proper mass difference from $\nu_e$ to be the sterile neutrino responsible for simultaneously solving all the neutrino puzzles; (3) the fact that the same range of $\zeta$ gives an appropriate amount of dark matter to give an overall $\Omega_M$ in the range 0.2 to 0.3; and (4) the fact that the same range of $\zeta$ gives an explanation of the MACHO microlensing events as being caused by mirror black holes of about $M_{odot}/2$ mass. [Acknowledgments]{} We appreciate a helpful communication from Tom Siegfried. The work of RNM is supported by a grant from the National Science Foundation under grant number PHY-9802551 and the work of V. L. T. is supported by the DOE under grant no. DE-FG03-95ER40908. [99]{} For a review, see M. Rees, astro-ph/9701162 ; Tsvi Piran, astro-ph/9810256. A. Dar, J. Goodman and S. Nussinov, Ap. J. [314]{}, L7 (1987). D. Eichler, M. Livio, T. Piran and D. Schramm, Nature [340]{}, 126 (1989). A. Dar, R. Kozlowsky, R. Ramaty and S. Nussinov, Ap. J. [388]{}, 164 (1992). W. Kluzniak, astro-ph/9807224. S. I. Blinnikov, astro-ph/9902305. D.O. Caldwell and R.N. Mohapatra, Phys. Rev. [D 48]{}, 3259 (1993); J. Peltoniemi and J. W. F. Valle, Nucl. Phys. [B 406]{}, 409 (1993); S. Bilenky, C. Giunti and W. Grimus, Eur. Phys. J. [C 1]{}, 247 (1998); hep-ph/9805368. R. Volkas and Y. Wong, astro-ph/9907161. R. Foot and R. Volkas, Phys. Rev. [D52]{}, 6595 (1995). Z. Berezhiani and R. N. Mohapatra, Phys. Rev. [ D 52]{}, 6607 (1995); Z. Berezhiani, A. Dolgov and R. N. Mohapatra, Phys. Lett. [B 375]{}, 26 (1996). Z. Silagadze, hep-ph/9908208 which also provides an extensive list of references to early literature on the mirror matter models. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [B 429]{}, 263 (1998). R. N. Mohapatra, hep-ph/9903261. T. Han, J. Lykken and R. Zhang, Phys. Rev. [D 59]{}, 105006 (1999). R. Foot, H. Lew and R. Volkas, Phys. Lett. [B 272]{}, 67 (1991). R. N. Mohapatra and V. L. Teplitz, Phys. Lett. [B]{} (to appear), astro-ph/9902085. Z. Berezhiani et al. Ref.[@berezhiani]; Z. Berezhiani, hep-ph/9602326. E. W. Kolb, D. Seckel and M. Turner, Nature, Nature [514]{}, 415 (1985); Z. Berezhiani, A. Dolgov and R. N. Mohapatra, Ref.[@berezhiani]. R. N. Mohapatra and V. L. Teplitz, Ap. J. [478]{}, 29 (1998); S. I. Blinnikov and M. Y. Khlopov, Sov. J. Nucl. Phys. [36]{}, 472 (1982). [^1]: e-address:Rmohapat@physics.umd.edu [^2]: Beaming reduces this by $\frac{\Delta \Omega}{4\pi}$ and increases the required burst rate by $\frac{4\pi}{\Delta \Omega}$ over the few per day seen in the universe. [^3]: Both $\nu\rightarrow \nu_s$ and $\nu_s\rightarrow \nu$ are quenched by dense matter if $\Delta m^2\leq 10^4$ eV$^2$[@yv]. [^4]: Disklike (and beamed) geometry may partially alleviate this problem. [^5]: In the p-Brane construction, ordinary and mirror matter could reside on two sets of branes[@moha] with a relatively large (compared to $\Lambda^{-1}\approx ~(TeV)^{-1}$) separation $r_0$. The gauge group is of the form $G\rightarrow G_{matter}\times G_{mirror}$ where each $G= SU(3)\times SU(2)_L\times U(1)_Y$. The detailed model implementing this scenario will have to be such that it can lead to enhanced amplitude for the four Fermi operators that lead to familiar particle production via the collision of mirror particles whereas suppressed coefficient for the ones that lead to neutrino mixing. The latter in general involve exchange of fermions and the desired suppression is therefore not implausible. We thank Markus Luty for discussions on this point.
--- abstract: 'In the experimental verification of Bell’s inequalities in real photonic experiments, it is generally believed that the so-called fair sampling assumption (which means that a small fraction of results provide a fair statistical sample) has an unavoidable role. Here, we want to show that the interpretation of these experiments could be feasible, if some different alternative assumptions other than the fair sampling were used. For this purpose, we derive an efficient Bell-type inequality which is a CHSH-type inequality in real experiments. Quantum mechanics violates our proposed inequality, independent of the detection-efficiency problems.' author: - \| A. Shafiee[^1] $^{\text{(1)}}$ and M. Golshani$^{\text{(2,3)% }}\bigskip $\ [ ]{}$\stackrel{1)}{}$ [Department of Chemistry, Sharif University of Technology,]{}\ [ P.O.Box 11365-9516, Tehran, Iran]{}\ [ ]{}$\stackrel{2)}{}$ [Department of Physics, Sharif University of Technology,]{}\ [ P.O.Box 11365-9161, Tehran, Iran.]{}\ [ ]{}$\stackrel{3)}{}$ [Institutes for Studies in Theoretical Physics & Mathematics,]{}\ [ P.O.Box 19395-5531, Tehran, Iran.]{} title: On The Relevance of Fair Sampling Assumption in The Recent Bell Photonic Experiments --- Introduction ============ In his celebrated 1964 paper, John Bell considered a system consisting of two spin-$\frac{1}{2}$ particles in a singlet state \[1\]. He showed that the correlation between the results of two experiments done on such spatially separated particles cannot be reproduced by a local hidden-variable theory. Experiments done since 1972 indicate that the spin correlations of two particles in a singlet state violate Bell’s inequality, as quantum mechanics requires. These experiments have been usually done for photons (see ref. \[2\] and the references therein), and recently they were performed for massive particles \[3\]. But, there has been two general loopholes in the standard interpretation of these experiments which are not yet taken care of simultaneously in a single experiment. They make the present interpretations inconclusive. These are known as detection loophole and locality loophole. The detection loophole \[4\] refers to the fact that in Bell-type experiments, due to the low efficiency of detectors and collimators, a large number of photons may be undetected, and the resulting correlation is obtained on the basis of detected photons. Consequently, it is always possible to construct a local hidden variable model which can reproduce the experimental results \[4, 5, 6\]. So, the interpretation of these experiments is only feasible if one makes the fair sampling (FS) assumption \[7\]. P. Grangier describes the detection-efficiency loophole as “Achill’s heel of experimental tests of Bell’s inequalities” \[8\]. In the experiments done with the massive particles, this problem is solved, because the detection of these particles could be done efficiently. Yet, the second loophole, i.e., the locality loophole (according to which there exist the possibility of (sub)luminal communication between two spatially separated particles) is still there. (see, e.g., ref. \[9\].) In this paper, we consider the problem of fair sampling in the experiments done with photons. (A complete review of these experiments can be found in ref. \[2\].) There have been considerable discussions in the literature on this subject. But, the main issue in all of them is either to obtain a new limit for the detector inefficiency in CH \[10\] and/or CHSH \[7\] inequalities (see, e.g., ref. \[11\]) or to include the detector inefficiency directly into the original Bell inequality (see, e.g., ref. \[12\]). Recently, the authors have also proposed a new CH-type inequality which can be violated by some quantum mechanical predictions independent of the efficiency factors \[13, 14\]. What we want to argue here is that, contrary to what is assumed so far, detection loophole could be neglected in the interpretations of the recent photonic experiments, since there are other independent assumptions which are reasonable by themselves and could be used. To show this, we use, in part 2 of this paper, an efficient type of CHSH inequality which has been tested in recent photonic experiments and is shown to have been violated. We argue that for observing the violation of this inequality in real experiments, there are at least three possible solutions (other than the FS assumption) which have different physical basis and can be used independently for deriving the inequality. In section 3, we show that quantum mechanical predictions violate this inequality for real experiments, independent of the efficiency of detectors and collimators. Thus, we find another way for the justification of the entanglement criteria in the microphysical Bell states. An Alternative Bell-type Inequality =================================== FS assumption means that unrecorded data do not have a weighty role in calculating the polarization correlations of the two entangled photons. This is the most common view about the FS assumption. That is what P. Pearle described as Data Rejection Hypothesis in his 1970 paper \[4\]: “Suppose that each particle has three responses to a spin-measuring apparatus instead of two....Then, instead of four possible experimental outcomes of the measurement of the spins of two particles, there are nine possible outcomes. In one of these outcomes, neither particle is detected, and so the experimenter is unaware that a decay has taken place. In four of these outcomes one of the particles is not detected. If the experimenter rejects these data (in the belief that the apparatus is not functioning properly and that if it had been functioning properly, the data recorded would have been representative of the accepted data), he is left with the usual four possible outcomes.” Similarly, P. Grangier describes the meaning of fair sampling assumption as \[8\]: “The detection-efficiency loophole argues that, in most experiments, only a very small fraction of the particles generated are actually detected....So, to extract a meaningful conclusion from the observed data, it was necessary to assume that a small fraction of data provides a fair statistical sample”. Considering FS assumption, however, there is no reason why the data recorded are representative of the accepted data, what is the nature of recording probabilities and how one can interpret the efficiencies. These questions are irrelevant when one refers to the FS assumption. In contrast, if we negate such a hypothesis, i.e. if we believe that the rejected data may have a significant role in calculating the correlations, it would be a crucial task to elucidate the above points. This is our main concern in the following. (See the appendix for a more concrete discussion about the FS assumption.) Let us consider an actual double-channel Bell experiment where for each emitted photon a binary event (i.e., passage or non-passage corresponding to two polarization eigenvalues) occurs at each analyzer. We assume a stochastic local hidden-variable (SLHV) theory, in which $\lambda $ represents a collection of hidden variables, belonging to a space $\Lambda $. To have a complete physical description of the whole system, the hidden variables $\lambda $ are assumed to include the underlying variables of both the particles and devices. At this level, $p_{r}^{(1)}(\widehat{a},\lambda )$ is the probability that the result $r$ is detected for the polarization of the first photon along $\widehat{a}$, where $r=\pm 1$ corresponds to two eigenvalues of photon’s polarization and the angle $\widehat{a}$ is an angle from the $x$-axis to the transmission axis of the first photon’s polarization filter. The detection probability $p_{r}^{(1)}(\widehat{a}% ,\lambda )$ can be defined as the following: $$p_{r}^{(1)}(\widehat{a},\lambda )=p_{r,id}^{(1)}(\widehat{a},\lambda )\ \eta _{1r}(\widehat{a},\lambda ) \tag{1}$$ Here, $p_{r,id}^{(1)}(\widehat{a},\lambda )$ is the probability that if the first photon encounters a polarizer at angle $\widehat{a}$, it will then be detected in channel $r$ in an ideal experiment. In an analogous real experiment, we assume that $\eta _{1r}(\widehat{a},\lambda )$ denotes the overall efficiency of detecting the first photon with polarization along $\widehat{a}$ in channel $r$. It contains, e.g., the probability that the first photon reaches its detector and then will be detected with a definite chance. One can define $p_{q}^{(2)}(\widehat{b},\lambda )$ in a similar fashion for the second photon with $q=\pm 1$. The probability of non-detection of photons 1 and 2, along $\widehat{a}$ and $\widehat{b}$ respectively, are represented by $% p_{0}^{(1)}(\widehat{a},\lambda )$ and $p_{0}^{(2)}(\widehat{b},\lambda )$, where the index zero denotes non-detection. According to relation (1), $% p_{0}^{(1)}(\widehat{a},\lambda )$ can be defined as $p_{0}^{(1)}(\widehat{a}% ,\lambda )=1-\alpha (\widehat{a},\lambda )$, where $\alpha (\widehat{a}% ,\lambda )=\stackunder{r=\pm 1}{\sum }\ p_{r}^{(1)}(\widehat{a},\lambda )$ is a representative function of the overall hidden efficiency$\ \eta _{1r}(% \widehat{a},\lambda )$. If one assumes that $\eta _{1r}(\widehat{a},\lambda ) $ is independent of the measured value of the polarization $r$ (which means that all the efficiencies are the same at two channels $+$ and $-$), then $\alpha (\widehat{a},\lambda )=\ \eta _{1}(\widehat{a},\lambda )$ can be interpreted as an overall measure of the efficiency at the hidden-variable level. I.e., $$p_{0}^{(1)}(\widehat{a},\lambda )=1-\eta _{1}(\widehat{a},\lambda ) \tag{2}$$ A similar relation can be considered for $p_{0}^{(2)}(\widehat{b},\lambda )$. The joint probability for detection of the two photons with the outcomes $r$ and $q$ corresponding to the polarizations along $\widehat{a}$ and $\widehat{% b}$, respectively, is assumed to be: $$p_{rq}^{(12)}(\widehat{a},\widehat{b},\lambda )=p_{r}^{(1)}(\widehat{a}% ,\lambda )\ p_{q}^{(2)}(\widehat{b},\lambda ) \tag{3}$$ This is known as Bell’s locality condition \[10\]. Relations similar to (3) hold for the joint probabilities concerning non-detections. In a SLHV theory, the average value of the outcomes of polarizations of two photons along $\widehat{a}$ and $\widehat{b}$ is given by $$\begin{aligned} \varepsilon ^{(12)}(\widehat{a},\widehat{b},\lambda ) &=&\stackunder{r,q=\pm 1}{\sum }rq\ p_{rq}^{(12)}(\widehat{a},\widehat{b},\lambda ) \nonumber \\ &=&\varepsilon ^{(1)}(\widehat{a},\lambda )\ \varepsilon ^{(2)}(\widehat{b}% ,\lambda ) \tag{4}\end{aligned}$$ where $\varepsilon ^{(1)}(\widehat{a},\lambda )$ and $\varepsilon ^{(2)}(% \widehat{b},\lambda )$ are the average values of the outcomes of polarizations for photons 1 and 2 along $\widehat{a}$ and $% \widehat{b}$, respectively. Assuming that the above probabilities are normalized to one, we have: $$\stackunder{j=\pm 1,0}{\sum }\ p_{j}^{(1)}(\widehat{a},\lambda )=\stackunder{% j=\pm 1,0}{\sum }\ p_{j}^{(2)}(\widehat{b},\lambda )=1 \tag{5}$$ Now, it is obvious that $$0\leq p_{r}^{(1)}(\widehat{a},\lambda )\leq \alpha (\widehat{a},\lambda ) \tag{6}$$ and $$0\leq p_{q}^{(2)}(\widehat{b},\lambda )\leq \beta (\widehat{b},\lambda ) \tag{7}$$ where $\alpha (\widehat{a},\lambda )=\stackunder{r=\pm 1}{\sum }\ p_{r}^{(1)}(\widehat{a},\lambda )$ and $\beta (\widehat{b},\lambda )=% \stackunder{q=\pm 1}{\sum }\ p_{q}^{(2)}(\widehat{b},\lambda )$. The constraints (6) and (7) are actual constraints for the detection of single particles at the hidden-variable level. In the ideal limit, where $\alpha (% \widehat{a},\lambda )\rightarrow 1$ and $\beta (\widehat{b},\lambda )\rightarrow 1$, the probability of detection lies within the interval $% \left[ 0,1\right] $. Using the aforementioned constraints, one gets: $$\mid \varepsilon ^{(1)}(\widehat{a},\lambda )\mid \leq \alpha (\widehat{a}% ,\lambda ) \tag{8}$$ and $$\mid \varepsilon ^{(2)}(\widehat{b},\lambda )\mid \leq \beta (\widehat{b}% ,\lambda ) \tag{9}$$ In the following, we introduce three independent solutions which include some plausible assumptions about the nature of non-detection probabilities at the level of hidden variables as well as the relation of the empirical correlations with the predictions of a SLHV theory. These assumptions provide alternative ways for deriving an efficient type of CHSH inequality. Then, one can argue for the soundness of the recent photonic experiments. Yet, there are some important points which should be noted here. Our solutions I and II below involve assumptions about the probabilities of non-detection. The non-detection probabilities are unobservable and it has been usually recommended to avoid them. Thus, the earlier works in this area involved constraints about the probabilities of detection, rather than non-detection \[7, 10\]. Nevertheless, it is reasonable to think that a more plausible approach with weaker assumptions is achieved when one takes into account the non-detection events. This is the main point of the present work in which the nature of the auxiliary assumptions are completely different with the so-called fair sampling or no-enhancement assumptions in CHSH or CH inequalities. As indicated before, what we are proposing here is that the non-detection probabilities do have an important role in calculating the photonic correlations. But, we shall argue that there are situations in which one can define an effective correlation function only based on detected events and derive an inequality which only contains the so-called effective correlations. This is our purposed inequality. Here, we survey these situations in the context of the following solutions. Solution I. This is based on the assumption that at the level of hidden variables, the probability of non-detection of each individual photon is independent of the direction of its polarization filter, i.e., $$p_{0}^{(1)}(\widehat{a},\lambda )=p_{0}^{(1)}(\widehat{a^{\prime }},\lambda ); \tag{10-a}$$ $$p_{0}^{(2)}(\widehat{b},\lambda )=p_{0}^{(2)}(\widehat{b^{\prime }},\lambda ) \tag{10-b}$$ According to relation (1), this means also that for each individual photon, the hidden probabilities for reaching a detector and detecting by it should be independent of the earlier preparation made by choosing a definite polarization angle. Now, let us consider the set of polarization directions $\widehat{a},$ $% \widehat{a^{\prime }}$ for the first photon and $\widehat{b},$ $\widehat{% b^{\prime }}$ for the second one. Furthermore, we define the function $u$ as $$u:=x(y-y^{\prime })+x^{\prime }(y+y^{\prime }) \tag{11}$$ where $x:=\varepsilon ^{(1)}(\widehat{a},\lambda ),$ $x^{\prime }:=\varepsilon ^{(1)}(\widehat{a^{\prime }},\lambda ),$ $y:=\varepsilon ^{(2)}(\widehat{b},\lambda )$ and $y^{\prime }:=\varepsilon ^{(2)}(\widehat{% b^{\prime }},\lambda )$. We have also $\mid x\mid \leq \alpha ,$ $\mid x^{\prime }\mid \leq \alpha ^{\prime },$ $\mid y\mid \leq \beta $ and $\mid y^{\prime }\mid \leq \beta ^{\prime }$, in which for example $\alpha :=\alpha (\widehat{a},\lambda )=1-p_{0}^{(1)}(\widehat{a},\lambda )$, $\beta :=\beta (\widehat{b},\lambda )=1-p_{0}^{(2)}(\widehat{b},\lambda )$ and similar definitions hold for $\alpha ^{\prime }$ and $\beta ^{\prime }.$ Considering the relations (10-a) and (10-b), we have $\alpha =\alpha ^{\prime }=\alpha (\lambda )$ and $\beta =\beta ^{\prime }=\beta (\lambda )$. So, the limits of $\mid x\mid $ and $\mid x^{\prime }\mid $ as well as $% \mid y\mid $ and $\mid y^{\prime }\mid $ are the same. Since $u$ is a linear function of the variables $x,$ $x^{\prime },$ $y$ and $y^{\prime },$ its upper and lower bounds are determined by the limits of these variables. The bounds are tabulated in the Table 1. This table shows that $u$ is confined by the limits $2\alpha \beta $ and $-2\alpha \beta $. ---------------------------------------------------------------------------------------------------------------------------------------------------------- $Rows$ $x$ $x^{\prime }$ $y$ $y^{\prime }$ $u$ -------- ------------ ---------------------- ----------- --------------------- --------------------------------------------------------------------------- 1 $-\alpha $ $-\alpha ^{\prime }$ $-\beta $ $-\beta ^{\prime }$ $% \alpha (\beta -\beta ^{\prime })+\alpha ^{\prime }(\beta +\beta ^{\prime })=2\alpha \beta $ 2 $\alpha $ $-\alpha ^{\prime }$ $-\beta $ $-\beta ^{\prime }$ $% -\alpha (\beta -\beta ^{\prime })+\alpha ^{\prime }(\beta +\beta ^{\prime })=2\alpha \beta $ 3 $-\alpha $ $\alpha ^{\prime }$ $-\beta $ $-\beta ^{\prime }$ $% \alpha (\beta -\beta ^{\prime })-\alpha ^{\prime }(\beta +\beta ^{\prime })=-2\alpha \beta $ 4 $-\alpha $ $-\alpha ^{\prime }$ $\beta $ $-\beta ^{\prime }$ $% -\alpha (\beta +\beta ^{\prime })-\alpha ^{\prime }(\beta -\beta ^{\prime })=-2\alpha \beta $ 5 $-\alpha $ $-\alpha ^{\prime }$ $-\beta $ $\beta ^{\prime }$ $% \alpha (\beta +\beta ^{\prime })+\alpha ^{\prime }(\beta -\beta ^{\prime })=2\alpha \beta $ 6 $\alpha $ $\alpha ^{\prime }$ $-\beta $ $-\beta ^{\prime }$ $% -\alpha (\beta -\beta ^{\prime })-\alpha ^{\prime }(\beta +\beta ^{\prime })=-2\alpha \beta $ 7 $\alpha $ $-\alpha ^{\prime }$ $\beta $ $-\beta ^{\prime }$ $% \alpha (\beta +\beta ^{\prime })-\alpha ^{\prime }(\beta -\beta ^{\prime })=2\alpha \beta $ 8 $\alpha $ $-\alpha ^{\prime }$ $-\beta $ $\beta ^{\prime }$ $% -\alpha (\beta +\beta ^{\prime })+\alpha ^{\prime }(\beta -\beta ^{\prime })=-2\alpha \beta $ 9 $-\alpha $ $\alpha ^{\prime }$ $\beta $ $-\beta ^{\prime }$ $% -\alpha (\beta +\beta ^{\prime })+\alpha ^{\prime }(\beta -\beta ^{\prime })=-2\alpha \beta $ 10 $-\alpha $ $\alpha ^{\prime }$ $-\beta $ $\beta ^{\prime }$ $% \alpha (\beta +\beta ^{\prime })-\alpha ^{\prime }(\beta -\beta ^{\prime })=2\alpha \beta $ 11 $-\alpha $ $-\alpha ^{\prime }$ $\beta $ $\beta ^{\prime }$ $% -\alpha (\beta -\beta ^{\prime })-\alpha ^{\prime }(\beta +\beta ^{\prime })=-2\alpha \beta $ 12 $\alpha $ $\alpha ^{\prime }$ $\beta $ $-\beta ^{\prime }$ $% \alpha (\beta +\beta ^{\prime })+\alpha ^{\prime }(\beta -\beta ^{\prime })=2\alpha \beta $ 13 $\alpha $ $\alpha ^{\prime }$ $-\beta $ $\beta ^{\prime }$ $% -\alpha (\beta +\beta ^{\prime })-\alpha ^{\prime }(\beta -\beta ^{\prime })=-2\alpha \beta $ 14 $\alpha $ $-\alpha ^{\prime }$ $\beta $ $\beta ^{\prime }$ $% \alpha (\beta -\beta ^{\prime })-\alpha ^{\prime }(\beta +\beta ^{\prime })=-2\alpha \beta $ 15 $-\alpha $ $\alpha ^{\prime }$ $\beta $ $\beta ^{\prime }$ $% -\alpha (\beta -\beta ^{\prime })+\alpha ^{\prime }(\beta +\beta ^{\prime })=2\alpha \beta $ 16 $\alpha $ $\alpha ^{\prime }$ $\beta $ $\beta ^{\prime }$ $% \alpha (\beta -\beta ^{\prime })+\alpha ^{\prime }(\beta +\beta ^{\prime })=2\alpha \beta $ ---------------------------------------------------------------------------------------------------------------------------------------------------------- Table 1: The limits of $\QTR{sl}{u}$. Thus, under these conditions, we have: $$\mid u\mid \leq 2\alpha \beta \tag{12}$$ In the ideal limit we have $\mid u\mid \leq 2$. Now, we assume that the empirical correlation functions have a definite relation with the averages of the outcomes of polarizations of the two photons along certain directions in a SLHV theory. For example, for the two polarization directions $\widehat{a}$ and $\widehat{b}$, we define: $$E^{(12)}(\widehat{a},\widehat{b})=\int_{\Lambda }\varepsilon ^{(1)}(\widehat{% a},\lambda )\ \varepsilon ^{(2)}(\widehat{b},\lambda )\ \rho (\lambda )\ d\lambda \tag{13}$$ where, $E^{(12)}(\widehat{a},\widehat{b})$ is the correlation function of the polarization measurements of the two photons along $\widehat{a}$ and $% \widehat{b},$ and $\rho (\lambda )$ is the normalized probability density of $\lambda $ over $\Lambda .$ Using the definitions of $\alpha $ and $\beta ,$ we have: $$\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}=\int_{\Lambda }\alpha (\lambda )\ \beta (\lambda )\ \rho (\lambda )\ d\lambda \tag{14}$$ where $P_{rq}^{(12)}$ is the probability of the simultaneous detection of the outcome $r$ for the first photon and $q$ for the second photon, with polarizations along two arbitrary directions, in a real experiment. The relation (14) is independent of the polarization directions. But, this does not mean that the total number of photons recorded by each detector is independent of the directions of the polarization filters, because the number of undetected photons has a weighty role in the definition of the detection probabilities. Using the relations (11), (13) and (14), the inequality (12) takes the following form: $$\mid U\mid \leq M \tag{15}$$ where $$U=E^{(12)}(\widehat{a},\widehat{b})-E^{(12)}(\widehat{a},\widehat{b^{\prime }% })+E^{(12)}(\widehat{a^{\prime }},\widehat{b})+E^{(12)}(\widehat{a^{\prime }}% ,\widehat{b^{\prime }}) \tag{16}$$ and $M=2\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}$. In an ideal case, we have $M=2$ where by ideal we mean an experiment in which the probabilities of non-detection are zero. In view of the fact that in general $M\leq 2$, one can infer from (15) that $$\mid U\mid \leq 2 \tag{17}$$ The inequality (17) is known as CHSH inequality in the literature. Now, we define the effective correlation functions measured in the photonic experiments as $$E_{eff}^{(12)}(\widehat{a},\widehat{b}):=\frac{E^{(12)}(\widehat{a},\widehat{% b})}{\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}(\widehat{a},\widehat{b})}=% \frac{\stackunder{r,q=\pm 1}{\sum }rq\ N_{rq}^{(12)}(\widehat{a},\widehat{b})% }{\stackunder{r,q=\pm 1}{\sum }N_{rq}^{(12)}(\widehat{a},\widehat{b})} \tag{18}$$ where $N_{rq}^{(12)}(\widehat{a},\widehat{b})$ is the number of photons that are detected with the outcomes $r$ and $q$ along $\widehat{a}$ and $\widehat{% b}$, respectively. Assuming that $P_{00}^{(12)}(\widehat{a},\widehat{b}% )=P_{0}^{(1)}(\widehat{a})P_{0}^{(2)}(\widehat{b})$, we have$\stackunder{% r,q=\pm 1}{\sum }P_{rq}^{(12)}(\widehat{a},\widehat{b})=\left( 1-P_{0}^{(1)}(% \widehat{a})\right) \left( 1-P_{0}^{(2)}(\widehat{b})\right) ,$where $% P_{0}^{(1)}(\widehat{a})$ ($P_{0}^{(2)}(\widehat{b})$) is the probability of non-detection of photon 1 (2) with a polarization along $% \widehat{a}$ ($\widehat{b}$) and $P_{00}^{(12)}(\widehat{a},\widehat{b})$ is the joint probability of non-detection for both photons. Using the definition (18), the inequality (15) is reduced to $$\mid U_{eff}\mid \leq 2 \tag{19}$$ where $$U_{eff}=E_{eff}^{(12)}(\widehat{a},\widehat{b})-E_{eff}^{(12)}(\widehat{a},% \widehat{b^{\prime }})+E_{eff}^{(12)}(\widehat{a^{\prime }},\widehat{b}% )+E_{eff}^{(12)}(\widehat{a^{\prime }},\widehat{b^{\prime }}) \tag{20}$$ The inequality (19) is our proposed Bell-type inequality for a real experiment. This is the inequality which has been tested in recent photonic experiments and is shown to have been violated. Solution II. The first solution was based on assumptions that are used at the hidden-variable level. In the second solution, however, both the experimental and hidden-variable levels are under consideration. To derive (19), we assume that non-detection probabilities for each individual photon are the same at both levels, i.e., $$P_{0}^{(1)}(\widehat{a})=p_{0}^{(1)}(\widehat{a},\lambda );\quad P_{0}^{(1)}(% \widehat{a^{\prime }})=p_{0}^{(1)}(\widehat{a^{\prime }},\lambda ) \tag{21-a}$$ $$P_{0}^{(2)}(\widehat{b})=p_{0}^{(2)}(\widehat{b},\lambda );\quad P_{0}^{(2)}(% \widehat{b^{\prime }})=p_{0}^{(2)}(\widehat{b^{\prime }},\lambda ) \tag{21-b}$$ Here, one can argue that non-detection probabilities are hidden, as is the case at the hidden-variable level. Because, there is no way for their detection. The necessary condition for the acceptance of above relations is the assumption that the non-detection probability for each individual photon, at the hidden-variable level, is independent of $\lambda $. Or, equivalently, this means that the hidden efficiencies for reaching a detector and detection by it are the same as the experimental ones (see relation (1)). Subsequently, One can define an effective average value at the level of hidden variables, as $$\begin{aligned} \varepsilon _{eff}^{(12)}(\widehat{a},\widehat{b},\lambda ) &=&\stackunder{% r,q=\pm 1}{\sum }rq\ (\frac{p_{r}^{(1)}(\widehat{a},\lambda )}{1-p_{0}^{(1)}(% \widehat{a},\lambda )})(\frac{p_{q}^{(2)}(\widehat{b},\lambda )}{% 1-p_{0}^{(2)}(\widehat{b},\lambda )}) \nonumber \\ &=&\varepsilon _{eff}^{(1)}(\widehat{a},\lambda )\ \varepsilon _{eff}^{(2)}(% \widehat{b},\lambda ) \tag{22}\end{aligned}$$ where $\varepsilon _{eff}^{(1)}(\widehat{a},\lambda )=\stackunder{r=\pm 1}{% \sum }r(\frac{p_{r}^{(1)}(\widehat{a},\lambda )}{1-p_{0}^{(1)}(\widehat{a}% ,\lambda )})$ and $\varepsilon _{eff}^{(2)}(\widehat{b},\lambda )=% \stackunder{q=\pm 1}{\sum }q(\frac{p_{q}^{(2)}(\widehat{b},\lambda )}{% 1-p_{0}^{(2)}(\widehat{b},\lambda )})$. Using (6) and (7), we get: $$\left\| \varepsilon _{eff}^{(1)}(\widehat{a},\lambda )\right\| \leq 1 \tag{23}$$ $$\left\| \varepsilon _{eff}^{(2)}(\widehat{b},\lambda )\right\| \leq 1 \tag{24}$$ Using relations (22)-(24) and integrating over $\lambda $, one can prove (19), in a fashion similar to the proof of CHSH inequality. Based on the relations (21-a) and (21-b), the function $E_{eff}^{(12)}(\widehat{a},% \widehat{b})$ in (18) has the following relation with the hidden variables level: $$E_{eff}^{(12)}(\widehat{a},\widehat{b})=(\frac{1}{1-P_{0}^{(1)}(\widehat{a})}% )(\frac{1}{1-P_{0}^{(2)}(\widehat{b})})\int_{\Lambda }\varepsilon ^{(1)}(% \widehat{a},\lambda )\ \varepsilon ^{(2)}(\widehat{b},\lambda )\ \rho (\lambda )\ d\lambda \tag{25}$$ Solution III. Unlike the first and second solutions, here, we do not make any assumption about the nature of the non-detection probabilities. Instead, we make a conjecture that one can replace (13) by $$E_{eff}^{(12)}(\widehat{a},\widehat{b})=\int_{\Lambda }\varepsilon _{eff}^{(1)}(\widehat{a},\lambda )\ \varepsilon _{eff}^{(2)}(\widehat{b}% ,\lambda )\ \rho (\lambda )\ d\lambda \tag{26}$$ where $\varepsilon _{eff}^{(1)}(\widehat{a},\lambda )$ and $\varepsilon _{eff}^{(2)}(\widehat{b},\lambda )$ are defined as before and $% E_{eff}^{(12)}(\widehat{a},\widehat{b})$ is defined in as (18). One can prove the inequality (19) by using (22)-(24) and (26). The relations (13) and (26) are identical in the ideal limit, but they have different predictions for the real experiments. The physical content of the relation (26) is that one can always reproduce experimental results using the predictions of a SLHV theory, whereas relations like (13) indicate that in real experiments one cannot reproduce the predictions of quantum mechanics without making extra assumptions. Our three solutions for reproducing the inequality (19) involve compatible assumptions. The conjunction of the first two solutions means that the probability of non-detection for a given particle should be merely a function of instrumental efficiencies. Then, the relations (25) and (26) are obtained by dividing both sides of (13) by a detection constant. In such a case, it is assumed that non-detection is only an instrumental problem which is present but does not depend on what a microphysical theory is aimed to describe. The Predictions of Quantum Mechanics ==================================== What are the predictions of quantum mechanics for the inequality (19)? In a real double-channel experiment, the respective quantum mechanical joint probability for detecting two photons is nearly equal to \[15\]: $$P_{rq,QM}^{(12)}(\widehat{a},\widehat{b})\approx \frac{1}{4}\eta _{1}\eta _{2}f_{12}\left[ 1+rq\ F\cos 2(\widehat{a}-\widehat{b})\right] \tag{27}$$ In this relation, $\eta _{k}$ is the efficiency of detecting the $k$th photon ($k=1,2$). The function $f_{12}=f_{1}f_{2}$ shows the probability that both photons reach their detectors, where $f_{1}$ denotes the probability for the first photon reaching its corresponding detector and $% f_{2}$ is the same probability for the second photon. They are indicating the efficiencies of the two corresponding collimators for photons 1 and 2. The function $F$ is a measure of the correlation of the two emitted photons. In the relation (27), the efficiencies of the analyzers are assumed to be approximately perfect, which is the case in all recent photonic experiments. In an ideal experiment, all of the above efficiencies are equal to one. Here, for simplicity, we assume that $\eta _{1}\approx \eta _{2}\approx \eta $. Then, using (27), we obtain: $$\stackunder{r,q=\pm 1}{\sum }P_{rq,QM}^{(12)}(\widehat{a},\widehat{b}% )\approx \eta ^{2}f_{12} \tag{28}$$ which is independent of polarization directions. Since $\left( 1-P_{0,QM}^{(1)}\right) \approx \eta f_{1}$ and $\left( 1-P_{0,QM}^{(2)}\right) \approx \eta f_{2},$ the relation (28) is also equal to $\left( 1-P_{0,QM}^{(1)}\right) \left( 1-P_{0,QM}^{(2)}\right) $. Now, using the fact that $E_{QM}^{(12)}(\widehat{a},\widehat{b})=\stackunder{% r,q=\pm 1}{\sum }rq\ P_{rq,QM}^{(12)}(\widehat{a},\widehat{b}),$ the quantum correlation function for the polarization directions $\widehat{a}$ and $% \widehat{b}$ is: $$E_{QM}^{(12)}(\widehat{a},\widehat{b})\approx \eta ^{2}f_{12}F\cos 2(% \widehat{a}-\widehat{b}) \tag{29}$$ and subsequently, $$E_{QM,eff}^{(12)}(\widehat{a},\widehat{b})=\dfrac{E_{QM}^{(12)}(\widehat{a},% \widehat{b})}{\stackunder{r,q=\pm 1}{\sum }P_{rq,QM}^{(12)}(\widehat{a},% \widehat{b})}\approx F\cos 2(\widehat{a}-\widehat{b}) \tag{30}$$ If we choose $\mid \widehat{a}-\widehat{b}\mid =$ $\mid \widehat{a^{\prime }}% -\widehat{b}\mid =$ $\mid \widehat{a^{\prime }}-\widehat{b^{\prime }}\mid =\varphi $ and $\mid \widehat{a}-\widehat{b^{\prime }}\mid =3\varphi $, then (19) yields $$F\mid 3\cos \varphi -\cos 3\varphi \mid \leq 2 \tag{31}$$ For $\varphi =\frac{\pi }{4},$ we have $F\sqrt{2}\leq 1$. In real experiments where the entangled photon pairs are produced through spontaneous parametric down-conversion, $F$ is about $0.95$ or more \[16, 17\]. Since the inequality (31) is independent of the efficiency of detectors and collimators (two main facts responsible for the FS assumption), the predictions of quantum mechanics violate (19) and thus (31), without using the FS assumption. This may be the reason why in spite of the low efficiencies in Bell’s photonic experiments, the value of $U_{eff}$ in (19) agrees so well with predictions of the standard quantum mechanics and why this value is nearly the same in different experiments with different efficiency factors. Appendix Here, we want to elucidate the meaning of FS assumption more clearly. The CHSH inequality can be expressed as $$\mid U\mid \leq 2 \tag{A-1}$$ where $U$ is a linear combination of some empirical correlation functions along different directions defined in (16). The main issue of FS assumption is that one can use the inequality (19) $\mid U_{eff}\mid \leq 2$ instead of (A-1) where $U_{eff}$ is defined in (20). But, how is it possible to obtain (19) from (A-1)* and what is the role of FS assumption in deriving (19)? To answer these two questions, we first remember that an effective correlation function measured in the photonic experiments can be defined as (18). Now, it is obvious that for every $\widehat{k}=\widehat{a}$ or $\widehat{a^{\prime }}$ and $\widehat{l}=% \widehat{b}$ or $\widehat{b^{\prime }}$, we should have $\left\| E_{eff}^{(12)}(\widehat{k},\widehat{l})\right\| \geq \left\| E^{(12)}(\widehat{% k},\widehat{l})\right\| $, because $\stackunder{r,q=\pm 1}{\sum }% P_{rq}^{(12)}(\widehat{k},\widehat{l})\leq 1$. We reformulate $E^{(12)}(% \widehat{k},\widehat{l})$ as $E^{(12)}(\widehat{k},\widehat{l}% )=E_{eff}^{(12)}(\widehat{k},\widehat{l})-\epsilon _{kl}$, where $\epsilon _{kl}:=\frac{E^{(12)}(\widehat{k},\widehat{l})\left( 1-\stackunder{r,q=\pm 1% }{\sum }P_{rq}^{(12)}(\widehat{k},\widehat{l})\right) }{\stackunder{r,q=\pm 1% }{\sum }P_{rq}^{(12)}(\widehat{k},\widehat{l})}$. Consequently, one can begin with the CHSH inequality (A-1) and reach the following one $$-2+\epsilon \leq U_{eff}\leq 2+\epsilon \tag{A-2}$$ where $\epsilon =\epsilon _{ab}-\epsilon _{a^{\prime }b}+\epsilon _{ab^{\prime }}+\epsilon _{a^{\prime }b^{\prime }}$. Since, there is no way to measure $E^{(12)}(\widehat{k},\widehat{l})$ in real photonic experiments, the empirical value of $\epsilon $ cannot be determined. Considering the predictions of quantum mechanics (see section 3), however, one can show that $\epsilon _{QM}=(1-\eta ^{2}f_{12})U_{eff}$, where $\eta $ and $f_{12}$ are some efficiencies defined in section 3. Thus, what quantum mechanics predicts is that $\left\| U_{eff,QM}\right\| \leq \dfrac{2}{\eta ^{2}f_{12}}$ which is far from violation in actuality. Yet, there are two situations in which $\epsilon $ can be assumed to be zero: > 1- The experiment is performed under ideal* conditions, that is $% > \stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}$ reaches one actually. > > 2- The statistics of the experiment can be fairly constructed on the basis of the accessible data , that is all the predicted values will remain valid when $\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}$ is renormalized to unity (FS assumption). There is no way to obtain (19) from (A-1) except for the above conditions. No assumption about the nature of non-detection probabilities does help. The renormalization of $\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}$ as well as the validness of the predictions under new conditions are the key points. Nevertheless, it is logically possible to derive (19) by a different approach. One possible way is to begin with an alternative inequality $\mid U\mid \leq 2\stackunder{r,q=\pm 1}{\sum }P_{rq}^{(12)}$ (relation (15)) which is the basis of our first solution. In our two other solutions, we suppose some alternative relations for $E_{eff}^{(12)}(\widehat{a},\widehat{b% })$ corresponding to the hidden-variable level, to derive (19). So, our three suggested solutions impose more stringent conditions on a SLHV theory for being compatible with the experiments. [99]{} J. S. Bell, Physics 1, 195 (1964). W. Tittel and G. Weihs, Q. Inf. Comp., 1, No. 2, 3 (2001). M. A. Rowe et al., Nature 409, 791 (2001). P. Pearle, Phys. Rev. D 2, 1418 (1970). E. Santos, Phys. Lett. A 212, 10 (1996). N. Gisin and B. Gisin, Phys. Lett. A 260, 323 (1999). J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). P. Grangier, Nature 409, 774 (2001). E. Santos, Phys. Lett. A 200, 1 (1995). J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974). A. Garg and D. Mermin, Phys. Rev. D 35, 3831 (1987). J. Larsson, Phys. Rev. A 57, 3304 (1998). A. Shafiee, J. Mod. Opt., 51, 1073 (2004), quant-ph/0401022. A. Shafiee and M. Golshani, Fortschritte Phys. 53, 105 (2005), quant-ph/0406042. J. F. Clauser and A. Shimony, Rep. Prog. Phys., 41, 1881 (1978). G. Weihs et al., Phys. Rev. Lett. 81, 5039 (1998). W. Tittel et al., Phys. Rev. A 59, 4150 (1999). [^1]: E-mail: shafiee@sharif.edu
--- abstract: 'In this work a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then applied to the set of spectral densities. Trimming techniques and restrictions on the scatter within groups reduce the effect of noise in the data and help to prevent the identification of spurious clusters. The procedure is tested in a simulation study, and is also applied to a real data set.' author: - 'Rivera-García D.' - 'García-Escudero L.A.' - 'Mayo-Iscar A.' - 'Ortega, J.' bibliography: - 'references.bib' title: Robust Clustering for Time Series Using Spectral Densities and Functional Data Analysis --- Introduction ============ Time series clustering has become a very active research area in recent times, with applications in many different fields. However, most methods developed so far do not take into account the possible presence of contamination by outliers or spurious information in the sample. In this work, we propose a clustering algorithm for stationary time series that is based on considering the estimated spectral density functions as functional data. This procedure has robust features that mitigate the effect of noise in the data and help to prevent the identification of spurious clusters. [@Liao05], @Caiado15 and [@Aghabozorgi201516] provide comprehensive revisions of the area (see also [@Fu11]). [@Mont13] present a package in R for time series clustering with a wide range of alternative methods. According to [@Liao05], there are three approaches to clustering of time series: methods that depend on the comparison of the raw data, methods based on the comparison of models fitted to the data and, finally, methods based on features derived from the time series. Our proposal falls within the third approach and the spectral density is the characteristic used to gauge the similarity between time series in the sample. Several authors have previously considered spectral characteristics as the main tool for time series clustering. [@Caiado06; @Caiado09] considered the use of periodogram and normalized periodogram ordinates for clustering time series. [@maharaj2011fuzzy] propose a fuzzy clustering algorithm based on the estimated cepstrum, which is the spectrum of the logarithm of the spectral density function. [@olasTVD] and [@HSM2016] consider the use of the total variation distance on normalized estimates of the spectral density as a dissimilarity measure for clustering time series. A brief description of the last two algorithms will be given in Sect. 2. Other works have focused on developing robust clustering algorithms for time series clustering. [@WuYu2006] use Independent Component Analysis to obtain independent components for multivariate time series and develop a clustering algorithm, known as ICLUS to group time series. [@D'Urso2015107] use a fuzzy approach to propose a robust clustering model for time series based on autoregressive models. A partition around medoids scheme is adopted and the robustness of the method comes from the use of a robust metric between time series. [@D'Urso20161] present robust fuzzy clustering schemes for heteroskedastic tine series based on GARCH parametric models, using again a partition around medoids approach. Three different robust models are proposed following different robustification approaches, metric, noise and trimming. [@bahadori2015functional] propose a clustering framework for functional data, known as Functional Subspace Clustering, which is based on subspace clustering [@vidal2011subspace] and can be applied to time series with warped alignments. Our approach considers the use of functional data clustering as a tool for grouping stationary time series, but the functional object considered is not the time series but its spectral density. The use of Functional Data Analysis is Statistics can be reviewed in the following two monographs: [@RS2006] and [@FV2006]. Several clustering methods for functional data have been already proposed in the literature as, for instance, [@jamessugar2003], [@JP2013] and [@BJ2011] but these methods are not aimed at dealing with outlying curves. A survey of functional data clustering can be found in [@MR3253859] Our proposal for robust time series clustering is based on the use of spectral densities, considered as functional data, and the application of the clustering algorithm recently developed in [@RFC2017], which will described in more detail in Sect \[sec3\]. Trimming techniques for robust clustering have been already applied in [@Efuncional2005] and [@cuestafraiman2007]. The rest of the paper is organized as follows: Section \[sec2\] considers time series clustering and describes the idea behind our proposal. Section \[sec3\] gives a brief description of the robust clustering algorithm for functional data that supports the time series clustering algorithm. Section \[Simustudy\] presents a simulation study designed to compare the performance of the algorithm with existing alternatives and Sect. \[sec5\] gives an application to a real data set. The paper ends with some discussion of the results and some ideas about future work. Time Series Clustering {#sec2} ====================== Consider a collection of $n$ stationary time series $X_{1,t},X_{2,t},\dots ,X_{n,t}$ with $1\leq t \leq T$. For ease of notation we take all series to have the same length, but this is not a requirement of the procedure. For each time series the corresponding spectral density $\varphi_i, 1\leq i\leq n$ is estimated by one of the many procedures available. As mentioned in the Introduction, previous clustering methods based on the use of spectral densities relied on similarity measures for discriminating between them. In this work, the spectra are considered as functional data to which the robust clustering procedure developed in [@RFC2017] and described briefly in the next section, is applied. The resulting clusters correspond to time series whose spectral densities have similar shapes, and therefore similar oscillatory behavior. The procedure is able to detect outliers in the collection of spectral densities, which correspond to time series having atypical oscillatory characteristics. Two methods recently proposed in the literature, that are based on the use estimated spectral densities as the characteristic feature of each time series, are presented in [@olasTVD] and [@HSM2016]. We describe these two procedures in more detail since they will be used for comparison purposes in the next sections, and we refer to them as “TVDClust" and “HSMClust", respectively. In both cases the total variation distance (TVD) is used to measure similarity between spectral densities. TVD is a frequently-used distance between probability measures that, in the case of probabilities having a density, measures the complement of the common area below the density curves. Thus, the more alike the densities are, the larger this common area and the smallest the TV distance. To use this distance to compare spectral densities, a prerequisite is that they have to be normalized so that the total area below the curve is equal to 1, which is equivalent to normalizing the original time series so that it has unit variance. Thus, it is the oscillatory behavior of the series, and not the magnitude of the oscillations that is taken into account in these clustering algorithms. For the first method, “TDVClust", a dissimilarity matrix is built up by measuring the TVD distance between all pairs of normalized estimated spectral densities. This matrix is then fed to a hierarchical agglomerative algorithm with the complete or average linkage functions. The result is a dendrogram which can be cut to obtain the desired number of groups. To decide on the number of clusters an external criteria such as the Silhouette or Dunn’s index is used. More details on this algorithm can be found in [@olasTVD]. The second method, “HSMClust", is a modification of the previous one in which every time two clusters are joined together, all the information in them is used to obtain a representative spectrum for the new cluster. There are two ways to do this, either all the spectral densities are averaged to obtain a representative spectral density for the new group, which is the average option in the algorithm, or else all the time series in the two groups are concatenated and a new spectral density is estimated, which corresponds to the single option. Under the assumption that the series in the same cluster have common oscillatory characteristics, either of this procedures will give a more accurate estimation of the common spectral density for the whole group. This algorithm, which comprises the two options described, is known as the Hierarchical Spectral Merger (HSM) algorithm, and its implementation in R is available at http://ucispacetime.wix.com/spacetime\#!project-a/cxl2. Every time two clusters are joined together, the dissimilarity matrix reduces its size. In the previous algorithm, “TVDClust", the dissimilarity matrix remains the same throughout the procedure and the distances between clusters are calculated using linear combinations of the distances of the individual points in each cluster. The linear combination used is determined by the linkage function employed. [@HSM2016] present two methods for determining the number of clusters. One of them is based on the value of the distance between the closest clusters at each step, and the other one is based on bootstrap procedures. More details can be found in the reference. Robust Clustering for Functional Data {#sec3} ===================================== In this section we give a brief description of the algorithm proposed in [@RFC2017], where more details can be found. Let $X$ be a random variable taking values in the Hilbert space $ L^{2}([0,T])$ of functions with inner product given by ${\langle}f,g {\rangle}= \int f(t)g(t)\, dt$. If $\mu(t)=E\{X(t)\}$ and $\Gamma(s,t)=\text{cov}\{X(s),X(t)\}$, then it is usual to represent $X$ through its Karhunen-Loève expansion $ X(t)=\mu(t)+\sum_{j=1}^{\infty}C_{j}(X)\psi_{j}(t) $. In that expansion, the $\psi_{j}$ are an orthonormal system of functions obtained as eigenfunctions of the covariance operator $\Gamma$, i.e. ${\langle}\Gamma(\cdot,t),\psi_{j} {\rangle}=\lambda_{j}\psi_{j}(t)$, and the eigenvalues $\lambda_j$ are taken in decreasing order and assumed to satisfy $\sum_{j=1}^{\infty}\lambda_{j}<\infty$. The principal component scores $C_{j}(X)={\langle}X-\mu, \psi_{j}{\rangle}$ are uncorrelated univariate random variables with zero mean and variance equal to $\lambda_{j}$. [@DH2010] show that $ \log P(\|\|X-x\|\|\leq h)$ can be approximated by $ \sum_{j=1}^{p}\log f_{C_{j}}(c_{j}(x))$, for any $x\in L_{2}([0,T])$ and small $h$, where $f_{C_{j}}$ corresponds to the probability density function of $C_{j}(X)$ and $c_{j}(x)={\langle}x, \psi_{j} {\rangle}$. This approximation entails a kind of “small-ball pseudo-density" approach for Functional Data Analysis by taking into account that probability density functions in the finite dimensional case can be seen as the limit of $P(\|\|X-x\|\|\leq h)/h$ when $h\rightarrow 0$. In the particular case of $X$ being a Gaussian process, the $C_{j}(X)$ are independent normally distributed random variables with mean equal to 0 and variance equal to $\lambda_j$. With these previous ideas in mind, [@JP2013] propose a “model-based" approach for clustering of functional data, where a finite number of independent normally distributed principal component scores are assumed and different variances are also allowed for each cluster. To simplify this largely parameterized problem, [@BJ2011] consider an alternative approach, where a certain fraction of the smallest variances are constrained to be equal for each cluster. [@RFC2017], starting from [@BJ2011], propose a robust functional clustering procedure where a proportion $\alpha$ of curves are allowed to be trimmed and constraints on the variances are considered. To be more precise, if $\{x_1,...,x_n\}$ is a set of curves in $L^2([0,T])$, we consider the maximization of a trimmed mixture-loglikelihood defined as $$\begin{aligned} \label{eq11} \sum_{i=1}^{n} \eta(x_{i}) \log \left(\sum_{g=1}^{K} \pi_{g} \left[\prod_{j=1}^{q_{g}} \frac{1}{\sqrt{2 \pi a_{jg}}} \exp\left(\frac{-c_{ijg}^{2}}{2a_{jg}}\right)\prod_{j=q_{g}+1}^{p} \frac{1}{\sqrt{2 \pi b_{g}}} \exp\left(\frac{-c_{ijg}^{2}}{2 b_{g}}\right)\right]\right)\end{aligned}$$ where $c_{ijg}=c_{jg}(x_{i})$ is the $j$-th principal component score of curve $x_{i}$ in group $g$, $g=1,...,K$, and, $\eta(\cdot)$ is an indicator function with $\eta(x_{i})=0$ if the $x_{i}$ curve is trimmed and 1 if it is not. A proportion $\alpha$ of curves is trimmed, so that $\sum_{i=1}^{n}\eta(x_{i})=[n(1-\alpha)]$. The main cluster variances are modeled through $a_{1g}$,..., $a_{q_gg}$, for the $g$-th cluster, while $b_{g}$ serves to model the “residual" variance. Notice that we take an equal number of principal components scores $p$ in every cluster but the number of main variance components $q_g$ may vary across clusters. Finally, to prevent the detection of spurious clusters, two constants $d_1\geq 1$ and $d_2\geq 1$ were fixed such that the maximization of (\[eq11\]) is done under the constraints: $$\frac{\max_{g=1,...,K;j=1,...,q_j}a_{jg}}{\min_{g=1,...,K;j=1,...,q_j}a_{jg}}\leq d_1$$ and $$\frac{\max_{g=1,...,K}b_{g}}{\min_{g=1,...,K}b_{g}}\leq d_2.$$ A feasible algorithm for performing the constrained maximization was detailed in [@RFC2017]. This algorithm is a modification of the traditional EM algorithm used in model-based clustering where a “trimming" step (T-step) is also added. In the trimming step, those curves with smallest contributions to the trimmed likelihood are temporarily not taken into account in each iteration of the algorithm. That trimming step is similar to that applied in the “concentration" steps applied when performing the fast-MCD algorithm [@RVD99]. To enforce the required constraints on the variances, optimally truncated variances as done in [@Fritz2013] are adopted if needed. With respect to the estimation of the $q_g$ dimensions in each cluster, a BIC approach was proposed in [@RFC2017] as a sensible way to estimate those dimensions. Simulation Study {#Simustudy} ================ In order to evaluate the performance of the methodology proposed here, a simulation study was carried out. We now describe the different scenarios and contamination types. As in [@HSM2016], the simulations are based on combinations of autoregressive processes of order 2. AR(2) processes are defined as $$\begin{aligned} X_{t} = u_{1}X_{t-1}+ u_{2}X_{t-2}+\epsilon_{t}\end{aligned}$$ where $\epsilon_{t}$ is a white noise process. The characteristic polynomial associated with this model is $h(y)=1-u_{1}y-u_{2}y^{2}$ and its roots, denoted by $y_{1}$ and $y_{2}$ are related to the oscillatory properties of the corresponding time series. If the roots are complex-valued, then they must be conjugate, i.e., $y_{1}=\overline{y_{2}}$ and their polar representation is $$\begin{aligned} \label{coef1} \|y_{1}\|=\|y_{2}\|=M \quad \text{and} \quad \arg(y_{i})=\frac{2\pi\nu}{w_{s}}\end{aligned}$$ where $w_{s}$ is the sampling frequency in Hertz; $M$ is the magnitude of the root ($M>1$ for causality) and $\nu$ the frequency index, $\nu \in (0, w_{s}/2)$. The spectrum of the AR(2) process with roots defined as above will have modal frequency in $\nu$. The modal frequency will be sharper when $M\rightarrow \infty$ and narrower when $M\rightarrow 1^{+}$. Then, given ($\nu, M, w_{s}$) we have $$\begin{aligned} \label{coef2} u_{1}=\frac{2\cos(\omega_{0})}{M} \quad \text{and} \quad u_{2}=-\frac{1}{M^2}\end{aligned}$$ where $\omega_{0}=\frac{2\pi\nu}{w_{s}}$. Two groups of 50 time series each were simulated, with parameters $\nu_1 = 0.21$, $\nu_2 = 0.22$, $M_1=M_2 = 1.15$, $w_s=1$ and length $T=1000$. From the simulated time series, the spectral densities were estimated using a smoothed lag-window estimator with a Parzen window and bandwidth 100/T. The estimated spectral densities for both clusters are shown in Figure \[fig:simolas\](a). The functional form of the estimated spectral densities was recovered using a B-Spline basis of degree 3 with $14$ equispaced nodes and smoothing parameter $\lambda=0.000003$ (see e.g. [@RS2006], Ch. 3) We want to test the performance of the different algorithms in recovering these two groups, even in the presence of contaminating data. In the absence of contamination we have 100 observations divided into two groups. Before describing the contamination schemes considered, we introduce the mixtures of AR(2) processes that will be used in some cases. Let $Y_t^i, i=1,2$ be two AR(2) processes with parameters $M_i$ and $\nu _i$, $i=1,2$. Their mixture is given by $$X_t = a_1 Y_t^1 + a_2 Y_t^2 + \epsilon_t$$ where the $a_i, i=1,2$ are the weights and $\epsilon_t$ is a white noise process. This mixture of AR(2) processes creates a signal that combines the oscillatory behavior of the original processes $Y_t^i, i=1,2$. Starting from the two groups of 50 AR(2) time series described in the beginning of this section, which are considered as the clean data, we added another 11 time series (around 10% contamination level). We consider the following schemes for generating these additional time series: - Using the previously described simulation procedure, simulate 11 AR(2) processes with parameters $\nu_{i}$ chosen randomly with uniform distribution in the interval $(.20,.25)$, denoted as $U(.20,.25)$, $i=1,\dots,11$; $M=1.2$ and $w_{s}=1$. This means that the contaminating curves have less energy than the series in the clusters. See Fig. \[fig:simolas\](b). - A mixture of two AR(2) processes having parameters $\nu_{i}=.20$ and $.25$; $M_{i}=1.05, 1.1$, $i=1,2$ y $w_{s}=1$. See Fig. \[fig:simolas\](c). - A mixture of two $AR(2)$ processes with random parameters $\nu_{1}=U(.19,.22)$ y $\nu_{2}=U(.24,.26)$; $M_{i}=1.05, 1.1$, $i=1,2$ and $w_{s}=1$, See Fig. \[fig:simolas\](d). Figures \[fig:simolas\](b), (c) and (d) show the spectral density for the simulated time series with the three contamination schemes described. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Spectral density of the simulated time series: (a) No contamination, (b) Contamination type (i), (c) Contamination type (ii) and (d) Contamination type (iii)[]{data-label="fig:simolas"}](SimNOC.pdf "fig:") ![Spectral density of the simulated time series: (a) No contamination, (b) Contamination type (i), (c) Contamination type (ii) and (d) Contamination type (iii)[]{data-label="fig:simolas"}](SimCRAM.pdf "fig:") ![Spectral density of the simulated time series: (a) No contamination, (b) Contamination type (i), (c) Contamination type (ii) and (d) Contamination type (iii)[]{data-label="fig:simolas"}](Sim2M.pdf "fig:") ![Spectral density of the simulated time series: (a) No contamination, (b) Contamination type (i), (c) Contamination type (ii) and (d) Contamination type (iii)[]{data-label="fig:simolas"}](Sim2CRAM.pdf "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In order to test the performance of the robust functional clustering (RFC) methodology proposed here, the simulated proceses and their estimated spectral densities were used to compare with the results obtained when using the “Funclust" algorithm [@JP2013] and hierarchical methods using the total variation distance: “HSMClust" [@HSM2016] and “TVDClust" [@olas; @olasTVD]. It is important to recall that we assume the $q_g$ dimensions in the RFC procedure to be unknown parameters and that the BIC criterion is used to estimate them when applying this algorithm. The results in [@RFC2017] already show the importance of trimming. Trimming levels $\alpha=0$ and $\alpha=0.1$ are used. As regards the constraints, we are assuming $d_1=d_2$ to simplify the simulation study. Values of $d_{1}=d_{2}=3$, $d_{1}=d_{2}=10$ and $d_{1}=d_{2}=10^{10}$ (i.e., almost unconstrained in this last case) were used. We always return the best solution in terms of the highest BIC value for each combination of all those fixed values of trimming level and constraints. We use 100 random initializations with 20 iterations. For the “Funclust" method we have used the library `Funclustering` [@funclustR] in R where the EM algorithm has been initialized with the best solutions out of 20 “short" EM algorithms with only 20 iterations and threshold values of $\varepsilon=0.001, 0.05, 0.1$ in the Cattell test. In case of the agglomerative methods we use the library `HSMClust` in R for “HSMClust" and “TVDClust" by means of the algorithm described in [@olas; @olasTVD]. Figure \[fig:Mths\] shows the results for the simulation study. This figure is composed of a matrix of graphs, where the rows correspond to the different contamination schemes (uncontaminated in the first row) while the columns correspond to the methodologies tested. The first column corresponds to “Funclust", the second to “HSMClust", the third shows the results for the robust functional clustering (RFC) procedure with trimming levels $\alpha=0$ (untrimmed) and $\alpha=0.1$ and three constraint levels $d_{i}=3$, $d_{i}=10$ and $d_{i}=10^{10}, i=1,2$ (i.e., almost unconstrained in this last case). Finally the fourth column shows the results corresponding to “TVDClust". The $x$-axis corresponds to the threshold applied in the Cattell test for “Funclust", the procedure in “HSMClust", the constraint level for RFC and the linkage function for the agglomerative method “TVDClust", while the $y$-axis corresponds to the correct classification rate (CCR). ![Correct classification rate (CCR) for the four methods considered, represented in different columns. Rows correspond to the different contamination schemes described previously in this section, starting with no contamination in the first row and following with contamination schemes (i), (ii) and (iii) described in the text. Constraint levels $d_1=d_2=3$, $10$ and $10^{10}$, trimming levels $\alpha=0$ and $0.1$ were used for the RFC method. Threshold values $\varepsilon= 0.001, 0.05$ and $0.1$ are used for the “Cattell" procedure in “Funclust". Single and average version were used for “HSMClust" while average and complete linkage functions were used for “TVDClust". []{data-label="fig:Mths"}](BoxplotTotald31.pdf) Results show that the hierarchical methods, “HSMClust” and “TVDClust” are better in the absence of contamination, giving very consistent results. However, their performance degrades sharply in the presence of noise. This is not susprising since these procedures were not designed to handle contamination in the sample. The joint use of trimming and constraints in RFC improve the results (CCR) substantially. Results are very good for moderate $(d_{1}=d_{2}=10)$ and small $(d_{1}=d_{2}=3)$ values of the constraint constants, while for high values the results are poor. Very high values for these constants are equivalent to having unconstrained parameters. The use of trimming also turns out to be very useful in all the contaminated cases while the results are not affected by trimming in the uncontaminated case. In the presence of contamination, the results for “Funclust", “HSMClust" and “TVDClust" fall below those of RFC when applying the $\alpha=0.1$ trimming and small/moderate values $d_1$ and $d_2$ for the variance parameters. Analysis of Real Data {#sec5} ===================== In this section we consider wave-height data measured by a buoy located in Waimea Bay, Hawaii, at a water depth of approximately 200 m. This buoy is identified as number 106 (51201 for the National Data Buoy Centre). The data was collected in June 2004 and has previously been analyzed by [@olas] where more details can be found. The data corresponds to 72.5 hours divided into 30-minute intervals. In their work, for each of these 145 30-minute intervals, the spectral density of the corresponding time series was estimated using a lag window estimator with a Parzen window. These densities were normalized, so the total area below the curve is equal to one, and the total variation distance between all spectral densities was used to build a dissimilarity matrix, that was fed into a hierarchical agglomerative clustering algorithm. For more details on this procedure see [@olasTVD]. The 145 normalized densities are shown in figure \[fig:olasO\](a). ![Spectral densities for the sea wave data after normalization so that the integral of the densities are all equal to one. (a) Original data. (b) Original data plus 22 additional densities in black color, considered as noise. []{data-label="fig:olasO"}](olasGG.pdf "fig:") ![Spectral densities for the sea wave data after normalization so that the integral of the densities are all equal to one. (a) Original data. (b) Original data plus 22 additional densities in black color, considered as noise. []{data-label="fig:olasO"}](olasoutGGB.pdf "fig:") The RFC method was applied to this data set considering the spectral densities as functional data in order to obtain an alternative clustering. The functional form of the data was recovered using a B-splines of order $3$ with $31$ equispaced nodes. We use $100$ initializations with $20$ iterations each. The constraint level considered was $d_{1}=d_{2}=3$, and the trimming level $\alpha=0.13$. In [@olas] two different clusterings were obtained, depending on the linkage function used: 4 clusters for the complete linkage and 3 for average. We will only consider the clustering into 4 groups for comparison purposes in what follows. To compare with the results obtained using the RFC method, the Adjusted Rand Index [@Arand] was used. This Adjusted Rand Index (ARI) is an improvement with respect to the original Rand Index [@rand] and it measures the similarity between two partitions of the same set having value equal to 1 when there is exact coincidence and close to 0 in the case of considering a completely “random" partition of data into clusters. First of all, we will see that the effect of trimming and constraints is not harmful even in the case where no contaminating time series were added. For instance, we can see that the ARI of RFC with $d_1=d_2=3$ and $\alpha=0.13$ is equal to 0.513 with respect to the “reference" partition which is obtained when applying [@olas] with 4 groups. To compute this ARI index we assign all the time series (trimmed and non-trimmed) to clusters by using “posterior" probabilities from the fitted mixture model that was described in Sect 3. The two rows in Fig. \[fig:olask4\] show the clusters found when using the total variation distance and RFC, respectively. Even though in this case the groups have differences in membership and size, it is possible to see from the figures that the shape of the functions in the corresponding clusters are very similar and the mean value functions look alike. The variations are probably due to the different clustering techniques employed, but the similarity in the groups obtained point to consistent results for both clustering methods. Observe that the trimmed curves for the RFC method are different from the rest of the functions in their cluster. For “HMSClust" , both versions, single and average, gave a value of 0.723 for the ARI, higher than that obtained with RFC, while for “Funclust" , values were lower, with a maximum of 0.315 with a threshold value of 0.01 or 0.1 in the “Cattell" test. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G1TVDNC1.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G2TVDNC1.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G3TVDNC1.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G4TVDNC1.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G1RFCNCalp13.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G2RFCNCalp13.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G3RFCNCalp13.pdf "fig:") ![(top) Clusters found using the total variation distance using the complete linkage function. (bottom) Clusters found using the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Each panel corresponds to spectral densities in each cluster, grey lines represent the means and black lines represent the trimmed observations.[]{data-label="fig:olask4"}](G4RFCNCalp13.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In order to test the performance of the different clustering methods with real data and in the presence of contamination, 22 time series were added to the sample. These series are measurements recorded at the same location and during the same month of June, but during different days. The corresponding estimated spectral densities are shown in black in Fig. \[fig:olasO\](b). Some of these densities are bimodal while others are unimodal but have lower modal frequency than those in the original sample. The four clustering procedures that are being considered were applied to this contaminated sample and the obtained results were compared using again as “reference" the clustering obtained in [@olas] with 4 groups applied to the clean (i.e., before adding the contaminating curves). The ARI was computed by taking only into account the classification of the non-contaminating densities to evaluate the performance of the different methodologies. In the case of the RFC methodology, the assignments based on “posterior" probabilities were considered for the wrongly trimmed observations. Table \[tab:randW\] shows the results for the RFC method with constraints $d_1=d_2=3$ and different trimming levels. The associated ARI when using Funclust are always below 0.21 for all three Cattell thresholds tested (0.001, 0.05 and 0.1) as this method is not designed to cope with outlying curves. The other methods tested, “linkage TVD" and “Linkage HSWClust", have even worst results in this contaminated case reaching ARI values equal to 0 in both case. Therefore, the best results overall were obtained using RFC with a trimming level $\alpha=13\%$ while the other methods show poor results in the presence of contaminating data. $\alpha=0$ $\alpha=0.13$ $\alpha=0.2$ ------------ --------------- -------------- 0.167 [0.723]{} 0.596 : Adjusted Rand Index for partitions with four clusters in [@olas] in the presence of contamination. The values correspond to the RFC method with constraint levels $d_1=d_2=3$ and three different trimming levels. \[tab:randW\] To reinforce previous claims, Figure \[fig:olask5\] shows in the first row, the partition obtained in [@olas] with four clusters before adding the contaminating time series. In the second row, the results when using RFC with four clusters, $d_1=d_2=3$ and trimming level $\alpha=13\%$ to the “contaminated" data set. In the third row the results obtained with “TDVClust", then “HMSClust" and “Funclust", also in case that the contaminating time series were added. Once again, we can see that the clusters obtained with RFC differ slightly from those obtained in [@olas] but, in spite of the presence of contamination, the shape of the spectral densities in the corresponding clusters are very similar and the four average densities are very close. The trimmed functions when using level $\alpha=13\%$ are shown in black in the second row. The last three rows show the poor results obtained with the other three clustering methods. For instance, in the third row, which corresponds to “TVDClust" , all the original sample is clustered together in a single group in the leftmost panel, while the other three groups only contain contaminating functions that were added as noise. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G1TVDNC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G2TVDNC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G3TVDNC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G4TVDNC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G1RFCalp13d.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G2RFCalp13d.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G3RFCalp13d.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G4RFCalp13d.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G1TVDC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G2TVDC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G3TVDC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G4TVDC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G1HSMC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G2HSMC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G3HSMC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G4HSMC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G1JPC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G2JPC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G3JPC1.pdf "fig:") ![Clusters found with the different procedures when $K=4$. Each panel corresponds to a cluster of spectral densities found. Gray lines represent the means and black lines represent the trimmed observations. First row: Original clusters for the total variation distance using the complete linkage function before adding contaminating time series. Second row: Clusters for the RFC method for $K=4$ with constrains $d_{1}=d_{2}=3$ and trimming $\alpha=0.13$. Third row: Clusters for the “TVDClust" method with complete linkage. Fourth row: Clusters for “HSMClust" method. Fifth row: Clusters for “Funclust"[]{data-label="fig:olask5"}](G4JPC1.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Given that $\alpha=0.13$, there were 19 trimmed curves when applying the RFC procedure. Most of these trimmed densities come from the contaminating series that were added to the original sample. Finally, it is also important to point out that trimming and clustering are performed simultaneously in the RFC approach. Conclusions =========== A feasible methodology of robust clustering for stationary time series has been proposed and illustrated. The key idea behind the algorithm presented is the use of estimated spectral densities of the time series, that are then considered as functional data. A robust model-based algorithm based on approximation of the “density" for functional data, together with the simultaneous use of trimming and constraints is then used to cluster the original time series. The use of trimming tools protects the estimation of the parameters against effect of outlying curves, while the constraints avoid the presence of spurious clusters in the solution and improve the performance of the algorithms. The simulation study shows that the joint use of constraints and trimming tools improve results of the clustering algorithm in the presence of outliers, in comparison to some other procedures for time series and functional data clustering, not designed to work with contamination. The real data example shows that the proposed RFC method for time series clustering has a good performance, with or without the presence of outlying curves. The trimmed curves often correspond to curves with different characteristics to the rest. Moreover, in the presence of contamination, the RFC method is able to detected almost all the outliers in the data. In fact, we conclude that the proposed robust methodology can be a useful tool to detected contamination and groups in a time serie data set simultaneously. However, this methodology has some limitations. The choice of trimming level $\alpha$ and the choice of the scatter constraints constants $d_1$ and $d_2$, can be subjective and sometimes depend on the final purpose of the cluster analysis. For this reason, we always recommend the use of different values of trimming and constraint, monitoring the effect in the clustering partition of these choices. The development of more automated selection procedures for these values may be considered as an open problem for future research. Acknowledgements ================ Data for station 160 were furnished by the Coastal Data Information Program (CDIP), Integrative Oceanographic Division, operated by the Scripps Institution of Oceanography, under the sponsorship of the U.S. Army Corps of Engineers and the California Department of Boating and Waterways (http://cdip.ucsd.edu/). Research by DRG and JO was partially supported by Conacyt, Mexico Project 169175 Análisis Estadístico de Olas Marinas, Fase II, Research by LA G-E and A M-I was partially supported by the Spanish Ministerio de Economía y Competitividad y fondos FEDER, grant MTM2014-56235-C2-1-P, and by Consejería de Educación de la Junta de Castilla y León, grant VA212U13
--- author: - \| Elliott Ginder$^{*}$ (Meiji University)\ Ayumu Katayama (Hokkaido University) title: \| On the inclusion of damping terms in\ the hyperbolic MBO algorithm --- Introduction ============ We develop an approximation method for computing the damped motion of interfaces under hyperbolic mean curvature flow (HCMF): $$\begin{aligned} \label{hmcf} \alpha {\bm{x}}_{tt}(t,s)+\beta{\bm{x}}_{t}(t,s) = -\gamma\kappa(t,s)\nu (t,s).\end{aligned}$$ In the above, ${\bm{x}}: [0,T)\times I \rightarrow {\bf{R}}^2$ denotes a closed curve in ${\bf{R}}^2$ (parameterized over an interval $I$), $T>0$ is a final time, $\kappa$ denotes the curvature of the interface, and $\nu$ is the outward unit normal of the interface. The nonnegative parameters $\alpha,\beta,$ and $\gamma$, designate mass, damping, and surface tension coefficients, respectively. The subscripts signify differentiation with respect to their variables, so that ${\bm{x}}_{tt}$ refers to the normal acceleration of the interface, and ${\bm{x}}_t$ denotes the normal velocity. We remark that the presence of the inertial term signifies that the HMCF is an oscillatory interfacial motion. The equation of motion (\[hmcf\]) is accompanied by two initial conditions: one for the initial shape of the interface, and another prescribing the initial velocity field along the interface. It can be shown [@lefloch] that, when the initial velocity field is normal to the interface, the velocity field of the interface remains normal for the remainder of the flow. Although tangental velocities can be used to impart features such as rotation into the interfacial dynamics, our study assumes the initial velocity field to act in the normal direction of the interface. A generalized HMBO algorithm ============================ The original threshold dynamical (TD) algorithm (the so-called MBO algorithm, see [@MBO]) is a method for approximating motion by mean curvature flow (MCF). Borrowing on such ideas, a TD algorithm for hyperbolic mean curvature flow was introduced in [@label:jjiam]. Whereas previous TD algorithms utilize properties of the diffusion equation to approximate MCF, properties of wave propagation (along with a particular choice of initial condition) were used to design an approximation method for HMCF. For a time step size $\tau>0$, the error of the approximation was shown to be of the order $O(\tau).$ In this study, we will use properties of wave propagation, together with a suitable initial velocity field, to incorporate damping terms into the HMCF. Let time be discretized with a step size $\tau>0$, and $n$ be a non-negative integer. For the sake of simplicity in the exposition, let $V_n$ denote the normal velocity of the interface at the time step $n$, $\dot{V}_n$ be the normal acceleration, and $\kappa_n$ be the corresponding curvature of the interface. For the time being, we take the mass, damping, and surface tension coefficients to be unity, and proceed to construct an approximation method for the following interfacial dynamics: $$\begin{aligned} \dot{V}_{n}-V_{n}=-\kappa _{n} . \label{label:damp}\end{aligned}$$ Our approach is to observe the propagation of interfaces under the wave equation: $$\begin{aligned} \begin{cases} u_{tt}=c^{2}\Delta u,\ &\text{in}\ (0,\tau)\times \Omega \\ u(0,\bm{x})=u_{0}(\bm{x}),\ &\text{in}\ \Omega \\ u_{t}(0,\bm{x})=-v_{0}(\bm{x}),\ &\text{in}\ \Omega\\ \partial_{{\bm{n}}}u =0 &\text{on}\ (0,\tau)\times\partial\Omega, \end{cases}\end{aligned}$$ where $\Omega$ is a given domain with smooth boundary, $c^2$ sets the wave speed, $u_0$ is an initial profile, $v_0$ designates the initial velocity, and $\tau$ is the time step. Although we have prescribed a Neumann boundary condition, $\partial_{{\bm{n}}}u =0$, we will only focus on the motion of interfaces located away from the boundary of the domain. In particular, away from the boundary, the short-time solution of the wave equation can be expressed using the Poisson formula: $$\begin{aligned} u(t,\bm{x})=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{u_{0}(\bm{y})+\nabla u_{0}(\bm{y})\cdot (\bm{y}-\bm{x})-tv_{0}(\bm{y})}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y}, \label{label:poisson}\end{aligned}$$ where $B(\bm{x},ct)$ denotes the ball centered at ${\bm{x}}$ with radius $ct$. Let $\Gamma^{n}$ be the closed curve at time step $n$, described as the boundary of a set $S_n$, and denote its signed distance function by $$\begin{aligned} d_n({\bm{x}})=\begin{cases} \inf_{{\bm{y}}\in \Gamma^n}\|\|{\bm{x}}-{\bm{y}}\|\| \hspace{30pt} {\bm{x}}\in S_n\\ -\inf_{{\bm{y}}\in \Gamma^n}\|\|{\bm{x}}-{\bm{y}}\|\| \hspace{21pt}\text{otherwise.} \end{cases}\end{aligned}$$ We remark that $d_0$ is constructed from the given initial configuration of the interface, and that $d_{-1}$ can be constructed using the initial velocity field along the interface. This allows us to define $u_0({\bm{x}})$ as follows, for any non-negative integer: $$u_{0}(\bm{x})=2d_{n}(\bm{x})-d_{n-1}(\bm{x}).$$ By taking $v_{0}(\bm{x})=0$ in (\[label:poisson\]) and $c^2=2$, it can be shown (see [@label:jjiam]) that $$\begin{aligned} \delta _{n}=\delta _{n-1}-(2\kappa _{n}-\kappa _{n-1})\tau ^{2}+O(\tau ^{3}), \label{label:deltan}\end{aligned}$$ where $\delta_{n}$ denotes the distance traveled in the normal direction at time step $n$ (see figure \[fig:test\]). ![Motion of a single point of the interface in the normal direction．The point moves a distance $\delta_{n}$ at step $n.$ Without loss of generality, the direction of motion at the $n^{th}$ step is in the $x_2$ direction.[]{data-label="fig:test"}](./coordinate_system.eps) Denoting the average velocity within the time interval $[(n-1)\tau, n\tau]$ by $\bar{V}_{n}$, one has $$\begin{aligned} \delta _{n}=\bar{V}_{n}\tau,\ \ \ \ \delta _{n-1}=\bar{V}_{n-1}\tau, \notag\end{aligned}$$ and hence equation (\[label:deltan\]) can be written: $$\begin{aligned} \bar{V}_{n}\tau =\bar{V}_{n-1}\tau -(2\kappa _{n}-\kappa _{n-1})\tau ^{2}+O(\tau ^{3}). \notag\end{aligned}$$ Formally assuming $\|\kappa _{n}-\kappa _{n-1}\|<C\tau$ for some non-negative $C$, one obtains $$\begin{aligned} \bar{V}_{n}\tau =\bar{V}_{n-1}\tau -\kappa _{n}\tau ^{2}+O(\tau ^{3}), \label{label:avev}\end{aligned}$$ and hence $$\begin{aligned} \dot{V}_{n} =-\kappa _{n}+O(\tau )\hspace{30pt}\text{(as $\tau \rightarrow 0$)}. \label{label:approximate}\end{aligned}$$ The damping term in equation (\[label:damp\]) can be included by prescribing the initial velocity of the wave equation to be $v_{0}(\bm{x})=d_{n}(\bm{x})$. This can be seen by expanding $d_{n}(\bm{x})$ in a Taylor series about $\bm{x}={\bf{0}}$ (see [@label:sdf]): $$\begin{aligned} d_{n}(x_{1},x_{2})=x_{2}+\dfrac{1}{2}\kappa _{n}x_{1}^{2}+\dfrac{1}{6}(\kappa _{n})_{x_{1}}x_{1}^{3}-\dfrac{1}{2}\kappa _{n}^{2}x_{1}^{2}x_{2}+O(\|\bm{x}\|^{4}),\end{aligned}$$ and appealing to Poisson’s formula (\[label:poisson\]): $$\begin{aligned} u_{v}(t,\bm{x})=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{-tv_{0}(\bm{y})}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y}.\end{aligned}$$ Making the change of variables: $$\begin{aligned} \bm{y}-\bm{x}&=ct\bm{z}, \label{label:hh}\end{aligned}$$ we note that $$\begin{aligned} O(\|\bm{y}\|^{4})=O(t^{4})\hspace{30pt}\text{(as $t\rightarrow 0$)}. \label{label:order}\end{aligned}$$ We thus investigate the contribution of the first four terms in the Taylor expansion, $u_{v}^{1},u_{v}^{2}, u_{v}^{3},$ and $u_{v}^{4}$. We begin with the lowest order term: $$\begin{aligned} u_{v}^{1}(t,\bm{x})&=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{-ty_{2}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{-t}{2\pi ct}\int _{B(0,1)} \dfrac{ctz_{2}+x_{2}}{ct\sqrt{1-\|\bm{z}\|^{2}}}(ct)^{2}d\bm{z}. \notag\end{aligned}$$ Appealing to function parity we have: $$\begin{aligned} \int _{B(0,1)} \dfrac{z_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}=0,\end{aligned}$$ and it follows that $$\begin{aligned} u_{v}^{1}(t,\bm{x})&=\dfrac{-t}{2\pi }\int _{B(0,1)} \dfrac{x_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \end{aligned}$$ By making the change of variables: $$\begin{aligned} z_{1}=r\text{cos}\theta, \hspace{15pt} z_{2}=r\text{sin}\theta, \label{label:varch1}\end{aligned}$$ one arrives at $$\begin{aligned} u_{v}^{1}(t,\bm{x})&=\dfrac{-tx_{2}}{2\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{1}{\sqrt{1-r^{2}}}rd\theta dr \notag \\ &=\dfrac{-tx_{2}}{2\pi }\int _{0}^{1} \dfrac{2\pi r}{\sqrt{1-r^{2}}}dr \notag \\ &=-tx_{2}\int _{0}^{1} \dfrac{r}{\sqrt{1-r^{2}}}dr. \notag \end{aligned}$$ Composite function integration yields $$\begin{aligned} u_{v}^{1}(t,\bm{x})&=-tx_{2}\int _{0}^{1} (1-r^{2})^{-\frac{1}{2}}(-2r)\left( -\dfrac{1}{2} \right) dr =-tx_{2}. \label{label:v1}\end{aligned}$$ We next consider the influence of the second term: $$\begin{aligned} u_{v}^{2}(t,\bm{x})&=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{-\dfrac{1}{2}t\kappa _{n}y_{1}^{2}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{-t\kappa _{n}}{4\pi ct}\int _{B(\bm{x},ct)} \dfrac{y_{1}^{2}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{-t\kappa _{n}}{4\pi }\int _{B(0,1)} \dfrac{c^{2}t^{2}z_{1}^{2}+2ctx_{1}z_{1}+x_{1}^{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \notag \end{aligned}$$ As before, function parity yields $$\begin{aligned} \int _{B(0,1)} \dfrac{z_{1}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}=0,\end{aligned}$$ and hence $$\begin{aligned} u_{v}^{2}(t,\bm{x})&=\dfrac{-t\kappa _{n}}{4\pi }\int _{B(0,1)} \dfrac{c^{2}t^{2}z_{1}^{2}+x_{1}^{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \notag \end{aligned}$$ Making the change of variables (\[label:varch1\]), we have $$\begin{aligned} u_{v}^{2}(t,\bm{x})&=\dfrac{-t\kappa _{n}}{4\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{c^{2}t^{2}r^{3}\text{cos}^{2}\theta +x_{1}^{2}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{-t\kappa _{n}}{4\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{c^{2}t^{2}r^{3}\dfrac{1+\text{cos}2\theta }{2} +x_{1}^{2}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{-t\kappa _{n}}{4\pi }\int _{0}^{1} \dfrac{\pi \left(c^{2}t^{2}r^{3} +2x_{1}^{2}r\right)}{\sqrt{1-r^{2}}}dr. \notag \end{aligned}$$ Using another change of variables: $$\begin{aligned} r=\text{cos}\theta, \label{label:varch2}\end{aligned}$$ allows one to obtain: $$\begin{aligned} u_{v}^{2}(t,\bm{x})&=\dfrac{-t\kappa _{n}}{4}\int _{0}^{\frac{\pi}{2}} \dfrac{c^{2}t^{2}\text{cos}^{3}\theta +2x_{1}^{2}\text{cos}\theta}{\sqrt{1-\text{cos}^{2}\theta}}\text{sin}\theta d\theta \notag \\ &=\dfrac{-t\kappa _{n}}{4}\int _{0}^{\frac{\pi}{2}} \left( c^{2}t^{2}\dfrac{\text{cos}3\theta +3\text{cos}\theta }{4} +2x_{1}^{2}\text{cos}\theta \right) d\theta \notag \\ &=\dfrac{-t\kappa _{n}}{4}\left( c^{2}t^{2}\left( -\dfrac{1}{12} +\dfrac{3}{4} \right) +2x_{1}^{2} \right) \notag \\ &=-t\kappa _{n}\left( \dfrac{c^{2}t^{2}}{6} +\dfrac{x_{1}^{2}}{2}\right) . \label{label:v2}\end{aligned}$$ The third term is similar: $$\begin{aligned} u_{v}^{3}(t,\bm{x})&=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{-\dfrac{1}{6}t(\kappa _{n})_{x_{1}}y_{1}^{3}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi ct}\int _{B(\bm{x},ct)} \dfrac{y_{1}^{3}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi }\int _{B(0,1)} \dfrac{z_{1}^{3}+3c^{2}t^{2}x_{1}z_{1}^{2}+3ctx_{1}^{2}z_{1}+x_{1}^{3}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \notag \end{aligned}$$ Again appealing to function parity: $$\begin{aligned} \int _{B(0,1)} \dfrac{z_{1}^{3}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}=0,\end{aligned}$$ and therefore $$\begin{aligned} u_{v}^{3}(t,\bm{x})&=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi }\int _{B(0,1)} \dfrac{3c^{2}t^{2}x_{1}z_{1}^{2}+x_{1}^{3}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \notag\end{aligned}$$ The change of variables (\[label:varch1\]) gives $$\begin{aligned} u_{v}^{3}(t,\bm{x})&=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{3c^{2}t^{2}x_{1}r^{3}\text{cos}^{2}\theta +x_{1}^{3}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{3c^{2}t^{2}x_{1}r^{3}\dfrac{1+\text{cos}2\theta }{2} +x_{1}^{3}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12\pi }\int _{0}^{1} \dfrac{\pi \left(3c^{2}t^{2}x_{1}r^{3} +2x_{1}^{3}r\right)}{\sqrt{1-r^{2}}}dr, \notag\end{aligned}$$ while (\[label:varch2\]) allows one to express: $$\begin{aligned} u_{v}^{3}(t,\bm{x})&=\dfrac{-t(\kappa _{n})_{x_{1}}}{12}\int _{0}^{\frac{\pi}{2}} \dfrac{3c^{2}t^{2}x_{1}\text{cos}^{3}\theta +2x_{1}^{3}\text{cos}\theta}{\sqrt{1-\text{cos}^{2}\theta}}\text{sin}\theta d\theta \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12}\int _{0}^{\frac{\pi}{2}} \left( 3c^{2}t^{2}x_{1}\dfrac{\text{cos}3\theta +3\text{cos}\theta }{4} +2x_{1}^{3}\text{cos}\theta \right) d\theta \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{12}\left( 3c^{2}t^{2}x_{1}\left( -\dfrac{1}{12} +\dfrac{3}{4} \right) +2x_{1}^{3} \right) \notag \\ &=\dfrac{-t(\kappa _{n})_{x_{1}}}{6}\left( c^{2}t^{2}x_{1} +x_{1}^{3}\right). \label{label:v3}\end{aligned}$$ The final term follows the same approach: $$\begin{aligned} u_{v}^{4}(t,\bm{x})&=\dfrac{1}{2\pi ct}\int _{B(\bm{x},ct)} \dfrac{\dfrac{1}{2}t\kappa _{n}^{2}y_{1}^{2}y_{2}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4\pi ct}\int _{B(\bm{x},ct)} \dfrac{y_{1}^{2}y_{2}}{\sqrt{c^{2}t^{2}-\|\bm{y}-\bm{x}\|^{2}}}d\bm{y} \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4\pi }\int _{B(0,1)} \dfrac{c^{3}t^{3}z_{1}^{2}z_{2}+c^{2}t^{2}x_{2}z_{1}^{2}+2c^{2}t^{2}x_{1}z_{1}z_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}\notag\\&+\dfrac{2ctx_{1}x_{2}z_{1}+ctx_{1}^{2}z_{2}+x_{1}^{2}x_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z} . \notag \end{aligned}$$ Function parity tells us that $$\begin{aligned} \int _{B(0,1)} \dfrac{z_{1}^{2}z_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}=0,\end{aligned}$$ and hence $$\begin{aligned} u_{v}^{4}(t,\bm{x})&=\dfrac{t\kappa _{n}^{2}}{4\pi }\int _{B(0,1)} \dfrac{c^{2}t^{2}x_{2}z_{1}^{2}+2c^{2}t^{2}x_{1}z_{1}z_{2}+x_{1}^{2}x_{2}}{\sqrt{1-\|\bm{z}\|^{2}}}d\bm{z}. \notag\end{aligned}$$ Applying the change of variables (\[label:varch1\]) and computing gives $$\begin{aligned} u_{v}^{4}(t,\bm{x})&=\dfrac{t\kappa _{n}^{2}}{4\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{c^{2}t^{2}x_{2}r^{3}\text{cos}^{2}\theta +2c^{2}t^{2}x_{1}r^{3}\text{cos}\theta \text{sin}\theta +x_{1}^{2}x_{2}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4\pi }\int _{0}^{1}\int _{0}^{2\pi } \dfrac{c^{2}t^{2}x_{2}r^{3}\dfrac{1+\text{cos}2\theta }{2}+c^{2}t^{2}x_{1}r^{3}\text{sin}2\theta +x_{1}^{2}x_{2}r}{\sqrt{1-r^{2}}}d\theta dr \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4\pi }\int _{0}^{1} \dfrac{\pi \left(c^{2}t^{2}x_{2}r^{3} +2x_{1}^{2}x_{2}r\right)}{\sqrt{1-r^{2}}}dr. \notag\end{aligned}$$ Using the change of variables (\[label:varch2\]) leads us to the expession: $$\begin{aligned} &=\dfrac{t\kappa _{n}^{2}}{4}\int _{0}^{\frac{\pi}{2}} \dfrac{c^{2}t^{2}x_{2}\text{cos}^{3}\theta +2x_{1}^{2}x_{2}\text{cos}\theta}{\sqrt{1-\text{cos}^{2}\theta}}\text{sin}\theta d\theta \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4}\int _{0}^{\frac{\pi}{2}} \left( c^{2}t^{2}x_{2}\dfrac{\text{cos}3\theta +3\text{cos}\theta }{4} +2x_{1}^{2}x_{2}\text{cos}\theta \right) d\theta \notag \\ &=\dfrac{t\kappa _{n}^{2}}{4}\left( c^{2}t^{2}x_{2}\left( -\dfrac{1}{12} +\dfrac{3}{4} \right) +2x_{1}^{2}x_{2} \right) \notag \\ &=t\kappa _{n}^{2}\left( \dfrac{c^{2}t^{2}x_{2}}{6} +\dfrac{x_{1}^{2}x_{2}}{2}\right). \label{label:v4}\end{aligned}$$ Equations (\[label:v1\])，(\[label:v2\])，(\[label:v3\])，and (\[label:v4\]) together express $$\begin{aligned} u_{v}(t,\bm{x})=&-t\left( x_{2}+\kappa _{n}\left( \dfrac{c^{2}t^{2}}{6} +\dfrac{x_{1}^{2}}{2}\right)+\dfrac{(\kappa _{n})_{x_{1}}}{6}\left( c^{2}t^{2}x_{1} +x_{1}^{3}\right) \right)\\ &+t \kappa _{n}^{2}\left( \dfrac{c^{2}t^{2}x_{2}}{6} +\dfrac{x_{1}^{2}x_{2}}{2}\right). \notag\end{aligned}$$ Upon taking $t=\tau$ and $\bm{x}=(0,\delta _{n})$, we arrive at $$\begin{aligned} 0&=-\tau \left( \delta _{n}+\kappa _{n}\dfrac{c^{2}\tau ^{2}}{6}-\kappa _{n}^{2}\dfrac{c^{2}\tau ^{2}\delta _{n}}{6} \right) =-\delta _{n}\tau +O(\tau ^{3}). \label{label:uv} \end{aligned}$$ Combining this equation with our previous results yields: $$\begin{aligned} &\delta _{n}-\delta _{n-1}-\delta _{n}\tau =-(2\kappa _{n}-\kappa _{n-1})\tau ^{2}+O(\tau ^{3}). \label{label:delv}\end{aligned}$$ Writing $\delta _{n}=\bar{V}_{n}\tau $ and $\delta _{n-1}=\bar{V}_{n-1}\tau$ in equation (\[label:delv\]) expresses $$\begin{aligned} \bar{V}_{n}\tau -\bar{V}_{n-1}\tau -\bar{V}_{n}\tau ^{2}=-(2\kappa _{n}-\kappa _{n-1})\tau ^{2}+O(\tau ^{3}). \notag\end{aligned}$$ Formally assuming $\|\kappa _{n}-\kappa _{n-1}\|<C\tau$, for some constant $C$, and dividing both sides by $\tau ^{2}$ gives $$\begin{aligned} \dfrac{\bar{V}_{n}-\bar{V}_{n-1}}{\tau } -\bar{V}_{n}=-\kappa _{n}+O(\tau ). \notag\end{aligned}$$ It follows that the damping term enters the equation of motion: $$\begin{aligned} \dot{V}_{n} -V_{n}=-\kappa _{n}+O(\tau ). \label{label:dampe}\end{aligned}$$ By linearity, taking $u_{0}(\bm{x})=a(2d_{n}(\bm{x})-d_{n-1}(\bm{x}))$ and $v_{0}(\bm{x})=bd_{n}(\bm{x})$ one can obtain the interfacial motion: $$\begin{aligned} a\dot{V}_{n} -bV_{n}=-\dfrac{ac^{2}}{2}\kappa _{n}+O(\tau ), \label{label:dampp}\end{aligned}$$ where $a$ and $b$ are real parameters. Therefore, one can rewrite the parameters: $$\begin{aligned} \alpha =a, \hspace{15pt} \beta =-b, \hspace{15pt} \gamma =\dfrac{ac^{2}}{2}, \label{label:para} \end{aligned}$$ to approximate a prescribed interfacial motion: $$\begin{aligned} \alpha\dot{V} +\beta V=-\gamma \kappa+O(\tau ). \label{label:gdamp}\end{aligned}$$ The previous results show that the wave equation’s initial velocity can be used in the HMBO algorithm to impart damping terms. In the next section, by choosing parameters, we will make a numerical investigation into using the HMBO to approximate interfacial motion by the standard mean curvature flow. The HMBO approximation of mean curvature flow ============================================= An approximation method for mean curvature flow can be obtained by returning to equation (\[label:poisson\]) and choosing appropriate initial conditions. For a predetermined time step $\tau>0$, we take $u_{0}(\bm{x})=0$, $v_{0}(\bm{x})=d_{n}(\bm{x})$, and $c^{2}= \lambda / \tau $. Then equation (\[label:uv\]) gives $$\begin{aligned} 0&=-\tau \left( \delta _{n}+\kappa _{n}\dfrac{\lambda \tau}{6}-\kappa _{n}^{2}\dfrac{\lambda \delta _{n}\tau}{6} \right)+O(\tau ^{3}), \notag \\ &=\delta _{n}+\kappa _{n}\dfrac{\lambda \tau}{6}-\kappa _{n}^{2}\dfrac{\lambda \delta _{n}\tau}{6}+O(\tau ^{2}). \notag\end{aligned}$$ Preceeding as in the previous section, we obtain $$\begin{aligned} \bar{V}_{n}=-\dfrac{\lambda }{6}\kappa _{n}+O(\tau ). \notag\end{aligned}$$ Since $\lambda $ is a free parameter, we find that the corresponding threshold dynamics can approximate curvature flow with a parameter $\gamma$: $$\begin{aligned} V_{n}=-\gamma\kappa _{n}+O(\tau )\hspace{30pt}(\text{as $\tau \rightarrow 0$}). \label{label:mboh}\end{aligned}$$ Numerical investigation ======================= We will now perform a numerical error analysis of the HMBO approximation of MCF. The numerical method’s performance will be compared to the case of a circle evolving by MCF. In such a setting, the evolution of the circle’s radius is governed by the solution of the following ordinary differential equation: $$\begin{aligned} \begin{cases} \dot{r}(t)=-\dfrac{1}{r(t)} \ \ \ \ t>0, \\ r(0)=r_{0}, \label{label:exr} \end{cases}\end{aligned}$$ where $r_0$ is the initial radius of the circle. We remark that the radius decreases until its extinction time $t_{e}= r_{0}^{2}/2$, and that $r(t)= \sqrt{r_{0}^{2}-2t}$. The HMBO approximation method solves the following wave equation for a small time $\tau>0$: $$\begin{aligned} \begin{cases} u_{tt}=c^{2}\Delta u\ &\text{in}\ (0,\tau)\times \Omega \\ u(0,\bm{x})=0\ &\text{in}\ \Omega \\ u_{t}(0,\bm{x})=d_{k}(\bm{x})\ & \text{in}\ \Omega \\ \partial_{\boldsymbol{n}}u=0 \ &\text{on}\ \partial \Omega, \end{cases} \label{label:ehmbo}\end{aligned}$$ where $\Omega =(-2,2)\times (-2,2)$ and $k$ denotes the $k^{th}$ step of the HMBO algorithm. We choose the initial interface to be a circle with radius one, so that $$\begin{aligned} &d_{0}(\bm{x})=\|\|\bm{x}\|\|-1.\end{aligned}$$ The initial velocity at the $k^{th}$ step is then defined as the signed distance function to the zero level set of the solution to the wave equation: $$\begin{aligned} d_k({\bm{x}})=\begin{cases} \inf_{{\bm{y}}\in \partial\{u({\bm{x}},\tau)>0\}}\|\|{\bm{x}}-{\bm{y}}\|\| \hspace{30pt} {\bm{x}}\in \{u({\bm{x}},\tau)>0\}\\ -\inf_{{\bm{y}}\in \partial\{u({\bm{x}},\tau)>0\}}\|\|{\bm{x}}-{\bm{y}}\|\| \hspace{21pt}\text{otherwise.} \end{cases}\end{aligned}$$ Since the extinction time $t_e$ depends on $r_0$, we set $\tau=t_{e}/N_{\tau}$. Here $r_{0}=1$ (hence $t_{e}=0.5$), and we set $N_{\tau}=150$ to ensure a level of precision. The time step is then $\tau =3.33\times 10^{-3}$. The target problem (\[label:exr\]) corresponds to $\gamma=1$ in equation (\[label:mboh\]), and we thus set $c^{2}=6/\tau $. Finite differences are used to numerically solve the wave equation with a time step $\Delta t=2.22\times 10^{-6}$. The grid spacing in the $x$ and $y$ directions are equal to $\Delta x=2/(N-1)$, where $N$ is a natural number. We examine the numerical error when $N=2^{j},$ for $j=4,5,6,7,8$. The numerical results are shown in figure \[label:figte\], where the radius of the numerical solution is defined to be the average distance $\tilde{r}(t)$ of the level set’s point cloud to the origin. ![Convergence of the approximation method as $N$ is increased.[]{data-label="label:figte"}](./te_unkey_new2.pdf) The error is measured using the quantity: $$\begin{aligned} Err(t)=\int _{0}^{T}\| r(t)-\tilde{r}(t)\|dt. \label{label:err}\end{aligned}$$ Since the extinction time of the numerical solution differs from the exact solution, the actual error is computed as follows: $$\begin{aligned} Err(t)\approx \displaystyle\sum_{i=0}^{N_{s}}\|r(i\tau )-\tilde{r}(i\tau )\|\tau, \label{label:erra}\end{aligned}$$ where $N_{s}$ denotes the number of time steps until the numerical solution’s radius disappears (the corresponding time is $N_{s}\tau$). Our results are summarized in table (\[label:erra\]), where we observe the convergence of our method to the exact solution. $N$ $N_{s}\tau $ $Err$ ----- -------------- ---------- 16 0.223333 0.044613 32 0.343333 0.039463 64 0.436667 0.022746 128 0.473333 0.008509 256 0.486667 0.003907 : Error Table with respect to $\Delta x.$[]{data-label="label:taberr1"} Acknowledgments =============== E. Ginder would like to acknowledge the support of JSPS Kakenhi Grant Number 17K14229, as well as that from the Presto Research Program of the Japan Science and Technology Agency. [99]{} Y. G. Chen, Y. Giga, and S. Goto. “Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations" Journal of Differential Geometry, Vol. 33, Number 3 (1991), 749-786. S. Essedoglu, S. Ruuth, R. Tsai. “Diffusion generated motion using signed distance functions" J. Comp. Phys., 229, 4 (2010), 1017-1042. E. Ginder, K. Svadlenka. “Wave-type threshold dynamics and the hyperbolic mean curvature flow" Japan Journal of Industrial Applied Mathematics, doi 10.1007/s13160-016-0221-0, (2016). P. G. LeFloch, K. Smoczyk. “The hyperbolic mean curvature flow" Journal de Math$\acute{\text{e}}$matiques Pures et Appliqu$\acute{\text{e}}$es, Vol. 90, Issue 6 (2008), 591-614. B. Merriman, J. Bence, S. Osher. “Diffusion Generated Motion by Mean Curvature” UCLA CAM, (1992), 1-11. S. Osher, R. Fedkiw. “Level Set Methods and Dynamic Implicit Surfaces" Applied Mathematical Science, (2003). R. C. Reilly. “Mean Curvature, The Laplacian, and Soap Bubbles" The American Mathematical Monthly, Vol. 89, No. 3 (1982), 180-188. S. Shin, D. Juric. “High Order Level Contour Reconstruction Method" Journal of Mechanical Science and Technology, Vol. 21, (2007), 311-326. ifdraftMine
--- abstract: 'We solve Einstein’s constraint equations in the conformal thin-sandwich decomposition to model thin shells of non-interacting particles in circular orbit about a non-rotating black hole. We use these simple models to explore the effects of some of the freely specifiable quantities in this decomposition on the physical content of the solutions. Specifically, we adopt either maximal slicing or Kerr-Schild slicing, and make different choices for the value of the lapse on the black hole horizon. For one particular choice of these quantities the resulting equations can be solved analytically; for all others we construct numerical solutions. We find that these different choices have no effect on our solutions when they are expressed in terms of gauge-invariant quantities.' author: - Keith Matera - 'Thomas W. Baumgarte' - Eric Gourgoulhon date: 8 November 2007 title: 'Shells around black holes: the effect of freely specifiable quantities in Einstein’s constraint equations' --- Introduction ============ A 3+1 decomposition of Einstein’s equations results in a set of constraint equations, which constrain the gravitational fields at all instants of coordinate time, and a set of evolution equations, which propagate the fields forward in time (e.g. ). The four constraint equations can constrain only a subset of the gravitational fields. Therefore, the constraint equations can be solved, for example for the construction of initial data, only after the constrained variables have been separated from freely specifiable ones, and after suitable choices have been made for the latter (see e.g. [@Pfe04; @Gou07b] for reviews). The constrained variables are separated from the freely specifiable ones by choosing a decomposition of the constraint equations. The conformal thin-sandwich decomposition [@Yor99; @PfeY03] has been particularly popular for the construction of quasiequilibrium data; it has been used extensively, for example, to model compact binaries containing black holes or neutron stars (see, e.g., for reviews). In the conformal thin-sandwich formalism, the spatial metric is conformally decomposed into a conformal factor and the conformally related metric, and the extrinsic curvature into its trace and a traceless part. In the so-called extended version [@PfeY03], the freely specifiable variables are the conformally related metric and the trace of the extrinsic curvature together with their time derivatives (which we may set to zero to construct equilibrium data), and the constrained variables are the lapse, the shift, and the conformal factor. Black holes may be constructed within the conformal thin-sandwich formalism by excising the black hole interior, and imposing suitable inner boundary conditions. In particular, these boundary conditions may be chosen so that the black hole is momentarily isolated, or in equilibrium (see [@Coo02; @CooP04; @JarGM04], also compare the isolated horizon formalism laid out in [@AshK04; @DreKSS03; @GouJ06] and references therein). As discussed in detail in [@CooP04], these geometric conditions lead to boundary conditions on some of the constrained variables in the thin-sandwich formalism, namely the conformal factor and the shift vector. The boundary condition for the lapse, however, remains arbitrary. Some of the choices in this formalism will clearly have an effect on the physical content of the solution. We can expect to find equilibrium solutions only if we set the time derivatives of the conformally related metric and the trace of the extrinsic curvature to zero. Also, a conformally flat solution is physically distinct from solutions that are not conformally flat. The choice of the trace of the extrinsic curvature, or mean curvature, is usually associated with an initial temporal gauge, and the lapse plays a similar role. It is less clear, then, whether or how the mean curvature and the boundary condition on the lapse affect the solutions. In [@CooP04], the authors found that sequences of binary black holes, and in particular their innermost stable circular orbit, do depend on the horizon lapse for their example of a non-maximal slicing, i.e. non-zero mean curvature. This finding, however, may be an artifact of their particular choice of the mean curvature, namely a superposition of two copies of its analytical value for a single Schwarzschild black hole expressed in Kerr-Schild coordinates (see (\[K\_KS\]) below), one for each companion in the binary. As the authors caution, the resulting background geometry then depends on binary separation, making the physical meaning of these sequences somewhat arguable. We consider a very simple physical system in order to analyze whether, at least in this context, the choice of the mean curvature and the horizon lapse affect the physical content of the solutions. Specifically, we solve the constraint equations in the thin-sandwich decomposition to construct thin shells of non-interacting, isotropic particles in circular orbit about a Schwarzschild black hole (compare [@SkoB02], whose results we generalize to account for the black hole). For one particular choice of the mean curvature and the horizon lapse the equations can be solved analytically (see Appendix \[appA\]), and we construct numerical solutions for many others. We find that these different choices have no effect on our solutions when they are expressed in terms of gauge-invariant quantities. Basic equations =============== Constraint equations -------------------- We write the spacetime metric $g_{ab}$ in the form $$g_{ab} dx^a dx^b = - \alpha^2 dt^2 + \gamma_{ij} (dx^i + \beta^i dt) (dx^j + \beta^j dt),$$ where $\alpha$ is the lapse function, $\beta^i$ the shift vector, and $\gamma_{ij}$ the spatial metric. We further decompose the latter as $$\label{metric} \gamma_{ij} = \psi^4 \bar \gamma_{ij},$$ where $\psi$ is a conformal factor and $\bar \gamma_{ij}$ a conformally related metric. We then solve Einstein’s constraint equations in the conformal thin-sandwich decomposition (see , as well as [@Coo00; @BauS03; @Gou07b] for reviews). Specifically, the Hamiltonian constraint becomes $$\begin{aligned} \label{psiConstraint} \bar D^2 \psi = \frac{1}{8} \psi \bar{R} + \frac{1}{12}\psi^5 K^2 - \frac{1}{8} \psi^{-7} \bar A_{ij} \bar A^{ij} - 2 \pi \psi^5 \rho_N.\end{aligned}$$ Here $\rho_N = T_{ab} n^a n^b $ is the energy density as measured by a normal observer, $\bar D^2 = \bar \gamma^{ij} \bar D_i \bar D_j$, and $\bar D_i$ and $\bar{R}$ are the covariant derivative and the Ricci scalar associated with the metric $\bar \gamma_{ij}$. We have also split the extrinsic curvature $K_{ij}$ into its trace $K$ and a traceless part $A_{ij}$ according to $$K_{ij} =A_{ij} +\frac{1}{3} \gamma_{ij} K =\psi^{-2} \bar A_{ij} + \frac{1}{3} \gamma_{ij} K.$$ From the evolution equation for the spatial metric we can express $ \bar A^{ij}$ as $$\label{matrixA} \bar A^{ij} = \frac{1}{2\bar \alpha} \left( \left( \bar L \beta \right)^{ij}-\bar u^{ij} \right).$$ Here $\bar \alpha = \psi^{-6} \alpha$ and $\bar{u}^{ij} = \partial_t \bar \gamma_{ij}$, and the conformal Killing operator $\bar{L}$ is defined as $$\label{Lbar} \left( \bar L \beta \right) ^{ij} \equiv \bar D^i \beta^j + \bar D^j \beta^i -\frac{2}{3} \bar \gamma^{ij} \bar D_k \beta^k .$$ The momentum constraint can now be written as $$\begin{aligned} \label{betaConstraint} \left( \bar \Delta_L \beta \right)^i &=& \left( \bar L \beta \right)^ {ij} \bar D_j \ln \left( \bar \alpha \right)+ \bar \alpha \bar D_j \left( \bar \alpha^{-1} \bar u^{ij} \right) \nonumber \\ &&+\frac{4}{3} \bar \alpha \psi^6 \bar D^i K + 16 \pi \bar \alpha \psi^{10} j^i,\end{aligned}$$ where $(\bar \Delta_L \beta)^i = D_j (\bar L \beta)^{ij}$ is a vector Laplacian, and $j^i = - \gamma^{ia} n^b T_{ab}$ is the mass current as measured by a normal observer. Finally, the trace of the evolution equation for $K_{ij}$, combined with the Hamiltonian constraint, results in $$\begin{aligned} \label{alphaConstraint} \bar{D}^2 \left( \alpha \psi \right) & = & \alpha \psi \Big( \frac{7} {8} \psi^{-8} \bar A_{ij} \bar A^{ij} + \frac{5}{12} \psi^4 K^2 + \frac{1}{8}\bar{R} \\ &&+ 2 \pi \psi^4 \left( \rho + 2S \right) \Big) - \psi^5 \partial_t K + \psi^5 \beta^i \bar{D}_i K, \nonumber\end{aligned}$$ where $S=\gamma^{ij} T_{ij}$ the trace of the spatial stress. The above equations form a set of equations for the lapse $\alpha$, the shift $\beta^i$ and the conformal factor $\psi$. Before these equations can be solved, however, we have to make choices for the freely specifiable quantities $\bar \gamma_{ij}$, $\bar u_{ij} = \partial_t \bar \gamma_{ij}$, $K$ and $\partial_t K$. For the construction of quasiequilibrium data it is natural to choose $\bar u_ {ij} =0$ and $\partial_t K = 0$. We will also restrict our analysis to spherical symmetry, where we may assume conformal flatness, $\bar \gamma_{ij} = \eta_{ij}$, without loss of generality. Here $\eta_{ij} $ is the flat metric in whatever coordinate system. We will, however, experiment with different choices for $K$ (see equations (\[K\]) below), as well as with different boundary conditions for the lapse $\alpha$ (see (\[alphaHorBound\]) below). With these choices, and in spherical symmetry, the above equations simplify dramatically. We write the spatial metric as $$\gamma_{ij} dx^i dx^j = \psi^4 \left( dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2) \right),$$ where $r$ is the isotropic radial coordinate. The shift $\beta^i$ is now purely radial, and we abbreviate $\beta \equiv \beta^r$. We may evaluate (\[Lbar\]) to find $$\left( \bar L \beta \right)^{ij} = \displaystyle - \frac{2}{3r} \left( \begin{array}{ccc} -2 r^2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \displaystyle \frac{1}{\sin^2\theta} \\ \end{array} \right) \partial_r \left(\dfrac{\beta}{r}\right),$$ so that $\bar{A}_{ij}\bar{A}^{ij}$ becomes $$\bar{A}_{ij}\bar{A}^{ij} = \frac{2}{3 \bar{\alpha}^2}r^2\left (\partial_r \frac{\beta}{r}\right)^2.$$ The Hamiltonian constraint (\[psiConstraint\]) can then be written as \[DiffEq\] $$\begin{aligned} \label{psiConstraintMod} r \partial_r^2 \psi & + & 2 \partial_r \psi + r \frac{\psi^5}{12} \left( \frac{ \left( \partial_r \beta -\beta / r \right)^2}{\alpha^2} - K^2 \right) \nonumber \\ & = & -2 \pi \psi^5 \rho_N,\end{aligned}$$ the momentum constraint (\[betaConstraint\]) as $$\begin{aligned} \label{betaConstraintMod} \partial_r^2 \beta & + & \left( \displaystyle \frac{2}{r} - \frac {\partial_r \alpha}{\alpha} + 6 \frac{\partial_r \psi}{\psi} \right) \left(\partial_r \beta - \displaystyle \frac{\beta}{r}\right) \nonumber \\ & = & \alpha \partial_r K + 12 \pi \psi^4 \alpha j^i,\end{aligned}$$ and the lapse equation (\[alphaConstraint\]) as $$\begin{aligned} \label{alphaConstraintMod} \partial_r^2 \left( \alpha \psi \right) &=& \alpha\psi \Big(\frac{7 \psi^4}{12\alpha^2} \big( \partial_r \beta - \frac{\beta}{r} \big)^2 + \frac{5}{12} \psi^4 K^2 \nonumber \\ && + 2 \pi \psi^4 \left(\rho + 2 S \right) + \psi^5 \beta \partial_r K \Big).\end{aligned}$$ Here we have expressed all quantities in terms of those variables that are used in our code. In the above equations the trace of the extrinsic curvature $K$ can still be chosen arbitrarily. Following [@CooP04] we consider two different possibilities, namely maximal slicing \[K\] $$K_{\rm MS} = 0$$ and Kerr-Schild slicing $$\label{K_KS} K_{\rm KS} = \frac{2 M_{\rm BH}}{R^2} \left(1+\frac{2M_{\rm BH}}{R} \right)^{-3/2} \left(1+\frac{3M_{\rm BH}}{R} \right).$$ Here $R$ is the areal radius, $R=\psi^2 r$, and we identify $M_{\rm BH}$ with the black hole’s irreducible mass (see (\[Mirr\]) below). Kerr-Schild coordinates are identical to ingoing Eddington-Finkelstein coordinates. Throughout this paper, we use the subscript $\rm_{BH}$ to refer to the black hole, and $\rm_{SH}$ to indicate a property of the shell. Boundary conditions {#sec:BoundCond} ------------------- At spatial infinity we impose asymptotic flatness, which results in the boundary conditions $$\label{infBound} \psi \rightarrow 1, ~~~~ \alpha \rightarrow 1, ~~~~ \beta \rightarrow 0$$ as $r \rightarrow \infty$. We excise the black hole interior inside an isotropic radius $r_{\rm BH}$ and impose the black hole equilibrium boundary conditions of @CooP04 on the resulting excision surface ${\mathcal S}$ (compare the notion of isolated horizons laid out in ). In particular, the condition $$\label{psiHorBound} \left. m^{ab} \nabla_a k_b \right\|_{\mathcal S} = \left. m^{ij} \left( D_i s_j - K_{ij} \right) \right\|_{\mathcal S} = 0,$$ where $s^i$ is the outward-pointing unit normal to the horizon, ensures that this surface corresponds to a marginally trapped surface (apparent horizon), while the condition $$\label{betaHorBound} \left. \beta_{\perp} \right\|_{\mathcal S} \equiv \left. \beta^r s_r \right\|_{\mathcal S} = \left. \alpha \right\|_{\mathcal S}$$ ensures that the coordinate system tracks the horizon. (The tangential components of the shift vanish identically in spherical symmetry.) In our case, (\[psiHorBound\]) becomes \[InBoundCond\] $$\label{psiHorBoundMod} \left( \partial_r \psi + \frac{\psi}{2r} + \frac{\psi^3}{6} \left (\frac{\partial_r \beta - \beta/r}{\alpha} - K \right) \right)_ {\mathcal S} = 0$$ and (\[betaHorBound\]) $$\label{betaHorBoundMod} \left. \beta \psi^2 \right\|_{\mathcal S} = \left. \alpha \right\|_ {\mathcal S}.$$ As discussed in [@CooP04], the boundary condition for the lapse $ \alpha$ is arbitrary. We will experiment with the Dirichlet boundary condition $$\label{alphaHorBound} \left. \alpha \right\|_{\mathcal S} = \alpha_{\rm AH},$$ and will compare results for values of $\alpha_{\rm AH}$ ranging from zero to unity in increments of 0.1. Matter equations ---------------- We consider a spherically symmetric shell of isotropic, non-interacting particles in circular orbit about the black hole (compare [@SkoB02]). The rest energy (baryon mass) of a spherically symmetric matter source may be written as $$M_{\rm SH} = \int \rho_0 u^t \sqrt{-g}d^3x = 4 \pi \int \rho_0 W \psi^6 r^2 dr,$$ where $W \equiv - n_{\alpha}u^{\alpha} = \alpha u^t$ is the Lorentz factor between a normal observer $n^a$ and an observer comoving with the fluid $u^a$. For an infinitesimally thin shell we can then identify the rest energy density (baryon density) as $$\label{restEnergyDensity} \rho_0 = \frac{M_{\rm SH}}{4 \pi W \psi^6 r^2 } \delta( r-r_{\rm SH}),$$ where $r_{\rm SH}$ is the (isotropic) radius of the shell. Since the particles are non-interacting, their stress-energy tensor is that of a pressureless fluid (dust): $T_{ab} = \rho_0 u_a u_b$. The matter sources $\rho_N $, $S$ and $j^r$ in equations (\[psiConstraintMod\]), (\[betaConstraintMod\]) and (\[alphaConstraintMod\]) can therefore be expressed as $$\begin{aligned} \rho_N & = & \rho_0 W^2, \nonumber \\ S & = & \rho_0 (W^2 - 1), \\ j^r & = & \rho_0 \beta W^2 / \alpha. \nonumber\end{aligned}$$ The delta function in these matter sources leads to a discontinuity in the derivatives of the solutions. We can find the jump in these derivatives by integrating equations (\[betaConstraint\]), (\[psiConstraint\]) and (\[alphaConstraint\]) from $r_{\rm SH} - \epsilon$ to $r_{\rm SH} + \epsilon$, which, in the limit $\epsilon \rightarrow 0$, results in the jump conditions $$\begin{aligned} \label{jumpConditions} \partial_r \beta_+ - \partial_r \beta_- &=& \frac{3WM_{\rm SH}\beta} {r_{\rm SH}^2 \psi^2}, \nonumber \\ \partial_r \psi_+ - \partial_r \psi_- &=& \frac{-M_{\rm SH}W}{2r_{\rm SH}^2 \psi}, \\ \partial_r \left( \alpha \psi \right)_+ - \partial_r \left( \alpha \psi \right)_- &=& \alpha \psi \left( \frac{M_{\rm SH} \left(3 W^2-2 \right)}{2r_{\rm SH} ^2 W \psi^2}\right). \nonumber\end{aligned}$$ Since the particles are non-interacting, their 4-velocity $u^a$ obeys the geodesic equation $$du^a/d\tau+\Gamma^a_{bc}u^bu^c=0,$$ Assuming circular orbits with $u^r = 0$ and $d u^r/d\tau = 0$ and, without loss of generality, focussing on a particle in the equatorial plane, we find $$\label{geodesicMod} \Gamma^r_{tt}\left(u^t\right)^2+\Gamma^r_{\phi\phi}\left(u^{\phi} \right)^2=0.$$ Using the normalization condition $u_a u^a=-1$, which is equivalent to $1=W^2- \gamma_{ij} u^i u^j$, we evaluate (\[geodesicMod\]) to find the geodesic condition $$\label{geodesicCondition} 1-W^{-2}= \left( \frac{\beta^2}{r^2} - \dfrac{\psi^3\beta\left(\psi\partial_r \beta+2\beta\partial_r\psi\right)-\alpha\partial_r\alpha} {\left(2r\partial_r\psi+\psi\right)\psi^3r} \right) \frac{ \psi^4 r^2} {\alpha^2}.$$ for circular orbits. To compute the Christoffel symbols we averaged the derivatives of the gravitational field variables inside and outside the shell (compare [@SkoB02]). Diagnostics ----------- We compute the ADM and Komar masses using the expressions (see e.g. Chap. 7 of [@Gou07a]) $$M_{\rm ADM} = -\frac{1}{2\pi} \int_{\partial\Sigma} \bar D_i \psi \, d \bar S^i$$ and $$M_{\rm K} = \frac{1}{4\pi} \int_{\partial \Sigma} \left( D_i \alpha - \beta^j K_{ij} \right) dS^i \, ,$$ where $dS^i$ is the outward pointing unit surface element of a closed surface at infinity. In our case, these expressions reduce to $$\label{ADM} M_{\rm ADM} = \lim_{r\rightarrow\infty} -2 r^2 \partial_r \psi$$ and $$\label{Komar} %M_{K} = \psi^2 r^2 \left( \partial_r \alpha - \frac{\psi^4 \beta^r} {3} \left(\frac{2}{\alpha} \left( \partial_r \beta^r - \frac{\beta^r}{r} \right)+K\right)\right) \, , M_{\rm K} = \lim_{r\rightarrow\infty} r^2 \partial_r \alpha \, .$$ For all configurations considered in this paper, the two mass expressions are found to be equivalent to within the accuracy of our code when the geodesic equation (\[geodesicCondition\]) is used to force the particles into circular orbit (compare [@SkoB02]). This is in agreement with a general theorem about the equality of ADM and Komar masses established by Beig [@Bei78] and Ashtekar & Magnon-Ashtekar [@AshM79]. We also define the binding energy as $$\label{Mbind} E_{\rm B} = \frac{M_{\rm ADM/K}}{M_{\rm BH} + M_{\rm SH}} - 1,$$ either terms of the ADM or Komar mass. Finally, we compute the black hole’s irreducible mass from the area ${\mathcal A}$ of the apparent horizon, $$\label{Mirr} M_{\rm irr} = \left( \frac{\mathcal A}{16 \pi} \right)^{1/2} = \frac {R}{2} = \frac{r_{\rm BH} \psi_{\rm BH}^2}{2}.$$ Numerics ======== Code ---- We developed a pseudo-spectral code to solve the differential equations (\[DiffEq\]) subject to the boundary conditions (\[infBound\]) and (\[InBoundCond\]) as well as the jump conditions (\[jumpConditions\]), using Chebyshev polynomials as basis functions (see [@GraN07] for a recent review of spectral methods). Equations (\[betaConstraintMod\]) and (\[alphaConstraintMod\]) can be solved directly, while equation (\[psiConstraintMod\]) has to be linearized and then solved iteratively. One complication arises as a consequence of the jump conditions (\[jumpConditions\]). Representing the solution functions across these jumps as a linear combination of the continuous Chebyshev polynomials would result in undesirable Gibbs phenomena. To avoid this problem, we solve the equations in two separate domains inside and outside the shell, each one represented by $N$ Chebyshev polynomials. The $N$ coefficients can then be determined by evaluating the equation at $N$ collocation points in each domain, and the jump conditions (\[jumpConditions\]) can then be imposed exactly as matching conditions between the two sets of Chebyshev polynomials. Each set of Chebyshev polynomials is $T_n(s)$ with $0\leq n\leq N-1$ and $s\in[-1,1]$. We map the inner region into the interval $[-1,1]$ with the transformation $$\label{transI} s_{\rm I}=\frac{2 r_{\rm BH}r_{\rm SH}/r-r_{\rm BH}-r_{\rm SH}}{r_ {\rm BH}-r_{\rm SH}}$$ and the outer region with $$\label{transO} s_{\rm O}=-2r_{\rm SH}/r +1.$$ Our computational domain therefore extends to $r = \infty$, and we can evaluate the masses (\[ADM\]) and (\[Komar\]) exactly. In addition to the choices for $\alpha_{\rm AH}$ and $K$, our solution depends on the parameters $M_{\rm SH}$ and $r_{\rm SH}$ in (\[restEnergyDensity\]), and the excision radius $r_{\rm BH}$. To construct a solution of given black hole mass $M_{\rm BH}$, we need to iterate over $r_{\rm BH}$ until the resulting irreducible black hole mass (\[Mirr\]) agrees with the desired black hole mass $M_ {\rm BH}$ to within a certain pre-determined tolerance. For a given shell radius $r_{\rm SH}$ a further iteration is needed to fix the Lorentz factor $W$ in such a way that the solution satisfies the geodesic condition (\[geodesicCondition\]), and the particles are in circular orbit. In practice, we instead fix $W$ and then iterate over $r_{\rm SH}$ until (\[geodesicCondition\]) is satisfied. For each case we start the iteration from the analytical solution for $K = 0$ and $\alpha_{\rm AH} = 0$ provided in Appendix \[appA\], and continue until the solution has converged. Specifically, our convergence criterion requires that the relative change between iteration steps in any of the fields is less then $10^{-10}$ at all collocation points. Tests ----- We tested our program for a number of different known vacuum solutions expressing the Schwarzschild geometry in different coordinate systems, as well as an analytical solution describing thin shells around black holes that we derive in Appendix \[appA\]. As a first vacuum test (for which we set $M_{\rm SH} = 0$ and $r_{\rm SH} = 10M_{\rm BH}$), we considered Schwarzschild in “standard" isotropic coordinates, representing the symmetry plane in a Carter-Penrose diagram. In our code, we can produce the solution $$\label{SSiso} \psi = 1 + \frac{M}{2r},~~~~ \alpha \psi = 1 - \frac{M}{2r},~~~~ \beta =0$$ by choosing $K = K_{\rm MS} = 0$ and $\alpha_{\rm AH} = 0$. After applying the transformations (\[transI\]) and (\[transO\]), these solutions become linear in our code’s coordinates $s_{\rm I}$ and $s_ {\rm O}$, meaning that the solution can be represented exactly in terms of the first two Chebyshev polynomials. For any $N \geq 1$, our code therefore converges to the correct solution within the predetermined tolerances. ![The average error over all collocation points $E^N$ as a function of the number of collocation points $N$ for two analytic representations of the Schwarzschild geometry in isotropic coordinates. The label $AP$ denotes the analytic puncture solution presented in [@BauN07], while the label $KS$ denotes the Kerr- Schild solution in [@CooP04]. The errors drop off exponentially as a function of the number of collocation points $N$, until a ‘floor’ of specified tolerance has been reached.[]{data-label="APKSError"}](Fig1){width="3in"} More interesting are the two isotropic representations of the Schwarzschild geometry presented in [@CooP04] and [@BauN07]. The former is a transformation of Kerr-Schild (Eddington-Finkelstein) coordinates to isotropic coordinates, keeping the same time slicing, which we can produce by choosing $K = K_{\rm KS}$ and $\alpha_{\rm AH} = 1/\sqrt{2}$ in our code. The latter is an isotropic representation of a maximal slice (with the critical parameter $C = 3 \sqrt{3}M^2/4$, see [@EstWCDST73; @Rei73]), which has recently attracted interest as an analytic “puncture" solution (compare ). We can produce this solution by choosing $K = K_{\rm MS} = 0$ and $\alpha_{\rm AH} = 3\sqrt{3} / 16$ in our code. To measure the deviation from an analytic solution, we compute the average of the absolute error $\epsilon_n$ at all $N$ collocation points in each of the two domains, $$E^{N} = \frac{1}{2N} \sum_{n=1}^{2N} \| \epsilon_n \|.$$ where the collocation points $N+1 \leq n \leq 2N$ are in the region outside of the shell. In Fig. \[APKSError\] we graph this error as a function of the number of collocation points $N$. As expected, the errors fall off exponentially for all variables, until they reach a floor corresponding to the predetermined tolerance. In addition to these vacuum solutions we also consider an analytic solution describing thin shells of non-interacting particles around a static black hole. As we demonstrate in Appendix \[appA\], we can solve the differential equations (\[DiffEq\]) subject to the boundary conditions (\[infBound\]) and (\[InBoundCond\]) as well as the jump conditions (\[jumpConditions\]) analytically for maximal slicing and $\alpha_{\rm AH} = 0$. As for the solution (\[SSiso\]), the field variables become linear in our code’s variables $s_{\rm I}$ and $s_{\rm O}$ (see (\[analyticPsiSH\]) and (\[analyticAlphaSH\]) below), so that they can be represented exactly for any $N \geq 1$. In addition to testing the solution of the field equations, however, this test also verifies that our code correctly solves the jump conditions at the shell. As in example, we show in Fig. \[shellPlot\] the analytic and numerical solutions for the lapse $\alpha$ and the conformal factor $\psi$ as a function of areal radius $R$ for a Lorentz factor of $W=1.20$ and a mass ratio $M_{BH}/M_{SH}=1$. Our numerical solutions agree with the analytical ones to within better than $10^{-10}$, making them indistinguishable in the plot. ![The conformal factor $\psi$ and the lapse $\alpha$ as a function of areal radius for the analytic shell solution ($K = 0$ and $\alpha_{\rm EH}=0$) for a Lorentz factor of $W=1.20$ and a mass ratio of $M_{\rm BH}/M_{\rm SH}=1$, corresponding to an areal shell radius of about $8.013 M_{\rm BH}$. The field variables are continuous across the shell, but, according to the jump conditions (\[jumpConditions\]) their derivatives are not. Our numerical solutions agree with the analytical ones to within better than $10^{-10}$, making the error far smaller than the line width in the graph.[]{data-label="shellPlot"}](Fig2){width="3in"} Results ======= ![The ADM binding energy (\[Mbind\]) $E_{\rm B}$ as a function of the shell’s areal radius $R$ for an extreme-mass-ratio sequence with $M_{BH}/M_{SH} = 1000$. Values of the horizon lapse $ \alpha_{\rm AH}$ range from zero to unity in increments of $0.1$. The graph includes eleven solid lines (representing an evaluation of the binding energy (\[Mbind\]) in maximal slicing) and eleven dashed lines (for Kerr-Schild slicing). All 22 lines coincide within our numerical error. We computed these sequences using $N = 26$ collocation points in each domain.[]{data-label="largeRatio"}](Fig3){width="3in"} We now construct constant-mass sequences, meaning sequences of varying shell radius $r_{\rm SH}$ but constant shell rest mass $M_ {\rm SH}$ and black hole irreducible mass $M_{\rm BH}$. Specifically, we focus on “extreme-mass-ratio" sequences with $M_ {BH}/M_{\rm SH} = 1000$ and equal-mass sequences $M_{\rm BH}/M_{\rm SH}=1$. For both choices of the mass ratio we construct sequences for our two choices of the extrinsic curvature (\[K\]), maximal slicing and Kerr-Schild slicing, and for the horizon lapse (\[alphaHorBound\]) ranging from $\alpha_{\rm AH} = 0$ to $\alpha_{\rm AH} = 1$ in increments of 0.1. In the following, we will graph the binding energy (\[Mbind\]) as a function of the areal radius. Typically, the binding energy $M_{\rm B}$ is only a small fraction of the involved masses, meaning that the relative error in the binding energy is larger than that for the masses – and hence the fields – themselves. We found that, to achieve similar accuracy, the extreme-mass-ratio sequences (for which the binding energy is much smaller than for the equal-mass sequences) required slightly more collocation points than the equal-mass sequences. In Fig. \[largeRatio\] we show the binding energy for extreme-mass- ratio sequences with $M_{\rm BH}/M_{\rm SH} = 1000$. The graph represents 22 plots, corresponding to the eleven different values of $ \alpha_{\rm AH}$ and to evaluating the ADM binding energy (\[Mbind\]) for both maximal slicing and Kerr-Schild slicing. To within the accuracy of our code, all 22 lines agree with each other, so that they all lie on top of each other and appear as one line in Fig. \[largeRatio\]. The minimum of the binding energy corresponds to the innermost stable circular orbit (ISCO). In the extreme-mass-ratio limit we may neglect the particles’ self-gravity, so that we are effectively solving for a test-particle in the Schwarzschild geometry. As expected, we find that the ISCO is located at $R=6M_{\rm BH}$, representing another independent test of our code. ![The ADM binding energy $E_{\rm B}$ as a function of the shell’s areal radius $R$ for an equal mass sequence, $M_{BH}/M_{SH} = 1$. As in Fig. \[largeRatio\], values of $\alpha_{\rm AH}$ again range from zero to unity in increments of $0.1$, and the graph again contains eleven lines each for Maximal and Kerr-Schild slicing. All 22 lines again coincide within our numerical error. We obtained these results with $N=20$ collocation points in each domain.[]{data-label="equalRatio"}](Fig4){width="3in"} In Fig. \[equalRatio\] we show the equivalent graphs for equal-mass sequences with $M_{\rm BH}/M_{\rm SH} = 1$. As for the extreme-mass- ratio sequences in Fig. \[largeRatio\] all 22 graphs coincide to within our numerical accuracy. We note that the ISCO is now located at a larger radius of about $9.367 M_{\rm BH}$. Even in this case, we do not find any evidence that the choice of the slicing condition (\[K\]), or the choice of the boundary condition for the lapse on the horizon (\[alphaHorBound\]), have any effect on the physical content of our solutions. Clearly, these different choices lead to different solutions for the conformal factor $\psi$, the lapse $\alpha$ and the shift $\beta$. When expressed in terms of gauge-invariant quantities, however, all our solutions become indistinguishable to within the accuracy of our numerical code. Summary ======= We solve Einstein’s constraint equations in the extended conformal thin-sandwich decomposition to construct spherical shells of non-interacting, isotropic particles in circular orbit about a non-rotating black hole. We construct these solutions for both maximal slicing and Kerr-Schild slicing (see equations (\[K\])), and for a number of different choices for the horizon lapse. These different choices lead to very different solutions for the lapse $\alpha$, the shift $\beta^i$ and the conformal factor $\psi$. However, when expressed in terms of gauge-invariant quantities – for example the binding energy as a function of the shell’s areal radius for given shell and black hole masses – our solutions become indistinguishable. At least in the limited context of our spherically symmetric solutions, these findings provide no evidence that the choices for the mean curvature and horizon lapse affect the physical content of the solutions. KM gratefully acknowledges support from the Coles Undergraduate Research Fellowship Fund at Bowdoin College and from the Maine Space Grant Consortium. KM and TWB would like to thank the Laboratoire Univers et Theories at the Observatoire de Paris, Meudon, for its hospitality. This work was supported in part by NSF grant PHY-0456917 to Bowdoin College and by the ANR grant 06-2-134423 Méthodes mathématiques pour la relativité générale. An analytical solution {#appA} ====================== For maximal slicing ($K = 0$) and the lapse boundary condition $ \alpha_{\rm AH} = 0$, we can find an analytic solution to the differential equations (\[DiffEq\]), subject to the boundary conditions (\[infBound\]) and (\[InBoundCond\]) as well as the jump conditions (\[jumpConditions\]). The solution depends on the input parameters $M_{\rm SH}$, $W$, $r_ {\rm BH}$, and $r_{\rm SH}$. We may then enforce the geodesic condition (\[geodesicCondition\]) to eliminate one of these variables, and compute the mass of the black hole from (\[Mirr\]). The solution presented here represents a generalization of the solutions of [@SkoB02], who considered the same system but without a black hole. We begin with the momentum constraint (\[betaConstraintMod\]). For $K = 0$ and $\alpha_{\rm AH} = 0$ we find that $$\label{analyticBetaSH} \beta = 0$$ is a self-consistent solution to both the equation and its boundary conditions. This implies that Eqs. (\[psiConstraintMod\]) and (\[alphaConstraintMod\]) for respectively the conformal factor $\psi$ and the combination $ \alpha \psi$ reduce to flat Laplace equations in the vacuum regions away from the shell. In spherical symmetry, the only possible solutions are of the form $k_1 + k_2/r$, where $k_1$ and $k_2$ are arbitrary constants that have to be determined from the boundary conditions. For each function we need four conditions to determine these constants both in the interior and the exterior of the shell. These four conditions arise from the outer boundary conditions (\[infBound\]), the inner boundary conditions (\[InBoundCond\]), continuity of the functions at the shell, and the jump conditions (\[jumpConditions\]) for their first derivatives. Using these conditions, we find $$\label{analyticPsiSH} \psi = \left\{ \begin{array}{ll} a \left(1+\dfrac{r_{\rm BH}}{r}\right) \, , & r \leq r_{\rm SH} \\[3mm] 1+ \dfrac{a \left(r_{\rm SH}+r_{\rm BH} \right) - r_{\rm SH} } {r} \, ,& r \geq r_{\rm SH} \end{array} \right.$$ for the conformal factor and $$\label{analyticAlphaSH} \alpha = \left\{ \begin{array}{ll} \displaystyle \dfrac{c \left(1-r_{\rm BH}/r \right)}{a \left(1+r_{\rm SH}/r \right)} \, , & r < r_{\rm SH} \\[3mm] \displaystyle \dfrac{1+\left( c \left( r_{\rm BH} - r_{\rm SH} \right) - r_{\rm SH} \right)/r}{1+\left(a \left(r_{\rm SH} +r_{\rm BH} \right)-r_{\rm SH} \right)/r } \, , & r \geq r_{\rm SH} \end{array} \right.$$ for the lapse. Here the constants $a$ and $c$ are given by $$\label{aExpression} a = 1 + \frac{M_{\rm SH}W}{2r_{\rm SH} \psi_{\rm SH}}$$ and $$\label{cExpression} c = 1 - \frac{M_{\rm SH} \alpha_{\rm SH} \left( 3 W^2 -2 \right)}{r_ {\rm SH} \left( 2 W \psi_{\rm SH} \right)}.$$ Inserting these constants, which themselves depend on the values of the conformal factor and the lapse at the shell, into (\[analyticPsiSH\]) and (\[analyticAlphaSH\]), we find $$\label{psiHorizon} \psi_{\rm SH} = \frac{1}{2} \left(1+\frac{r_{\rm BH}}{r_{\rm SH}} \right) \left[ 1+ \left( 1+ \frac{2 M_{\rm SH} W}{r_{\rm BH}+r_{\rm SH}} \right)^{1/2} \right]$$ and $$\label{alphaHorizon} \alpha_{\rm SH} = \left( p + \frac{\left(3+p\right)W^2-2}{2r_{\rm SH} \psi_{\rm SH}W} \right)^{-1}.$$ Here we have abbreviated $p\equiv(r_{\rm SH}+r_{\rm BH})/(r_{\rm SH}- r_{\rm BH})$. So far these solutions depend on all four parameters $M_{\rm SH}$, $W $, $r_{\rm BH}$, and $r_{\rm SH}$. We now find a relation between these parameters by inserting (\[analyticPsiSH\]) and (\[analyticAlphaSH\]) into the geodesic equation (\[geodesicMod\]), which yields $$\begin{aligned} \label{omegaAnalytic} && \Omega^2 = \left(\dfrac{u^{\phi}}{u^t}\right)^2 = -\dfrac{^{(4)} \Gamma^r_{tt}}{^{(4)}\Gamma^r_{\phi\phi}} \\ && = \dfrac{r_{\rm SH}^3 c \left(r_{\rm BH} - r_{\rm SH}\right)\left (r_{\rm SH}\left(a-c\right)+r_{\rm BH}\left(a+c+2ac\right)\right)} {2\left(a\left(r_{\rm SH}+r_{\rm BH}\right)\right)^6\left (ar_{\rm BH}-r_{\rm SH}\right)}. \nonumber\end{aligned}$$ Unfortunately, this expression is not very useful for our purposes in this form. We find an alternative form by evaluating (\[geodesicCondition\]) for our solution (\[analyticPsiSH\]) and (\[analyticAlphaSH\]), which results in a fifth order polynomial for $W$, $$\begin{aligned} \label{geoAnalytic} 0 &=& 4(r_{\rm BH}-r_{\rm SH})^3 r_{\rm SH} \nonumber \\ && + M_{\rm SH} (r_{\rm BH}-r_{\rm SH})^2(r_{\rm BH}+r_{\rm SH})W \nonumber \\ &&- 2r_{\rm SH}(r_{\rm BH}-r_{\rm SH})(5r_{\rm BH}^2-14r_{\rm BH}r_ {\rm SH}+5r_{\rm SH}^2)W^2 \nonumber \\ &&- 2M_{\rm SH}(r_{\rm BH}-2r_{\rm SH})(r_{\rm BH}-r_{\rm SH})(r_{\rm BH}+r_{\rm SH})W^3 \nonumber \\ &&+ 6r_{\rm SH}(r_{\rm BH}-r_{\rm SH})(r_{\rm BH}^2-4r_{\rm BH}r_{\rm SH}+r_{\rm SH}^2)W^4 \nonumber \\ &&+ M_{\rm SH} (r_{\rm BH}-2r_{\rm SH})^2(r_{\rm BH}+r_{\rm SH})W^5\end{aligned}$$ In the limit of a zero-mass black hole we have $r_{\rm BH} = 0$ and (\[geoAnalytic\]) reduces to $$\begin{aligned} 0&=&-4r_{\rm SH} + M_{\rm SH} W + 10 r_{\rm SH} W^2 - 4 M_{\rm SH} W^3 \nonumber \\ &&- 6 r_{\rm SH} W^4 + 4 M_{\rm SH} W^5,\end{aligned}$$ which agrees with the corresponding equation (26) of [@SkoB02]. Instead of parameterizing the solution by $r_{\rm BH}$, it is more desirable to fix the black hole mass $M_{\rm BH} = M_{\rm irr}$. Towards this end we combine (\[geoAnalytic\]) with (\[analyticPsiSH\]) and (\[Mirr\]), which results in a quadratic equation for $r_{\rm SH}$. The solutions to this equation are $$\begin{aligned} \label{constAnalytic} r_{\rm SH} &=& \Big[4M_{\rm BH}r_{\rm BH}^2 + 2 M_{\rm BH} M_{\rm SH} r_{\rm BH} W - 2 M_{\rm BH}^2 r_{\rm BH} \nonumber \\ &&\pm 2\sqrt{2} \big(M_{\rm BH} M_{\rm SH}^2 r_{\rm BH}^3 W^2\big)^ {1/2} \Big]/ \nonumber \\ && \Big[ 2(M_{\rm BH}^2-2M_{\rm BH} r_{\rm BH}) \Big] .\end{aligned}$$ We henceforth ignore the “$-$" solution, as only the “$+$" solution refers to stable solutions. Finally, we may insert (\[constAnalytic\]) back into (\[geoAnalytic\]), which results into a polynomial for $r_{\rm BH}$ that can be factored into two cubic polynomials $$\begin{aligned} \label{cubicPol1} 0 & = & \left( a_3 \bar r_{\rm BH}^3 + a_2 \bar r_{\rm BH}^2 + a_1 \bar r_{\rm BH} + a_0 \right) \times \nonumber \\ & & \left( b_3 \bar r_{\rm BH}^3 + b_2 \bar r_{\rm BH}^2 + b_1 \bar r_ {\rm BH} + b_0 \right).\end{aligned}$$ Here we have abbreviated $\bar r_{\rm BH} \equiv r_{\rm BH}/M_{\rm SH} $, and the coefficients $a_i$ and $b_i$ are, in terms of the mass ratio $q \equiv M_ {\rm BH}/M_{\rm SH}$, $$\begin{aligned} a_3 & = & b_3 = 32 q^2(-2+3W^2)^2 \\ a_2 & = & -16 q(-2+3W^2)[-4 q^2 + 6 q W \nonumber \\ && + (6q^2-1)W^2 -9qW^3 +2W^4] \nonumber \\ b_2 & = & -4q(-2+3W^2)^2(8q^2-8qW+5W^2) \nonumber \\ a_1 & = & 2[16 q^4 - 48 q^3 W + (44 q^2 - 48 q^4) W^2 \nonumber \\ && +(144 q^3-12 q) W^3+(36q^4-136q^2 \nonumber \\ && +1)W^4 +(38 q-108 q^3)W^5+(105 q^2 \nonumber \\ && - 2) W^6 -30qW^7+W^8] \nonumber \\ b_1 & = & 2(-2+3W^2)[-8q^4+16q^3W \nonumber \\ && +(12q^4-20q^2) W^2 +(10q-24q^3)W^3 \nonumber \\ && +(26q^2-2) W^4-12qW^5+3W^6] \nonumber \\ a_0 & = & b_0 = -qW^2(-2q+W+3qW^2-2W^3)^2 \nonumber\end{aligned}$$ We can now construct a solution for given masses $M_{\rm BH}$ and $M_ {\rm SH}$ and Lorentz factor $W$ as follows. Given the mass ratio $q $ and $W$ we first find the six solutions for $\bar r_{\rm BH}$ from the two polynomials in (\[cubicPol1\]). We then insert the corresponding values $r_{\rm BH}$ into (\[constAnalytic\]), which yields six solutions for $r_{\rm SH}$. The largest real solution is the solution of interest; we keep only this solution as well as its corresponding value of $r_{\rm BH}$ and disregard all others. These values can then be inserted into (\[analyticPsiSH\]) and (\[analyticAlphaSH\]), which determines the solution completely. [24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , **, (). , , , in , edited by (, ), p. . , in , edited by (, ), p. . , in (), . , in , edited by (, ), . , , (). , , (). , , (). , , (). , , (). , , (). , , , , (). , , (). , , , , , (). , , (). , , (). , (), , . , **, (). , , (). , (), , . , , (). , , , , , , , (). , **, (). , , , , , (), .
--- abstract: \| We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Möbius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) cross-ratio of the circular lattice. We derive the analogous condition for our generalized isthermic lattices using Steiner’s projective structure of conics and we present basic geometric constructions which encode integrability of the lattice. In particular we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.\ [Keywords:]{} discrete geometry; integrable systems; multidimensional quadrilateral lattices; isothermic surfaces; Darboux transformation\ \ address: 'Wydzia[ł]{} Matematyki i Informatyki, Uniwersytet Warmińsko-Mazurski w Olsztynie, ul. Żo[ł]{}nierska 14, 10-561 Olsztyn, Poland' author: - 'Adam Doliwa$^\ddagger$' title: Generalized Isothermic Lattices --- [^1] Introduction ============ Isothermic surfaces ------------------- In the year 1837 Gabriel Lamé presented [@Lame-isoth] results of his studies on distribution of temperature in a homogeneous solid body in thermal equilibrium. He was interested, in particular, in description of the isothermic surfaces, i.e. surfaces of constant temperature within the body; notice that his definition makes sense only for families of surfaces, and not for a single surface. Then he found a condition under which one parameter family of surfaces in (a subset of) ${{\mathbb E}}^3$ consists of isothermic surfaces, and showed (for details see [@Lame] or [@DarbouxOS]) that the three families of confocal quadrics, which provide elliptic coordinates in ${{\mathbb E}}^3$, meet that criterion. Subsequently, he proposed to determine all triply orthogonal systems composed by three isothermic families (triply isothermic systems). Such a program was fulfiled by Gaston Darboux [@DarbouxOS] (see also [@Eisenhart-SV]). Another path of research was initiated by Joseph Bertrand [@Bertrand] who showed that the surfaces of triply isothermic systems are divided by their lines of curvature into “infinitesimal squares”, or in exact terms, they allow for conformal curvature parametrization. This definition of isothermic surfaces (or surfaces of isothermic curvature lines), which can be applied to a single surface, was commonly accepted in the second half of the XIX-th century (see [@Bianchi; @DarbouxIV]). We mention that the minimal surfaces and the constant mean curvature surfaces are particular examples of the isothermic surfaces. The theory of isothermic surfaces was one of the most favorite subjects of study among prominent geometers of that period. Such surfaces exhibit particular properties, for example there exists a transformation, described by Gaston Darboux in [@Darboux-transf], which produces from a given isothermic surface a family of new surfaces of the same type. The Gauss-Mainardi-Codazzi equations for isothermic surfaces constitute a nonlinear system generalizing the $\mathrm{sinh}$-Gordon equation (the latter governs the constant mean curvature surfaces), and the Darboux transformation can be interpreted as Bäcklund-type transformation of the system. Soon after that Luigi Bianchi showed [@Bianchi-isoth] that two Darboux transforms of a given isothermic surface determine in algebraic terms new isothermic surface being their simultaneous Darboux transform. The Bianchi permutability principle can be considered as a hallmark of integrability (in the sense of soliton theory) of the above-mentioned system. Indeed, the isothermic surfaces were reinterpreted by Cieśliński, Goldstein and Sym [@CGS] within the theory of soliton surfaces [@Sym]. More information on isothermic surfaces and their history the Reader can find in the paper of Klimczewski, Nieszporski and Sym [@KNS], where also a more detailed description of the relation between the “ancient” differential geometry and the soliton theory is given, and in books by Rogers and Schief [@RogersSchief] and by Hertrich-Jeromin [@UHJ]. Discrete isothermic surfaces and discrete integrable geometry ------------------------------------------------------------- In the recent studies of the relation between geometry and the integrable systems theory a particular attention is payed to discrete (difference) integrable equations and the corresponding discrete surfaces or lattice submanifolds. Also here the discrete analogs of isothermic surfaces played a prominent role in the development of the subject. Bobenko and Pinkall [@BP2] introduced the integrable discrete analogoue of isothermic surfaces as mappings built of “conformal squares”, i.e., maps ${{\boldsymbol x}}:{{\mathbb Z}}^2\to{{\mathbb E}}^3$ with all elementary quadrilaterals circular, and such that the complex cross-ratios (with the plane of a quadrilateral identified with the complex plane ${{\mathbb C}}$) $$q(m,n) = {\mathrm{cr}}\big({{\boldsymbol x}}(m,n),{{\boldsymbol x}}(m+1,n+1);{{\boldsymbol x}}(m+1,n),{{\boldsymbol x}}(m,n+1) \big)_{{\mathbb C}}$$ are equal to $-1$. Soon after that it turned out [@BP-V] that it is more convenient to allow for the cross-rations to satisfy the constraint $$\label{eq:isoth-cr} q(m,n) q(m+1,n+1) = q(m+1,n)q(m,n+1).$$ Then the cross-ratio is a ratio of functions of single variables, which corresponds to allowed reparametrization of the curvature coordinates on isothermic surfaces. After the pioneering work of Bobenko and Pinkall, which was an important step in building the geometric approach to integrable discrete equations (see also [@DS-AL; @BP1; @DCN] and older results of the difference geometry (Differenzengeometrie) summarized in Robert Sauer’s books [@Sauer2; @Sauer]), the discrete isothermic surfaces and their Darboux transformations were studied in a number of papers [@HeHP; @Ciesl; @Schief-C]. Distinguished integrable reductions of isothermic lattices are the discrete constant mean curvature surfaces or the discrete minimal surfaces [@BP-V; @UHJ]. It should be mentioned that the complex cross-ratio condition was extended to circular lattices of dimension three [@BP-V; @Ciesl] placing the Darboux transformations of the discrete isothermic surfaces on equal footing with the lattice itself. In the present-day approach to the relation between discrete integrable systems and geometry [@DS-EMP; @BobSur] the key role is played by the integrable discrete analogue of conjugate nets – multidimensional lattices of planar quadrilaterals (the quadrilateral lattices) [@MQL]. These are maps $x:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ of $N$-dimensional integer lattice in $M\geq N$-dimensional projective space with all elementary quadrilaterals planar. Integrability of such lattices (for $N>2$) is based on the following elementary geometry fact (see Figure \[fig:TiTjTkx\]). ![The geometric integrability scheme[]{data-label="fig:TiTjTkx"}](TiTjTkx.eps) \[lem:gen-hex\] Consider points $x_0$, $x_1$, $x_2$ and $x_3$ in general position in ${{\mathbb P}}^M$, $M\geq 3$. On the plane $\langle x_0, x_i, x_j \rangle$, $1\leq i < j \leq 3$ choose a point $x_{ij}$ not on the lines $\langle x_0, x_i \rangle$, $\langle x_0,x_j \rangle$ and $\langle x_i, x_j \rangle$. Then there exists the unique point $x_{123}$ which belongs simultaneously to the three planes $\langle x_3, x_{13}, x_{23} \rangle$, $\langle x_2, x_{12}, x_{23} \rangle$ and $\langle x_1, x_{12}, x_{13} \rangle$. Various constraints compatible with the geometric integrability scheme define integrable reductions of the quadrilateral lattice. It turns out that such geometric notion of integrability very often associates integrable reductions of the quadrilateral lattice with classical theorems of incidence geometry. We advocate this point of view in the present paper. Among basic reductions of the quadrilateral lattice the so called quadratic reductions [@q-red] play a distinguished role. The lattice vertices are then contained in a hyperquadric (or in the intersection of several of them). Such reductions of the quadrilateral lattice can be often associated with various subgeometries of the projective geometry, when the quadric plays the role of the absolute of the geometry (see also corresponding remarks in [@q-red; @W-cong]). In particular, when the hyperquadric is the Möbius hypersphere one obtains, after the stereographic projection, the circular lattices [@Bobenko-O; @CDS; @DMS; @KoSchief2], which are the integrable discrete analogue of submanifolds of ${{\mathbb E}}^M$ in curvature line parametrization. Because of the Möbius invariance of the complex cross-ratio it is also more convenient to consider (discrete) isothermic surfaces in the Möbius sphere (both dimensions of the lattice and of the sphere can be enlarged) keeping the “cross-ratio definition”. For a person trained in the projective geometry it is more or less natural to generalize the Möbius geometry approach to discrete isothermic surfaces (lattices) in quadrics replacing the Möbius sphere by a quadric, and correspondingly, the complex cross-ratio by the Steiner cross-ratio of four points of a conic being intersection of the quadric by the plane of elementary quadrilateral of the quadrilateral lattice. However the “cross-ratio point of view” doesn’t answer the crucial question about integrability (understood as compatibility of the constraint with the geometric integrablity scheme) of such discrete isothermic surfaces in quadrics. Our general methodological principle in the integrable discrete geometry, applied successfuly earlier, for example in [@DS-sym; @DNS-Bianchi], which we would like to follow here is (i) to isolate basic reductions of the quadrilateral lattice and then (ii) to incorporate other geometric systems into the theory considering them as superpositions of the basic reductions. In this context we would like to recall another equivalent characterization of the classical isothermic surfaces which can be found in the classical monograph of Darboux [@DarbouxIV vol. 2, p. 267]: [Les cinq coordonnées pentasphériques d’un point de toute surface isothermique considérées comme fonctions des paramètres $\rho$ et $\rho_1$ des lignes de courbure satisfont à une équation linéaire du second ordre dont les invariants sont égaux. Inversement, si une équation de la forme $$\label{eq:Moutard-diff} \frac{\partial^2\theta}{\partial \rho \partial \rho_1} = \lambda \theta$$ ou, plus généralement, une équation à invariants égaux, admet cinq solutions particulières $x_1$, $x_2$, …, $x_5$ liées par l’équation $$\sum_{1}^{5} x_i^2 = 0,$$ les quantités $x_i$ sont les coordonnées pantasphériques qui définissent une surface isothermique rapportée à ses lignes de courbure. ]{} In literature there are known two (closely related) discrete integrable versions (of Nimmo and Schief [@NiSchief] and of Nieszporski [@Nieszporski-dK]) of the Moutard equation . It turns out that for our purposes it suits the discrete Moutard equation proposed in [@NiSchief]. Its projectively invariant geometric characterization has been discovered [@DGNS] only recently (for geometric meaning of the adjoint Moutard equation of Nieszporski in terms of the so called Koenigs lattice see [@Dol-Koe]). Indeed, it turns out that the generalized isothermic lattices can be obtained by adding to the quadratic constraint the projective Moutard constraint. Finally, the quadratic reduction and the Moutard reduction, when applied simultaneously, give a posteriori the cross-ratio condition . In fact, the direct algebraic discrete counterpart of the above description of the isothermic surfaces, i.e. existence of the light-cone lift which satisfies the (discrete) Moutard equation, appeared first in a preprint by Bobenko and Suris [@BobSur]. However, the pure geometric characterization of the discrete isothermic surfaces was not given there. Because integrability of the discrete Moutard equation can be seen better when one considers a system of such equations for multidimensional lattices, there was a need to find the projective geometric characterization of the system. The corresponding reduction of the quadrilateral lattice was called in [@BQL], because of its connection with the discrete BKP equation, the B-quadrilateral lattice (BQL). In fact, research in this direction prevented me from publication of the above mentioned generalization of discrete isothermic surfaces, announced however in my talk during the Workshop “Geometry and Integrable Systems” (Berlin, 3-7 June 2005). I suggested also there that the discrete S-isothermic surfaces of Tim Hoffmann [@Hoffmann] (see also [@BobSur]) should be considered as an example of the generalized isothermic lattices where the quadric under consideration is the Lie quadric. The final results of my research on generalized isothermic lattices were presented on the Conference “Symmetries and Integrability of Difference Equations VII” (Melbourne, 10-14 July 2006). When my paper was almost ready there appeared the preprint of Bobenko and Suris [@BobSur-gen-isoth] where similar ideas were presented in application to the sphere (Möbius, Laguerre and Lie) geometries. I would like also to point out a recent paper by Wallner and Pottman [@WP] devoted, among others, to discrete isothermic surfaces in the Laguerre geometry. Plan of the paper ----------------- As it often happens, the logical presentation of results of a research goes in opposite direction to their chronological derivation. In Section \[sec:BQL+QQL\] we collect some geometric results from the theory of the B-quadrilateral lattices (BQLs) and of quadrilateral lattices in quadrics (QQLs). Some new results concerning the relation between (Steiner’s) cross-ratios of vertices of elementary guadrilaterals of elementary hexahedrons of the QQLs are given there as well. Then in Section \[sec:gen-isoth-latt\] we define generalized isothermic lattices and discuss their basic properties. In particular, we give the synthetic-geometry proof of a basic lemma (the half-hexahedron lemma) which immediately gives the cross-ratio characterization of the lattices. We also present some algebraic consequences (some of them known already [@BobSur]) of the system of Moutard equations supplemented by a quadratic constraint. In Section \[sec:Darboux\] we study in more detail the Darboux transformation of the generalized isothermic lattices and the corresponding Bianchi permutability principle. Finally, in Section \[sec:isothermic\] we consider two dimensional generalized isothermic lattices. In two Appendices we recall necessary information concerning the cross-ratio of four points on a conic curve and we perform some auxilliary calculations. The B-quadrilateral lattices and the quadrilateral lattices in quadrics {#sec:BQL+QQL} ======================================================================= It turns out that compatibility of both BQLs and QQLs with the geometric integrablity scheme follows from certain classical geometric facts. We start each section, devoted to a particular lattice, from the corresponding geometric statement. The B-quadrilateral lattice [@BQL] ---------------------------------- \[lem:BKP-hex\] Under hypothesess of Lemma \[lem:gen-hex\], assume that the points $x_0$, $x_{12}$, $x_{13}$, $x_{23}$ are coplanar, then the points $x_1$, $x_2$, $x_3$, and $x_{123}$ are coplanar as well (see Figure \[fig:moutard\]). ![Elementary hexahedron of the B-quadrilateral lattice[]{data-label="fig:moutard"}](moutard) As it was discussed in [@BQL] the above fact is equivalent to the Möbius theorem (see, for example [@Coxeter]) on mutually inscribed tetrahedra. Another equivalent, but more symmetric, formulation of Lemma \[lem:BKP-hex\] is provided by the Cox theorem (see [@Coxeter]): Let $\sigma_1$, $\sigma_2$, $\sigma_3$, $\sigma_4$ be four planes of general position through a point $S$. Let $S_{ij}$ be an arbitrary point on the line $\langle \sigma_i, \sigma_j \rangle$. Let $\sigma_{ijk}$ denote the plane $\langle S_{ij}, S_{ik}, S_{jk} \rangle$. Then the four planes $\sigma_{234}$, $\sigma_{134}$, $\sigma_{124}$, $\sigma_{123}$ all pass through one point $S_{1234}$. \[def:BQL\] A quadrilateral lattice $x:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ is called the B-quadrilateral lattice if for any triple of different indices $i,j,k$ the points $x$, $x_{(ij)}$, $x_{(jk)}$ and $x_{(ik)}$ are coplanar. Here and in all the paper, given a fuction $F$ on ${{\mathbb Z}}^N$, we denote its shift in the $i$th direction in a standard manner: $F_{(i)}(n_1,\dots, n_i, \dots , n_N) = F(n_1,\dots, n_i + 1, \dots , n_N)$. One can show that a quadrilateral lattice $x:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ is a B-quadrilateral lattice if and only if it allows for a homogoneous representation ${{\boldsymbol x}}:{{\mathbb Z}}^N\to{{\mathbb R}}^{M+1}_{}$ satisfying the system of discrete Moutard equations (the discrete BKP linear problem) $$\label{eq:BKP-lin} {{\boldsymbol x}}_{(ij)} - {{\boldsymbol x}}= f^{ij} ({{\boldsymbol x}}_{(i)} - {{\boldsymbol x}}_{(j)}) , \quad 1\leq i< j\leq N,$$ for suitable functions $f^{ij}:{{\mathbb Z}}^N\to{{\mathbb R}}$. The compatibility condition of the system implies that the functions $f^{ij}$ can be written in terms of the potential $\tau:{{\mathbb Z}}^N\to{{\mathbb R}}$, $$\label{eq:tau} f^{ij} = \frac{\tau_{(i)}\tau_{(j)}}{\tau \, \tau_{(ij)}}, \qquad i\ne j ,$$ which satisfies Miwa’s discrete BKP equations [@Miwa] $$\label{eq:BKP-nlin} \tau\, \tau_{(ijk)} = \tau_{(ij)}\tau_{(k)} - \tau_{(ik)}\tau_{(j)} + \tau_{(jk)}\tau_{(i)}, \quad 1\leq i< j < k \leq N.$$ The trapezoidal lattice [@BobSur] is another reduction of the quadrilateral lattice being algebraically described by the discrete Moutard equations . Geometrically, the trapezoidal lattices are characterized by parallelity of diagonals of the elementary quadrilaterals, thus they belong to the affine geometry. Moreover, because the trapezoidal constraint is imposed on the level of elementary quadrilaterals then, from the point of view of the geometric integrability scheme, one has to check its three dimensional consistency. In contrary, the BQL constraint is imposed on the level of elementary hexahedrons, and to prove geometrically its integrability one has to check four dimensional consistency. The quadrilateral lattices in quadrics -------------------------------------- \[lem:qred-hex\] Under hypotheses of Lemma \[lem:gen-hex\], assume that the points $x_0$, $x_1$, $x_2$, $x_3$, $x_{12}$, $x_{13}$, $x_{23}$ belong to a quadric $\mathcal{Q}$. Then the point $x_{123}$ belongs to the quadric $\mathcal{Q}$ as well. The above fact is a consequence of the classical eight points theorem* (see, for example [@Coxeter]) which says that seven points in general position determine a unique eighth point, such that every quadric through the seven passes also through the eighth. In our case the point $x_{123}$ is contained in the three (degenerate) quadrics being pairs of opposite facets of the hexahedron. Lemma \[lem:BKP-hex\] can be considered as a “reduced” resion of Lemma \[lem:qred-hex\] when the quadric $\mathcal{Q}$ degenerates to a pair of planes. \[def:QQL\] A quadrilateral lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ contained in a hyperquadric $\mathcal{Q}$ is called the $\mathcal{Q}$-reduced quadrilateral lattice (QQL). Integrability of the QQLs was pointed out in [@q-red], where also the corresponding Darboux-type transformation (called in this context the Ribaucour transformation) was constructed in the vectorial form. When the quadric is irreducible then generically it cuts the planes of the hexahedron along conics. A quadrilateral lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ in a hyperquadric $\mathcal{Q}$ such that the intersection of the planes of elementary quadrilaterals of the latice with the quadric are irreducible conic curves is called locally irreducible. The following result, which will not be used in the sequel and whose proof can be found in Appendix \[sec:aux-calc\], generalizes the relation between complex cross-ratios of the opposite quadrilaterals of elementary hexahedrons of the circular lattices [@Bobenko-O]. \[prop:qred-cr\] Given locally irreducible quadrilateral lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ in a hyperquadric $\mathcal{Q}$, denote by $$\lambda^{ij} = {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)}), \qquad 1\leq i<j\leq N,$$ the cross-ratios (defined with respect to the corresponding conic curves) of the four vertices of the quadrilaterals. Then the cross-ratios are related by the following system of equations $$\label{eq:qred-cross-r} \lambda^{ij}\lambda^{ij}_{(k)}\lambda^{jk}\lambda^{jk}_{(i)} = \lambda^{ik}\lambda^{ik}_{(j)}, \quad 1\leq i<j<k\leq N.$$ The system can be considered as the gauge invariant integrable difference equation governing QQLs. Generalized isothermic lattices {#sec:gen-isoth-latt} =============================== Because simultaneous application of integrable constraints preserves integrability we know a priori that the following reduction of the quadrilateral lattice is integrable. A B-quadrilateral lattice in a hyperquadric $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ satisfying the local irreducibility condition is called a generalized isothermic lattice. The half hexahedron lemma and its consequences ---------------------------------------------- We start again from a geometric result, which leads to the cross-ratio characterization of the generalized isothermic lattice. \[lem:half-hex\] Under hypotheses of Lemmas \[lem:BKP-hex\] and \[lem:qred-hex\] and assuming irreducibility of the conics of the intersection of the corresponding planes with the quadric we have $$\label{eq:product-cr} {\mathrm{cr}}(x_1,x_3;x_0,x_{13}) = {\mathrm{cr}}(x_1,x_2;x_0,x_{12}) \, {\mathrm{cr}}(x_2,x_3;x_0,x_{23}),$$ where the cross-ratios are defined with respect to the corresponding conics. Denote (see Figure \[fig:hex-lem\]) ![Geometric proof of Lemma \[lem:half-hex\][]{data-label="fig:hex-lem"}](hex-lem) the plane $\sigma=\langle x_1, x_2, x_3 \rangle$, and represent points of the conic $\mathcal{C}_{ij}={{\mathcal Q}}\cap\langle x_0, x_i, x_j \rangle$, $1\leq i < j \leq 3$ by points of the line $\langle x_i, x_j \rangle$, via the corresponding planar pencil with the base at $x_0$. In this way the projective structure of the conics conicides with that of the corresponding lines. By $\ell_{ij}$ denote the intersection of the tangent line to $\mathcal{C}_{ij}$ at $x_0$ with $\sigma$. Notice that the points $\ell_{ij}$ belong to the intersection line of the tangent plane to the quadric at $x_0$ with $\sigma$, and all they represent $x_0$ but from the point of view of different conics. Denote by $d_{ij}$ the intersection point of the line $\langle x_0, x_{ij} \rangle$ with the plane $\sigma$. Notice that coplanarity of the points $x_0$, $x_{12}$, $x_{23}$ and $x_{23}$ is equivalent to collinearity of $d_{12}$, $d_{23}$ and $d_{13}$. Moreover, by definition of the cross-ratio on conics, we have $$\label{eq:cr-c-l} {\mathrm{cr}}(x_i,x_j;x_0,x_{ij}) = {\mathrm{cr}}(x_i,x_j;\ell_{ij},d_{ij}).$$ To find a relation between the cross-ratios consider perspectivity between the lines $\langle x_1, x_2 \rangle$ and $\langle x_1, x_3 \rangle$ with the center $\ell_{23}$. It transforms $x_2$ into $x_3$, $x_1$ into $x_1$, $d_{12}$ into $\tilde{d}_{12}$ (this is just definition of $\tilde{d}_{12}$) and $\ell_{12}$ into $\ell_{13}$, therefore $${\mathrm{cr}}(x_1,x_2;\ell_{12},d_{12}) = {\mathrm{cr}}(x_1,x_3;\ell_{13},\tilde{d}_{12}).$$ Smilarly, considering perspectivity between the lines $\langle x_2, x_3 \rangle$ and $\langle x_1, x_3 \rangle$ with the center $\ell_{12}$ we obtain $${\mathrm{cr}}(x_2,x_3;\ell_{23},d_{23}) = {\mathrm{cr}}(x_1,x_3;\ell_{13},\tilde{d}_{23}),$$ where again $\tilde{d}_{23}$ is the projection of $d_{23}$. The comparison of Figures \[fig:hex-lem\] and \[fig:multiplication\] gives $$\label{eq:cr-l-pr} {\mathrm{cr}}(x_1,x_3;\ell_{13},\tilde{d}_{12}) {\mathrm{cr}}(x_1,x_3;\ell_{13},\tilde{d}_{23})= {\mathrm{cr}}(x_1,x_3;\ell_{13},d_{13}),$$ which because of equations - implies the statement. For those who do not like synthetic geometry proofs we give the algebraic proof of the above Lemma in Appendix \[sec:aux-calc\]. Equation can be written in a more symmetric form $$\label{eq:product-cr-symm} {\mathrm{cr}}(x_1,x_2;x_0,x_{12}) \, {\mathrm{cr}}(x_2,x_3;x_0,x_{23}) \, {\mathrm{cr}}(x_3,x_1;x_0,x_{13}) = 1.$$ The cross-ratio of the four (coplanar) points $x_0$, $x_{12}$, $x_{13}$ and $x_{23}$ can be expressed by the other cross-ratios as $${\mathrm{cr}}(x_0,x_{12};x_{13},x_{23})={\mathrm{cr}}(x_0,x_{2};x_{3},x_{23}) {\mathrm{cr}}(x_1,x_{0};x_{3},x_{13}).$$ Consider the line $\langle d_{13}, d_{23} \rangle$, which is the section of the planar pencil containing lines $\langle x_0, x_{13} \rangle$ and $\langle x_0, x_{23} \rangle$ with the plane $\sigma$. Denote by $\ell$ the intersection point of the line with the line $\langle \ell_{13}, \ell_{23} \rangle$, then $${\mathrm{cr}}(x_0,x_{12};x_{13},x_{23}) = {\mathrm{cr}}(\ell,d_{12};d_{13},d_{23}).$$ After projection from $\ell_{12}$ we have, in notation of Figure \[fig:hex-lem\], $${\mathrm{cr}}(\ell,d_{12};d_{13},d_{23}) = {\mathrm{cr}}(\ell_{13},x_1;d_{13},\tilde{d}_{23})= {\mathrm{cr}}(1,\infty;\lambda\nu,\nu).$$ Then the standard permutation properties of the cross-ratio give the statement. \[cor:hex\] Under assumption of Lemma \[lem:half-hex\] the cross-ratios on opposite quadrilaterals of the hexahedron are equal, i.e. $$\begin{aligned} \nonumber {\mathrm{cr}}(x_1,x_2;x_0,x_{12}) & = {\mathrm{cr}}(x_{13},x_{23};x_3,x_{123}) , \\ \label{eq:equalities-cr} {\mathrm{cr}}(x_2,x_3;x_0,x_{23}) & = {\mathrm{cr}}(x_{12},x_{13};x_1,x_{123}) , \\ \nonumber {\mathrm{cr}}(x_1,x_3;x_0,x_{13}) & = {\mathrm{cr}}(x_{12},x_{23};x_2,x_{123}) . \end{aligned}$$ Equation written for the three quadrilaterals meeting in $x_3$ reads $${\mathrm{cr}}(x_0,x_{12};x_3,x_{1}) \, {\mathrm{cr}}(x_{13},x_{23};x_3,x_{123}) \, {\mathrm{cr}}(x_{23},x_0;x_3,x_{2}) = 1,$$ which compared with gives, after using elementary properties of the cross-ratio, the first equation of . Others can be obtained similarly. By symmetry we have also $${\mathrm{cr}}(x_0,x_{12};x_{13},x_{23}) = {\mathrm{cr}}(x_{1},x_{2};x_{3},x_{123}).$$ Notice that two neighbouring facets of the above hexahedron determine the whole hexahedron via construction visualized on Fig. \[fig:twotothree\]. ![Construction of the point $x_{13}$ from points $x_0$, $x_{1}$, $x_{2}$, $x_{3}$, $x_{12}$ and $x_{23}$. It belongs to the intersection line of two planes $\langle x_0, x_{1}, x_{3} \rangle$ with the plane $\langle x_0, x_{12}, x_{23} \rangle$. Because the line intersects the quadric at $x_0$, it must have also the second intersection point.[]{data-label="fig:twotothree"}](twotothree) It is easy to see that, unlike in the case of isothermic lattices, three vertices of a quadrilateral of trapezoidal lattice in a quadric [@BobSur] determine the forth vertex. \[prop:cross-ratio-gen\] A quadrilateral lattice in a quadric $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ satisfying the local irreducibility condition is a generalized isothermic lattice if and only if there exist functions $\alpha^{i}:{{\mathbb Z}}\to{{\mathbb R}}$ of single arguments $n_i$ such that the cross-ratios $\lambda^{ij} = {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)})$ can be factorized as follows $$\label{eq:isot-cross-res} \lambda^{ij} = \frac{\alpha^i}{\alpha^j}, \qquad 1\leq i<j\leq N.$$ Equations and can be rewritten as $$\label{eq:izot-cross-r-1} \lambda^{ij}\lambda^{jk} = \lambda^{ik}, \quad 1\leq i<j<k\leq N,$$ and $$\label{eq:izot-cross-r-op} \lambda^{ij}=\lambda^{ij}_{(k)}, \quad \lambda^{jk} = \lambda^{jk}_{(i)}, \quad \lambda^{ik}=\lambda^{ik}_{(j)}, \quad 1\leq i<j<k\leq N;$$ notice their consistency with the general system . Equations - imply that cross-ratios of two dimensional sub-lattices of the generalized isothermic lattice satisfy condition of the form , i.e., $$\lambda^{ij}_{(ij)}\lambda^{ij} = \lambda^{ij}_{(i)}\lambda^{ij}_{(j)}, \quad 1\leq i<j\leq N.$$ For a fixed pair $i,j$, the above relation can be resolved as in (the first equation in asserts that the functions $\alpha^i$ and $\alpha^j$ are the same for all $i,j$ sublattices). Finally, equations imply the the functions $\alpha$ can be defined consistently on the whole lattice. For convenience of the Reader we present also the algebraic proof of the above properties of the generalized isothermic lattice (see also [@BobSur] for analogous results concerning T-nets in a quadric). Assume that solutions of the system of the discrete Moutard equations satisfy the quadratic constraint $$\label{eq:q-constr} ({{\boldsymbol x}}\| {{\boldsymbol x}}) = 0,$$ where $(\cdot \| \cdot ) $ is a symmetric nondegenerate bilinear form. Then the coefficients of the Moutard equations should be of the form $$\label{eq:f-isoth} f^{ij} = \frac{({{\boldsymbol x}}\| {{\boldsymbol x}}_{(i)} - {{\boldsymbol x}}_{(j)})}{({{\boldsymbol x}}_{(i)} \| {{\boldsymbol x}}_{(j)})}, \quad 1\leq i<j\leq N.$$ Moreover by direct calculations one shows that $$({{\boldsymbol x}}_{(i)} \| {{\boldsymbol x}})_{(j)} = ({{\boldsymbol x}}_{(i)} \| {{\boldsymbol x}}), \quad 1\leq i<j\leq N,$$ which implies that the products $({{\boldsymbol x}}_{(i)} \| {{\boldsymbol x}}) $, which we denote by $\alpha_i$, are functions of single variables $n_i$. Consider the points $x, x_{(i)}, x_{(j)}$ as the (projective) basis of the plane $\langle x, x_{(i)}, x_{(j)} \rangle$. Then the homogeneous coordinates of points of the plane can be written as $${{\boldsymbol y}}= t {{\boldsymbol x}}+ t_i {{\boldsymbol x}}_{(i)} + t_j {{\boldsymbol x}}_{(j)}, \qquad (t,t_i,t_j) \in {{\mathbb R}}^3_,$$ modulo the standard common proportionality factor. In particular, the line $\langle x, x_{(i)} \rangle$ is given by equation $t_j = 0$, and the line $\langle x, x_{(j)} \rangle$ is given by equation $t_i = 0$. Due to the discrete Moutard equation the line $\langle x, x_{(ij)} \rangle$ is given by equation $t_i + t_j = 0$. To find the cross-ratio $ {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)})=\lambda^{ij} $ via lines of the planar pencil with the base point $x$ we need equation of the tangent to the conic $({{\boldsymbol y}}\| {{\boldsymbol y}}) =0$ at that point. It is easy to check that the conic is given by $$t t_i \alpha_{i} + t t_j \alpha_{j} + t_i t_j ({{\boldsymbol x}}_{(i)} \| {{\boldsymbol x}}_{(j)}) =0.$$ The tangent to the conic at $x$ is then given by $$t_i \alpha_{i} + t_j \alpha_{j} =0,$$ which implies equation . It should be mentioned the a similar quadratic reduction of the discrete Moutard equation (for $N=2$) appeared in a paper of Wolfgang Schief [@Schief-C] under the name of discrete vectorial Calapso equation, as an integrable discrete vectorial analogue of the Calapso equation [@Calapso], which is of the fourth order and describes isothermic surfaces. It turns out that the discrete Calapso equation describes also the so called Bianchi reduction of discrete asymptotic surfaces [@DNS-Bianchi-ass]. Isothermic lattices in the Möbius sphere and the so called Clifford configuration --------------------------------------------------------------------------------- In [@KoSch-Clifford] Konopelchenko and Schief observed that the “complex cross-ratio definition” of the discrete isothermic surfaces, when extended to three dimensional lattices, is related with the so called Clifford configuration of circles (see Figure \[fig:Clifford\]). ![The so called Clifford configuration of circles (the second Miquel configuration)[]{data-label="fig:Clifford"}](Clifford.eps){width="8cm"} In this Section we would like to explain that fact geometrically. Our point of view on the Clifford configuration is closely related to the geometric definition of the isothermic lattice in the Möbius sphere. Therefore we start with another, less restrictive, configuration of circles on the plane, the Miquel configuration (see Figure \[fig:Miquel\]), which provides geometric explanation of integrability of the circular lattice [@CDS]. When the quadric in Lemma \[lem:qred-hex\] is the standard sphere then the intersection curves of the planes of the quadrilaterals with the sphere are circles. After the stereographic projection from a generic point of the sphere we obtain the classical Miquel theorem [@Miquel], which can be stated as follows (given three distinct points $a$, $b$ and $c$, by $C(a,b,c)$ we denote the unique circle-line passing through them). \[th:Miquel\] Given four coplanar points $s_0$, $s_i$, $i=1,2,3$. On each circle $C(s_0,s_i, s_j)$, $1\leq i < j \leq 3$ choose a point, denoted correspondingly by $s_{ij}$. Then there exists the unique point $s_{123}$ which belongs simultaneously to the three circles $C(s_1, s_{12}, s_{13})$, $C(s_2, s_{12}, s_{23})$ and $C(s_3, s_{13}, s_{23})$. ![The (first) Miquel configuration of circles[]{data-label="fig:Miquel"}](Miquel.eps){width="8cm"} The additional assumption about coplanarity of the points $x_0$, $x_{12}$, $x_{13}$, $x_{23}$ on the sphere is then equivalent to the additional assumption about concircularity of the corresponding points $s_0$, $s_{12}$, $s_{13}$, $s_{23}$. In view of Lemma \[lem:BKP-hex\] we obtain therefore another configuration of circles, the so called Clifford configuration, which can be described as follows. \[th:Clifford\] Under hypotheses of Theorem \[th:Miquel\] assume that the points $s_0$, $s_{12}$, $s_{13}$, $s_{23}$ are concircular, then the points $s_1$, $s_{2}$, $s_{3}$ and $s_{123}$ are concircular as well. The original Clifford’s formulation of the above result was more symmetric. Its relation to Theorem \[th:Clifford\] is analogous to relation of the Cox theorem to Lemma \[lem:BKP-hex\]. We remark that although the above theorem is usually attributed (see for example [@Coxeter]) to Wiliam Clifford [@Clifford] it appeared in much earlier paper [@Miquel] of Auguste Miquel, where we read as Thèorème II* the following statement: Lorsqu’un quadrilatère complet curviligne ABCDEF est formé par quatre arcs de cercle AB, BC, CD, DA, qui se coupent tous quatre en un même point P, si l’on circonscrit des circonférences de cercle à chacun des quatre triangles curvilignes que forment les côtés de ce quadrilatère, les circonférences de cercle AFB, EBC, DCF, DAE ainsi obtenues se couperont toutes quatre en un même point G. Points on Figure \[fig:Clifford\] are labelled in double way to visualize simultaneously the configuration in formulation of Theorem \[th:Clifford\] and in Miquel’s formulation. The Darboux transformation of the generalized isotermic lattice {#sec:Darboux} =============================================================== The fundamental, Moutard and Ribaucour transformations ------------------------------------------------------ Usually, on the discrete level there is no essential difference between integrable lattices and their transformations. The analogue of the fundamental transformation of Jonas for quadrilateral lattices is defined as construction of a new level of the lattice [@TQL] keeping the basic property of planarity of elementary quadrilaterals. Below we recall the relevant definitions of the fundamental transformation and its important reductions – the BQL reduction [@BQL] (algebraically equivalent to the Moutard transformation [@NiSchief]), and the QQL reduction [@q-red] called the Ribaucour transformation. The fundamental transform of a quadrilateral lattice $x:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ is a new quadrilateral lattice $\hat{x}:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ constructed under assumption that for any point $x$ of the lattice and any direction $i$, the four points $x$, $x_{(i)}$, $\hat{x}$ and $\hat{x}_{(i)}$ are coplanar. The fundamental transformation of a B-quadrilateral lattice $x:{{\mathbb Z}}^N\to{{\mathbb P}}^M$ constructed under additional assumption that for any point $x$ of the lattice and any pair $i,j$ of different directions, the four points $x$, $x_{(ij)}$, $\hat{x}_{(i)}$ and $\hat{x}_{(j)}$ are coplanar is called the BQL (Moutard) reduction of the fundamental transformation of $x$. Algebraic description of the above transformation is given as follows [@NiSchief]. Given solution ${{\boldsymbol x}}$ of the system of discrete Moutard equations and given its scalar solution $\theta$, then the solution $\hat{{{\boldsymbol x}}}$ of the system $$\label{eq:Moutard-transf} \hat{{\boldsymbol x}}_{(i)} - {{\boldsymbol x}}= \frac{\theta}{\theta_{(i)}}\left( \hat{{\boldsymbol x}}- {{\boldsymbol x}}_{(i)}\right),$$ satisfies equations with the new potential $$\hat{f}^{ij} = f^{ij}\frac{\theta_{(i)}\theta_{(j)}}{\theta\theta_{(ij)}}, \qquad i<j,$$ and new $\tau$-function $$\label{eq:transf-Mout-tau} \hat\tau = \theta\tau.$$ The fundamental transformation of a quadrilateral lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ in a quadric, constructed under additional assumption that also $\hat{x}$ satifies the same quadratic constraint is called the Riboucour transformation of $x$. The Darboux transformation -------------------------- The fundamental transformation of a generalized isothermic lattice which is simultaneously the Ribaucour and the Moutard transformation is called the Darboux transformation. Notice that there is essentially no difference between the Darboux transformation and construction of a new level of the generalized isothermic lattice. Therefore, given points $x$, $x_{(i)}$, $x_{(j)}$, $x_{(ij)}$, $i\ne j$, of the initial lattice, and given points $\hat{x}$, $\hat{x}_{(j)}$ of its Darboux transform then the point $\hat{x}_{(i)}$ is determined by the “half-hexahedron construction” visualized on Figure \[fig:twotothree\], i.e., $\hat{x}_{(i)}$ is the intersection point of the line $\langle x, x_{(i)}, \hat{x} \rangle \cap \langle x, x_{(ij)}, \hat{x}_{(j)} \rangle$ with the quadric. Moreover, Lemma \[lem:half-hex\] implies $$\label{eq:cr-prod-transf} {\mathrm{cr}}(x_{(i)},\hat{x};x,\hat{x}_{(i)}) = {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)}) {\mathrm{cr}}(x_{(j)},\hat{x};x,\hat{x}_{(j)}),$$ while Corollary \[cor:hex\] gives $${\mathrm{cr}}(\hat{x}_{(i)},\hat{x}_{(j)};\hat{x},\hat{x}_{(ij)})= {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)}).$$ The algebraic derivation of the above results is given below. The Darboux transformations of the discrete isothermic surfaces in the light-cone description were discussed in a similar spirit in [@BobSur]. \[prop:Darboux-prod\] If $\hat{x}:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ is a Darboux transform of the generalized isothermic lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$ then the product $(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}})$ of their homogeneous coordinates in the gauge of the linear problem and the corresponding Moutard transformation , with respect to the bilinear form $(\cdot\|\cdot)$ defining the quadric $\mathcal{Q}$, is constant. The homogeneous coordinates ${{\boldsymbol x}}$ and $\hat{{{\boldsymbol x}}}$ in the Moutard transformation satisfy equation of the form $$\hat{{{\boldsymbol x}}}_{(i)} - {{\boldsymbol x}}= f^i(\hat{{{\boldsymbol x}}} - {{\boldsymbol x}}_{(i)}), \qquad 1\leq i \leq N,$$ with appropriate functions $f^i:{{\mathbb Z}}^N\to{{\mathbb R}}$. The quadratic condition $(\hat{{{\boldsymbol x}}}_{(i)}\|\hat{{{\boldsymbol x}}}_{(i)})=0$ together with other quadratic conditions give $$\label{eq:fi-isoth} f^i = \frac{({{\boldsymbol x}}\| \hat{{{\boldsymbol x}}} -{{\boldsymbol x}}_{(i)})}{(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}}_{(i)})},$$ which implies $(\hat{{{\boldsymbol x}}}_{(i)}\|{{{\boldsymbol x}}}_{(i)}) = (\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}})$. Notice the above proof goes along the corresponding reasoning in the first part of the algebraic proof of Proposition \[prop:cross-ratio-gen\]. The analogous reasoning as in its second part gives the following statement. Under hypothesis of Proposition \[prop:Darboux-prod\] denote $\zeta=(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}})$ and $\lambda^i = {\mathrm{cr}}(\hat{x},x_{(i)};x,\hat{x}_{(i)})$ then equations and should be replaced by $$\label{eq:cr-lambda-prod} \lambda^i = \frac{\zeta}{\alpha_i}, \qquad \lambda^i \lambda^{ij} = \lambda^j.$$ The above reasoning can be reversed giving the algebraic way to find the Darboux transform of a given generalized isothermic lattice. \[th:Darboux\] Given a solution ${{\boldsymbol x}}:{{\mathbb Z}}^N\to {{\mathbb R}}^{M+1}_$ of the system of Moutard equations satisfying the constraint $({{\boldsymbol x}}\|{{\boldsymbol x}})=0$, considered as homogeneous coordinates of generalized isothermic lattice $x:{{\mathbb Z}}^N\to\mathcal{Q}\subset{{\mathbb P}}^M$, denote $\alpha_i = ({{\boldsymbol x}}_{(i)}\|{{\boldsymbol x}})$. Given a point $[\hat{{{\boldsymbol x}}}_0]=\hat{x}_0\in\mathcal{Q}$, denote $\zeta = (\hat{{{\boldsymbol x}}}_0\|{{\boldsymbol x}}(0))$. Then there exists unique solution of the linear system $$\label{eq:Darboux} \hat{{{\boldsymbol x}}}_{(i)} = {{\boldsymbol x}}+ \frac{\zeta - \alpha_i}{(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}}_{(i)})} (\hat{{{\boldsymbol x}}} - {{\boldsymbol x}}_{(i)}), \qquad, \quad 1\leq i \leq N,$$ with initial condition $\hat{{{\boldsymbol x}}}(0) = \hat{{{\boldsymbol x}}}_0$ which gives the Darboux transform of the lattice $x$. In particular $$\begin{aligned} \label{eq:cr-zeta-alpha} {\mathrm{cr}}(\hat{x},x_{(i)};x,\hat{x}_{(i)}) & = \frac{\zeta}{\alpha_i} \\ {\mathrm{cr}}(\hat{x}_{(i)},\hat{x}_{(j)};\hat{x},\hat{x}_{(ij)}) & = {\mathrm{cr}}(x_{(i)},x_{(j)};x,x_{(ij)}) = \frac{\alpha^i}{\alpha^j}.\end{aligned}$$ Before proving the Theorem let us state a Lemma relating the parameter $\zeta$ of the Darboux transformation with the functional parameter $\theta$ of the Moutard transformation . \[eq:zeta-theta\] Under hypotheses of Theorem \[th:Darboux\] the solution $\theta$ of the system $$\theta_{(i)}=\theta\frac{(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}}_{(i)})}{\zeta - \alpha_i}, \qquad \quad 1\leq i \leq N,$$ satisfies the system of discrete Moutard equations with the coefficients given by . By direct verification. Notice that both ways to calculate $\theta_{(ij)}$, $i\ne j$, from $\theta$ give the same result, and to do that we do not use compatibility of the system . By direct calculation one can check that the system preserves the constraints $(\hat{{{\boldsymbol x}}}\|\hat{{{\boldsymbol x}}}) =0$ and $(\hat{{{\boldsymbol x}}}\|{{\boldsymbol x}})=\zeta$, moreover $(\hat{{{\boldsymbol x}}}_{(i)}\|\hat{{{\boldsymbol x}}}) =({{\boldsymbol x}}_{(i)}\|{{\boldsymbol x}})$. Compatibility of the system can be checked by direct calculation, but in fact it is the consequence of Lemma \[eq:zeta-theta\] and properties of the Moutard transformation . Notice that because there is essentially no diffrence between the lattice directions and the transformation directions, the tranformation equations can be guessed by keeping the Moutard-like form supplementing it by calculation of the coefficient $f^i$ from the quadratic constraint. We will use this observation in the next Section where we consider the permutability principle for the Darboux transformations of generalized isothermic lattices. The Bianchi permutability principle ----------------------------------- The original Bianchi superposition principle for the Darboux transformations of the isothermic surfaces reads as follows [@Bianchi-isoth]: Se dalla superficie isoterma $S$ si ottengono due nuove superficie isoterme $S_1$, $S_2$ mediante le trasformazioni di Darboux $D_{m_1}$, $D_{m_2}$ a costanti $m_1$, $m_2$ differenti, esiste una quarta superficie isoterma $\overline{S}$, pienamente determinata e costruibile in termini finiti, che è legata alla sua volta alle medesime superficie $S_1$, $S_2$ da due trasformazioni di Darboux $\overline{D}_{m_2}$, $\overline{D}_{m_1}$ colle costanti invertite $m_2$, $m_1$.* ![Geometric construction of the superposition of two Darboux transformations[]{data-label="fig:bianchi"}](bianchi) Its version for generalized isothermic lattices can be formulated analogously. \[prop:superp\] When from given generalized isothermic lattice $x$ there were constructed two new isothermic lattices $\hat{x}^1$ and $\hat{x}^2$ via the Darboux transformations with different parameters $\zeta_1$ and $\zeta_2$, then there exists the unique forth generalized isothermic lattice $\hat{x}^{12}$, determined in algebraic terms from the three previous ones, which is connected with two intermediate lattices $\hat{x}^1$ and $\hat{x}^2$ via two Darboux transformations with reversed parameters $\zeta_2$, $\zeta_1$. The algebraic properties of the B-reduction of the fundamental transformation (the discrete Moutard transformation) imply that in the gauge of the linear problem and of the transformation equations the superposition of two such transformations reads $$\label{eq:-transf-sup} \hat{{{\boldsymbol x}}}^{12} - {{\boldsymbol x}}= f (\hat{{{\boldsymbol x}}}^1 - \hat{{{\boldsymbol x}}}^2),$$ where $f$ is an appropriate function [@NiSchief; @BQL]. Because of the additional quadratic constraints the function is given by (compare also equations and ) $$\label{eq:f-isoth-sup} f = \frac{({{\boldsymbol x}}\| \hat{{{\boldsymbol x}}}^1 - \hat{{{\boldsymbol x}}}^2)}{(\hat{{{\boldsymbol x}}}^1 \| \hat{{{\boldsymbol x}}}^2)}.$$ The lattice $\hat{x}^{12}$ with homogeneous coordinates given by and is superposition of two Darboux transforms. Finally, direct calculation shows that $$(\hat{{{\boldsymbol x}}}^{12} \| \hat{{{\boldsymbol x}}}^2) = (\hat{{{\boldsymbol x}}}^1 \| {{\boldsymbol x}}) = \zeta_1, \qquad (\hat{{{\boldsymbol x}}}^{12} \| \hat{{{\boldsymbol x}}}^1)= (\hat{{{\boldsymbol x}}}^2 \| {{\boldsymbol x}})= \zeta_2 .$$ The final algebraic superposition formula reads $$\hat{{{\boldsymbol x}}}^{12} - {{\boldsymbol x}}= \frac{\zeta_1 - \zeta_2}{(\hat{{{\boldsymbol x}}}^1 \| \hat{{{\boldsymbol x}}}^2)} (\hat{{{\boldsymbol x}}}^1 - \hat{{{\boldsymbol x}}}^2),$$ while the cross-ratio of the four corresponding points calculated with respect to the conic intersection of the plane $\langle x,\hat{x}^1,\hat{x}^2 \rangle$ and the quadric is given by $$\label{eq:cr-zeta-sup} {\mathrm{cr}}(\hat{x}^1,\hat{x}^2;x,\hat{x}^{12}) = \frac{\zeta_1}{\zeta_2}.$$ To find the lattice $\hat{x}^{12}$ geometrically we can use again the “half-hexahedron construction” in the new context visualized on Figure \[fig:bianchi\] (compare with Figure \[fig:twotothree\]). Moreover, Lemma \[lem:half-hex\] gives $$\label{eq:cr-prod-sup} {\mathrm{cr}}(\hat{x}^1,\hat{x}^2;x,\hat{x}^{12}) = {\mathrm{cr}}(\hat{x}^1, x_{(i)};x,\hat{x}^1_{(i)}) \, {\mathrm{cr}}(x_{(i)}, \hat{x}^2;x,\hat{x}^2_{(i)}),$$ which, due to equation , is in agreement with . Two dimensional generalized isothermic lattice {#sec:isothermic} ============================================== In the previous Sections we were mainly interested in generalized isothermic lattices of dimension greater then two. However, simultaneous application of the B-constraint and the quadratic constraint lowers dimensionality of the lattice (in the sense of the initial boundary value problem). One can see it from Figure \[fig:twotothree\], which implies that two intersecting strips made of planar quadrialterals with vertices in a quadric (see Figure \[fig:isoth-init\]) can be extended to a two dimensional quadrilateral lattice in the quadric. Because of Lemma \[lem:half-hex\] such lattice satisfies Steiner’s version of the cross-ratio constraint . ![Two intersecting initial strips of a two dimensional generalized isothermic lattice allow to build all the lattice. The additional transverse quadrilateral allows to build the lattice (together wih its Darboux transform) in a three dimensional fashion[]{data-label="fig:isoth-init"}](isoth-init) One can define however geometrically two dimensional generalized isothermic lattices (generalized discrete isothermic surfaces) without using the three dimensional construction. An important tool here is the projective interpretation of the discrete Moutard equation [@DGNS] as representing quadrilateral lattice with additional linear relation between any of its points $x$ and its four second-order neighbours $x_{(\pm 1 \pm 2)}$. Geometrically, such five points of a two dimensional B-quadrilateral lattice are contained in a subspace of dimension three; for generic two dimensional quadrilateral lattice such points are contained in a subspace of dimension four. To exclude further degenerations we assume that no of the four points $x_{\pm 1}$, $x_{\pm 2}$ belongs to that three dimensional subspace. A two dimensional B-quadrilateral lattice in a hyperquadric $x:{{\mathbb Z}}^2\to\mathcal{Q}\subset{{\mathbb P}}^M$ satisfying the local irreducibility condition is called a generalized discrete isothermic surface. Notice that the above Definition gives (the conclusion was drawn by Alexander Bobenko) a geometric characterization of the classical discrete isothermic surfaces of Bobenko and Pinkall. Mainly, the non-trivial intersection of a three dimensional subspace with the Möbius sphere is a two dimensional sphere. After the stereographic projection, which preserves co-sphericity of points, a discrete isothermic surface in the Möbius (hyper)sphere gives circular two dimensional lattice $s:{{\mathbb Z}}^2\to{{\mathbb E}}^M$ such that for any of its points $s$ there exists a sphere containig the point and its four second-order neighbours $s_{(\pm 1 \pm 2)}$. Notice that, actually, all calculations where we used simultaneously both the discrete Moutard equation and the quadratic constraint (algebraic proofs of Proposition \[prop:cross-ratio-gen\], Theorem \[th:Darboux\] and Proposition \[prop:superp\]) remain true for $N=2$. Therefore the corresponding results on the cross-ratio characterization of generalized discrete isothermic surfaces, their Darboux transformation and the Bianchi superposition principle are still valid. To complete this Section let us present the geometric construction of a generalized discrete isothermic surface (see Figure \[fig:geom-constr-surf\]). ![Geometric construction of generalized discrete isothermic surfaces[]{data-label="fig:geom-constr-surf"}](geom-constr-surf) The basic step of the construction, which allows to build the generalized discrete isothermic surface from two initial quadrilateral strips in a quadric in a two dimensional fashion can be discribed as follows. Consider the four dimensional subspace $V_4 = \langle x, x_{(1)}, x_{(2)}, x_{(-1)}, x_{(-2)} \rangle$, where the basic step takes place. Denote by $V_3 = \langle x, x_{(-1-2)}, x_{(-12)}, x_{(1-2)} \rangle$ its three dimensional subspace passing through the points $x$, $x_{(-1-2)}$, $x_{(-12)}$ and $x_{(1-2)}$, and by $V_2 = \langle x, x_{(1)}, x_{(2)} \rangle$ the plane of the elementary quatrilateral whose fourth vertex $x_{12}$ we are going to find. In the construction of the two dimensional B-quadrilateral lattice the vertex must belong to the line $V_1 = V_3 \cap V_2$. In our case it should also belong to the conic $\mathcal{C}=V_2 \cap \mathcal{Q}$. Because the conic contains already one point $x$ of the line $V_1$, the second point is unique. Notice that although the points $x_{(-1)}$ and $x_{(-2)}$ do not play any role in the construction, they can be easily recovered in a similar way as above. Conclusions and discussion ========================== In the paper we defined new integrable reduction of the lattice of planar quadrilaterals, which contains as a particular example the discrete isothermic surfaces. We studied, by using geometric and algebraic means, various aspects of such generalized isothermic lattices. In particular, we defined the (analogs of the) Darboux transformations for the lattices and we showed the corresponding permutablity principle. The theory of integrable systems is deeply connected with results of geometers of the turn of XIX and XX centuries. The relation of integrability and geometry is even more visible on the discrete level, where into the game there enter basic results of the projective geometry. In our presentation of the generalized isothermic lattices the basic geometric results were a variant of the Möbius theorem and the generalization of the Miquel theorem to arbitrary quadric, which combined together gave the corresponding generalization of the Clifford theorem (known already to Miquel). An important tool in our research was also Steiner’s description of conics and the geometric properties of von Staudt’s algebra (see Appendix \[sec:cr\]). Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank to Jaros[ł]{}aw Kosiorek and Andrzej Matraś for discusions cencerning incidence geometry and related algebraic questions. The main part of the paper was prepared during my work at DFG Research Center MATHEON in Institut für Mathematik of the Technische Universität Berlin. The paper was supportet also in part by the Polish Ministry of Science and Higher Education research grant 1 P03B 017 28. Finally, it is my pleasure to thank the organizers of the SIDE VII Conference for support. The cross-ratio and the projective structure of a conic {#sec:cr} ======================================================= For convenience of the Reader we have collected some facts from projective geometry (see, for example [@Samuel; @VeblenYoung]) used in the main text of the paper. Let $a,b,c$ be distinct points of the projective line $D$ over the field ${{\mathbb K}}$. Given $d\in D$, the cross-ratio ${\mathrm{cr}}(a,b;c,d)$ of the four points $a,b,c,d$ is defined as $h(d)\in \hat{{\mathbb K}}={{\mathbb K}}\cup\{\infty\}$ where $h$ is the unique projective transformation $D\to\hat{{\mathbb K}}$ that takes $a$, $b$ and $c$ to $\infty$, $0$ and $1$, respectively. For $D=\hat{{\mathbb K}}$, with the usual conventions about operations with $0$ and $\infty$, the cross-ratio is given by $${\mathrm{cr}}(a,b;c,d) = \frac{(c-a)(d-b)}{(c-b)(d-a)}.$$ Denote by $\boldsymbol{a}$ and $\boldsymbol{b}$ homogenous coordinates of the points $a$ and $b$. If the homogenous coordinates of the points $c$ and $d$ collinear with $a$, $b$ are, respectively, $\boldsymbol{c}=\alpha\boldsymbol{a}+ \beta\boldsymbol{b}$ and $\boldsymbol{d}=\gamma\boldsymbol{a}+ \delta\boldsymbol{b}$, then $$\label{eq:cr-proj} {\mathrm{cr}}(a,b;c,d) = \frac{\beta\gamma}{\alpha\delta}.$$ Let $D$ and $D^\prime$ be projective lines, $a,b,c,d$ distinct points on $D$, and $a^\prime,b^\prime,c^\prime,d^\prime$ distinct points on $D^\prime$. There exists a projective transformation $u:D\to D^\prime$ taking $a,b,c,d$ into $a^\prime,b^\prime,c^\prime,d^\prime$, respectively, if and only if the cross-ratios ${\mathrm{cr}}(a,b;c,d)$ and ${\mathrm{cr}}(a^\prime,b^\prime;c^\prime,d^\prime)$ are equal. ![Multiplication of cross-ratios on the line[]{data-label="fig:multiplication"}](multiplication) Following von Staudt one can perform geometrically algebraic operations on cross-ratios (see [@VeblenYoung] for details). We will be concerned with geometric multiplication, which can be considered as “projectivization” of the Thales theorem – see the self-explanatory Figure \[fig:multiplication\]. The planar pencil of lines has the natural projective structure inherited from any line not intersecting its base. Let $\mathcal{C}$ be an irreducible conic in a projective plane, and $a\in\mathcal{C}$ a point. To each line $D$ of the pencil $F_a$ of the base $a$, we associate the second point where $D$ intersects $\mathcal{C}$ (see Figure \[fig:conic\]); we denote this point by $j_a(D)$. When $D$ is the tangent to $\mathcal{C}$ at $a$, let $j_a(D)$ be the point $a$. Thus $j_a$ is a bijection from $F_a$ to $\mathcal{C}$. If $b$ is another point on $\mathcal{C}$, the composition $j_b^{-1}\circ j_a$ is a projective transformation from $F_a$ to $F_b$. ![The projective structure of a conic[]{data-label="fig:conic"}](conic) Thus the bijection from $F_a$ to $\mathcal{C}$ allows us to transport to $\mathcal{C}$ the projective structure of $F_a$. This structure does not depend on the point $a$. Conversely, any projective transformation between two pencils defines a conic. Finally we present the relation between the complex cross-ratio of the Möbius geometry, and Steiner’s conic cross-ratio. \[prop:Moebius-and-Steiner\] Four points $a,b,c,d\in\hat{{\mathbb C}}$ are cocircular or collinear if and only if their cross-ratio ${\mathrm{cr}}(a,b;c,d)_{{\mathbb C}}$ computed in $\hat{{\mathbb C}}$, is real. The cross-ratio ${\mathrm{cr}}(a,b;c,d)_{{\mathbb C}}$ is equal to the cross-ratio ${\mathrm{cr}}(a,b;c,d)$ computed using the real projective line structure of the line or the circle considered as a conic. Auxiliary calculations {#sec:aux-calc} ====================== In this Appendix we would like to prove the “cross-ratio” characterization (Proposition \[prop:qred-cr\]) of locally irreducible quadrilateral lattices in quadrics, and then to give algebraic proof of the basic Lemma \[lem:half-hex\]. It turns out that in the course of our calculations will give also algebraic “down to earth” proofs of the basic Lemmas \[lem:gen-hex\], \[lem:BKP-hex\] and \[lem:qred-hex\]. Proposition \[prop:qred-cr\] is immediate consequence of the following result. \[lem:qred-hex-cr\] Under hypotheses of Lemma \[lem:qred-hex\] and irreducibility of the conics, being intersections of the planes of the quadrilaterals with the quadric, the cross-ratios (defined with respect to the conics) of points on opposite sides of the hexahedron are connected by the following equation $$\label{eq:qred-cross-ratios} \begin{split} {\mathrm{cr}}(x_1,x_2;x_0,x_{12})& \, {\mathrm{cr}}(x_{13},x_{23};x_3,x_{123}) {\mathrm{cr}}(x_2,x_3; x_0,x_{23}) \, {\mathrm{cr}}(x_{12},x_{13};x_1,x_{123}) = \\ &={\mathrm{cr}}(x_1,x_2;x_0,x_{13}) \, {\mathrm{cr}}(x_{13},x_{23};x_2,x_{123}). \end{split}$$ Let us choose the points $x_0$, $x_1$, $x_2$ and $x_3$ as the basis of projective coordinate system in the corresponding three dimensional subspace, i.e., $${{\boldsymbol x}}_0 = [1:0:0:0], \quad {{\boldsymbol x}}_1 = [0:1:0:0], \quad {{\boldsymbol x}}_2 = [0:0:1:0],\quad {{\boldsymbol x}}_3 = [0:0:0:1],$$ and denote by $[t_0 : t_1 : t_2 : t_3 ]$ the corresponding homogeneous coordinates. For generic points $x_{ij}\in\langle x_0, x_{i}, x_{j} \rangle$, $1\leq i < j \leq 3$, with homogeneous coordinates $${{\boldsymbol x}}_{12} = [a_0:a_1:a_2:0], \quad {{\boldsymbol x}}_{13} = [b_0:b_1:0:b_3], \quad {{\boldsymbol x}}_{23} = [c_0:0:c_2:c_3],$$ one obtains, via the standard linear algebra, equations of the planes $\langle x_1, x_{12}, x_{13} \rangle$, $\langle x_2, x_{12}, x_{23} \rangle$, $\langle x_3, x_{13}, x_{23} \rangle$ respectively, $$\begin{aligned} a_2 b_3 t_0 = & a_0 b_3 t_2 + b_0 a_2 t_3,\\ a_1 c_3 t_0 = & a_0 c_3 t_1 + c_0 a_1 t_3,\\ \label{eq:til_pi_12} b_1 c_2 t_0 = & b_0 c_2 t_1 + c_0 b_1 t_2.\end{aligned}$$ The intersection point $x_{123}$ of the planes has the following coordinates ${{\boldsymbol x}}_{123}=[y_0:y_1:y_2:y_3]$ $$\label{eq:x_123-gen} \begin{split} y_0 = & a_0 b_0 c_0 \left( \frac{1}{a_2 b_1 c_3} + \frac{1}{a_1 b_3 c_2} \right), \\ y_1 = & \frac{b_0 c_0}{b_3 c_2} + \frac{a_0 c_0}{a_2 c_3} - \frac{c_0^2}{c_2 c_3} , \\ y_2 = & \frac{a_0 b_0}{a_1 b_3} + \frac{b_0 c_0}{b_1 c_3} - \frac{b_0^2}{b_1 b_3} , \\ y_3 = & \frac{a_0 c_0}{a_1 c_2} + \frac{a_0 b_0}{a_2 b_1} - \frac{a_0^2}{a_1 a_2} . \end{split}$$ Up to now we have not used the additional quadratic restriction, and what we have done was just the algebraic proof of Lemma \[lem:gen-hex\]. Any quadric ${{\mathcal Q}}$ passing through $x_0$, $x_1$, $x_2$ and $x_3$ must have equation of the form $$\label{eq:quadric} a_{01}t_0t_1 + a_{02}t_0t_2 + a_{03}t_0t_3 + a_{12}t_1t_2 + a_{13}t_1t_3 + a_{23}t_2t_3 = 0.$$ The homogeneous coordinates of the points $x_{12}$, $x_{23}$ and $x_{13}$ can be parametrized in terms of the corresponding cross-ratios $\lambda = {\mathrm{cr}}(x_1,x_2;x_0,x_{12})$, $\nu = {\mathrm{cr}}(x_2,x_3;x_0,x_{23})$ and $\mu = {\mathrm{cr}}(x_1,x_3;x_0,x_{13})$ as $$\begin{aligned} \label{eq:x12} {{\boldsymbol x}}_{12} &= \Big[\, \frac{\lambda a_{12}}{1-\lambda} \,:\, -\lambda a_{02} \,:\, a_{01}\,: \,0 \, \Big], \\ \label{eq:x23} {{\boldsymbol x}}_{23} &= \Big[\, \frac{\nu a_{23}}{1-\nu}\,: \, 0\,: \, -\nu a_{03} \,: \,a_{02} \,\Big],\\ \label{eq:x_13-quadr} {{\boldsymbol x}}_{13} &= \Big[\, \frac{\mu a_{13}}{1-\mu} \,: \, -\mu a_{03}\,: \, 0 \,: \, a_{01} \,\Big].\end{aligned}$$ We will only show how to find the homogeneous coordinates of $x_{12}$ in terms of $\lambda$. Let us parametrize points of the conic $$\label{eq:conic} a_{01}t_0t_1 + a_{02}t_0t_2 + a_{12}t_1t_2 = 0,$$ being intersection of the quadric with the plane $t_3=0$, by the planar pencil with base at $x_0$. The point $x_0$ corresponds to the tangent to the conic at $x_0$ $$a_{01}t_1 + a_{02}t_2 = 0,$$ while the points $x_1$ and $x_2$ correspond to lines $t_2=0$ and $t_1 = 0$, respectively. The line $\langle x_0, x_{12} \rangle$ must have equation (see equation in Appendix \[sec:cr\]) of the form $$a_{01}t_1 + \lambda a_{02}t_2 = 0,$$ which inserted into equation of the conic gives, after exluding the point $x_0$, the homogeneous coordinates of the point $x_{12}$. Inserting expressions , and into formulas we obtain homogeneous coordinates $[y_0:y_1:y_2:y_3]$ of the point $x_{123}$ parametrized in terms of the cross-ratios $\lambda$, $\nu$, $\mu$ $$\label{eq:x_123-quadr} \begin{split} y_0 = & \frac{a_{12}a_{23}a_{13}}{a_{01} a_{02} a_{03}} \frac{\lambda \nu - \mu}{(1-\lambda)(1-\mu)(1-\nu)}, \\ y_1 = & \frac{a_{23}}{1-\nu} \left( \frac{a_{13}}{a_{01}a_{03}}\frac{\mu}{1-\mu} - \frac{a_{23}}{a_{02}a_{03}}\frac{\nu}{1-\nu} - \frac{a_{12}}{a_{01}a_{02}}\frac{\lambda\nu}{1-\lambda} \right) , \\ y_2 = & \frac{a_{13}}{1-\mu} \left( -\frac{a_{13}}{a_{01}a_{03}}\frac{\mu}{1-\mu} + \frac{a_{23}}{a_{02}a_{03}}\frac{\nu}{1-\nu} + \frac{a_{12}}{a_{01}a_{02}}\frac{\mu}{1-\lambda} \right) , \\ y_3 = & \frac{a_{12}}{1-\lambda} \left( \frac{a_{13}}{a_{01}a_{03}}\frac{\lambda}{1-\mu} - \frac{a_{23}}{a_{02}a_{03}}\frac{1}{1-\nu} - \frac{a_{12}}{a_{01}a_{02}}\frac{\lambda}{1-\lambda} \right) . \end{split}$$ One can check that such expressions do satisfy the quadric equation , i. e., we have obtained the direct proof of Lemma \[lem:qred-hex\] under additional assumption of ireducibility of the conics. Further calculations give the cross-ratios on remaining three sides of the hexahedron $$\begin{aligned} \label{eq:qred-cross-ratios-opposite} {\mathrm{cr}}(x_{13}, x_{23}; x_3, x_{123}) & = - \frac{\mu (1-\nu) a_{13} y_1}{\nu (1-\mu) a_{23} y_2} , \nonumber \\ {\mathrm{cr}}(x_{12}, x_{13}; x_1, x_{123}) & = - \frac{(1-\mu) a_{12} y_2}{(1-\lambda) a_{13} y_3}, \\ {\mathrm{cr}}(x_{12}, x_{23}; x_2, x_{123}) & = \lambda \frac{(1-\nu) a_{12} y_1}{(1-\lambda) a_{23} y_3}, \nonumber\end{aligned}$$ with $y_i$ given by , which implies equation . We can express coplanarity of four points in ${{\mathbb P}}^3$ as vanishing of the determinant of the matrix formed by their homogeneous coordinates. 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--- abstract: \| [Background]{} Mathematical modeling approaches are becoming ever more established in clinical neuroscience. They provide insight that is key to understand complex interactions of network phenomena, in general, and interactions within the migraine generator network, in particular. [Purpose]{} In this study, two recent modeling studies on migraine are set in the context of premonitory symptoms that are easy to confuse for trigger factors. This causality confusion is explained, if migraine attacks are initiated by a transition caused by a tipping point. [Conclusion]{} We need to characterize the involved neuronal and autonomic subnetworks and their connections during all parts of the migraine cycle if we are ever to understand migraine. We predict that mathematical models have the potential to dismantle large and correlated fluctuations in such subnetworks as a dynamical network biomarker of migraine. author: - 'Markus A. Dahlem' - Jürgen Kurths - 'Michel D. Ferrari' - Kazuyuki Aihara - Marten Scheffer - Arne May title: Understanding migraine using dynamical network biomarkers --- Bullet points {#bullet-points .unnumbered} ============= - Article highlights the use of mathematical models in migraine research. - It explains causality confusion between triggers and premonitory symptoms by tipping points. - This explanation makes specific predictions of large scale correlated fluctuations that need to be tested by noninvasive imaging. Introduction {#introduction .unnumbered} ============ Although migraine sufferers often are convinced that certain food, stress, bright light, neck pain, and other factors may trigger attacks, under controlled experimental conditions, there is very little if any evidence that these putative trigger factors can actually provoke attacks (1,2). Instead of being the trigger initiating an attack, craving for certain food, perceiving normal events as stressful or normal light intensities as too bright, and experiencing neck pain in the few hours to days prior to the clinical manifestation of an migraine attack more likely are early premonitory symptoms of an attack. Premonitory symptoms are actually expected as early–warning signs of an imminent transition and they are easy to confuse for trigger factors, if migraine attacks are initiated by a transition caused by a tipping point and therefore exhibiting universal behavior (3) (see Fig. 1). We addressed this question with mathematical migraine models in two recent articles (4, 5). ![Illustration of tipping point behavior: A changing landscape with potential wells and unique (violet landscape) or two (all other) stable attractors, representing the pain state (left) and attack–free state (right). The balls represent the current state and its variability (yellow dashed–line) under constant influence of noise as putative trigger factors (hammer). In this schematic illustration, a gradual increase of excitability over time lowers the potential well and the local landscape of the current state becomes shallower. The curvature of the well is inversely proportional to the system–immanent time scale ($\tau$) that determines the response to natural noise. The incipient loss of the threshold separating the attack–free minimum state from the attack state, that is, the prodromal phase, is indicated by both large amplitudes and critical slowing down (large $\tau$) of the fluctuations of the ball. These large amplitudes and critical slowing down of fluctuations can lead to confusing this for trigger factors, when in fact even purely internal noise (absence of hammer) will cause these eventually. Figure modified from Refs.(3,4).](tippingPointFig.pdf){width="\columnwidth"} Universal tipping point behavior explains premonitory symptoms {#universal-tipping-point-behavior-explains-premonitory-symptoms .unnumbered} ============================================================== The theoretical concept of tipping points, if transferred to migraine research, addresses the role of reduced resilience (4). As the brain comes to a tipping point, a small stimulus can trigger a slow cortical wave experienced as aura symptoms (6). The most important ‘cause’ or ‘explanation’ of migraine as a chronic disease with episodic manifestations is therefore the dynamic change in excitability bringing the brain to this tipping point, rather than the small perturbation that finally tipped the balance leading to the attack as such. Although perturbations can be reduced, for example using exercise, stress relaxation techniques or cognitive behavioral therapy, they can not be excluded entirely near tipping points (Fig. 1). When the state of the brain gets again farther away from the tipping point, i.e., in a remissions phase of the migraine cycle, even very large perturbations—as recently published (1)—will not be able to trigger episodes. This refractoriness of hyperexcitability to external stimuli in the postictal and interictal phase can also be predicted as a general feature of chronic disorders with episodes caused by recurrently passing through a tipping point. The neural correlates of early–warning signs caused by tipping points can be described as dynamical network biomarkers (DNB) (7). A DNB describes a certain behavior in a subnetwork of a complex disease, namely signals that announce the reduced resilience at an imminent tipping point by large and correlated amplitudes and critical slowing down of fluctuations in this subnetwork. In Fig. 1, this subnetwork is represented by the cross–section within a higher–dimensional landscape. Only in this particular cross–section the well is becoming shallower, while in all other directions perpendicular to this cross–section the well keeps its steep depth profile. DNB have been found for lung injury disease, liver cancer, and lymphoma cancer (7, 8). The theoretical concept and established methods of DNB can be transferred from such sudden deterioration diseases to chronic disorders with paroxysmal episodic manifestations like migraine (5). Tipping points as a transdisciplinary concept {#tipping-points-as-a-transdisciplinary-concept .unnumbered} ============================================= Tipping points may be better known for the earth climate system (3). However, tipping points can be found in medicine, financial markets, traffic, power grid systems to which a large amount of renewable energy is introduced and that may fail therefore, ecosystems where wildlife populations may be threatened, and in the global climate system (9)—not too surprising, as all these are complex systems that exhibit nonlinear behavior and therefore are very likely to show tipping points. Although mere coincidence, there are metaphors about migraine and the brain’s climate, migraine being a thunderstorm or lightning in the head. It can be fruitful to see beyond such metaphors the consequences of tipping points and the related common structure of causal misinterpretation, both schematically illustrated in Fig. 1. Consider the statement of Kleinen et al. (10) referring to the North Atlantic currents : “[It is becoming increasingly evident that there are critical thresholds in the Earth system, where the climate may change dramatically \[...\]. The exact positions of these thresholds are, however, still unclear and it might be doubted whether they can be determined with enough precision to give concrete information on the threat of crossing the threshold. Therefore, additional independent methods for assessing the closeness of the system to these thresholds are needed. These methods could contribute to an early warning system for assessing the danger of crossing a threshold and possibly provide the information necessary for controlling the system]{}” (10). One arrives at a central question in migraine research that will profit from complex systems theory, when in this citation ‘Earth’ is replaced with ‘brain’ and ‘climate’ with ‘neural dynamics’ and when we start to use established concepts in climate research: How can we assess the proximity of a migraine threshold and, if the risk is large, control in this early stage the imminent migraine attack? Climate change over decadal time scales probably involves changes in the ocean’s conveyor belt, the thermohaline circulation, which was modeled in Ref. (10). Dahlem et al. (5) proposed to consider migraine pain caused by central sensitization in analogy as an overturning circulation in nerve traffic of the brain’s migraine generator network (MGN) (11). What are early warning signals of this overturning circulation? Again, let us consider the climate system in analogy: It is easy to mistake cold winters for contradicting global warming, while in fact, severe winters like the ones of 2005-06 and 2009 do not conflict with the global warming picture, but rather supplement it as an integral part of the large and correlated amplitude fluctuations (12). Neural correlate of premonitory symptoms in a subnetwork {#neural-correlate-of-premonitory-symptoms-in-a-subnetwork .unnumbered} ======================================================== There are no simple answers to simple questions in nonlinear systems, in particular a causality interpretation is difficult. Reduced resilience and consequently large and slow fluctuations can explain the abovementioned situations where events that belong to the natural variability are mistaken near tipping points for triggers even if this is not intuitive for patients concerned. It was actually also suggested that migraine patients are driven or have the urge to exercise as a premonitory symptom (13). Unchallenged is that excessive yawning is a well know prodromal symptom in migraine, the same holds true for rapid mood changes, fatigue and craving for certain foods to name but a few. However, active coping, such as biofeedback (14) including contingent negative variation (15) but also behavioral treatments including relaxation trainings, stress–management training and cognitive–behavior therapy (16) clearly showed that the pre-transition state is in principle reversible, at least in some of the attacks. Where in the proposed subnetworks would such a behavioral therapy take its effects? Brainstem activation is thought to be specific for migraine attacks and specifically the dorsolateral pons has been repeatedly demonstrated by imaging data (17-19), while it also was suggested that this area alone cannot be the migraine generator (20). Given that the premonitory (21) and the subsequent attack symptoms (22) are characterized by interdependent networks which explains many of the facets of each event, the possible interplay between these networks has been coined the MGN (11). If a DNB can be found in migraine it will identify the neural correlate for multiple early–warning signs as a common subnetwork of the MGN. In fact, there is a unitary hypothesis that identifies such a subnetwork—but only for multiple triggers causing migraine pain and strain (23). Therefore, if indeed triggers and symptoms are often mistaken at the incipient tipping point, this unitary hypothesis would suggest that large and correlated fluctuations in this subnetwork are crucial, that is, given the clinical picture, in the limbic system as well as the pre- and postganglionic parasympathetic neurons that control the sympathestic/parasympathestic balance. We predict therefore large and correlated fluctuations in this subnetwork as a DNB of migraine (5). Conclusion {#conclusion .unnumbered} ========== To summarize, quantitative modeling approaches are becoming ever more established as a transdisciplinary research field. At the same time, the clinical research audience faces the difficult task, if not to penetrate mathematical concepts, at least to take away the message relevant for their own research. The particular message for clinical research is that our prediction must be tested: We need to characterize these neuronal, i.e., cortical, subcortical and autonomic subnetworks and their connections in the prodromal phase and the cortical slow wave during the aura phase if we are ever to understand the true beginnings of an attack. The general message from complex systems theory is that migraine is an inherently dynamical disease (24) with a complex network generating interdependent events. 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Weiller C, May A, Limmroth V, Jüptner M, Kaube H, Schayck R v, Coenen HH, Diener HC. Brain stem activation in spontaneous human migraine attacks; Nature Medicine, 1995; 1: 658- 660. 18. Stankewitz A, Aderjan D, Eippert F, May A. Trigeminal nociceptive transmission in migraineurs predicts migraine attacks. J. Neurosci., 2011; 31:1937-1943. 19. Goadsby PJ, Fields HL. On the functional anatomy of migraine.; Ann Neurol. 1998 43:272. 20. Borsook D, Burstein R. The enigma of the dorsolateral pons as a migraine generator. Cephalalgia 2012; 32: 803-812. 21. Maniyar FH, Sprenger T, Monteith T, Schankin C, Goadsby PJ. Brain activations in the premonitory phase of nitroglycerin-triggered migraine attacks. Brain. 2014 137:232-241. 22. Stankewitz A, Aderjan D, Eippert F, May A. Trigeminal nociceptive transmission in migraineurs predicts migraine attacks. J. Neurosci 2011; 31:1937-1943. 23. Burstein R, Jakubowski M. Unitary hypothesis for multiple triggers of the pain and strain of migraine. J. 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--- abstract: 'We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský [@ball:rainbow] by showing that if such a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd, then it also does not contain a rainbow cycle of length $m$ for all $m$ greater than $2n^2$. In addition, we present two examples which demonstrate that this result does not hold for even $n$. Finally, we state several open problems in the area.' author: - Boris Alexeev date: 'November 8, 2005' title: On Lengths of Rainbow Cycles --- Introduction ============ A rainbow cycle within an edge-colored graph is a cycle all of whose edges are colored with distinct colors. Rainbow cycles, sometimes called colorful or totally multicolored cycles in other sources, have been introduced in many different contexts. For example, Burr, Erdős, Sós, Frankl, and Graham [@MR1000076; @MR1170778] studied which graphs have the property that regardless of how their edges are $r$-colored (for a fixed integer $r$), the coloring contains a rainbow subgraph, in particular a rainbow cycle of a certain length. From another perspective, Erdős, Simonovits, and Sós [@MR0379258] investigated a function $f(n, C_k)$, defined as the maximum number of colors in which the edges of the complete graph $K_n$ on $n$ vertices may be colored so that the coloring contains no rainbow $k$-cycles; they conjectured that $f(n, C_k) = n\cdot\bigl((k - 2)/2 + 1/(k - 1)\bigr) + O(1)$ for $n \ge k \ge 3$ and proved the case $k=3$. Alon [@MR693025] proved the conjecture for $k = 4$ and derived an upper bound for general $k$; Jiang and West [@MR2037072] further improved these bounds and mentioned that the conjecture has been proven for all $k \le 7$. Finally, Montellano-Ballesteros and Neumann-Lara [@MR2190794] recently proved the conjecture completely. More research has occurred in related areas; these references are not intended as a comprehensive survey of the area, but rather only a small sample. Within this paper, we build on the research of Ball, Pultr, and Vojtěchovský [@ball:rainbow], who studied rainbow cycles within edge-colored complete graphs. In particular, they asked when the existence of a rainbow cycle of a certain length forces the existence of a rainbow cycle of another length, approaching the problem with applications for distributive lattices. To begin, it is easy to see that within such a coloring, the absence of rainbow $n$-cycles and rainbow $m$-cycles implies the absence of rainbow $(n+m-2)$-cycles. (For example, by this fact alone, the absence of rainbow $3$-cycles implies there are no rainbow cycles at all, while the absence of rainbow $4$-cycles implies there are no even-length rainbow cycles. In general, if there are no rainbow $n$-cycles, then there are no cycles with lengths congruent to $2\bmod (n-2)$.) Ball et al. asked whether or not the restrictions implied by this observation were the only restrictions on the lengths of rainbow cycles in a coloring. We answer this question in the negative by showing that there are more, and much stronger, restrictions on the possible lengths. In particular, our main result states that if a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd, then it also does not contain a rainbow cycle of length $m$ for all $m \ge 2n^2$. These results follow from the observation above combined with intermediate results that show that the absence of a rainbow $n$-cycle (again, with $n$ odd) implies the absence of rainbow cycles of lengths $\binom{n}{2}$ and $3n-6$. However, the absence of a rainbow cycle of even length does not put the same restrictions on the lengths of longer rainbow cycles. For example, Ball et al. showed that there are colorings with no even-length rainbow cycles, but with rainbow cycles of all odd lengths up to the number of vertices in the graph. We present these colorings, as well as original colorings in which there are rainbow cycles of all lengths except those congruent to $2\bmod 4$. We briefly describe the organization of the paper. We begin with preliminaries, including definitions and the proof of the observation above, allowing us to formulate the original question as well as the main result formally. In order to provide a flavor of colorings that avoid rainbow cycles, we present the “even-length” examples next. We then prove that the absence of a rainbow $n$-cycle (for $n$ odd) implies the absence of rainbow $\binom{n}{2}$-cycles, and show how this prohibits all sufficiently large cycles (with lengths on the order of a cubic in $n$). Next, we show the stronger result that the absence of a rainbow $n$-cycle (for $n$ odd) also prohibits rainbow $(3n-6)$-cycles, implying the absence of all sufficiently large cycles (with lengths on the order of a quadratic in $n$, in particular $2n^2$, the main result). At the end of the paper, we state some still-open problems in the area and present some computer-obtained results. Preliminaries ============= We begin by introducing some conventions and definitions. In this, we mostly follow the earlier work of Ball et al. [@ball:rainbow], where they introduced the following definitions, as well as proved Lemma \[lemma:obvious\] (and its corollary) and Claim \[claim:even\]. In the context of this paper, a coloring is an edge-coloring of an undirected complete graph; the colors may come from an arbitrary set and there is no restriction that the coloring be proper. As above, a rainbow $n$-cycle (again, sometimes called colorful or totally multicolored in other sources) within a coloring is a cycle consisting of $n$ distinct vertices, all of whose edges are colored with distinct colors; in this case, we will also say that the coloring contains a rainbow $n$-cycle (that is, the coloring restricted to the edges of the cycle is a bijection). As notation, we write $(v_1, \dotsc, v_n)$ for the cycle that visits vertices $v_1, \dotsc, v_n$ in order (and then returns to $v_1$). Notice that although we allow infinite graphs, all cycles are of course finite. A simple lemma and an immediate corollary guide us in our study of rainbow cycles. \[lemma:obvious\] If a coloring contains no rainbow $n$-cycles nor rainbow $m$-cycles, then it contains no rainbow $(n+m-2)$-cycles. \[corollary:obvious\] In particular, if a coloring contains no rainbow $n$-cycle, then it contains no rainbow cycles of length $\ell$, where $\ell \equiv 2 \pmod{n-2}$. The proof is by contradiction: assume that the coloring contains a rainbow $(n+m-2)$-cycle. It can be divided, by a single chord, into an $n$-cycle and an $m$-cycle (see Figure \[figure:obvious\]). Consider the color of this chord. On the one hand, it must agree with one of the other edges of the $n$-cycle to avoid a rainbow $n$-cycle; on the other hand, it must similarly agree with one of the other edges of the $m$-cycle to avoid a rainbow $m$-cycle. This, however, is a contradiction, as we assumed the outer $(n+m-2)$-cycle was rainbow. If an $n$-cycle is prohibited, then so are $(2n-2)$-cycles. By induction, we obtain precisely all lengths congruent to $2\bmod n-2$. (-1.5,-1.5)(1.5,1.5) (1.5;15) (1.5;45) (1.5;75) (1.5;105)(1.5;135)(1.5;165) (1.5;195)(1.5;225)(1.5;255) (1.5;285)(1.5;315)(1.5;345) (1.5;15)(1.5;165) ( 0.9;90)[$n$]{} (-0.5;90)[$m$]{} ( 0.9;90)[0.3]{}[270]{}[180]{} (-0.5;90)[0.3]{}[270]{}[180]{} Original Question and the Main Result {#section:question} ===================================== The original question [@ball:rainbow] can be stated succinctly as follows: Is the restriction in Lemma \[lemma:obvious\] the only restriction on what lengths of rainbow cycles a coloring can contain? In other words, can any set of lengths that does not contradict the lemma be obtained as the lengths of rainbow cycles of some coloring? As we shall see later in the proof of Lemma \[lemma:monoid\], this question can be stated formally as follows: Is it true that for any submonoid $S\subset {\ensuremath{\mathbb{N}}}$ the set of lengths $\{ s+2 \mid s\not\in S\}$, may be obtained as the lengths of rainbow cycles of some coloring? Note also that if one only considers finite graphs, then it may be necessary to add in a restriction prohibiting sets containing arbitrarily large cycle lengths. We may also formulate a weaker question based on Corollary \[corollary:obvious\]. \[question:weak\] If $m \not\equiv 2 \pmod{n-2}$, does there exist a coloring of $K_m$ with a rainbow (Hamiltonian) $m$-cycle but no rainbow $n$-cycle? However, both questions can be answered in the negative; indeed, they both contradict our main theorem, which is somewhat of an opposite, Ramsey-type result. We state it here but prove it later. \[theorem:main\] Suppose $n$ is an odd integer. If a coloring does not contain a rainbow $n$-cycle, it also does not contain a rainbow $m$-cycle for all sufficiently large $m$; in particular, $m \ge 2n^2$ suffices. In particular, we shall show that if there is no rainbow $5$-cycle, there is also no rainbow $10$-cycle, answering Question \[question:weak\] in the negative with $m=10$ and $n=5$. The Even Case ============= Before proving our result, it is instructive to consider examples of colorings which contain some lengths of rainbow cycles, but yet do not contain many other lengths. In particular, we will construct colorings that show that Theorem \[theorem:main\] is not true for even $n$. Of the following two results, Claim \[claim:even\] was proven in [@ball:rainbow] but Claim \[claim:2mod4\] is original. \[claim:even\] There exists a coloring $\operatorname{col}$ of an infinite complete graph that contains rainbow cycles of all odd lengths, but no even-length rainbow cycles. Furthermore, for each $n$, taking appropriate finite subgraphs (and their induced colorings) yields colorings that contain rainbow cycles of all odd lengths up to $n$, but still no even-length rainbow cycles. We construct $\operatorname{col}$ first. Let the vertex set be the positive integers ${\ensuremath{\mathbb{Z}}}^{\mathord +}$ and the colors the nonnegative integers ${\ensuremath{\mathbb{N}}}$ and define the color of the edge joining distinct vertices $x$ and $y$ to be $$\operatorname{col}(x,y) = \begin{cases} 0 & \text{if $y - x$ is even}, \\ \min(x,y) & \text{if $y - x$ is odd}. \end{cases}$$ First, we must show that there exist rainbow cycles of all odd lengths. But this is easy! Consider the cycle $(1, 2, 3, \dotsc, k)$ for $k$ odd. For $1 \le i < k$, the color of the edge joining $i$ and $i+1$ is $\operatorname{col}(i, i+1) = i$; finally, the color of the edge joining $k$ and $1$ is $\operatorname{col}(k,1) = 0$. These are all distinct, so it remains to show that there are no even-length cycles. However, by Corollary \[corollary:obvious\], we need only show that there are no rainbow $4$-cycles. Suppose, by contradiction, that there is a rainbow cycle $(a, b, c, d)$. How many times is the case “$\operatorname{col}(x,y) = 0$ if $y - x$ is even” used along the edges? It cannot be used more than once because otherwise the cycle would contain a repeated color, $0$. But $(b-a) + (c-b) + (d-c) + (a-d) = 0$ and thus, by parity, an even number of $b-a$, $c-b$, $d-c$, and $a-d$ are odd and this case cannot be used exactly once. Therefore, all of the edges use the “$\operatorname{col}(x,y) = \min(x,y)$ if $y - x$ is odd” case of the above definition. Now, without loss of generality, assume $a$ is the smallest-numbered vertex of the four; a contradiction is immediate: since $a$ is the smallest-numbered vertex, $\operatorname{col}(a,b) = \operatorname{col}(a,d) = a$. Therefore there are no rainbow $4$-cycles and thus no even-length rainbow cycles at all. Finally, taking the induced subgraph on the vertices from $1$ to $n$ accomplishes the second statement of the claim. Indeed, all of the necessary odd-length rainbow cycles mentioned above still exist, and a rainbow $4$-cycle still doesn’t exist. \[claim:2mod4\] There exists another similar coloring $\operatorname{col}'$ of an infinite complete graph that contains rainbow cycles of all lengths, except those lengths congruent to $2\bmod 4$. Furthermore, taking appropriate finite subgraphs has the same effect as before. Again, let the vertex set be the positive integers ${\ensuremath{\mathbb{Z}}}^{\mathord +}$ and the colors the nonnegative integers ${\ensuremath{\mathbb{N}}}$; then for distinct vertices $x$ and $y$, define $$\operatorname{col}'(x,y) = \begin{cases} 0 & \text{if $y - x \equiv \hphantom{-}0 \pmod 2$}, \\ x & \text{if $y - x \equiv \hphantom{-}1 \pmod 4$}, \\ y & \text{if $y - x \equiv - 1 \pmod 4$}. \end{cases}$$ Note that $\operatorname{col}'$ is well-defined, as $\operatorname{col}'(x,y) = \operatorname{col}'(y,x)$ for all $x$ and $y$. We show the existence of rainbow $k$-cycles, for all $k\not\equiv 2 \pmod 4$; luckily, the cycle $(1, 2, 3, \dotsc, k)$ accomplishes the task. For $1 \le i < k$, the color of the edge joining $i$ and $i+1$ is $\operatorname{col}'( i, i+1 ) = i$; we must only analyze the color of the edge joining $k$ and $1$. If $k$ is odd, then $\operatorname{col}'( k, 1 ) = 0$; if $k$ is divisible by $4$, the color $\operatorname{col}'( k, 1 ) = k$. In either case, the cycle is rainbow. It remains to show that there is no $6$-cycle (once again, Corollary \[corollary:obvious\] implies we need only check this length). We proceed by contradiction: assume there is a rainbow cycle $(a, b, c, d, e, f)$. As before, the rule “$\operatorname{col}'(x, y) = 0$ if $y - x \equiv 0 \pmod 2$” cannot be used at all. We may also assume $b - a \equiv -1 \pmod 4$: if it is $1 \bmod 4$, simply change the direction of the cycle $(a, b, c, d, e, f) \mapsto (b, a, f, e, d, c)$. It follows that $c - b \equiv -1$; indeed, if $c - b \equiv 1 \pmod 4$, then $\operatorname{col}( a , b ) = b = \operatorname{col}( b, c )$. Similarly, we may conclude that $b - a \equiv c - b \equiv d - c \equiv \dotsb \equiv a - f \equiv -1 \pmod 4$. This is a contradiction; clearly $(b - a) + (c - b) + \dotsb + (a - f) = 0$, but the former would imply it were equal to $2 \bmod 4$. Of course, taking the corresponding induced subgraphs achieves the finite results. Building up to the Main Result {#section:weaker} ============================== We now prove a theorem slightly weaker than the main result. \[theorem:weaker\] Suppose $n$ is an odd integer. If a coloring does not contain a rainbow $n$-cycle, it also does not contain a rainbow $m$-cycle for all sufficiently large $m$; in particular, a cubic bound $m \ge n^3/2$ suffices. The proof will consist of two intermediate lemmas. \[lemma:hard\] If a coloring does not contain a rainbow $n$-cycle, where $n = 2k+1$ is odd, it also does not contain a rainbow $m$-cycle, where $m = \binom{n}{2} = k \cdot (2k+1)$. The $k=2$ case yields the result involving $5$ and $10$ mentioned in Section \[section:question\]. We prove the contrapositive. Assume that we have a coloring, $\operatorname{col}$, of $K_m$ such that there is a rainbow (Hamiltonian) $m$-cycle but no rainbow $n$-cycle. Without loss of generality, we may number the vertices of the $K_m$ by residues modulo $m$ and insist that $\operatorname{col}( i, i+1 ) = i$ for $i \in {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$; in other words, we assume that our rainbow $m$-cycle is given by $(0, 1, \dotsc, m-1)$. Consider the following $k+1$ different $(2k+1)$-cycles: (see Figure \[figure:hard\] for a visual accompaniment) $$\begin{array}{@{($ $}r@{,$ $}r@{,$ $}r@{,$ $\dotsc,$ $}r@{}l@{$ $)}} 0 & 1 & 2 & 2k &\\ 2k & 2k+1 & 2k+2 & 2\cdot 2k & \\ \multicolumn{5}{c}{\vdots} \\ (k-1)\cdot 2k & (k-1)\cdot 2k+1 & (k-1)\cdot 2k+2 & k\cdot 2k & \\ k\cdot 2k & k\cdot 2k+1 & k\cdot 2k+2, & (k+1)\cdot 2k &{}\equiv k \end{array}$$ (-1.5,-1.5)(1.5,1.5) (0;0)[1.5]{} (1.5;110)(1.5;30)(1.5;-50) (1.5;70)(1.5;150)(1.5;230) (1.5;-50)(1.5;230) (1.5;110)[$0$]{} (1.5;70)[$k$]{} (1.5;30)[$2k$]{} (1.5;-50)[$4k$]{} (1.5;150)[$-k$]{} (1.5;230)[$-3k$]{} By assumption, each of them must have a repeated color, so defining $c_i = \operatorname{col}( i\cdot k, (i+2)\cdot k)$ for $i\in {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$, we may conclude that $$\begin{array}{r@{=\operatorname{col}($ $}r@{,$ $}r@{$ $)\in\{$ $}r@{,$ $}r@{,$ $\dotsc,$ $}r@{\}}l} c_0 & 0 & 2k & 0 & 1 & 2k-1 & , \\ c_2 & 2k & 2\cdot 2k & 2k & 2k+1 & 2\cdot 2k - 1 & , \\ \multicolumn{7}{c}{\vdots} \\ c_{2(k-1)} & (k-1) \cdot 2k & k\cdot 2k & (k-1)\cdot 2k & (k-1)\cdot 2k+1 & k\cdot 2k - 1 & , \\ c_{-1} = c_{2k} & k \cdot 2k & k & k\cdot 2k & k\cdot 2k+1 & k - 1 & . \end{array}$$ Now consider the $(2k+1)$-cycle $(0, 2k, 4k, \dotsc, (k-1)\cdot 2k, k\cdot 2k, k, k-1, k-2, \dotsc, 1)$; it has colors $\{0, 1, \dotsc, k-1\}\cup \{ c_0, c_2, \dotsc, c_{2k}\}$. This collection must have a repeated color. But none of $c_2, \dotsc, c_{2(k-1)}$ can contribute a repeated color, so we can conclude that one of $c_0$ and $c_{-1} = c_{2k}$ is a member of $\{0, 1, \dotsc, k-1\}$. Notice, now, that we may translate this argument to also conclude that “either $c_i$ or $c_{i-1}$ is a member of $\{i \cdot k, i\cdot k+1, \dotsc, (i+1)\cdot k - 1\}$”. By symmetry, assume that $c_0 \in \{0, 1, \dotsc, k-1\}$. It follows that $c_1 \in \{k, k+1, \dotsc, 2k-1\}$ and, in general, $c_i \in \{i\cdot k, i\cdot k+1, \dotsc, (i+1)\cdot k - 1\}$. Now for the contradiction: consider the cycle $(0, 2k, \dotsc, k\cdot 2k, k, 3k, \dotsc, k\cdot(2k-1))$, illustrated in Figure \[figure:star\]. The colors along the edges are precisely the $c_i$, no two of which can be equal! (-1.5,-1.5)(1.5,1.5) (0;0)[1.5]{} (1.5;30) (1.5;110)(1.5;190) (1.5;270)(1.5;350)(1.5;70) (1.5;150)(1.5;230)(1.5;310) (1.5;110)[$0$]{} (1.5;70)[$k$]{} (1.5;30)[$2k$]{} (1.5;150)[$-k$]{} \[lemma:monoid\] If a coloring contains no rainbow $n$-cycles, where $n = 2k+1$, and no rainbow $m$-cycles, where $m = k\cdot(2k+1)$, then it contains no rainbow $M$-cycles for all $M\ge 4k^3 - 2k^2 - 8k + 8 = n^3/2 - 2n^2 - 3n/2 + 11$. Let ${\ensuremath{\mathbb{N}}}(2)$ denote the set $\{2, 3, 4, \dotsc\}$. With the operation $m\circ n = m + n - 2$ of Lemma \[lemma:obvious\], ${\ensuremath{\mathbb{N}}}(2)$ becomes a monoid; moreover, by the map $n \mapsto n-2$, it is isomorphic to ${\ensuremath{\mathbb{N}}}$ under addition. This observation is useful because of its interaction with “absent” rainbow cycle lengths. Let the spectrum of a coloring $\operatorname{col}$ be the set of absent lengths of rainbow cycles, that is, $\{ n\ge 2 \mid \operatorname{col}$ does not contain a rainbow $n$-cycle$ \}$. Notice that by definition, all spectra contain $2$. Then by Lemma \[lemma:obvious\], the spectrum of a coloring is a submonoid of ${\ensuremath{\mathbb{N}}}(2)$. This allows us to apply the well-known Claim \[claim:monoid\], perhaps first published by Sylvester, to complete the proof. \[claim:monoid\] If a submonoid of ${\ensuremath{\mathbb{N}}}$ under addition contains the relatively prime integers $a$ and $b$, then it contains all integers greater than or equal to $(a-1) (b-1)$. (In other words, every sufficiently large integer can be written as a nonnegative integer linear combination of $a$ and $b$.) By the isomorphism between $({\ensuremath{\mathbb{N}}}(2),\mathord{\circ})$ and $({\ensuremath{\mathbb{N}}}, \mathord{+})$, we can check that $(2k+1) - 2$ and $k\cdot(2k+1) - 2$ are relatively prime (indeed, this is true because $k\cdot(2k+1) - 2 \equiv -1 \pmod{(2k+1)-2}$, a fact we will use later as well) and obtain the desired bound $4k^3 - 2k^2 - 8k + 8$ after a basic computation. Clearly, Lemma \[lemma:hard\] and Lemma \[lemma:monoid\] together complete the proof of Theorem \[theorem:weaker\], since $n^3/2 \ge n^3/2 - 2n^2 - 3n/2 + 11$ for $n \ge 2$. Proof of the Main Result {#section:main} ======================== We now prove the main result, which is restated here for convenience: Suppose $n$ is an odd integer. If a coloring does not contain a rainbow $n$-cycle, it also does not contain a rainbow $m$-cycle for all sufficiently large $m$; in particular, $m \ge 2n^2$ suffices. As in Section \[section:weaker\], the proof will consist of two intermediate lemmas. \[lemma:harder\] If a coloring does not contain a rainbow $n$-cycle, where $n = 2k+1 > 3$ is odd, it also does not contain a rainbow $m$-cycle, where $m = 3n-6 = 6k-3$. The $k=2$ case yields the result that the absence of rainbow $5$-cycles implies the absence of rainbow $9$-cycles. The proof of this lemma has the same basic structure as that of Lemma \[lemma:hard\], but the cycles we consider are different. As before, we prove the contrapositive. Assume that we have a coloring $\operatorname{col}$ of $K_m$ such that there is a rainbow (Hamiltonian) $m$-cycle but no rainbow $n$-cycle. Without loss of generality, we may number the vertices of the $K_m$ by residues modulo $m$ and insist that $\operatorname{col}( i, i+1 ) = i$ for $i \in {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$. Consider the following three $n$-cycles: (see Figure \[figure:harder1\] for a visual accompaniment — all of the chords of the cycles are either along drawn edges or along the arcs of the outside cycle) $$\begin{array}{lrrrrrcrr} ( & 0, & 1, & 2, & 3, & 4, & \dotsc, & n-1 & )\\ ( & 1, & 0, & -1, & -2, & -3, & \dotsc, & 2n-4 & ) \\ ( & 1, & 0, & n-1, & n, & n+1, & \dotsc, & 2n-4 & ) \end{array}$$ (-1.5,-1.5)(1.5,1.5) (0;0)[1.5]{} (1.5;76.66)(1.5;223.33) (1.5;103.33)(1.5;316.66) (1.5;343.33)(1.5;196.66) (1.5;103.33)[$0$]{} (1.5;76.66)[$1$]{} (1.5;343.33)[$n-2$]{} (1.5;316.66)[$n-1$]{} (1.5;223.33)[$2n-4$]{} (1.5;196.66)[$2n-3$]{} (0.430205;30)[$x$]{} (0.430205;150)[$z$]{} Letting $x = \operatorname{col}(0, n-1)$ and $z = \operatorname{col}(1, 2n-4)$, the colors along the edges of these cycles are as follows: $$\begin{array}{lrrrrrcrrr} ( & 0, & 1, & 2, & 3, & 4, & \dotsc, & n-2, & x )\\ ( & 0, & -1, & -2, & -3, & -4, & \dotsc, & 2n-4, & z ) \\ ( & 0, & x, & n-1, & n, & n+1, & \dotsc, & 2n-3, & z ) \end{array}$$ Because each of these cycles must contain a repeated color, it follows that either $x$ or $z$ is $0$. (Assume there are no rainbow cycles. Then if $x$ is not $0$, it is between $1$ and $n-2$; similarly, if $z$ is not $0$, it is between $-1$ and $-(n-2)$. It follows that if neither $x$ nor $z$ is $0$, the third cycle is a rainbow cycle, a contradiction.) Furthermore, let $y = \operatorname{col}(n-2,2n-3)$; by symmetry, we can also apply this argument to $x$ and $y$ to deduce that either $x$ or $y$ is $n-2$, and similarly for $y$ and $z$. Altogether, either $x=0$, $y=n-2$, and $z=2n-4$ (call this “orientation $+$”); or $x=n-2$, $y=2n-4$, and $z=0$ (call this “orientation $-$”). In a sense, our previous argument was “centered” on the edge $(0,1)$. By additive symmetry, we may also use the same argument centered at $(i,i+1)$ for any residue $i\in {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$; thus, we can assign an “orientation $\pm$” to every residue $i$ (by construction, though, the orientations of $i$, $i + m/3$, and $i + 2m/3$ are the same). Let $\Delta$ be a fixed integer; since $m$ is odd, there exists a residue $x$ such that $x$ and $x+\Delta$ have the same orientation. Without loss of generality, we may assume that $x = 0$ and that the orientation is “orientation $+$.” In the following, we let $\frac{n-3}2$ be our particular choice of $\Delta$ (note that since $n>3$, $\Delta > 0$). Figure \[figure:harder2\] illustrates all of the edge colors that we may assume without any loss of generality. Now let $t = \operatorname{col}(0,n-2)$; we will show that $t\in\{0, n-2\}$. Before continuing, let us define the shorthand notation $x{\ensuremath{\nearrow}}y$ to mean $x, x+1, x+2, \dotsc, y$ (read: “$x$ up to $y$”) and $x{\ensuremath{\searrow}}y$ to mean $x, x-1, x-2, \dotsc, y$ (read: “$x$ down to $y$”); Then in this notation, examine the following $n$-cycles and their corresponding edge colors, illustrated separately in Figure \[figure:harder3\]: $$\begin{array}{rll} & \multicolumn{2}{c}{\text{Respectively, the cycle and its edge colors}} \\ \text{(a)} & (0{\ensuremath{\nearrow}}\Delta, \Delta+n-1{\ensuremath{\searrow}}n-2) & (0{\ensuremath{\nearrow}}\Delta, \Delta+n-2{\ensuremath{\searrow}}n-2, t) \\ \text{(b)} & (0{\ensuremath{\nearrow}}\Delta+1, \Delta+2n-4{\ensuremath{\searrow}}2n-3, n-2) & (0{\ensuremath{\nearrow}}\Delta, \Delta+2n-4{\ensuremath{\searrow}}2n-3, n-2, t) \\ \multirow{2}{}{\text{(c)}} & (0,n-1{\ensuremath{\nearrow}}\Delta+n-2, & (0, n-1{\ensuremath{\nearrow}}\Delta+n-2, \\ & \multicolumn{1}{r}{\Delta+2n-3{\ensuremath{\searrow}}2n-3,n-2)} & \multicolumn{1}{r}{\Delta+2n-2{\ensuremath{\searrow}}2n-3, n-2, t)} \end{array}$$ Once more, because each of these cycles must contain a repeated color, $t$ can be only $0$ or $n-2$, as desired. (Simply note that one of these cycles would be a rainbow cycle unless $t$ repeats a color found on all of them: either $0$ or $n-2$.) That is, $t = \operatorname{col}(0,n-2)\in\{0,n-2\}$; by symmetry, it follows that $\operatorname{col}(n-2,2n-4) \in \{n-2,2n-4\}$ and $\operatorname{col}(2n-4,0) \in \{2n-4, 0\}$. By enumerating the cases, one can check that there necessarily exists an $i$ such that $\operatorname{col}\bigl(i(n-2), (i+1)(n-2)\bigr) + n-2 = \operatorname{col}\bigl((i+1)(n-2), (i+2)(n-2)\bigr)$. (See Figure \[figure:harder4\], particularly subfigure (a), for continued visual accompaniment.) Our final contradiction will be to show that this is impossible; by symmetry, we may assume that this happens when $i = 0$. We can rule out the possibility $\operatorname{col}(0,n-2) = 0$ and $\operatorname{col}(n-2,2n-4) = n-2$ by considering the $n$-cycle $(n-2,2n-4{\ensuremath{\nearrow}}0)$ which would then have colors $(n-2, 2n-4{\ensuremath{\nearrow}}0)$, a contradiction since no colors repeat and thus this cycle is rainbow. We can rule out the only other possibility $\operatorname{col}(0,n-2) = n-2$ and $\operatorname{col}(n-2,2n-4) = 2n-4$ by considering the $n$-cycle $(0{\ensuremath{\nearrow}}\Delta, \Delta+n-1{\ensuremath{\nearrow}}2n-4, n-2)$ which would then have colors $(0{\ensuremath{\nearrow}}\Delta, \Delta+n-1{\ensuremath{\nearrow}}2n-4, n-2)$, also a contradiction. We have thus obtained a contradiction in all cases! \[lemma:adhoc\] If a coloring contains no rainbow $n$-cycles, where $n = 2k+1 > 3$, and no rainbow $m_1$-cycles and $m_2$-cycles, where $m_1 = k\cdot(2k+1)$ and $m_2 = 6k-3$, then it contains no rainbow $M$-cycles for all $M\ge 8k^2 - 8k + 12 = 2n^2 - 13n + 23$. One approach is to simply proceed as in the proof of Lemma \[lemma:monoid\] and consider only $n$ and $m_2$ (since $n-2$ and $m_2-2$ are relatively prime). This approach, using Claim \[claim:monoid\], yields a bound of $12k^2 - 24k + 14 = 3n^2 - 18n + 29$, which is approximately a constant multiplicative factor worse than the desired bound. We shall use both $m_1$ and $m_2$ to derive a better result. The analog of Claim \[claim:monoid\] for three variables is well-studied, and many special cases and algorithms have been developed. For completeness, we use a self-contained ad-hoc argument applicable in our particular situation. \[claim:adhoc\] Suppose a submonoid of ${\ensuremath{\mathbb{N}}}$ under addition contains the integers $a < b < c$, with $a \ge 3$ odd, $b \equiv -2 \pmod a$, and $c \equiv -1 \pmod a$ (thus, $a$ and $b$ are relatively prime, as are $a$ and $c$). Then the submonoid contains all integers greater than or equal to $ab/2 - a -3b/2 + c + 1$. It suffices to prove the claim for the submonoid generated* by $a$, $b$, and $c$. For each residue $r$ modulo $a$, consider the smallest number in the submonoid congruent to $r$ (modulo $a$). The smallest number in this submonoid that is congruent to $r \equiv -(2x) \bmod a$, where $2x < a$, is $bx$; similarly, the smallest number congruent to $r \equiv -(2x+1) \bmod a$, where $2x+1 < a$, is $bx + c$. The largest of these numbers, ranging over all residues $r$ modulo $a$, occurs for $r\equiv 2 \equiv - (a-2) \equiv -(2 \frac{a-3}2 + 1) \bmod a$, in which case it is $y = b \frac{a-3}2 + c$. By choice of $y$, all numbers greater than $y-a$ are in the monoid, as desired. To finish the proof of the lemma, apply this claim with $a = n-2$, $b = m_2-2$, and $c = m_1-2$, and use the isomorphism between ${\ensuremath{\mathbb{N}}}$ and ${\ensuremath{\mathbb{N}}}(2)$. One obtains the desired bound of $8k^2 - 8k + 12 = 2n^2 - 13n + 23$. As before, Lemma \[lemma:harder\] and Lemma \[lemma:adhoc\] together complete the proof of Theorem \[theorem:main\], since $2n^2 \ge 2n^2 - 13n + 23$ for $n \ge 2$. Further Directions ================== This paper leaves open a few avenues of further experimentation which interest the author. We state some of these problems. Completely characterize when the existence of a rainbow $m$-cycle implies the existence of a rainbow $n$-cycle. Let $g(n)$ be the smallest value of $M$ such that if a coloring does not contain a rainbow $n$-cycle, where $n$ is odd, then it also does not contain a rainbow $m$-cycle for all $m \ge M$. Determine $g(n)$ for specific cases or in general. For example, computer experimentation (see Appendix \[appendix:computer\]) yields $g(5) = 8$, $g(7) = 11$, $g(9) = 15$, and $15 \le g(11) \le 34$. In general, Theorem \[theorem:main\] shows that $g(n)$ is at most quadratic in $n$. Is $g(n)$ actually subquadratic? Finally, there is evidence to support the following conjecture. The asymptotic behavior of a spectrum $S$ can be classified into three categories: either (a) $S$ contains all sufficiently large numbers, (b) $S$ contains all sufficiently large even numbers, or (c) $S$ contains all sufficiently large numbers congruent to $2\bmod 4$. In terms of monoids, the spectrum becomes regular either modulo one, two, or four. Acknowledgments {#acknowledgments .unnumbered} =============== The author wishes to thank Professor Daniel Kleitman of the Massachusetts Institute of Technology for guiding this work, as well as Jacob Fox, Matt Ince, and Petr Vojtěchovský for helpful conversations. [MBNL05]{} N. Alon, On a conjecture of [E]{}rd[ő]{}s, [S]{}imonovits, and [S]{}ós concerning anti-[R]{}amsey theorems, J. Graph Theory 7 (1983), no. 1, 91–94. S. A. Burr, P. Erd[ő]{}s, R. L. Graham, and V. T. S[ó]{}s, Maximal anti-[R]{}amsey graphs and the strong chromatic number, J. Graph Theory 13 (1989), no. 3, 263–282. S. A. Burr, P. Erd[ö]{}s, V. T. S[ó]{}s, P. Frankl, and R. L. Graham, Further results on maximal anti-[R]{}amsey graphs, Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., Wiley, New York, 1991, pp. 193–206. R. N. Ball, A. Pultr, and P. Vojt[ě]{}chovsk[' y]{}, Colored graphs without colorful cycles, Preprint, Jun 2005. P. Erd[ő]{}s, M. Simonovits, and V. T. S[ó]{}s, Anti-[R]{}amsey theorems, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, North-Holland, Amsterdam, 1975, pp. 633–643. Colloq. Math. Soc. János Bolyai, Vol. 10. T. Jiang and D. B. West, On the [E]{}rd[ő]{}s-[S]{}imonovits-[S]{}ós conjecture about the anti-[R]{}amsey number of a cycle, Combin. Probab. Comput. 12 (2003), no. 5-6, 585–598, Special issue on Ramsey theory. J. J. Montellano-Ballesteros and V. Neumann-Lara, An anti-[R]{}amsey theorem on cycles, Graphs Combin. 21 (2005), no. 3, 343–354. [MR ]{}[MR2190794 (2006h:05141)]{} J. J. Sylvester, Question 7382, Mathematical Questions from the Educational Times 41 (1884), 21. Computer Results {#appendix:computer} ================ An original C++ program was used to exhaustively determine (by searching through the space of all possible colorings by always coloring the most-constrained edge), for small $n$ and $m$, whether or not it is possible to color $K_m$ such that there exists a rainbow $m$-cycle but no rainbow $n$-cycle. These results are summarized in Table \[table:computer\]. Some of these results have been confirmed by Petr Vojtěchovský; in particular, the case $m = n+1 > 3$ is known (this corresponds to the top entry of each column). [@ball:rainbow] -------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -------- 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 -------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -------- : Data obtained from a computer program used to determine when the absence of a particular length of rainbow cycles forbids the existence of another length. For example, looking at column 5 and row 10 (the lowest ‘’ in that column), we see that the absence of rainbow $5$-cycles implies the absence of rainbow $10$-cycles. The meaning of the symbols is as follows: ‘’ means that Claim \[claim:even\] or Claim \[claim:2mod4\] implies that the existence of the larger length is possible, while ‘’ means that these Claims do not apply but the program produced a specific example. ‘’ means that the program determined that the larger length could not occur, while ‘’s are consequences of Lemma \[lemma:obvious\] and earlier ‘’s and ‘’s. An empty square means no results were obtained.[]{data-label="table:computer"}
--- abstract: 'In this paper, we discuss the constraints on the reheating temperature supposing an early post-reheating cosmological phase dominated by one or more simple scalar fields produced from inflaton decay and decoupled from matter and radiation. In addition, we explore the combined effects of the reheating and non-standard scalar field phases on the inflationary number of $e$-foldings.' author: - Alessandro Di Marco - Gianfranco Pradisi - Paolo Cabella bibliography: - 'apssamp.bib' nocite: '[@]' title: 'Inflationary scale, reheating Scale and pre-BBN Cosmology with scalar fields' --- Introduction ============ Before the Hot Big Bang (HBB) epoch, our Universe likely experienced an early quantum gravity phase (at the so called Planck scale) in which gravitational, strong, weak and electromagnetic interactions were unified in a single fundamental force [@1]. Due to expansion and cooling, at lower (GUT) scales the gravitational interaction decoupled and the Universe entered an hypothetical phase where matter and radiation can be described in terms of a Grand-Unified gauge theory [@2]. According to the inflationary paradigm, at a scale $M_{inf}$ ($<10^{16}$ GeV), after the spontaneous symmetry breaking to $SU (3)\times SU (2)\times U (1)$ (the gauge group of the Standard Model of particle physics), cosmological inflation is supposed to have taken place, in order to make the Universe almost flat, isotropic and homogeneous on large astronomical scales [@3]. In the simplest version, the inflationary mechanism was driven by a scalar field, called inflaton, minimally coupled to gravity and probing an almost flat region (a false vacuum) of the corresponding effective scalar potential. At the end of inflation, where the potential steepens, the inflaton field falls in the global minimum of the potential, oscillates, decays and “reheats" the Universe (see [@4] for detailed studies on the mechanism and [@5] for general constraints), giving rise to the standard HBB evolution characterized by an initial radiation-dominated phase. However, this last step is not necessarily the unique possible scenario. Indeed, there is of course room for a peculiar evolution in the history of the Universe immediately after the reheating. In particular, the expansion of the Universe could have been submitted to additional phases where, for instance, it was driven by one (or more) new simple scalar species, before the radiation-dominated era and, especially, well before the Big Bang Nucleosinthesys (BBN). Additional scalar fields, not necessarily directly interacting with the Standard Model degrees of freedom, are quite common in superstring theory with branes. They typically parametrize the brane positions along directions internal to the extra-dimensions transverse to the branes. Since their energy density exhibits a modified dilution law, they can give rise to a non-standard post-reheating phase. Scenarios of this type have been recently introduced to study modification on relics abundances and decay rates of dark matter [@6; @7], as well as to study enhancements in the inflationary number of $e$-foldings [@8; @9]. In this paper, we consider non-standard cosmologies inspired by string theory orientifold models [@10] with, generically, multiple sterile scalar fields entering a non-standard post-reheating phase and we analyze in details the constraints put on the reheating temperature by the additional fields. As a consequence, we can derive more stringent model independent predictions about the number of $e$-folds during inflation. The paper is organized as follows. In Sec. II, we derive general expressions for the energy density in the case of non-standard post-reheating cosmological evolution, given by one or more scalar fields. In Sec. III, we discuss how the features of the new species affect the reheating scale. In particular, we derive an upper limit to the reheating temperature. In Sec. IV, we study the relation between reheating and postinflationary scalar fields and we calculate the inflationary number of $e$-foldings, also constrained by the maximum reheating temperature. In Sec. V, we add our conclusions and some discussions. In the Appendixes, we show numerical examples of the consequences of the variation in the number of $e$-foldings on the inflationary predictions of $n_s$ and $r$, for various selected inflaton potentials. In this manuscript we use the particle natural units $c=\hbar=1$, unless otherwise stated. postinflationary scalar fields and cosmology ============================================ The cosmological history of early Universe immediately after the reheating should be characterized by a radiation-dominated era. In that phase, the corresponding evolution is well described by $$\begin{aligned} H^2(T)\simeq\frac{1}{3M^2_{Pl}}\rho_{rad}(T), \quad \rho_{rad}(T)=\frac{\pi^2}{30}g_{E}(T) T^4 , \label{matdom}\end{aligned}$$ where $H$ denotes the Hubble rate, $M_{Pl}$ is the reduced Planck Mass, $\rho_{rad}$ is the radiation energy density and $T$ indicates the temperature scale of the universe at a given (radiation-dominated) epoch. Finally, $g_{E}$ is the effective number of relativistic degrees of freedom turning out to be $$g_{E}(T)=\sum_{b} g_{b}\left( \frac{T_b}{T}\right)^4 + \frac{7}{8}\sum_{f} g_{f}\left(\frac{T_f}{T}\right)^4,$$ where $b$ and $f$ label contributions from bosonic and fermionic degrees of freedom, respectively, and $T_b$ and $T_f$ indicate the corresponding temperatures. In this Section, we would like to analyze a modification of the evolution of the early Universe after the reheating phase, realized through the presence of a set of scalar fields $\phi_i (i=1,...,k)$. They are assumed to dominate at different time scales until radiation becomes the most relevant component, well before the BBN era [@6]. The last assumption is crucial in order to not spoil the theoretical successes related to the prediction of light element abundances (see [@7]). Therefore, the total energy density after the inflaton decay can be assumed to be $$\label{eqn:totalenergydensity} \rho(T)=\rho_{rad}(T) + \sum_{i=1}^k \rho_{\phi_i}(T).$$ We introduce the scalar fields in such a way that, for $i>j$, $\rho_{\phi_{i}}$ hierarchically dominates at higher temperatures over $\rho_{\phi_{j}}$ when the temperature decreases. All the scalar fields, supposed to be completely decoupled from each other and from matter and radiation fields, can be described as perfect fluids diluting faster than radiation. In this respect, the dynamics is encoded in $$\dot{\rho}_{\phi_i} + 3H\rho_{\phi_i}( 1 + w_i ) = 0 ,$$ where $w_i=w_{\phi_i}$ is the equation of state (EoS) parameter of the field $i$. Integrating this equation one finds $$\rho_{\phi_i}(T)=\rho_{\phi_i}(T_i)\left(\frac{a(T_i)}{a(T)} \right)^{4+n_i}, \quad n_i=3w_i -1$$ where the index $n_i$, the “dilution" coefficient, is understood to satisfy the conditions $$n_i>0, \quad n_i<n_{i+1}.$$ $T_i$ can be conveniently identified with the transition temperature at which the contribution of the energy density of $\phi_i$ becomes subdominant with respect to the one of $\phi_{i-1}$. In other words, the scalar fields are such that $$\begin{aligned} \rho_{\phi_i}>\rho_{\phi_{i-1}} \mbox{ for } T>T_i\\ \rho_{\phi_i}=\rho_{\phi_{i-1}} \mbox{ for } T=T_i\\ \rho_{\phi_i}<\rho_{\phi_{i-1}} \mbox{ for } T<T_i.\end{aligned}$$ Using the conservation of the “comoving" entropy density $$g_S(T)a^3(T)T^3=g_S(T_i)a^3(T_i)T^3_i ,$$ being $g_S$, defined by $$g_{S}(T)=\sum_{b} g_{b}\left( \frac{T_b}{T}\right)^3 + \frac{7}{8}\sum_{f} g_{f}\left(\frac{T_f}{T}\right)^3,$$ the effective number of relativistic degrees of freedom associated with entropy, the energy density of the various fields at a temperature $T$ can be expressed in terms of the transition temperatures $T_i$ [@6; @9] $$\rho_{\phi_i}(T)=\rho_{\phi_i}(T_i)\left( \frac{g_S(T)}{g_S(T_i)}\right)^{\frac{4+n_i}{3}}\left(\frac{T}{T_i}\right)^{4+n_i}. \label{ratioofrhophi}$$ For the first scalar field $\phi_1$, by definition, the transition temperature is such that its energy density is identical to the one of the radiation fluid, so that $$\rho_{\phi_1}(T_1)=\rho_{rad}(T_1)=\frac{\pi^2}{30}g_{E}(T_1) T_1^4.$$ The second scalar field $\phi_2$ is subdominant compared to $\phi_1$ below the temperature $T_2$. Using Eq. and observing that $T_2$ is the transition temperature at which $\rho_{\phi_2}(T_2)=\rho_{\phi_{1}}(T_2)$, one gets $$\label{eqn: energy phi2} \rho_{\phi_2}(T)= \rho_{\phi_1}(T_1)\left( \frac{T_2 g_S^{1/3}(T_2)}{T_1 g_S^{1/3}(T_1)} \right)^{4+n_1} \left( \frac{T g_S^{1/3}(T)}{T_2 g_S^{1/3}(T_2)} \right)^{4+n_2}$$ This equation tells us that the energy density of the scalar field $\phi_2$ depends on the ratio between the two scales $T_1$ and $T_2$, where the $\phi_1$-dominance occurs. In the same way, we can derive the analogous expression for the other scalar fields $\phi_i$. The general expression for the energy density carried by $\phi_i$ turns out to be $$\label{eqn: general result} \rho_{\phi_i}(T)=\rho_{\phi_1}(T_1)\prod_{j=1}^{i-1}\left(\frac{T_{j+1} g_S^{1/3}(T_{j+1})}{T_j g_S^{1/3}(T_j)}\right)^{4+n_j} \left(\frac{T g_S^{1/3}(T)}{T_i g_S^{1/3}(T_i)}\right)^{4+n_i}, \quad i\ge 2.$$ Inserted in Eq. , the previous expressions provide the total energy density dominating the expansion of the Universe after the standard reheating phase, up to the beginning of the radiation-dominated epoch. In particular, the Hubble rate acquires the compact form $$\label{eqn: hubble rate evo} H^2(T)\simeq\frac{1}{3 M^2_{Pl}} \rho_{\phi_1}(T_1)\sum_{i=1}^{k} f_i(n_i,T,T_1,...,T_i),$$ where $f_i$ can be extracted by the previous equations. Nature of the scalar fields and reheating temperature ===================================================== In the previous section, we have discussed a modified post-reheating scenario, where several component species in the form of non-interacting (decoupled) scalar fields, are added to the relativistic plasma. Even if we do not specify their nature, it should be underlined that these kind of components are quite common both in scalar modifications of General Relativity and in theories with extra dimensions. In particular, in orientifold superstring models compactified to four dimensions and equipped with D-branes, the presence of additional scalars is an almost ubiquitous phenomenon [@10]. Indeed, the D-brane action is the sum of a DBI term and a Wess-Zumino term, generalizations of the familiar mass and charge terms of a particle action. The dynamical fluctuations of the D-branes in the transverse directions correspond to degrees of freedom that are described by scalar fields. Their coupling to the four-dimensional metric is induced on D-branes by the embedding inside the ten-dimensional space-time, and gives rise typically to a warp factor depending on the internal coordinates and to additional couplings entering the DBI action (disformal terms, see [@9] and references therein). The most important point, however, is that these scalar fields always interact with the inflaton, that can thus decay into them and the remaining components of the standard reheating fluid after inflation. In this paper, we neglect the interactions of the scalar fields with matter longitudinal to the D-branes. From Eq. we can easily argue that $\rho(T)$ increases with temperature, reaching the maximum value at $T=T_{reh}$. For instance, in the case of a single additional scalar field $\phi_1$, with a transition-to-radiation temperature $T_1$, one has $$\rho_{\phi_1}(T)=\rho_{\phi_1}(T_1)\left( \frac{g_S(T)}{g_S(T_1)}\right)^{\frac{4+n_1}{3}}\left(\frac{T}{T_1}\right)^{4+n_1}.$$ It is clear that there must be an upper bound to this energy density for $T=T_{reh}$. At this stage, whatever the nature of $\phi_1$ is, the energy density cannot assume arbitrary values, since it is at least limited by the presence of the Planck scale, $M_{Pl}$. In other words, we have to introduce a maximum scale $M$ (with $M\le M_{Pl}$) such that $$\rho_{\phi_1}(T_{reh})\leq M^4 ,$$ corresponding to an upper limit to the production scale of $\phi_1$. As a consequence, it turns out also to be an upper limit to the reheating temperature, once we set the scale $M$. Since $T_1$ is the transition-to-radiation temperature, i.e. $$\rho_{\phi_1}(T_1)=\rho_{rad}(T_1) ,$$ using $$\label{eqn: g_relation} g_{E}(T)\sim g_{S}(T)\sim 100 \mbox{ for } T>T_{QCD}$$ (where $T_{QCD}>150$ MeV is the QCD phase transition scale), the reheating temperature must satisfy the condition $$\label{eqn: temp bound} T_{reh}\leq \alpha_1 M \left(\frac{T_1}{M} \right)^{\frac{n_1}{4+n_1}}, \quad \alpha_1=\left(\frac{30}{\pi^2 g_E}\right)^{\frac{1}{4+n_1}}$$ with a resulting upper limit $$\label{eqn: templimit} T^{max}_{reh}=\alpha_1 M \left(\frac{T_1}{M} \right)^{\frac{n_1}{4+n_1}}.$$ In general, the scale $M$ could be the Planck scale $M_{Pl}$ but also, for instance, the GUT scale $M_{GUT}$ or a lower scale of the order of the string scale, $M_{s}$, that is unconstrained in orientifolds [@10; @11]. In particular, if the field $\phi_1$ is supposed to be produced by the inflaton decay or during the reheating phase, we can also assume $M=M_{inf}$, where $M_{inf}$ is the inflationary scale. It is interesting to analyze how the upper limit on $T_{reh}$ varies with the scale $M$. In Fig.($\ref{fig: 1}$) we plot the behaviour of $T^{max}_{reh}$ as a function of the model parameter $n=n_1$ for given values of the scale $M$. The transition-to-radiation temperature is chosen to be $T_1\sim 10^4$ GeV. As expected, the maximum reheating temperature is larger for larger values of $M$, while it decreases with the model parameter $n$. The region below each curve representing $T^{max}_{reh}(n)$ describes the possible reheating temperatures compatible with the chosen bound $M$. For example, for $n=2$ and $M=M_{Pl}$, we might have $T_{reh}\le 10^{13}$ GeV, while for $M=M_{inf}$ we might only have $T_{reh}\le 10^{11}$ GeV. For $n=4$, a reheating temperature of the order of $10^9$ GeV is compatible with $M=M_{inf}$ and [a fortiori]{} with Planckian or GUT bounds. In Fig.($\ref{fig: 2}$) we fix the scale $M$ to the inflationary scale ($\sim 10^{15}$ GeV) and plot the behaviour of $T^{max}_{reh}(n)$ for different values of the transition-to-radiation temperature. It happens that $T^{max}_{reh}(n)$ becomes smaller and smaller, for a fixed $n$, as the transition temperature decreases. For example, with $n=2$ and $T_1\sim 10^7$ GeV, we get $T_{reh}\le 10^{12}$ GeV, while with $n=4$ $T_{reh}\le 10^{11}$. We postpone the discussion of the case with more scalar fields to Section V. ![The maximum reheating temperature $T^{max}_{reh}$ as a function of the parameter $n$ for $T_1=10^4$ GeV. The maximum reheating temperature becomes larger as $M$ increases, while it decreases with $n$.[]{data-label="fig: 1"}](1.eps){width="8.5cm" height="6cm"} ![The maximum reheating temperature $T_{reh}^{max}$ as a function of the parameter $n$ for different values of the transition-to-radiation temperature and an inflationary scale $M_{inf}\sim 10^{15}$ GeV. The maximum reheating temperature becomes larger as the scale $T_1$ increases.[]{data-label="fig: 2"}](2.eps){width="8.5cm" height="6cm"} Inflationary $e$-foldings, Reheating and pre-BBN scalar fields ============================================================== In the case of standard post-reheating radiation-dominated Universe, the inflationary number of $e$-foldings $N_$ has been calculated and used in many works [@5]. In this Section, we would like to discuss how this number changes in the presence of a non-standard postinflationary scenario. As shown in [@9], in the non-standard case $N_$ acquires an additional $e$-folds term $\Delta N(\phi_i,T_{reh})$, that depends on the reheating temperature and on the features of the additional decoupled scalar fields discussed in Section II. Thus, we may write $$\begin{aligned} \label{eqn: efolds} N_&= \xi_* -\frac{1}{3(1+w_{reh})}\ln\left(\frac{\rho_{end}}{\rho_{reh}}\right)\\ \nonumber &+ \frac{1}{4}\ln\left(\frac{V^2_}{M^4_p \rho_{reh}}\right) + \Delta N(\phi_i,T_{reh}) , \end{aligned}$$ where $\rho_{end}$ is the energy density at the end of inflation, $\rho_{reh}$ is the energy density when the reheating is completely realized, $w_{reh}$ is the mean value of the EoS parameter of the reheating fluid, while $V_=M_{inf}^4$ is the inflationary energy density. In Eq. $$\xi_* = -\ln\left(\frac{k_}{a_0 H_0}\right) + \ln\left(\frac{T_0}{H_0}\right) + c \ ,$$ with $$c=- \frac{1}{12} \ln g_{reh} + \frac{1}{4}\ln\left(\frac{1}{9}\right) + \ln\left(\frac{43}{11}\right)^{\frac{1}{3}}\left(\frac{\pi^2}{30}\right)^{\frac{1}{4}} ,$$ where $k_$ is the pivot scale for testing the cosmological parameters, $a_0$ and $H_0$ are the scale factor and the Hubble rate at the current epoch, respectively, $T_0$ is the CMB photon temperature while $g_{reh}$ denotes the effective number of relativistic degrees of freedom at the end of reheating (we are using $g_E(T_{reh})=g_S(T_{reh})=g_{reh}$ because of Eq.($\ref{eqn: g_relation}$)). Assuming $k_=0.002$ Mpc$^{-1}$, $H_0=1.75\times 10^{-42}$ GeV, $T_0=2.3\times 10^{-13}$ GeV and $g_{reh}\sim 100$, we get $\xi_\sim 64$ and $c\sim 0.77$. The additional term comes out to be $$\Delta N(\phi_i,T_{reh})=\frac{1}{4}\ln\eta(n_i,T_i,T_{reh}),$$ where $\eta$ is the ratio of the total energy density to the energy density of radiation at the reheating temperature, $$\eta= 1 + \frac{\sum_i \rho_{\phi_i}(T_{reh})}{\rho_{rad}(T_{reh})}.$$ Using Eq.($\ref{eqn: general result}$) and expressing the radiation energy density in terms of $T_1$ $$\rho_{rad}(T_{reh})= \rho_{rad}(T_1) \frac{g_E(T_{reh})}{g_E(T_1)}\left(\frac{T_{reh}}{T_1}\right)^4$$ we can write $$\label{eqn: eta_general} \eta = 1 + \frac{g_E(T_1)}{g_E(T_{reh})}\left(\frac{T_1}{T_{reh}}\right)^4 \biggl\{\left[\frac{T_{reh} g_S^{1/3}(T_{reh})}{T_{1} g_S^{1/3}(T_{1})}\right]^{4+n_1} + \sum_{i=2}^k \prod_{j=1}^{i-1} \left[\frac{T_{reh} g_S^{1/3}(T_{reh})}{T_{i} g_S^{1/3}(T_{i})}\right]^{4+n_i} \ \left[\frac{T_{j+1} g_S^{1/3}(T_{j+1})}{T_{j} g_S^{1/3}(T_{j})}\right]^{4+n_j}\biggl\} .$$ It should be noticed that the more scalar fields we have, the larger the parameter $\eta$ is. Moreover, $N_$ is inflationary-model dependent due to the presence of the potential function in the second and third contributions of Eq.($\ref{eqn: efolds}$). However, by assuming $\rho_{end}\sim M_{inf}^4$, converting $\rho_{reh}$ in $T_{reh}$ and neglecting some small numerical factors, $N_$ can also be written as $$\begin{aligned} \label{eqn: efolds ind} N_&\sim \xi_ - \frac{1-3w_{reh}}{3(1+w_{reh})}\ln\left(\frac{M_{inf}}{T_{reh}}\right) \\ \nonumber &+ \ln\left(\frac{M_{inf}}{M_{Pl}}\right) + \frac{1}{3(1+w_{reh})}\ln\eta .\end{aligned}$$ We can distinguish three main contributions. The first $$A(w_{reh},T_{reh})=\frac{1-3w_{reh}}{3(1+w_{reh})}\ln\frac{M_{inf}}{T_{reh}} \label{eq:reheatingcontr}$$ is entirely related to the reheating phase, the second involves the ratio between the Planck scale and the inflationary scale while the last one is due to the fraction of energy carried by the scalar fields, namely to the $\eta$ factor. Let us provide a simple example considering a single scalar field post-reheating dominance. By using Eq.($\ref{eqn: g_relation}$), the general expression Eq.($\ref{eqn: eta_general}$) turns out to be $$\label{eqn: eta1} \eta = 1 + \left(\frac{T_1}{T_{reh}}\right)^4\left(\frac{T_{reh}}{T_1}\right)^{4+n_1}\simeq \left(\frac{T_{reh}}{T_1}\right)^{n_1} ,$$ and therefore $$\label{eqn: eta_term} \Delta N(\phi_1,T_{reh})=\frac{n_1}{3(1+w_{reh})}\ln\left(\frac{T_{reh}}{T_1}\right)$$ that, for the trivial $w_{reh}=0$ case, results into $$\Delta N(\phi_1,T_{reh})=\frac{n_1}{3}\ln\left(\frac{T_{reh}}{T_1}\right).$$ The reheating and the $\eta$ terms are strongly correlated. Indeed, in Section II we have shown that the reheating temperature is constrained by an upper bound dependent on a scale $M$, by the transition-to-radiation temperature $T_1$ and by the dilution coefficient $n=n_1$. As a consequence, we have a lower bound on the reheating contribution in Eq.. Using the bound in Eq.($\ref{eqn: temp bound}$), we get $$\label{eqn: reh_constraint} A(w_{reh},T_{reh})\ge\frac{1-3w_{reh}}{3(1+w_{reh})}\ln\frac{M_{inf}}{\alpha_{1}M}\left(\frac{M}{T_1}\right)^{\frac{n_1}{4+n_1}}.$$ In Fig.($\ref{fig: 3}$) we report the quantity $\Delta N(\phi_1,T_{reh})$ as a function of the transition-to-radiation temperature for some values of $n_1$, assuming an equation of state $w_{reh}=0$ and\ a reheating temperature $T_{reh}\sim 10^9$ GeV. Let us take a\ look that for $n=4$ and $T_1\sim 10^4$ GeV, we can easily\ extract more than $15$ extra $e$-folds, while for a larger $T_1\sim 10^6$ GeV we would have $\Delta N\sim 9$. In Fig.($\ref{fig: 4}$) we plot the complete result for the variable $N_$ as a function of the reheating equation of state parameter $w_{reh}$ for $n=1,2,3,4$, assuming $T_1\sim 10^4$ GeV. In general, the value of $N_$ increases with $w_{reh}$, as expected by the expression in Eq.($\ref{eqn: efolds ind}$). For $n=2$ and $w_{reh}=0$, we get $N_\sim 59$, while $N_\sim 67$ for $n=4$ and $w_{reh}=0$. In the next section we briefly discuss the multifield cases. The obtained results have non-trivial consequences on the theoretical predictions of the underlying inflationary models. The reason is that one usually infers the values of the two main inflationary parameters, the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$, assuming an $N_$ in the range between 50 and 60. Therefore, if we considered a different $N_$ we could have new predictions to compare with the current experimental bounds [@12]. In the Appendixes we will briefly examine how the non-trivial values of $N_$ affect some paradigmatic inflationary models. ![Number of extra $e$-folds with $w_{reh}=0$ and $T_{reh}\sim 10^9$ GeV. We have chosen this temperature because it is compatible with all values of $n$ from 1 to 4 and with all the transition temperatures $T_1>10^4$ GeV, as seen in Sec.II.[]{data-label="fig: 3"}](3.eps){width="8.5cm" height="6cm"} ![The inflationary number of $e$-folds $N_$ in a nonstandard postreheating cosmology as a function of $w_{reh}$. $N_$ increases with the value of the EoS parameter. The growth of $N_$ is also decreasing with $n$.[]{data-label="fig: 4"}](4.eps){width="8.5cm" height="6cm"} Summary and Discussion ====================== Inflation should have taken place at very high-energy scales. The accelerated expansion was followed by a “reheat" stage that produced the Standard Model radiation fluid and the observable large comoving entropy of the Universe. However, available data do not guarantee that the mentioned scenario is the correct one. For instance, a viable alternative is to have one (or more) additional field(s) that dominates the energy budget of the Universe at different phases after the reheating epoch. In particular, many authors have recently considered the inclusion of new scalar fields (quintessence, scalar field decoupled from matter and radiation or even scalar fields or moduli coupled to gravity) to approach some problems related to dark matter relics abundances or to the number of inflationary $e$-folds (see [@6; @7; @8; @9] and references therein). The relation between scalar decaying particle and black hole formation in GUT cosmology was also studied in the past [@13]. In general, nonstandard cosmological histories before the BBN are interesting possibilities whose signatures could be tested in the near future, for istance by gravity-waves experiments (see [@14] for details). In this paper, extending the approach of [@9], we have described a post-reheating era dominated by a collection of simple scalar fields $\phi_i, (i=1,...,k)$ completely decoupled from Standard Model matter and radiation. Their presence is described in terms of perfect fluids with energy densities that scale as $\rho_i\sim a^{-(4+n_i)}$, $n_i>0$. Each $\phi_i$ dominates at different times. In particular, $\phi_1$ is the field connecting the non-standard part of the post-reheating phase to the radiation-dominated era. In this scenario, it is mandatory to assume that the transition to radiation occurs well before the BBN, in order not to ruin the theoretical predictions about light element abundances [@6]. In Eqs. and the general expressions related to the total energy density and to the energy density of a single field $\phi_i$ have been derived, with the proviso of absence of entropy variation. The changes in the Hubble rate during the multifield driven evolution are regulated by Eqs. and . In Sec. III, we observed that the energy density after reheating must be at most Planckian. As a consequence, there exists an upper limit to the reheating temperature, as shown in Eq. ($\ref{eqn: templimit}$) and illustrated in Fig.($\ref{fig: 1}$) for different choices of the limiting scale and a transition temperature to HBB $\sim 10^4$ GeV. The upper bound depends on $T_1$ and also on the indices $n_i$, as shown in Fig.($\ref{fig: 2}$). Let us take a closer look to the multifield case, already mentioned in Sec. III. Of course, the upper bound on $T_{reh}$ is always present, but it depends on the intermediate temperatures $T_i$. For instance, in the presence of two scalar fields, with $\phi_2$ dominating at higher temperature $T>T_2$ on $\phi_1$, the condition becomes $\rho_{\phi_2}\leq M^4$ at $T=T_{reh}$. As a consequence, one gets $$T_{reh}<\alpha_2 M \left( \frac{T_1^{n_1} T_2^{n_2-n_1}}{M^{n_2}} \right)^{\frac{1}{4+n_2}} \label{eq:boundwithtwo}$$ where $\alpha_2=\left(30/\pi^2 g_E\right)^{1/4+n_2}$ (by assumption $n_2>n_1$). Using, for instance, $n_1=1$, $n_2=2$, $T_1\sim 10^4$ GeV and $T_2\sim 10^6$ GeV one has $T_{reh}<10^{12}$ GeV for $M<M_{inf}$, while $T_{reh}<10^{13.7}$ GeV for $M<M_{Pl}$. Note that in the first case the value of $T_{reh}^{max}$ is very close to the one found in the presence of a single scalar field with $n_1=2$ at a transition-to-radiation temperature $T_1\sim 10^7$ GeV (see Sec.II). The upper bound can obviously be computed for any number $k$ of scalar fields and it turns out to depend on $2k$ parameters, the $T_i$ temperatures and the $n_i$ dilution coefficients. A nonstandard cosmological epoch after reheating gives also rise to an extra term in the general expression of the inflationary number of $e$-foldings, $N_$ (see Eqs. and ). The upper bound on the energy density of the $k$-th scalar field leads to an additional constraint on the contribution to $N_$ coming from the reheating phase, as shown in Eq.($\ref{eqn: reh_constraint}$). As a result, we found the possibility of having an inflationary number of $e$-foldings well beyond $60$, as shown in Fig.($\ref{fig: 4}$). The higher is the number of scalar fields, the larger is the correction $\Delta N$ to $N_$, since the ratio of the total energy density to the radiation density at $T_{reh}$ is larger. For instance, with two scalar fields and using Eq.($\ref{eqn: g_relation}$), Eq.($\ref{eqn: eta_general}$) provides $$\eta\simeq 1 + \left(\frac{T_1}{T_{reh}}\right)^4\left(\frac{T_{reh}}{T_1}\right)^{4+n_1} + \left(\frac{T_1}{T_{reh}}\right)^4\left(\frac{T_2}{T_1}\right)^{4+n_1} \left(\frac{T_{reh}}{T_2}\right)^{4+n_2}.$$ By choosing the same data as after Eq. and $w_{reh}=0$, $T_{reh}\sim10^{13}$ GeV, one gets $\eta\sim 10^{16}$, $\Delta N(\phi_1,\phi_2)\sim 12$ and $N_\sim 70$. As expected, a nonstandard post-reheating phase produces a variety of enhancements in the inflationary number of $e$-foldings, depending on the the number of additional scalar fields and on the details of their dilution properties. Enhancements affect the theoretical predictions of the inflationary models, mainly in the bottom right portion of the familiar $(n_s,r)$ plane. In [@9], Maharana and Zavala have studied the functions $n_s(N_)$ and $r(N_)$. In Appendixes A and B, we report some results for typical classes of inflationary models, extending the range of parameters provided in [@9]. We deserve an extended analysis to a future publication [@15]. This work was supported in part by the MIUR-PRIN Contract 2015MP2CX4 “Non-perturbative Aspects Of Gauge Theories And String". P.C. would also like to thank the company “L’isola che non c’è S.r.l" for the support. Monomial Potentials =================== The first class of inflationary models we are going to analyze are those characterized by single monomial potentials of the form $$V(\varphi)=\lambda_p\varphi^p, \quad \lambda_p=M^4_{inf} M_{Pl}^{-p}.$$ In this class of models the inflaton field, in order to drive inflation, must exhibit a super-Planckian variation $\Delta\phi>M_{Pl}$. Historically, the most known scenarios are the ones with $p=2$ and $p=4$, that were introduced by Linde [@3]. Monomial potentials naturally occur also in superstring compactifications, where they are called “axion monodromy” [@16]. In these models, one uses a cover of the compactification manifold, with branes wrapping suitable internal cycles. As a result, even though the manifold is compact, the wrapping of branes around certain cycles weakly breaks the original shift symmetry, allowing for closed-string axions with super-Planckian excursions and the suppression of dangerous higher-dimensional operators. The involved inflationary potentials come out precisely of the form $V(\varphi)\sim \varphi^p$ with $p=2/5,2/3,1$ or $4/3$. The slow-roll parameters give rise to standard theoretical predictions for the spectral index and the tensor-to-scalar ratio in terms of the inflationary number of $e$-foldings: $$n_s\sim 1-\frac{p+2}{2 N_}, \quad r=\frac{4p}{N_}.$$ It should be noticed that both $n_s$ and $r$ depend on the model parameter $p$. In Tab. I and in Tab. II we report the theoretical predictions for some scenarios related to monomial potentials, assuming two possible non-standard post-reheating data. [lcdr]{} & &\ Axion model $p=2/5$ & 0.9821 & 0.0238\ Axion model $p=2/3$ & 0.9801 & 0.0398\ Axion model $p=1$ & 0.9776 & 0.0597\ Axion model $p=4/3$ & 0.9751 & 0.0796\ Linde model $p=2$ & 0.9701 & 0.1194\ Linde model $p=4$ & 0.9552 & 0.2388\ [lcdr]{} & &\ Axion model $p=2/5$ & 0.9830 & 0.0229\ Axion model $p=2/3$ & 0.9810 & 0.0381\ Axion model $p=1$ & 0.9785 & 0.0571\ Axion model $p=4/3$ & 0.9762 & 0.0762\ Linde model $p=2$ & 0.9714 & 0.1143\ Linde model $p=4$ & 0.9571 & 0.2286\ Exponential Potentials ====================== The second class of models we would like to consider is that of exponential potentials of the form $$V(\varphi)\sim M^4_{inf}\left(1-e^{-b\varphi}\right) , \quad b=\sqrt{\frac{2}{3\alpha}} , \label{exppot}$$ where $\alpha$ is a free parameter. These potentials arise in many contexts. Important examples are the well known Starobinsky model ($\alpha=1$), the Goncharov-Linde model ($\alpha=1/9$) and the Higgs Inflation model ($\alpha=\sqrt{2/3}$)[@17]. More recently, the so called $\alpha$-attractor models of inflation [@18] have also been considered, that fall in the same class of Eq. . Furthermore, other very interesting examples come out in superstring-inspired scenarios, like Kähler Moduli Inflation, Poly-instanton Inflation and Fiber Inflation [@19]. At first order, the theoretical predictions of this class of models result $$n_s\sim 1-\frac{2}{N_}, \quad r\sim \frac{12\alpha}{N^2_}.$$ In this case, the scalar spectral index does not depend on the value of $\alpha$. Therefore, for $N_=67$ one has $n_s\sim 0.9701$ while for $N_=70$, $n_s=0.9714$, independently on $\alpha$. On the contrary, the tensor-to-scalar ratio depends on $\alpha$ as shown in Tab. III, where we report its values for some choices of the parameters. 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--- abstract: 'We report the discovery of fast variability of $\gamma$-ray flares from blazar OJ 287. This blazar is known to be powered by binary system of supermassive black holes. The observed variability time scale $T_{\rm var}\lesssim 3-10$ hr is much shorter than the light crossing time of more massive ($1.8\times 10^{10}M_\odot$) black hole and is comparable to the light crossing time of the less massive ($1.3\times 10^8M_\odot$) black hole. This indicates that emission is produced by relativistic jet ejected by the black hole of smaller mass. Detection of s with energies in excess of 10 GeV during the fast variable flares constrains the Doppler factor of the jet to be larger than 4. Possibility of the study of orbital modulation of emission from relativistic jet makes a unique laboratory for the study of the mechanism(s) of formation of jets by black holes, in particular, of the response of the jet parameters to the changes of the parameters of the medium from which the black hole accretes and into which the jet expands.' author: - \| A. Neronov$^{1}$, Ie. Vovk$^{1}$\ $^{1}$ISDC Data Center for Astrophysics, Geneva Observatory, Chemin d’Écogia 16, 1290 Versoix, Switzerland date: 'Received $<$date$>$ ; in original form $<$date$>$ ' title: 'Fast variability of $\gamma$-ray emission from supermassive black hole binary OJ 287' --- \[firstpage\] [gamma-rays: observations, galaxies: nuclei, radiation mechanisms: non-thermal, black hole physics, BL Lacertae objects: individual: OJ 287]{} Introduction ============ Although large-scale jets ejected by Active Galactic Nuclei (AGN) were discovered almost a century ago, origin of this phenomenon remains obscure up to present days (see @jet_review for a recent review). It might be that the jets are accelerated via magneto-centrifugal force along twisted magnetic field lines above accretion disk around the black hole [@blandford-payne]. Otherwise, an outflow can be created via Blandford-Znajek mechanism of electromagnetic power extraction from a rotating BH, similar to the mechanism responsible for the generation of relativistic pulsar winds [@blandford-znajek]. Blazar ($z=0.306$ [@stickel89]) provides a unique laboratory for the study of the mechanism of AGN activity, because this is one of the few AGN known to host binary black hole system [@lehto96; @valtonen09]. In this system, a lighter black hole of the mass $M_{\rm BH1}\simeq 1.3\times 10^8M_\odot$ orbits a heavier black hole of the mass $M_{\rm BH2}\simeq 1.8\times 10^{10}M_\odot$ with a period $P_{\rm orb}\simeq 11.65$ yr [@valtonen09; @sillanpaa88]. Separation of the components of the system at periastron is just about 10 Schwarzschild radii of the heavier black hole, so that the orbital motion is strongly affected by relativistic gravity effects [@OJ_nature]. is known to belong to the BL Lac sub-class of AGNi, which means that it emits a relativistic jet whose direction is aligned with the line of sight. It is not clear a priori, which of the black holes ejects the observed relativistic jet. Following a naive argument, which does not take into account the relativistic beaming of the jet emission, one would assume that the observed relativistic jet is the one ejected by the heavier black hole, simply because the bigger black hole accretes more matter and, therefore, could produce more powerful jet. Most of the existing studies of multi-wavelength blazar activity of adopt this assumption (see e.g. @valtonen09). However, relativistic jets in BL Lacs are known to Doppler factors $\delta\gg 1$. This results in boosting the apparent luminosity of the jets by a factor $\delta^4$, when the jets are viewed face-on. Thus, if the less powerful jet emitted by the smaller black hole is aligned with the line of sight, while the jet from the larger black hole is not, the jet from the smaller black hole might give dominant contribution to the source flux. In what follows we show that variability properties of emission from the source indicate that the relativistically beamed emission comes from the jet produced by the smaller black hole. Independently of the value of the Doppler factor of the jet $\delta$, the shortest observed variability time scale $\Delta T_{min}$ imposes a constraint on the size of the jet’s “central engine”, $R_{\rm CE}\lesssim c\Delta T$ [@celotti98; @neronov09]. In the case of , $R_{\rm CE}$ turns out to be much smaller than the Schwarzschild radius of the more massive black hole, but compatible with the size of the smaller mass black hole. Observation of emission from the base of the jet of the smaller black hole in the system makes a unique laboratory for the study of mechanisms of jet production. Regular orbital modulation of the physical parameters of the ambient medium around the $1.3\times 10^8M_\odot$ black hole provides a unique possibility to study the response of the jet to the changes of the properties of accretion flow and of external medium in which the jet propagates. In this respect, the system provides a scaled-up analog of Galactic -loud binaries, in which orbital modulation of emission enables a study of response of relativistic outflow from a compact object (a neutron star or a black hole) to the changes of the properties of external medium (stellar wind and radiation field of companion star) (see e.g. [@zdz10]). [Fermi]{} observations ======================== ![August 2008 – January 2010 lightcurve of (black) and background region (grey). Orange bands mark the periods of passage of the Sun through the observation region. []{data-label="fig:lcurve_longterm"}](OJ_lc_longterm1.eps){width="\columnwidth"} ![Lightcurve of October-November 2009 flare of in $E>0.1$ GeV energy band, binned in time bins with S/N ratio equal to 3. Black curve shows model fit to the lightcurve, Eq. (\[eq:model\]) with parameters given in Table \[tab:model\].[]{data-label="fig:lcurve"}](OJ_lcurve_flare.eps){width="\linewidth"} In order to study variability of the signal during the flaring activity, we have processed publicly available data of the LAT instrument, using the [Fermi]{} Science Tools provided by the [Fermi]{} Science Support Centre. The data were selected using [gtselect]{} tool. The lightcurves were produced with the help of [gtbin]{} and [gtexposure]{} tools as it is explained in the [Fermi]{} data Analysis Threads [^1]. ![Images of sky region around . Panels [a]{} and [c]{} show 0.3-1 GeV and 1-300 GeV band images for the entire observation period. Panels [b]{} and [d]{} show 0.3-1 GeV and 1-300 GeV band images during the flaring activity of . Coordinate grid show ecliptic coordinates. Green circle shows the region from which the background lightcurve was extracted. Images in the 0.3-1 GeV band are smoothed with a gaussian of the width $1^\circ$. Images in the 1-300 GeV band are smoothed with the Gaussian with $0.5^\circ$ width. []{data-label="fig:image"}](OJ287_images.eps){width="\linewidth"} Fig. \[fig:lcurve\_longterm\] shows the longterm lightcurve of the source at the energies above 0.1 GeV over the August 2008 – September 2010 period binned to achieve signal to noise ratio S/N=5 per time bin. Photons were collected from the circle $2^\circ$ in radius and centered on . The flaring activity period, which started on October 5 2009, could be readily identified. The detailed lightcurve of the flaring period shown in Fig. \[fig:lcurve\] reveals several well separated flares. The brightest flare which happened on October 22, 2009 was reported by @fermi_atel. Followup observations of October 22 flare by [Swift]{}/XRT have revealed an increase of the X-ray flux accompanying the flare [@swift]. The direction toward is situated close to the plane of Ecliptic. The lightcurve of the source could be affected by the passage of the Sun through the field of view of the telescope. The periods of the Sun passage within the region of the radius $15^\circ$ centered on (the sky region chosen for the analysis) are shown by the orange-shaded strips in Fig. \[fig:lcurve\_longterm\]. Possible effect of the Sun passage on the source lightcurve is most clearly seen in the lightcurve collected from a region of the radius $2^\circ$ displaced by $\sim 5^\circ$ from the position of to the position RA=127.11 DEC=19.89. No source is detected at this position, so the lightcurve collected from this region (shown by grey data points in Fig. \[fig:lcurve\_longterm\]) could be considered as a measure of the diffuse background level close to the position of . One could notice a flare in the background lightcurve, associated with the Sun passage close to the background extraction region. The effect of the Sun passage on the lightcurve of is less pronounced (see Fig. \[fig:lcurve\_longterm\]), because the source is situated higher above the Ecliptic plane, compared to the background region. Analysis of the images of sky around show that, apart from , the region of the radius $15^\circ$ centered on contains several other sources (most clearly visible in the 1-300 GeV band image in the panel [c]{} of Fig. \[fig:image\]). However, these sources are rather weak, so that they are not detected on one month exposure time scale corresponding to the duration of the flaring activity of (see the right panel of Fig. \[fig:image\]). Absence of strong sources near ensures that the observed variability of the signal from the source during the flaring period is not affected by possible variable emission from a nearby source for which the tail of the point spread function overlaps with that of . The $E>0.1$ GeV lightcurve of the flare consists of several well separated pulses with rather sharp rise and decay. To find the rise and decay times we have fitted the lightcurve with a phenomenological model of a sum of exponentially rising and decaying pulses $$F(t)=B+\sum_{k=1}^3\left\{ \begin{array}{ll} A_k\exp\left((t-t_k)/t_{rk}\right), & t<t_k\\ A_k\exp\left(-(t-t_k)/t_{dk}\right), & t>t_k \end{array} \right. \label{eq:model}$$ where $B=const$ is the background level. The background level was found from the circle of the radius $2^\circ$ displaced by 5 degrees from the source position. Parameters of the model function (\[eq:model\]), derived from the fitting, are given in Table \[tab:model\]. --- ------------------------- ------------------------ ------------------------ ---------------------------------- k $t_k$ $t_{rk}$ $t_{dk}$ $A_k$ d d d $\times 10^{-6}$ cm$^{-2}s^{-1}$ 1 $1.4\pm 0.5$ $5.0_{-2.5}^{+10.0}$ $7.0^{+4.0}_{-3.5}$ $0.33_{-0.08}^{+0.07}$ 2 $5.6_{-0.3}^{+0.2}$ $0.15_{-0.14}^{+0.37}$ $1.0^{+0.6}_{-0.4}$ $1.0_{-0.3}^{+0.4}$ 3 $16.02_{+0.07}^{-0.05}$ $0.33_{-0.09}^{+0.19}$ $0.30_{-0.1}^{+0.15}$ $2.3^{+0.6}_{-0.6}$ 4 $17.6_{-0.15}^{+0.18}$ $0.5^{+0.4}_{-0.25}$ $0.46^{+0.25}_{-0.18}$ $1.5_{-0.5}^{+0.5}$ 5 $29.8_{-0.3}^{+0.4}$ $0.4^{+0.3}_{-0.23}$ $1.0^{+0.5}_{-0.4}$ $1.3_{+0.3}^{-0.3}$ 6 $37.2_{-1.0}^{+1.3}$ $2.0^{+1.8}_{-1.0}$ $1.9^{-0.9}_{-2.2}$ $0.41_{+0.12}^{-0.14}$ --- ------------------------- ------------------------ ------------------------ ---------------------------------- : Parameters of the model fit (see Eq. (\[eq:model\])) to the lightcurve of the flare of together with their $68\%$ confidence ranges. []{data-label="tab:model"} One can see from Fig. \[fig:lcurve\] and Table \[tab:model\] that brightest flares are characterized by the rather sharp rises and decays, with the rise/decay times of several hours. At this time scales measurement of the rise/decay times of the flares is complicated by the fact that [Fermi]{}/LAT telescope observes a given patch of the sky once in $3.2$ hours (once per two rotation periods of $\simeq 96$ min). This is clear from Fig. \[fig:lcurve\_zoom\] in which lightcurve for the time interval of several hours around the brightest flare from the source is shown in more details. The upper panel of the figure shows the $E\ge 0.1$ GeV lightcurve of the source. The lower panel shows the energies of photons collected from the circle of the radius $2^\circ$ centered on the source (black points) and photons collected from a background extraction circle displaced of the same radius displaced by 5 degrees from the source position. Photons from the source and background regions come only within periodic time intervals spaced by 3.2 hr marked by vertical grey strips in the two panels of the Figure. It is clear that [Fermi]{}/LAT pointing pattern does not allow to constrain the rise/decay time of the flares to better than 3.2 hr. The apparently abrupt end of the flare (12 photons from the source are detected between $t=$MJD55126.0$+3$ hr and $t=$MJD55126.0$+4$ hr, while no photon is detected within the subsequent observation period MJD55126.0$+6.2$ hr$<t<$MJD55126.0$+7.2$ hr) limits the decay time of the flare to be less than $3.2$ hr. Assuming that the source flux did not change in the time interval following the peak of the flare, one could estimate the chance probability of detecting zero photons in this time bin to be $7\times 10^{-4}$, taking into account the exposures in the two adjacent time intervals with maximal and zero count rates, $3.3\times 10^6$ cm$^2$s and $2\times 10^6$ cm$^2$s, respectively. To summarize, both fitting of the lightcurve profile with a phenomenological exponential rise / exponential decay model and and direct photon counting in individual 1 hr long [Fermi]{} exposures indicate that the flux from the source is significantly variable on time scales shorter or comparable to 3-10 hr. This fact has important implications for the physical model of the origin of emission from the system. ![Upper panel: lightcurve of brightest episode of October-November 2009 flare of in $E>0.1$ GeV energy band. Lower panel: energies and arrival times of from the source (black points) and from the background region (grey points). Vertical grey strips show the periods when the source was in the field of view of LAT telescope. []{data-label="fig:lcurve_zoom"}](OJ_flare_zoom.eps){width="\linewidth"} Origin of relativistic jet in ============================== Constraint on the variability time scale of the flares $T_{\rm var}=\mbox{min}(t_{rk},t_{dk})\le 3.2$ hr, derived above, enables to identify the emission site within the binary black hole system of . It is commonly accepted that emitting jets are generated by the AGN “central engines”, the supermassive black holes, on the distance scales of the order of gravitational radius of supermassive black hole $$\label{Rg} R_g=\frac{G_NM_{\rm BH}}{c^2}=2\times 10^{11}\left[\frac{M_{\rm BH}}{1.3\times 10^{8}M_\odot}\right]\mbox{ cm,}$$ where $M_{\rm BH}$ is the black hole mass. Minimal variability time scale of electromagnetic emission originating from the AGN central engine is expected to be not shorter than the light-crossing time of the supermassive black hole, $$\begin{aligned} \label{lc} T_{\rm lc}&=&(1+z)2R_{\rm BH}/c=2(1+z)\left(R_g+\sqrt{R_g^2-a^2}\right)/c\\ &\simeq&\left\{ \begin{array}{ll} 0.5\left[M_{\rm BH}/1.3\times 10^{8}M_\odot\right]\mbox{ hr,}&a=R_g\\ 0.9\left[M_{\rm BH}/1.3\times 10^{8}M_\odot\right]\mbox{ hr,}&a=0 \end{array} \right.\nonumber\end{aligned}$$ where $R_{\rm BH}$ is the size of the black hole horizon and $0<a<R_g$ is black hole rotation moment per unit mass. Variability of X-ray emission at time scale $T\sim T_{\rm lc}$ is observed in X-ray emission from Galactic sources powered by black holes with masses $M_{\rm BH}\sim 10M_\odot$ [@remillard06]. Variability at the time scale $T_{\rm var}\sim T_{\rm lc}$ is observed also in emission from blazars, a special type of AGN with jets aligned along the line of sight [@mrk421; @m87; @m87_theory; @mrk501; @pks2155; @neronov09]. ![Comparison of the upper bound on variability time scale of flares (grey shaded region) with the light crossing times of the two supermassive black holes in the system.[]{data-label="fig:OJ_jet_size"}](OJ_jet_size.eps){width="\linewidth"} Fig. \[fig:OJ\_jet\_size\] shows a comparison of the upper limit $T_{\rm var}\le 3.2$ hr on the variability time scale of the flares from with the light crossing time of the two supermassive black holes in the system. One can see that the upper bound is much lower than the light crossing time of the more massive black hole with $M_{\rm BH}\simeq 1.8\times 10^{10}M_\odot$, independently of the black hole rotation moment $a$. At the same time, the limit on the variability time scale is larger than the light crossing time of the smaller black hole. This implies that the observed emission is generated by the jet emitted by the lower mass companion black hole in the system. Note, that the above result does not depend on the assumptions about relativistic motion of the emission region. Indeed, the minimal possible time scale $\Delta T_{min}$ of emission from relativistic jet moving with bulk Lorentz factor $\Gamma$ does not depend on $\Gamma$ and is instead determined by the size of the non-moving “central engine” which ejected the jet [@celotti98; @neronov09; @neronov09a]. In the jet comoving frame, the minimal variability time scale $\Delta T_{min}'\sim \Gamma\Delta T_{min}$ is the light-crossing time of the smallest size emitting “blobs”, $\Delta T_{min}'\sim R_{min}'/c$. emitting blobs which were ejected by the jet’s “central engine” of the size $R_{\rm CE}$ could not have size smaller than $R_{min}'\ge R_{\rm CE}\Gamma$ in the comoving frame. This implies that $\Delta T_{min}\sim \Delta T_{min}'/\Gamma\sim R_{min}'/c\Gamma\ge R_{\rm CE}/c$, independently of $\Gamma$. At the first sight, such conclusion looks counter-intuitive. Indeed, since the two black holes accrete matter from the same reservoir, the accretion rate onto more massive black hole should be much higher. In general, higher accretion rate should lead to production of a more powerful jet. However, the above argument does not take into account possible relativistic beaming effects, which are normally very significant in BL Lacs. flux from relativistically beamed jet moving with bulk Lorentz factor $\Gamma$ at an angle $\theta$ with respect to the line of sight is boosted by a factor $\delta^4$, where $\delta=(\Gamma(1-\beta\cos\theta))^{-1}$ is the Doppler factor and $\beta$ is the bulk velocity of the jet. A constraint on the bulk Lorentz factor of the jet produced by the $M_{\rm BH}\simeq 1.3\times 10^8M_\odot$ black hole could be found under assumption that the observed X-ray emission from the system [@suzaku] originates from the same jet region as the GeV emission. The highest energy of photons coming from the source is $E_{\gamma,max}\ge 30$ GeV (see Fig. \[fig:lcurve\_zoom\]). s of such energies can produce $e^+e^-$ pairs in interactions with X-ray photons of the energies $$E_X\ge \frac{\Gamma^2m_e^2c^4}{(1+z)^2E_\gamma}\simeq 0.5\left[\frac{\Gamma}{4}\right]^2\left[\frac{E_{\gamma,max}}{30\mbox{ GeV}}\right]^{-1}\mbox{ keV}$$ where we have assumed that typical collision angles for photons emitted from the jet are $\alpha\simeq \Gamma^{-1}$. Apparent (relativistically beamed) luminosity of in the soft X-ray band is $L_{\rm app}\simeq 3\times 10^{44}$ erg/s, which corresponds to the observed flux $F_X\simeq 10^{-12}$ erg/cm$^2$s [@suzaku]. Calculating the optical depth of emission region of comoving size $R'\simeq\delta cT_{\rm var}/(1+z)$ and luminosity $L'\simeq \delta^{-4}(1+z)^2L_X$ one finds $$\tau_{\gamma\gamma}\simeq 0.5\left[\frac{\delta}{4}\right]^{-6}\left[\frac{L_X}{10^{44}\mbox{ erg/s}}\right]\left[\frac{T_{\rm var}}{3.2\mbox{ hr}}\right]^{-1}$$ High energy s could escape from the source if $\tau_{\gamma\gamma}< 1$. This condition imposes a restriction on the Doppler factor $\delta\gtrsim 4$. Thus, the observed flux from the jet is Doppler boosted by at least a factor $\delta^4\gtrsim 3\times 10^2$. It is interesting to note that even if emission from the higher mass black hole is not relativistically beamed toward observer on the Earth, it might be noticed in the spectrum of the source. Indeed, assuming a simple Eddington-like scaling of the accretion rate and jet luminosity with the black hole mass, $L_i\sim M_{{\rm BH}i}$, one finds that the relativistically beamed luminosity of the jet from the lighter black hole $L\sim \delta^4L_1\ge 10^2L_1$ could, in fact, be comparable to the overall luminosity of the heavier black hole, $L_2\sim (M_{\rm BH2}/M_{\rm BH1})L_1\simeq 1.4\times 10^2 L_1$. Fast variability of emission could therefore, serve as a tool for identification of the contribution of emission from the lighter black hole in the overall source spectrum. A common feature of all models of jet production by black holes is that matter ejection into the jet is associated with rotation of matter around the black hole and/or with rotation of the black hole [@blandford-znajek; @blandford-payne]. This implies that characteristic time scale at which the properties of the jet could change is given by the period of rotation of the black hole itself or of the accretion flow onto the black hole. Period of rotation around a circular orbit at a distance $r$ from the black hole is given by [@bardeen] $$\label{eq:P} P(r)=2\pi(1+z)\frac{r^{3/2}\pm aR_g^{1/2}}{cR_g^{1/2}}\;,$$ The $+$ ($-$) sign corresponds to the prograde (retrograde) orbit. Stable circular orbits exist only down to certain distance $r_{\rm ms}$ from the BH. The period of rotation along the last prograde stable orbit at the distance $r_{\rm ms}$ is $$\label{eq:period} P(r_{\rm ms})\simeq\left\{ \begin{array}{ll} 3\left[\frac{\displaystyle M_{\rm BH}}{\displaystyle 1.3\times 10^{8}M_\odot}\right] \mbox{ hr,}&a=R_g\\ 22 \left[\frac{\displaystyle M_{\rm BH}}{\displaystyle 1.3\times 10^{8}M_\odot}\right]\mbox{ hr,}&a=0 \end{array} \right.$$ Upper bound on the variability time scale $T_{\rm var}\le 3.2$ hr is much shorter than period of rotation around the non-rotating black hole and is comparable or smaller than the period of rotation around maximally rotating black hole with $a=R_g$. This means that relativistic ejections into the jet, responsible for the observed flares, are produced by the matter moving in the direct vicinity of the black hole horizon, well inside the $R=6R_g$ radius of the last stable orbit around non-rotating black hole. Conclusions {#CONCL} =========== To summarize, we find that observations of in the $E>0.1$ geV energy band constrain the minimal timescale of flux variations of the source to be shorter than 3.2 hr. The upper limit on the minimal variability timescale imposes a restriction on the size of the jet formation region. We find that the size of the jet formation region in the system is much smaller than the size of the horizon of the more massive black hole in the binary black hole system powering the source. This means that the observed emission is produced by the jet ejected from the smaller mass black hole. Higher apparent luminosity of the smaller mass companion is explained by the effect of relativistic beaming of emission. Combining X-ray and data, we find a restriction on the Doppler factor of the emitting part of the jet, $\delta\gtrsim 4$. The observed variability time scale indicates that relativistic jet from the smaller mass black hole is formed close to the black hole horizon, well inside the last stable orbit around non-rotating supermassive black hole. data provide a new insight in the physical model of the unique binary supermassive black hole system in . flaring activity is produced in connection with the passage of the smaller mass black hole through the accretion disk around the larger mass black hole during the periods of close approach of the two black holes in the periastron of the binary orbit. Interaction of the smaller mass black hole with the larger mass black hole accretion disk leads to the transient episodes of ejection into relativistic jet from the smaller mass black hole. It appears that the transient jet from the small mass black hole happens to be aligned along the line of sight, the fact responsible for the BL Lac type appearance of the source. It is not clear [a priori]{} if the jet from the smaller mass black hole forms only during the periastron passage or it exists throughout the binary orbit. If the jet is powered by the transient accretion onto the black hole, it would be natural to expect that the the small mass black hole jet (and the associated emission) should disappear soon after the periastron passage on characteristic time scale of accretion. Otherwise, if the jet is powered by the rotation energy stored in the small mass black hole, it is natural to expect that the jet and the from the jet would be persistent throughout the binary orbit. Systematic monitoring of the source evolution in s on the orbital (11.7 years) time scale, which is now possible with [Fermi]{}, might clarify this question. It is interesting to note that if the jet from the smaller mass black hole is directed along the black hole spin axis, its alignment with the line of sight might be destroyed by the precession of the black hole spin axis (see @valtonen06 for detailed discussion of the orbital evolution of the system). This would mean that the BL Lac appearance of of the source might be time-dependent. An immediate consequence of the mis-alignment of the smaller black hole jet with the line of sight should be the loss of the strong Doppler boosting of the flux. In the absence of Doppler boosting, the emission from the smaller mass black hole might become sub-dominant compared to the emission from the larger mass black hole. Study of the details of the overall time evolution and of the short time scale variability properties of the source along the orbit and from periastron to periastron should clarify the transient/permanent nature of the BL Lac appearance of the source. 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W., Kuehr, H., 1989, A&AS, 80, 103. Valtonen M.J. et al., 2006, Ap.J. 646, 36. Valtonen M.J. et al., 2009, Ap.J., 698, 781. Valtonen M.J. et al., 2008, Nature, 452, 851. Zdziarski A., Neronov A., Cheryakova M., MNRAS, 2010, 403, 1873 \[lastpage\] [^1]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/
--- abstract: 'The Mott-Hubbard transition is studied in the context of the two-dimensional Hubbard model. Analytical calculations show the existence of a critical value $U_{c}$ of the potential strength which separates a paramagnetic metallic phase from a paramagnetic insulating phase. Calculations of the density of states and double occupancy show that the ground state in the insulating phase contains always a small fraction of empty and doubly occupied sites. The structure of the ground state is studied by considering the probability amplitude of intersite hopping. The results indicate that the ground state of the Mott insulator is characterized by a local antiferromagnetic order; the electrons keep some mobility, but this mobility must be compatible with the local ordering. The vanishing of some intersite probability amplitudes at $U=U_{c}$ puts a constrain on the electron mobility. It is suggested that such quantities might be taken as the quantities which control the order in the insulating phase.' author: - 'F. Mancini[^1]' title: \| The Mott-Hubbard transition\ and the paramagnetic insulating state\ in the two-dimensional Hubbard model --- euromacr € There are several indications that the two-dimensional Hubbard model (HM) can describe a metal-insulator transition and that some kind of order is established in the paramagnetic insulating state. However, there is no clear picture about the structure of the ground state and no indication about the existence of an order parameter. In particular, there is a difficulty to conciliate the existence of a finite value for the doubly occupancy, which implies mobility of the electrons, and the existence of some order which would imply a localization of the electrons. In this article we study the Hubbard model by means of the composite operator method (COM) in the two-pole approximation. The main results can be so summarized: (i) a Mott- Hubbard transition does exist; (ii) a local antiferromagnetic (AF) order is present in the insulating state; (iii) a quantity which controls the order in the insulating state is individuated. According to the band model approximation several transition oxides should be metals. In practice one finds both metallic and insulating states, with a metal-insulator transition induced by varying the boundary conditions (pressure, temperature, compound composition). Mott \[1\] pointed out that for narrow bands the electrons are localized on the lattice ions and therefore the correlations among them cannot be neglected. A model to describe these correlations was proposed by Hubbard \[2\]. In a standard notation his Hamiltonian is given by $$H=\sum_{ij}(t_{ij} - \mu\delta_{ij}) c^\dagger(i) c(j) +U\sum_{i} n_{\uparrow}(i) n_{\downarrow}(i)$$ where $c(i)$ and $c^\dagger(j)$ are annihilation and creation operators of electrons at site $i$, in the spinor notation; $t_{ij}$ describes hopping between different sites and it is usually taken as $t_{ij}=-4t\alpha_{ij}$, where $\alpha_{ij}$ is the projection operator on the first neighbor sites; the $U$-term is the Coulomb repulsive interaction between two electrons at the same site with $n_{\sigma}(i) = c^\dagger_{\sigma}(i) c_{\sigma}(i)$; $\mu$ is the chemical potential. The magnitudes of the on-site Coulomb energy $U$ and the one-electron band width $W=8t$ control the properties of the system. In this competition between the kinetic and the potential energy the most difficult part of the model resides and exact solutions do not exist, except in some limiting cases. In particular, an adequate description of the ground state and elementary excitations is still missing. In the case of one dimension an exact solution is available \[3\] which shows that there is no Mott-Hubbard transition: a gap in the density of states is present for any value of U. In higher dimensions there are several results that indicate the existence of a Mott-Hubbard transition, in the sense that at half filling there is a critical value of the Coulomb potential $U_{c}$ which separates the metallic phase from the insulating phase; but no rigorous results. In Hubbard I approximation \[2\] and in the work by Roth \[4\] no transition is observed \[5\]. In Hubbard III approximation \[6\] an opening of the gap is observed for $U_{c}=W\sqrt{3/2}$. By using the Gutzwiller variational method \[7\] Brinkman and Rice \[8\] find $U_{c}=8\|\overline\epsilon \|\approx 1.65 W$, with $\overline\epsilon$ being the average kinetic energy per electron; the vanishing of the double occupancy D at this value induced them to propose the double occupancy as an order parameter to describe the metal-insulator transition. However, this result is based on the use of the Gutzwiller approximation, which becomes exact only for infinite dimensions \[9\]. For finite dimensions theoretical \[10\] and numerical analysis \[11\] show that the double occupancy tends to zero only in the limit $U\to \infty$. By using the dynamical mean-field approach, or $d\to\infty$ limit, Georges et. al. \[12\] find that at some critical $U$ a gap opens abruptly in the density of states, due to the disappearance of a Kondo-like peak. A recent calculation \[13\] of the HM in infinite dimensions shows a continuous Mott-Hubbard transition, with a gap opening at $U_{c}\approx W$. The same qualitative result has been found \[14\] by using Quantum Monte Carlo (QMC) simulation; working at the high $T=0.33\, t$ the authors observe a transition with a gap opening continuously at $U_{c}\approx W/2$. In conclusion, while there are several results indicating the existence of a Mott-Hubbard transition in the 2D Hubbard model, there is no unified picture; the mechanisms that lead to the transition are different; the value of the critical interaction strength varies from $U_{c}\approx 0.5W$ \[14\] up to $U_{c}\approx 1.65W$ \[8\]. A description of the structure of the ground state in the paramagnetic insulating state is also lacking. In the framework of the COM \[15,5\] the Hubbard model has been solved in the two pole approximation \[16\], where the operatorial basis is described by the doublet Heisenberg operator $$\psi (i) = \left(\begin{array}{c} \xi(i)\\ \eta(i) \end{array} \right)$$ and finite life-time effects are neglected. The fields $\xi (i)=[1-n(i)] c(i)$ and $\eta(i)=n(i) c(i)$, with $n(i)= c^\dagger (i) c(i)$, are the Hubbard operators \[2\]. In this framework the single-particle propagator is given by $$F.T. \langle R[\psi(i)\psi^\dagger(j)]\rangle =\sum^{2}_{i=1}\frac{\sigma^{(i)}({\bf k})}{\omega-E_i({\bf k})+i\eta}$$ where $F.T.$ means Fourier transform. The expressions of the spectral functions $\sigma^{(i)}(\mathbf{\mathrm{k}})$ and energy spectra $E_{i}(\mathbf{\mathrm{k}})$ have been reported in previous works \[15\]. These functions are calculated in a fully self-consistent treatment, where attention is paid to the conservation of relevant symmetries \[15,5,17\]. Differently from other approaches, one does not need to recur to different schemes in order to describe the weak- and strong-coupling regimes. [Both the limits $U\to 0$ and $U\to\infty$ are recovered by Eq. (3).]{} The result (3) has been derived by assuming a paramagnetic phase. It is an open question if this is the true ground state at half filling and zero temperature. Results of numerical simulation seem to indicate that the paramagnetic phase is unstable against a long range antiferromagnetic order. However, numerical analysis is severely restricted in cluster size, and it is very hard to conclude that the true solution has an infinite range AF order. As it will be shown later, the calculation of the probability amplitudes for electron transfers shows that in the paramagnetic insulating state a local AF order is established, with a correlation length of the order of few hundred time the lattice constant. At first, we observe that the Mott-Hubbard transition can be studied by looking at the chemical potential, which is the quantity which mostly controls the single-particle properties. Let us define $$-\mu_{1} = \left({\partial\mu\over \partial n}\right) _{n=1} = {1\over \kappa (1)}$$ where $\kappa (n) = (\partial n/\partial\mu)/n^{2}$ is the compressibility. Analytical calculations show that at zero temperature there is a critical value of the interaction, fixed by the equation $$U_{c} = 8t\sqrt{4p-1}$$ such that for $U>U_{c}$ the quantity $\mu_{1}$ diverges. The parameter $p$ describes a bandwidth renormalization and is defined by $$p\equiv {1\over 4} \langle n^\alpha_{\mu}(i) n_{\mu} (i)\rangle - \langle [ c_{\uparrow}(i) c_{\downarrow}(i)]^\alpha c_{\downarrow}^\dagger (i) c^\dagger_{\uparrow}(i) \rangle$$ where $n_{\mu}(i)=c^\dagger (i)\sigma_{\mu} c(i)$ is the charge ($\mu=0$) and spin ($\mu=1,2,3$) density operator. We use the notation $A^\alpha(i)=\sum_{j} \alpha_{ij} A(j)$ to indicate the operator $A$ on the first neighbor site of $i$. The quantities $p$ and $\mu_{1}$ are functions of the external parameters $n$, $T$, $U$ and are self-consistently calculated. Numerical solution of the self-consistent equation (5) shows $U_{c} \approx 1.68W$. In Fig. 1a $\mu_{1}$ is plotted versus $U/t$ for $k_{B} T/t=0,0.3,1$. At finite temperature $-\mu_{1}$ increases by increasing U and tends to $\infty$ in the limit $U\to \infty$. At zero temperature $\mu_{1}$ exhibits a discontinuity at $U=U_{c}$.When the intensity of the local interaction exceeds the critical value $U_{c}$, the chemical potential exhibits a discontinuity at half filling, showing the opening of a gap in the density of states and therefore a phase transition from the metallic to the insulating phase. Calculations show that the density of states (DOS) is made up by two bands: lower and upper band. When $U<U_{c}$ the two band overlap: metallic phase. The region of overlapping is given by $\Delta\omega = 16 tp-\sqrt{U^{2}+64t^{2} (2p-1)^{2}}$. When $U>U_{c}$ the two band do not overlap: insulating phase. In Fig. 1b the electronic DOS is reported for different values of $U$. We see that when $U$ increases the central peak opens in two peaks; some of the central weight is transferred to the two peaks, which correspond to the elementary excitations, described by the fields $\xi$ and $\eta$. When $U$ reaches the critical value $U_{c}\approx 1.68 W$ the central peak vanishes abruptly; a gap appears and the electronic density of states splits into two separate bands. This is seen in Fig. 1c, where the DOS calculated at the Fermi level is plotted versus $U$. We find that the gap develops continuously, following the law $\Delta \approx 1.5W (U/U_{c}-1)$. A more detailed study of the density of states can be obtained by considering the contributions of the different channels. Calculations show that both the fields $\xi$ and $\eta$ contribute to the two bands. Only in the limit $U\to\infty$ the two operators do not interact and separately contribute to the two bands. Although, the lower band is essentially made up by the contribution of “$\xi$-electron", there is always a contribution coming from the “$\eta$-electron". The viceversa is true for the upper band. Particularly, the cross contribution plays an important role in the region around the Fermi value, where $N_{\xi\eta} (\mu) \approx N_{\xi\xi}(\mu) \approx N_{\eta\eta} (\mu)$ \[for $U>0$\]. This result shows that in the insulating phase the ground state has a structure different from the simple one where all sites are singly occupied; the competition between the itinerant and local terms leads to a ground state characterized by a small fraction of empty and doubly occupied sites. Some questions arise: (1) what is the structure of the ground state? and in particular there exists any order?; (2) if an ordered state is established, why this order is not destroyed by the mobility of the electrons; (3) can we individuate an order parameter describing the transition at $U=U_{c}$? An important quantity for the comprehension of the properties of the system is the double occupancy $D=\langle n_{\uparrow} n_{\downarrow}\rangle$ which gives the average number of sites occupied by two electrons. Analytical calculations show that at zero temperature the double occupancy, as a function of $U$, exhibits a drastic change when the critical value is crossed, however remains finite for $U>U_{c}$ and tends to zero only in the limit of infinite $U$ as $\lim_{U\to\infty} D = J/ 8U$ where $J=4t^{2}/U$ is the AF exchange constant. In the case of one dimension our analytical results give $\lim_{U\to\infty} D = 3t^2/U^2$ which is very close to the Bethe Ansatz result $\lim_{U\to\infty} D = 4\ln 2t^2/ U^2$ \[18\]. Double occupite sites are used by the system in order to lower its energy. As a matter of fact this is precisely the origin of the effective spin-spin interaction in the $t-J$ model \[19\]. To better understanding the structure of the ground state, we have to study the matrix element $\langle c_{\sigma} (j) c^\dagger_{\sigma}(i)\rangle$. This quantity represents the probability amplitude that an electron of spin $\sigma$ is created at site $i$ and an electron of spin $\sigma$ is destroyed at site $j$. However, this quantity gives only a limited information about the occupation of the sites $i$ and $j$; there are four possible ways to realize the transition $j(\sigma)\to i(\sigma)$, and the quantity $\langle c_{\sigma}(j) c_{\sigma}^\dagger(i)\rangle$ cannot distinguish among them. By means of the decomposition $c_{\sigma }(i) = \xi_{\sigma} (i) + \eta_{\sigma}(i)$, the probability amplitude is written as the sum of four contributions $\langle c_{\sigma}(j) c_{\sigma}^\dagger(i)\rangle = \langle \xi_{\sigma} (j) \xi^\dagger_{\sigma} (i)\rangle + \langle\xi_{\sigma} (j) \eta^\dagger_{\sigma}(i)\rangle +\langle\eta_{\sigma} (j) \xi^\dagger_{\sigma}(i)\rangle + \langle\eta_{\sigma} (j) \eta^\dagger_{\sigma}(i)\rangle$ which correspond to the following transitions: $\begin{array}{cccccc} \langle \xi_{\sigma }(j) \xi^\dagger_{\sigma} (i) \rangle :\qquad & \framebox{0}\ &\ \framebox{$\sigma$}\ & \ \to\ & \framebox{$\sigma$}\ & \framebox{0}\\ & \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} && \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} \\ \langle \xi_{\sigma }(j) \eta^\dagger_{\sigma} (i) \rangle :\qquad & \framebox{$-\sigma$}\ &\ \framebox{$\sigma$}\ &\ \to\ & \framebox{$\sigma-\sigma$}\ & \framebox{0}\\ & \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} && \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} \\ \langle \eta_{\sigma }(j) \xi^\dagger_{\sigma} (i) \rangle :\qquad & \framebox{0}\ &\ \framebox{$\sigma-\sigma$}\ &\ \to\ & \framebox{$\sigma$}\ & \framebox{$-\sigma$}\\ & \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} && \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} \\ \langle \eta_{\sigma}(j) \eta^\dagger_{\sigma} (i) \rangle :\qquad & \framebox{$-\sigma$}\ &\ \framebox{$\sigma-\sigma$}\ &\ \to\ & \framebox{$\sigma-\sigma$}\ & \framebox{$-\sigma$}\\ & \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} && \raisebox{6pt}{\it i} & \raisebox{6pt}{\it j} \end{array}$ A study of the probability amplitudes $\langle\psi (j) \psi^\dagger (i)\rangle$ will give detailed information about the structure of the ground state. In Fig. 2a the amplitude $A= \langle\xi^\alpha (i) \xi^\dagger (i)\rangle = \langle\eta^\alpha (i) \eta^\dagger (i)\rangle$ is reported as a function of $U/t$ for two different temperatures $k_{B}T/t = 0,1$. We see that in the case of zero temperature this quantity vanishes for $U>U_{c}$. This can be easily seen also by analytical methods. The quantity $B=\langle\eta^\alpha (i) \xi^\dagger (i)\rangle = \langle\xi^\alpha (i) \eta^\dagger (i)\rangle$ is reported in Fig. 2b; we see that this probability amplitude does not vanish above $U_{c}$. Owing to this contribution, we have that for $U>U_{c}$ the hopping of electrons from site $i$ to the nearest neighbor is not forbidden, although restricted by the fact that $A=0$. The hopping amplitudes have been studied up to the third nearest neighbors, but the analysis is easily extended to any site, by symmetry considerations. The scheme that emerges from analytical and numerical calculations can be summarized in the following table. Putting all these results together, for $U>U_{c}$ the situation can be so summarized: 1. an electron $\sigma$ which singly occupies a site (a) can hop on first, third,.....neighboring sites if and only if these sites are already occupied by an electron $-\sigma$; (b) can hop on second, fourth,.....neighboring sites if and only if these sites are empty; 2. an electron $\sigma$ which doubly occupies a site (a) can hop on first, third,.....neighboring sites if and only if these sites are empty; (b) can hop on second, fourth,.....neighboring sites if and only if these sites are already occupied by an electron $-\sigma$. The picture that emerges by these results is that the paramagnetic ground state in the insulating phase is characterized by a finite-range antiferromagnetic order. Due to the fact that there are empty and doubly occupied sites, the electrons have some mobility, but there are strong constrains on this mobility, such that the local antiferromagnetic order is not destroyed. This result is consistent with the fact that there is a competition between the itinerant and localizing energy terms. A study of the kinetic and potential energies as functions of U shows that for any $t\neq 0$ there is always some contribution which comes from the kinetic energy which allows the hopping among sites. Only in the limit of infinite U, the double occupancy and all transition amplitudes go to zero. In conclusion, the two-dimensional single-band Hubbard model at half filling and zero temperature has been studied by means of the composite operator method. Analytical calculations show the existence of a critical value $U_{c}$ which separates the metallic and insulating phases. As soon as U increases from zero, a depletion appears in the density of states; some weight of the central region is transferred to the lower and upper Hubbard bands. For larger values of U, DOS develops three separated structures: part of the weight remains in the center around the Fermi value and discontinuously disappears at $U=U_c$. Similar results, although based on different mechanism, have been previously obtained in Ref. \[20\] for the case of infinite-dimensional Hubbard model, in Ref. \[21\] by means of standard perturbation expansions, in Ref. \[22\] by Monte Carlo simulations. For $U>U$ a gap opens and the density of states splits into two separated structures. Our calculations show that even for $U\gg U_{c}$, where the lower and upper bands are well separated, the two contributions coming from $\xi$ and $\eta$ do not separate. The ground state in the insulating phase contains always a small fraction of empty and doubly occupied sites. This result is confirmed by the study of the matrix element $\langle c_{\sigma} (j) c^\dagger _{\sigma}(i)\rangle$, which gives the probability amplitude of hopping from the site $j$ to the site $i$. When $j$ is an odd nearest neighboring site of $i$, this quantity is not zero for $U>U_{c}$ and vanishes only for infinite U. However, when we split $c=\xi+\eta$ and analyze $\langle c_{\sigma} (j_{odd}) c^\dagger_{\sigma}(i)\rangle$ in components, we find that for $U>U_{c}$ only the matrix elements $\langle \xi_{\sigma} (j_{odd}) \eta^\dagger_{\sigma}(i)\rangle$ and $\langle \eta_{\sigma} (j_{odd}) \xi^\dagger_{\sigma}(i)\rangle$ survive. The probability amplitudes $\langle \xi_{\sigma} (j_{odd}) \xi^\dagger_{\sigma}(i)\rangle$ and $\langle \eta_{\sigma} (j_{odd}) \eta^\dagger_{\sigma}(i)\rangle$ vanish at $U=U_{c}$ and remain zero for all $U>U_{c}$. On the other hand, the matrix element $\langle c_{\sigma} (j_{even}) c^\dagger_{\sigma}(i)\rangle$ is always zero for any value of U; the two contributions $\langle \xi_{\sigma} (j_{even}) \xi^\dagger_{\sigma}(i)\rangle$ and $\langle \eta_{\sigma} (j_{even}) \eta^\dagger_{\sigma}(i)\rangle$ compensating each other. Summarizing, our calculations suggest that the ground state of the Mott insulator has the following characteristics: (1) a small fraction of sites are empty or doubly occupied; the number of these sites depend on the value of U/t and tends to zero only in the limit $U\to \infty$; (2) a local antiferromagnetic order is established; (3) the electrons keep some mobility, but this mobility must be compatible with the local AF order; (4) the matrix elements $\langle \xi_{\sigma} (j_{odd}) \xi^\dagger_{\sigma}(i)\rangle$ and $\langle \eta_{\sigma} (j_{odd}) \eta^\dagger_{\sigma}(i)\rangle$ might be considered as the quantities which control the order in the insulating phase. The author wishes to thank Doctors Adolfo Avella and Dario Villani for valuable discussions. It is gratefully acknowledged an enlightening discussion with Professor Peter Fulde, that partly motivated the writing of this article. -12pt [99]{} N.F. Mott, Proc. Roy. Soc. London, A [62]{}, 416 (1949). J. Hubbard, Proc. Roy. 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Mancini, Physics Letters A [249]{}, 231 (1998). F. Mancini and A. Avella, Condensed Matter Physics [1]{}, 11 (1998). P. Fulde, Electron Correlations in Moleculus and Solids, Springer (1995) We wish to thank the referee for this remark. A. Georges and G. Kotliar, Phys. Rev. B [45]{}, 6479 (1992); [48]{}, 7167 (1993) H. Schweitzer and G. Czycholl, Z. Physik. B [83]{}, 93 (1991) S.R. White, D.J. Scalapino, R.L. Sugar and N.E. Bickers, Phys. Rev. Lett. [63]{}, 1523 (1989) [^1]: E-mail: mancini@vaxsa.csied.unisa.it
--- abstract: 'We prove in this paper that every $p$-local compact group is approximated by transporter systems over finite $p$-groups. To do so, we use unstable Adams operations acting on a given $p$-local compact group and study the structure of resulting fixed points.' author: - 'A. González' title: 'Unstable Adams operations acting on $p$-local compact groups and fixed points' --- The theory of $p$-local compact groups was introduced by C. Broto, R. Levi and B. Oliver in [@BLO3] as the natural generalization of $p$-local finite groups to include some infinite structures, such as compact Lie groups or $p$-compact groups, in an attempt to give categorical models for a larger class of $p$-completed classifying spaces. Nevertheless, when passing from a finite setting to an infinite one, some of the techniques used in the former case are not available any more. As a result, some of the more important results in [@BLO2] were not extended to $p$-local compact groups, and, roughly speaking, that $p$-local compact groups are not yet as well understood as $p$-local finite groups. It is then the aim of this paper to shed some light on the new theory introduced in [@BLO3]. The underlying idea of this paper can be traced back to work of E. M. Friedlander and G. Mislin, [@F1], [@F2], [@FM1] and [@FM2], where the authors use unstable Adams operations (or Frobenius maps in the algebraic setting) to approximate classifying spaces of compact Lie groups by classifying spaces of finite groups. More recently, C. Broto and J. M. M[ø]{}ller studied a similar construction for connected $p$-compact groups in [@BM]. Here, by an approximation of a compact Lie group $G$ by finite groups we mean the existence of a locally finite group $G^{\circ}$, together with a mod $p$ homotopy equivalence $BG^{\circ} \to BG$ (that is, this map induces a mod $p$ homology isomorphism). Since $G^{\circ}$ is locally finite, it can be described as a colimit of finite groups, and this allows then to extend known properties of finite groups to compact Lie groups. Of course, this argument is unnecessary in the classical setting of compact Lie groups, since other techniques are at hand. As we have mentioned, the works of Friedlander and Mislin depended on Frobenius maps and their analogues in topological $K$-theory, unstable Adams operations. For $p$-local compact groups, unstable Adams operations were constructed for all $p$-local compact groups in the doctoral thesis [@Junod], although in this work we will use the more refined version of unstable Adams operations from [@JLL]. One can then study the action of unstable Adams operations on $p$-local compact groups from a categorical point of view, focusing on the definition of a $p$-local compact group as a triple ${(S, {\mathcal{F}}, {\mathcal{L}})}$, which is in fact the approach that we have adopted in this paper, and which will lead to a rather explicit description of the fixed points of a $p$-local compact group under the action of an unstable Adams operation (of high enough degree). The following result, which is the main theorem in this paper, will be restated and proved as Theorems \[transpi\] and \[hocolim1\]. \[ThA\] Let ${\mathcal{G}}= {(S, {\mathcal{F}}, {\mathcal{L}})}$ be a $p$-local compact group, and let $\Psi$ be an unstable Adams operation on ${\mathcal{G}}$. Then, $\Psi$ defines a family of finite transporter systems $\{{\mathcal{L}}_i\}_{i \in {\mathbb{N}}}$, together with faithful functors $\Theta_i: {\mathcal{L}}_i \to {\mathcal{L}}_{i+1}$ for all $i$, such that there is a mod $p$ homotopy equivalence $$B{\mathcal{G}}\simeq_p \operatorname{hocolim} \|{\mathcal{L}}_i\|.$$ Each transporter system ${\mathcal{L}}_i$ is associated to an underlying fusion system ${\mathcal{F}}_i$, which is $Ob({\mathcal{L}}_i)$-generated and $Ob({\mathcal{L}}_i)$-saturated (see definition \[hgenhsat\]). The notation comes from [@BCGLO1]. The first property (generation) means that morphisms in ${\mathcal{F}}_i$ are compositions of restrictions of morphisms among the objects in $Ob({\mathcal{L}}_i)$, and the second property (saturation) means that the objects in $Ob({\mathcal{L}}_i)$ satisfy the saturation axioms. The saturation of the fusion systems ${\mathcal{F}}_i$ remains unsolved in the general case, but we study some examples in this paper where, independently of the operation $\Psi$, the triples ${\mathcal{G}}_i$ are always (eventually) $p$-local finite groups. To simplify the statements below, we will say that $\Psi$ induces an approximation of ${\mathcal{G}}$ by $p$-local finite groups if the triples ${\mathcal{G}}_i$ in Theorem A are $p$-local finite groups. This definition will be made precise in section §3. The following results correspond to Theorems \[rank1\] and \[Un\] respectively. Let ${\mathcal{G}}$ be a $p$-local compact group of rank $1$, and let $\Psi$ be an unstable Adams operation acting on ${\mathcal{G}}$. Then, $\Psi$ induces an approximation of ${\mathcal{G}}$ by $p$-local finite groups. Let ${\mathcal{G}}$ be the $p$-local compact group induced by the compact Lie group $U(n)$, and let $\Psi$ be an unstable Adams operation acting on ${\mathcal{G}}$. Then, $\Psi$ induces an approximation of ${\mathcal{G}}$ by $p$-local finite groups. As an immediate consequence of approximations of $p$-local compact groups by $p$-local finite groups, we prove in section §3 a Stable Elements Theorem for $p$-local compact groups (whenever such an approximation is available). Stable Elements Theorem has proved to be a rather powerful tool in the study of $p$-local finite groups, and one would of course like to have a general proof in the compact case. In this sense, our conjecture is that the constructions that we introduce in this paper yield approximations by $p$-local finite groups for all $p$-local compact groups. One could also choose a different approach to the study of fixed points in this $p$-local setting. Indeed, given an unstable Adams operation $\Psi$ acting on ${\mathcal{G}}$, one could consider the homotopy fixed points of $B{\mathcal{G}}$ under the natural map induced by an unstable Adams operation acting on ${\mathcal{G}}$, namely the homotopy pull-back $$\xymatrix@C=15mm{ X \ar[r] \ar[d] & B{\mathcal{G}}\ar[d]^{\Delta} \\ B{\mathcal{G}}\ar[r]_{(id, \|\Psi\|)} & B{\mathcal{G}}\times B{\mathcal{G}},\\ }$$ and apply the topological tools provided in [@BLO2] and [@BLO4] to study the homotopy type of $X$. This point of view is in fact closer to the work of Broto and M[ø]{}ller [@BM] mentioned above, and will constitute the main subject in a sequel of this paper, where we will relate the constructions introduced in this paper and homotopy fixed points. The paper is organized as follows. The first section contains the main definitions of discrete $p$-toral groups, (saturated) fusion systems, centric linking systems, transporter systems and $p$-local compact groups. This section also contains the definition of unstable Adams operations from [@JLL]. In the second section we study the effect of a whole family of unstable Adams operations acting on a fixed $p$-local compact group. This section is to be considered as a set of tools that we use in the following section. Indeed, the third section contains the construction of the triples ${\mathcal{G}}_i$ and the proof for Theorem A above. It also contains a little discussion about approximations of $p$-local compact groups by $p$-local finite groups, where we prove a Stable Elements Theorem (in this particular situation) for $p$-local compact groups. The last section is devoted to examples (Theorems B and C above). The author would like to thank professors C. Broto, R. Levi and B. Oliver for many interesting conversations and discussions on the contents of this paper, as well as for their help and encouragement. Also many thanks to A. Libman for (at least one) thorough early review on this paper, as well as many useful hints. Finally, the author would like to thank the Universitat Aut[ò]{}noma de Barcelona and the University of Aberdeen, where most of this work has been carried on. This work summarizes part of the doctoral thesis of the author, realized under the supervision of C. Broto and R. Levi, and as such the author would like to thank both of them for their continuous help. The author was partially supported by FEDER/MEC grant BES-2005-11029. Background on $p$-local compact groups ====================================== In this section, we review the definition of a $p$-local compact group and state some results that we will use later on. Mostly, the contents in this section come from [@BLO3]. When this is the case, we will provide a reference where the reader can find a proof, in order to simplify the exposition of this paper. Discrete $p$-toral groups and fusion systems -------------------------------------------- \[defidiscreteptoral and fusion systems\] A discrete $p$-torus is a group $T$ isomorphic to a finite direct product of copies of ${{\mathbb{Z}}/p^{\infty}}$. A discrete $p$-toral group $P$ is an extension of a finite $p$-group $\pi$ by a discrete $p$-torus $T$. For such a group, we call $T \cong ({{\mathbb{Z}}/p^{\infty}})^r$ the maximal torus of $P$, and define the rank of $P$ as $r$. Discrete $p$-toral groups were characterized in [@BLO3] (Proposition 1.2) as those groups satisfying the descending chain condition and such that every finitely generated subgroup is a finite $p$-group. In this paper we will deal with some infinite groups. For an infinite group $G$, we say that $G$ has Sylow $p$-subgroups if $G$ contains a discrete $p$-toral group $S$ such that any finite $p$-subgroup of $G$ is $G$-conjugate to a subgroup of $S$. For a group $G$ and subgroups $P,P' \leq G$, define $N_G(P,P') = \{g \in G \mbox{ } \| \mbox{ } g \cdot P \cdot g^{-1} \leq P'\}$ and $Hom_G(P,P') = N_G(P,P')/C_G(P)$. Fusion systems over discrete $p$-toral groups are defined just as they were defined in the finite case. \[defifusion\] A fusion system ${\mathcal{F}}$ over a discrete $p$-toral group $S$ is a category whose objects are the subgroups of $S$ and whose morphism sets $Hom_{{\mathcal{F}}}(P,P')$ satisfy the following conditions: (i) $Hom_S(P,P') \subseteq Hom_{{\mathcal{F}}}(P,P') \subseteq Inj(P,P')$ for all $P,P' \leq S$. (ii) Every morphism in ${\mathcal{F}}$ factors as an isomorphism in ${\mathcal{F}}$ followed by an inclusion. Given a fusion system ${\mathcal{F}}$ over a discrete $p$-toral group $S$, we will often refer to $T$ also as the maximal torus of ${\mathcal{F}}$, and the rank of ${\mathcal{F}}$ will then be the rank of the discrete $p$-toral group $S$. Two subgroups $P,P'$ are called ${\mathcal{F}}$-conjugate if $Iso_{{\mathcal{F}}}(P,P') \neq \emptyset$. For a subgroup $P \leq S$, we denote $${{P}^{{\mathcal{F}}}} = \{P' \leq S \mbox{ } \| \mbox{ } P' \mbox{ is } {\mathcal{F}}\mbox{-conjugate to } P\}.$$ For a discrete $p$-toral group $P$, the order of $P$ was defined in [@BLO3] as $\|P\| = (rk(P), \|P/T_P\|)$, where $T_P$ is the maximal torus of $P$. Thus, given two discrete $p$-toral groups $P$ and $Q$ we say that $\|P\| \leq \|Q\|$ if either $rk(P) < rk(Q)$, or $rk(P) = rk(Q)$ and $\|P/T_P\| \leq \|Q/T_Q\|$. \[defisaturation\] Let ${\mathcal{F}}$ be a fusion system over a discrete $p$-toral group $S$. A subgroup $P \leq S$ is called fully ${\mathcal{F}}$-normalized, resp. fully ${\mathcal{F}}$-centralized, if $\|N_S(P')\| \leq \|N_S(P)\|$, resp. $\|C_S(P')\| \leq \|C_S(P)\|$, for all $P' \leq S$ which is ${\mathcal{F}}$-conjugate to $P$. The fusion system ${\mathcal{F}}$ is called saturated if the following three conditions hold: (I) For each $P \leq S$ which is fully ${\mathcal{F}}$-normalized, $P$ is fully ${\mathcal{F}}$-centralized, $Out_{{\mathcal{F}}}(P)$ is finite and $Out_S(P) \in Syl_p(Out_{{\mathcal{F}}}(P))$. (II) If $P \leq S$ and $f \in Hom_{{\mathcal{F}}}(P,S)$ is such that $P' = f(P)$ is fully ${\mathcal{F}}$-centralized, then there exists $\widetilde{f} \in Hom_{{\mathcal{F}}}(N_f,S)$ such that $f = \widetilde{f}_{\|P}$, where $$N_f = \{g \in N_S(P) \| f \circ c_g \circ f^{-1} \in Aut_S(P')\}.$$ (III) If $P_1 \leq P_2 \leq P_3 \leq \ldots$ is an increasing sequence of subgroups of $S$, with $P = \cup_{n=1}^{\infty} P_n$, and if $f \in Hom(P,S)$ is any homomorphism such that $f_{\|P_n} \in Hom_{{\mathcal{F}}}(P_n,S)$ for all $n$, then $f \in Hom_{{\mathcal{F}}}(P,S)$. Let $(S, {\mathcal{F}})$ be a saturated fusion system over a discrete $p$-toral group. Note that, by definition, all the automorphism groups in a saturated fusion system are artinian and locally finite. The condition in axiom (I) of $Out_{{\mathcal{F}}}(P)$ being finite is in fact redundant, as was pointed out in Lemmas 2.3 and 2.5 [@BLO3], where the authors show that the set $Rep_{{\mathcal{F}}}(P,Q) = Inn(Q) \setminus Hom_{{\mathcal{F}}}(P,Q)$ is finite for all $P, Q \in Ob({\mathcal{F}})$. Given a discrete $p$-toral group $S$ and a subgroup $P \leq S$, we say that $P$ is centric in $S$, or $S$-centric, if $C_S(P) = Z(P)$. We next define ${\mathcal{F}}$-centric and ${\mathcal{F}}$-radical subgroups. \[deficentricrad\] Let ${\mathcal{F}}$ be a saturated fusion system over a discrete $p$-toral group. A subgroup $P \leq S$ is called ${\mathcal{F}}$-centric if all the elements of ${{P}^{{\mathcal{F}}}}$ are centric in $S$: $$C_S(P') = Z(P') \mbox{ for all } P' \in {{P}^{{\mathcal{F}}}}.$$ A subgroup $P \leq S$ is called ${\mathcal{F}}$-radical if $Out_{{\mathcal{F}}}(P)$ contains no nontrivial normal $p$-subgroup: $$O_p(Out_{{\mathcal{F}}}(P)) = \{1\}.$$ Clearly, ${\mathcal{F}}$-centric subgroups are fully ${\mathcal{F}}$-centralized, and conversely, if $P$ is fully ${\mathcal{F}}$-centralized and centric in $S$, then it is ${\mathcal{F}}$-centric. There is, of course, a big difference between working with finite $p$-groups and with discrete $p$-toral groups: the number of conjugacy classes of subgroups. Fortunately, in [@BLO3] the authors came out with a way of getting rid of infinitely many conjugacy classes while keeping the structure of a given fusion system. This construction will be rather important in this paper, and we reproduce it here for the sake of a better reading. Let then $(S, {\mathcal{F}})$ be a saturated fusion system over a discrete $p$-toral group, and let $T$ be the maximal torus of ${\mathcal{F}}$ and $W = Aut_{{\mathcal{F}}}(T)$. \[defibullet\] Let ${\mathcal{F}}$ be a saturated fusion system over a discrete $p$-toral group $S$, and let $e$ denote the exponent of $S/T$, $$e = exp(S/T) = min\{p^k \mbox{ } \| \mbox{ } x^{p^k} \in T \mbox{ for all } x \in S\}.$$ (i) For each $P \leq T$, let $$I(P) = \{t \in T \mbox{ } \| \mbox{ } \omega(t) = t \mbox{ for all } \omega \in W \mbox{ such that } \omega_{\|P} = id_P\},$$ and let $I(P)_0$ denote its maximal torus. (ii) For each $P \leq S$, set $P^{[e]} = \{x^{p^e} \mbox{ } \| \mbox{ } x \in P\} \leq T$, and let $$P^{\bullet} = P \cdot I(P^{[e]})_0 = \{xt \| x \in P, t \in I(P^{[e]})_0\}.$$ (iii) Set ${\mathcal{H}}^{\bullet} = \{P^{\bullet} \| P \in {\mathcal{F}}\}$ and let ${\mathcal{F}}^{\bullet}$ be the full subcategory of ${\mathcal{F}}$ with object set $Ob({\mathcal{F}}^{\bullet}) = {\mathcal{H}}^{\bullet}$. The following is a summary of section §3 in [@BLO3]. \[bulletprop\] Let ${\mathcal{F}}$ be a saturated fusion system over a discrete $p$-toral group $S$. Then, (i) the set ${\mathcal{H}}^{\bullet}$ contains finitely many $S$-conjugacy classes of subgroups of $S$; and (ii) every morphism $(f:P \to Q) \in Mor({\mathcal{F}})$ extends uniquely to a morphism $f^{\bullet}: P^{\bullet} \to Q^{\bullet}$. This makes ${(\underline{\phantom{B}})^{\bullet}}: {\mathcal{F}}\to {\mathcal{F}}^{\bullet}$ into a functor. This functor is an idempotent functor ($(P^{\bullet})^{\bullet} = P^{\bullet}$), carries inclusions to inclusions ($P^{\bullet} \leq Q^{\bullet}$ whenever $P \leq Q$), and is left adjoint to the inclusion ${\mathcal{F}}^{\bullet} \subseteq {\mathcal{F}}$. Finally, we state Alperin’s fusion theorem for saturated fusion systems over discrete $p$-toral groups. \[Alperin\] (3.6 [@BLO3]). Let ${\mathcal{F}}$ be a saturated fusion system over a discrete $p$-toral group $S$. Then, for each $f \in Iso_{{\mathcal{F}}}(P,P')$ there exist sequences of subgroups of $S$ $$\begin{array}{ccc} P = P_0, P_1, \ldots, P_k = P' & \mbox{and} & Q_1, \ldots, Q_k,\\ \end{array}$$ and elements $f_j \in Aut_{{\mathcal{F}}}(Q_j)$ such that (i) for each $j$, $Q_j$ is fully normalized in ${\mathcal{F}}$, ${\mathcal{F}}$-centric and ${\mathcal{F}}$-radical; (ii) also for each $j$, $P_{j-1}, P_j \leq Q_j$ and $f_j(P_{j-1}) = P_j$; and (iii) $f = f_k \circ f_{k-1} \circ \ldots \circ f_1$. It is also worth mentioning the alternative set of saturation axioms provided in [@KS], since it will be useful in later sections. Let ${\mathcal{F}}$ be a fusion system over a finite $p$-group $S$, and consider the following conditions: 1. $Out_S(S) \in Syl_p(Out_{{\mathcal{F}}}(S))$. 2. Let $f:P \to S$ be a morphism in ${\mathcal{F}}$ such that $P' = f(P)$ is fully ${\mathcal{F}}$-normalized. Then, $f$ extends to a morphism $\widetilde{f}: N_f \to S$ in ${\mathcal{F}}$, where $$N_f = \{g \in N_S(P)\|f \circ c_g \circ f^{-1} \in Aut_S(P')\}.$$ The following result is a compendium of appendix A in [@KS]. \[Stancuaxioms\] Let ${\mathcal{F}}$ be a fusion system over a finite $p$-group $S$. Then, ${\mathcal{F}}$ is saturated (in the sense of definition \[defisaturation\]) if and only if it satisfies axioms (I’) and (II’) above. A more general version of this result for fusion systems over discrete $p$-toral groups was proved in [@Gonza], but is of no use in the paper. Linking systems and transporter systems --------------------------------------- Linking systems are the third and last ingredient needed to form a $p$-local compact group. \[defilinking\] Let ${\mathcal{F}}$ be a saturated fusion system over a discrete $p$-toral group $S$. A centric linking system associated to ${\mathcal{F}}$ is a category ${\mathcal{L}}$ whose objects are the ${\mathcal{F}}$-centric subgroups of $S$, together with a functor $$\rho: {\mathcal{L}}\longrightarrow {\mathcal{F}}^c$$ and “distinguished” monomorphisms $\delta_P: P \to Aut_{{\mathcal{L}}}(P)$ for each ${\mathcal{F}}$-centric subgroup $P \leq S$, which satisfy the following conditions. (A) $\rho$ is the identity on objects and surjective on morphisms. More precisely, for each pair of objects $P, P' \in {\mathcal{L}}$, $Z(P)$ acts freely on $Mor_{{\mathcal{L}}}(P,P')$ by composition (upon identifying $Z(P)$ with $\delta_P(Z(P)) \leq Aut_{{\mathcal{L}}}(P)$), and $\rho$ induces a bijection $$Mor_{{\mathcal{L}}}(P,P')/Z(P) \stackrel{\cong} \longrightarrow Hom_{{\mathcal{F}}}(P,P').$$ (B) For each ${\mathcal{F}}$-centric subgroup $P \leq S$ and each $g \in P$, $\rho$ sends $\delta_P(g) \in Aut_{{\mathcal{L}}}(P)$ to $c_g \in Aut_{{\mathcal{F}}}(P)$. (C) For each $\varphi \in Mor_{{\mathcal{L}}}(P,P')$ and each $g \in P$, the following square commutes in ${\mathcal{L}}$: $$\xymatrix{ P \ar[r]^{\varphi} \ar[d]_{\delta_P(g)} & P' \ar[d]^{\delta_{P'}(h)} \\ P \ar[r]_{\varphi} & P', }$$ where $h = \rho(\varphi)(g)$. A $p$-local compact group is a triple ${\mathcal{G}}= (S, {\mathcal{F}}, {\mathcal{L}})$, where $S$ is a discrete $p$-toral group, ${\mathcal{F}}$ is a saturated fusion system over $S$, and ${\mathcal{L}}$ is a centric linking system associated to ${\mathcal{F}}$. The classifying space of ${\mathcal{G}}$ is the $p$-completed nerve $$B{\mathcal{G}}\stackrel{def} = \|{\mathcal{L}}\|^{\wedge}_p.$$ Given a $p$-local compact group ${\mathcal{G}}$, the subgroup $T \leq S$ will be called the maximal torus of ${\mathcal{G}}$, and the rank of ${\mathcal{G}}$ will then be the rank of the discrete $p$-toral group $S$. We will in general denote a $p$-local compact group just by ${\mathcal{G}}$, assuming that $S$ is its Sylow $p$-subgroup, ${\mathcal{F}}$ is the corresponding fusion system, and ${\mathcal{L}}$ is the corresponding linking system. As expected, the classifying space of a $p$-local compact group behaves “nicely”, meaning that $B{\mathcal{G}}= \|{\mathcal{L}}\|^{\wedge}_p$ is a $p$-complete space (in the sense of [@BK]) whose fundamental group is a finite $p$-group, as proved in Proposition 4.4 [@BLO3]. Next we state some properties of linking systems. We start with an extended version of Lemma 4.3 [@BLO3]. \[3.2OV\] Let ${\mathcal{G}}$ be a $p$-local compact group. Then, the following holds. (i) Fix morphisms $f \in Hom_{{\mathcal{F}}}(P,Q)$ and $f' \in Hom_{{\mathcal{F}}}(Q,R)$, where $P,Q,R \in {\mathcal{L}}$. Then, for any pair of liftings $\varphi '\in \rho^{-1}_{Q,R}(f')$ and $\omega \in \rho^{-1}_{P,R}(f' \circ f)$, there is a unique lifting $\varphi \in \rho^{-1}_{P,Q}(f)$ such that $\varphi ' \circ \varphi = \omega$. (ii) All morphisms in ${\mathcal{L}}$ are monomorphisms in the categorical sense. That is, for all $P,Q,R \in {\mathcal{L}}$ and all $\varphi_1, \varphi_2 \in Mor_{{\mathcal{L}}}(P,Q)$, $\psi \in Mor_{{\mathcal{L}}}(Q,R)$, if $\psi \circ \varphi_1 = \psi \circ \varphi_2$ then $\varphi_1 = \varphi_2$. (iii) For every morphism $\varphi \in Mor_{{\mathcal{L}}}(P,Q)$ and every $P_0, Q_0 \in {\mathcal{L}}$ such that $P_0 \leq P$, $Q_0 \leq Q$ and $\rho(\varphi)(P_0) \leq Q_0$, there is a unique morphism $\varphi_0 \in Mor_{{\mathcal{L}}}(P_0, Q_0)$ such that $\varphi \circ \iota_{P_0,P} = \iota_{Q_0,Q} \circ \varphi_0$. In particular, every morphism in ${\mathcal{L}}$ is a composite of an isomorphism followed by an inclusion. (iv) All morphisms in ${\mathcal{L}}$ are epimorphisms in the categorical sense. In other words, for all $P,Q,R \in {\mathcal{L}}$ and all $\varphi \in Mor_{{\mathcal{L}}}(P,Q)$ and $\psi_1, \psi_2 \in Mor_{{\mathcal{L}}}(Q,R)$, if $\psi_1 \circ \varphi = \psi_2 \circ \varphi$ then $\psi_1 = \psi_2$. Since the functor $\rho: {\mathcal{L}}\to {\mathcal{F}}^c$ is both source and target regular (in the sense of definition A.5 [@OV]) by axiom (A) of linking systems, the proof for Lemma 3.2 [@OV] applies in this case as well. Let ${\mathcal{G}}$ be a $p$-local compact group, and, for each $P \in {\mathcal{L}}$ fix a lifting of $incl^S_P:P \to S$ in ${\mathcal{L}}$, $\iota_{P,S} \in Mor_{{\mathcal{L}}}(P,S)$. Then, by the above Lemma, we may complete this to a family of inclusions $\{\iota_{P,P'}\}$ in a unique way and such that $\iota_{P,S} =\iota_{P',S} \circ \iota_{P,P'}$ whenever it makes sense. \[1.11BLO2\] Fix such a family of inclusions $\{\iota_{P,P'}\}$ in ${\mathcal{L}}$. Then, for each $P,P' \in {\mathcal{L}}$, there are unique injections $$\delta_{P,P'}:N_S(P,P') \longrightarrow Mor_{{\mathcal{L}}}(P,P')$$ such that (i) $\iota_{P',S} \circ \delta_{P,P'}(g) = \delta_S(g) \circ \iota_{P,S}$, for all $g \in N_S(P,P')$, and (ii) $\delta_P$ is the restriction to $P$ of $\delta_{P,P}$. The proof for Proposition 1.11 [@BLO2] applies here as well, using Lemma \[3.2OV\] (i) above instead of Proposition 1.10 (a) [@BLO2]. We now introduce transporter systems. The notion that we present here was first used in [@OV] as a tool to study certain extensions of $p$-local finite groups. In this sense, most of the results in [@OV] can be extended to the compact case without restriction, as proved in [@Gonza], but we will not make use of such results in this paper. More details can be found at [@Gonza]. Let $G$ be an artinian locally finite group with Sylow $p$-subgroups, and fix $S \in Syl_p(G)$. We define ${\mathcal{T}}_S(G)$ as the category whose object set is $Ob({\mathcal{T}}_S(G)) = \{P \leq S\}$, and such that $$Mor_{{\mathcal{T}}_S(G)}(P,P') = N_G(P,P') = \{g \in G \mbox{ } \| \mbox{ } gPg^{-1} \leq P'\}.$$ For a subset ${\mathcal{H}}\subseteq Ob({\mathcal{T}}_S(G))$, ${\mathcal{T}}_{{\mathcal{H}}}(G)$ denotes the full subcategory of ${\mathcal{T}}_S(G)$ with object set ${\mathcal{H}}$. \[defitransporter\] Let ${\mathcal{F}}$ be a fusion system over a discrete $p$-toral group $S$. A transporter system associated to ${\mathcal{F}}$ is a category ${\mathcal{T}}$ such that (i) $Ob({\mathcal{T}}) \subseteq Ob({\mathcal{F}})$; (ii) for all $P \in Ob({\mathcal{T}})$, $Aut_{{\mathcal{T}}}(P)$ is an artinian locally finite group; together with a couple of functors $${\mathcal{T}}_{Ob({\mathcal{T}})}(S) \stackrel{\varepsilon} \longrightarrow {\mathcal{T}}\stackrel{\rho} \longrightarrow {\mathcal{F}},$$ satisfying the following axioms: - $Ob({\mathcal{T}})$ is closed under ${\mathcal{F}}$-conjugacy and overgroups. Also, $\varepsilon$ is the identity on objects and $\rho$ is inclusion on objects. - For each $P \in Ob({\mathcal{T}})$, let $$E(P) = Ker(Aut_{{\mathcal{T}}}(P) \to Aut_{{\mathcal{F}}}(P)).$$ Then, for each $P, P' \in Ob({\mathcal{T}})$, $E(P)$ acts freely on $Mor_{{\mathcal{T}}}(P, P')$ by right composition, and $\rho_{P, P'}$ is the orbit map for this action. Also, $E(P')$ acts freely on $Mor_{{\mathcal{T}}}(P, P')$ by left composition. - For each $P, P' \in Ob({\mathcal{T}})$, $\varepsilon_{P,P'}: N_S(P,P') \to Mor_{{\mathcal{T}}}(P, P')$ is injective, and the composite $\rho_{P,P'} \circ \varepsilon_{P,P'}$ sends $g \in N_S(P, P')$ to $c_g \in Hom_{{\mathcal{F}}}(P,P')$. - For all $\varphi \in Mor_{{\mathcal{T}}}(P,P')$ and all $g \in P$, the following diagram commutes in ${\mathcal{T}}$: $$\xymatrix{ P \ar[r]^{\varphi} \ar[d]_{\varepsilon_{P,P}(g)} & P' \ar[d]^{\varepsilon_{P',P'}(\rho(\varphi)(g))} \\ P \ar[r]_{\varphi} & P'. }$$ - $Aut_{{\mathcal{T}}}(S)$ has Sylow $p$-subgroups, and $\varepsilon_{S,S}(S) \in Syl_p(Aut_{{\mathcal{T}}}(S))$. - Let $\varphi \in Iso_{{\mathcal{T}}}(P,P')$, and $P \lhd R \leq S$, $P' \lhd R' \leq S$ such that $$\varphi \circ \varepsilon_{P,P}(R) \circ \varphi^{-1} \leq \varepsilon_{P',P'}(R').$$ Then, there is some $\widetilde{\varphi} \in Mor_{{\mathcal{T}}}(R, R')$ such that $\widetilde{\varphi} \circ \varepsilon_{P,R}(1) = \varepsilon_{P', R'}(1) \circ \varphi$, that is, the following diagram is commutative in ${\mathcal{T}}$: $$\xymatrix{ P \ar[r]^{\varphi} \ar[d]_{\varepsilon_{P,R}(1)} & P' \ar[d]^{\varepsilon_{P',R'}(1)} \\ R \ar[r]_{\widetilde{\varepsilon}} & R'. }$$ - Let $P_1 \leq P_2 \leq \ldots$ be an increasing sequence of subgroups in $Ob({\mathcal{T}})$, and $P = \bigcup_{n=1}^{\infty} P_n$. Suppose in addition that there exists $\psi_n \in Mor_{{\mathcal{T}}}(P_n, S)$ such that $$\psi_n = \psi_{n+1} \circ \varepsilon_{P_n, P_{n+1}}(1)$$ for all $n$. Then, there exists $\psi \in Mor_{{\mathcal{T}}}(P, S)$ such that $\psi_n = \psi \circ \varepsilon_{P_n, P}(1)$ for all $n$. Given a transporter system ${\mathcal{T}}$, the classifying space of ${\mathcal{T}}$ is the space $B{\mathcal{T}}\stackrel{def} = \|{\mathcal{T}}\|^{\wedge}_p$. Note that, in axiom (III), $P$ is an object in $Ob({\mathcal{T}})$, since $Ob({\mathcal{T}})$ is closed under ${\mathcal{F}}$-conjugacy and overgroups. As in [@OV], the axioms are labelled to show their relation with the axioms for linking and fusion systems respectively. Note also that, whenever $S$ is a finite $p$-group, the above definition agrees with that in [@OV]. \[3.5OV\] Let ${\mathcal{G}}= (S,{\mathcal{F}}, {\mathcal{L}})$ be a $p$-local compact group. Then, ${\mathcal{L}}$ is a transporter system associated to ${\mathcal{F}}$. The usual projection functor $\rho: {\mathcal{L}}\to {\mathcal{F}}$ in the definition of a linking system plays also the role of the projection functor in the definition of transporter system. Also, in Lemma \[1.11BLO2\] we have defined a functor $\varepsilon: {\mathcal{T}}_{Ob({\mathcal{L}})}(S) \to {\mathcal{L}}$. It remains to check that ${\mathcal{L}}$ satisfies the axioms in definition \[defitransporter\]. (A1) This follows from axiom (A) on ${\mathcal{L}}$. (A2) By axiom (A) on ${\mathcal{L}}$, we know that, for all $P, P' \in {\mathcal{L}}$, $E(P) = Z(P)$ acts freely on $Mor_{{\mathcal{L}}}(P,P')$ and that $\rho_{P,P'}$ is the orbit map of this action. Thus, we have to check that $E(P') = Z(P')$ acts freely on $Mor_{{\mathcal{L}}}(P,P')$. Suppose $\varphi \in Mor_{{\mathcal{L}}}(P,P')$ and $x \in E(P')$ are such that $\varepsilon_{P'}(x) \circ \varphi = \varphi$. Then, $x$ centralizes $\rho(\varphi)(P)$, so $x = \rho(\varphi)(y)$ for some $y \in Z(P)$, since $P$ is ${\mathcal{F}}$-centric. Hence, $\varphi = \delta_{P'}(x) \circ \varphi = \varphi \circ \delta_P(y)$ by axiom (C) for linking systems, and thus by axiom (A) we deduce that $y=1$, $x=1$ and the action is free. \(B) By construction of the functor $\varepsilon$, we know that $\varepsilon_{P,P'}:N_S(P,P') \longrightarrow Mor_{{\mathcal{L}}}(P,P')$ is injective for all $P,P' \in {\mathcal{L}}$. Thus, we have to check that the composite $\rho_{P,P'} \circ \varepsilon_{P,P'}$ sends $g \in N_S(P,P')$ to $c_g \in Hom_S(P,P')$. Note that the following holds for any $P,P' \in {\mathcal{L}}$ and any $x \in N_S(P,P')$: $$\iota_{P'} \circ \varepsilon_{P,P'}(x) = \varepsilon_{P',S}(1) \circ \varepsilon_{P,P'}(x) = \varepsilon_{S}(x) \circ \varepsilon_{P,S}(1) = \delta_S(x) \circ \iota_P$$ and hence so does the following on ${\mathcal{F}}$: $$incl^S_{P'} \circ \rho_{P,P'}(\varepsilon_{P,P'}(x)) = \rho_{P,S}(\iota_{P'}\circ \varepsilon_{P,P'}(x)) = \rho_{P,S}(\delta_S(x) \circ \iota_P) = c_x.$$ \(C) This follows from axiom (C) for linking systems. \(I) The group $Aut_{{\mathcal{L}}}(S)$ has Sylow subgroups by Lemma 8.1 [@BLO3], since $\delta(S)$ is normal in it and has finite index prime to $p$. This also proves that $\delta(S)$ is a Sylow $p$-subgroup. \(II) Let $\varphi \in Iso_{{\mathcal{L}}}(P,P')$, $P \lhd R$, $P' \lhd R'$ be such that $\varphi \circ \varepsilon_{P,P}(R) \circ \varphi^{-1} \leq \varepsilon_{P',P'}(R')$. We want to see that there exists $\widetilde{\varphi} \in Mor_{{\mathcal{L}}}(R,R')$ such that $\widetilde{\varphi} \circ \varepsilon_{P,R}(1) = \varepsilon_{P',R'}(1) \circ \varphi$. Since $P'$ is ${\mathcal{F}}$-centric, it is fully ${\mathcal{F}}$-centralized. Then, we may apply axiom (II) for fusion systems to the morphism $f = \rho(\varphi)$, that is, $f$ extends to some $\widetilde{f} \in Hom_{{\mathcal{F}}}(N_f,S)$, where $$N_f = \{ g \in N_S(P) \| f c_g f^{-1} \in Aut_S(P')\},$$ and clearly $R \leq N_f$. Hence, $\widetilde{f}$ restricts to a morphism in $Hom_{{\mathcal{F}}}(R,S)$. Furthermore, $\widetilde{f}(R) \leq R'$ since $f$ conjugates $Aut_R(P)$ into $Aut_{R'}(P')$. Now, $(\iota_{P',R'} \circ \varphi) \in Mor_{{\mathcal{L}}}(P,R')$ is a lifting in ${\mathcal{L}}$ for $incl^{R'}_{P'} \circ f \in Hom_{{\mathcal{F}}}(P,R')$, and we can fix a lifting $\psi \in Mor_{{\mathcal{L}}}(R,R')$ for $\widetilde{f}$. Thus, by Lemma \[3.2OV\] (i) there exists a unique $\widetilde{\iota} \in Mor_{{\mathcal{L}}}(P,R)$, a lifting of $incl^R_P$, such that $\iota_{P',R'} \circ \varphi = \psi \circ \widetilde{\iota}$. Since $\rho(\widetilde{\iota}) = incl^R_P = \rho(\iota_{P,R})$, by axiom (A) it follows that there exists some $z \in Z(P)$ such that $\widetilde{\iota} = \iota_{P,R} \circ \delta_P(z) = \delta_R(\rho(\iota_{P,R})(z))\circ \iota_{P,R}$, where the second equality holds by axiom (C). Hence $\iota_{P',R'} \circ \varphi = (\psi \circ \delta_R(\rho(\iota_{P,R})(z))) \circ \iota_{P,R}$. \(III) Let $P_1 \leq P_2 \leq \ldots$ be an increasing sequence of objects in ${\mathcal{L}}$, $P = \cup P_n$, and $\varphi_n \in Mor_{{\mathcal{L}}}(P_n,S)$ satisfying $\varphi_n = \varphi_{n+1} \circ \iota_{P_n,P_{n+1}}$ for all $n$. We want to see that there exists some $\varphi \in Mor_{{\mathcal{L}}}(P,S)$ such that $\varphi_n = \varphi \circ \iota_{P_n,P}$ for all $n$. Set $f_n = \rho(\varphi_n)$ for all $n$. Then, by hypothesis, $f_n = f_{n+1} \circ incl^{P_{n+1}}_{P_n}$ for all $n$. Now, it is clear that $\{f_n\}$ forms a nonempty inverse system, and there exists $f \in Hom_{{\mathcal{F}}}(P,S)$ such that $f_n = f_{\|P_n}$ for all $n$ (the existence follows from Proposition 1.1.4 in [@RZ], and the fact that $f$ is a morphism in ${\mathcal{F}}$ follows from axiom (III) for fusion systems). Consider now the following commutative diagram (in ${\mathcal{F}}$): $$\xymatrix{ & P \ar[d]^{f} \\ P_1 \ar[ru]^{incl} \ar[r]_{f_1} & S\\ }$$ The same arguments used to prove that axiom (II) for transporter systems holds on ${\mathcal{L}}$ above apply now to show that there exists a unique $\varphi \in Mor_{{\mathcal{L}}}(P,S)$ such that $\varphi_1 = \varphi \circ \iota_{P_1,P}$. Combining this equality with $\varphi_1 = \varphi_2 \circ \iota_{P_1,P_2}$ and Lemma \[3.2OV\] (iv) (morphisms in ${\mathcal{L}}$ are epimorphisms in the categorical sense), it follows that $\varphi_2 = \varphi \circ \iota_{P_2,P}$. Proceeding by induction it now follows that $\varphi$ satisfies the desired condition. Finally we state Proposition 3.6 from [@OV], deeply related to Theorem A in [@BCGLO1]. These two results are only suspected to hold in the compact case, but yet no proof has been published. Before stating the result, we introduce some notation. For a finite group $G$, the subgroup $O_p(G) \leq G$ is the maximal normal $p$-subgroup of $G$. \[hgenhsat\] Let ${\mathcal{F}}$ be a fusion system over a finite $p$-group $S$, and let ${\mathcal{H}}\subseteq Ob({\mathcal{F}})$ be a subset of objects. Then, we say that ${\mathcal{F}}$ is ${\mathcal{H}}$-generated if every morphism in ${\mathcal{F}}$ is a composite of restrictions of morphisms in ${\mathcal{F}}$ between subgroups in ${\mathcal{H}}$, and we say that ${\mathcal{F}}$ is ${\mathcal{H}}$-saturated if the saturation axioms hold for all subgroups in the set ${\mathcal{H}}$. (3.6 [@OV]). Let ${\mathcal{F}}$ be a fusion system over a finite $p$-group $S$ (not necessarily saturated), and let ${\mathcal{T}}$ be a transporter system associated to ${\mathcal{F}}$. Then, ${\mathcal{F}}$ is $Ob({\mathcal{T}})$-saturated. If ${\mathcal{F}}$ is also $Ob({\mathcal{T}})$-generated, and if $Ob({\mathcal{T}}) \supseteq Ob({\mathcal{F}}^c)$, then ${\mathcal{F}}$ is saturated. More generally, ${\mathcal{F}}$ is saturated if it is $Ob({\mathcal{T}})$-generated, and every ${\mathcal{F}}$-centric subgroup $P \leq S$ not in $Ob({\mathcal{T}})$ is ${\mathcal{F}}$-conjugate to some $P'$ such that $$Out_S(P') \cap O_p(Out_{{\mathcal{F}}}(P')) \neq \{1\}.$$ Unstable Adams operations on $p$-local compact groups ----------------------------------------------------- To conclude this section, we introduce unstable Adams operations for $p$-local compact groups and their main properties. Basically, we summarize the work from [@JLL] in order to give the proper definition of such operations and the main properties that we will use in later sections. Let $(S, {\mathcal{F}})$ be a saturated fusion system over a discrete $p$-toral group, and let $\theta: S \to S$ be a fusion preserving automorphism (that is, for each $f \in Mor({\mathcal{F}})$, the composition $\theta \circ f \circ \theta^{-1} \in Mor({\mathcal{F}})$). The automorphism $\theta$ naturally induces a functor on ${\mathcal{F}}$, which we denote by $\theta_{\ast}$, by setting $\theta_{\ast}(P) = \theta(P)$ on objects and $\theta_{\ast}(f) = \theta \circ f \circ \theta^{-1}$ on morphisms. \[defiuao\] Let ${\mathcal{G}}= {(S, {\mathcal{F}}, {\mathcal{L}})}$ be a $p$-local compact group and let $\zeta$ be a $p$-adic unit. An unstable Adams operation on ${\mathcal{G}}$ of degree $\zeta$ is a pair $(\psi, \Psi)$, where $\psi$ is a fusion preserving automorphism of $S$, $\Psi$ is an automorphism of ${\mathcal{L}}$, and such the following is satisfied: (i) $\psi$ restricts to the $\zeta$ power map on $T$ and induces the identity on $S/T$; (ii) for any $P \in Ob({\mathcal{L}})$, $\Psi(P) = \psi(P)$; (iii) $\rho \circ \Psi = \psi_{\ast} \circ \rho$, where $\rho: {\mathcal{L}}\to {\mathcal{F}}$ is the projection functor; and (iv) for each $P, Q \in Ob({\mathcal{L}})$ and all $g \in N_S(P,Q)$, $\Psi(\delta_{P,Q}(g)) = \delta_{\psi(P), \psi(Q)}(\psi(g))$. In particular, $\Psi$ is an isotypical automorphism of ${\mathcal{L}}$ in the sense of [@BLO3]. For a $p$-local compact group ${\mathcal{G}}$, let $\operatorname{Ad}({\mathcal{G}})$ be the group of unstable Adams operations on ${\mathcal{G}}$, with group operation the composition and the indentity functor as its unit. Also, for a positive integer $m$, let $\Gamma_m(p) \leq ({{\mathbb{Z}}^\wedge_p})^{\times}$ denote the subgroup of all $p$-adic units $\zeta$ of the form $1 + p^m {{\mathbb{Z}}^\wedge_p}$. Next, we state the existence of unstable Adams operations for all $p$-local compact groups. The following result corresponds to the second part of Theorem 4.1 [@JLL]. \[existencepsi\] Let ${\mathcal{G}}$ be a $p$-local compact group. Then, for any sufficiently large positive integer $m$ there exists a group homomorphism $$\label{alpham} \alpha: \Gamma_m(p) \longrightarrow \operatorname{Ad}({\mathcal{G}})$$ such that, for each $\zeta \in \Gamma_m(p)$, $\alpha(\zeta) = (\psi, \Psi)$ has degree $\zeta$. There is an important property of unstable Adams operations which we will use repeatedly in the forthcoming sections. This was stated as Corollary 4.2 in [@JLL]. \[finitesetinv\] Let ${\mathcal{G}}$ be a $p$-local compact group, and let ${\mathcal{P}}\subseteq Ob({\mathcal{L}})$ and ${\mathcal{M}}\subseteq Mor({\mathcal{L}})$ be finite subsets. Then, for any sufficiently large positive integer $m$, and for each $\zeta \in \Gamma_m(p)$, the group homomorphism $\alpha$ from (\[alpham\]) satisfies $\alpha(\zeta)(P) = P$ and $\alpha(\zeta)(\varphi) = \varphi$ for all $P \in {\mathcal{P}}$ and all $\varphi \in {\mathcal{M}}$. Let $(\psi, \Psi)$ be an unstable Adams operation on a $p$-local compact group ${\mathcal{G}}$. By point (iv) in \[defiuao\], $\Psi \circ \delta_S = \delta \circ \psi: S \to Aut_{{\mathcal{L}}}(S)$, and hence the automorphism $\psi$ is completely determined by $\Psi$. Thus, for the rest of this paper we will make no mention of $\psi$ (unless necessary) and refer to the unstable Adams operation $(\psi, \Psi)$ just by $\Psi$. Families of operations and invariance ===================================== Let ${\mathcal{G}}$ be a $p$-local compact group, and let $\Psi$ be an unstable Adams operation on ${\mathcal{G}}$. The degree of $\Psi$ will not be relevant in any of the constructions introduced in this section, and thus we will make no reference to it. Let $S^{\Psi} \leq S$ be the subgroup of fixed elements of $S$ under the fusion preserving automorphism $\psi:S \to S$, that is, $$S^{\Psi} = \{x \in S \mbox{ } \| \mbox{ } \psi(x) = x\},$$ and more generally, for a subgroup $P \leq S$, let $P^{\Psi} = P \cap S^{\Psi}$. The action of $\Psi$ on the fusion system ${\mathcal{F}}$ is somehow too crude to allow us to see any structure on the fixed points, since for each $H \leq S^{\Psi}$, $$Aut_{{\mathcal{F}}}(H)^{\Psi} \stackrel{def} = \{f \in Aut_{{\mathcal{F}}}(H) \mbox{ } \| \mbox{ } \psi_{\ast}(f) = f\} = Aut_{{\mathcal{F}}}(H).$$ We look then for fixed points in ${\mathcal{L}}$. \[restrictmorph\] Let $\varphi: P \to Q$ be a $\Psi$-invariant morphism in ${\mathcal{L}}$. Then, $\rho(\varphi)$ restricts to a morphism $f: P^{\Psi} \to Q^{\Psi}$ in the fusion system ${\mathcal{F}}$. This follows by axiom (C) for linking systems, applied to each $\delta(x) \in \delta(P^{\Psi}) \leq Aut_{{\mathcal{L}}}(P)$, since then both $\delta(x)$ and $\varphi$ are $\Psi$-invariant morphisms in ${\mathcal{L}}$. The above Lemma justifies then defining the fixed points subcategory of ${\mathcal{L}}$ as the subcategory ${\mathcal{L}}^{\Psi}$ with object set $Ob({\mathcal{L}}^{\Psi}) = \{P \in Ob({\mathcal{L}}) \mbox{ } \| \mbox{ } \Psi(P) = P\}$ and with morphism sets $$Mor_{{\mathcal{L}}^{\Psi}}(P, Q) = \{\varphi \in Mor_{{\mathcal{L}}}(P, Q) \mbox{ } \| \mbox{ } \Psi(\varphi) = \varphi\}.$$ We can also define the fixed points subcategory of ${\mathcal{F}}$ as the subcategory ${\mathcal{F}}^{\Psi}$ with object set the set of subgroups of $S^{\Psi}$, and such that $$Mor({\mathcal{F}}^{\Psi}) = {\langle \{\rho(\varphi) \mbox{ } \| \mbox{ } \varphi \in Mor({\mathcal{L}}^{\Psi})\} \rangle}.$$ ${\mathcal{F}}^{\Psi}$ is, by definition, a fusion system over the finite $p$-group $S^{\Psi}$, but the category ${\mathcal{L}}^{\Psi}$ is far from being a transporter system associated to it. \[trouble\] This way of considering fixed points has many disadvantages. For instance, there is no control on the morphism sets $Hom_{{\mathcal{F}}^{\Psi}}(P,Q)$, since given a subgroup $H \leq S^{\Psi}$ there might be several subgroups $P \in Ob({\mathcal{L}}^{\Psi})$ such that $P \cap S^{\Psi} = H$. It becomes then rather difficult to check any of the saturation axioms on ${\mathcal{F}}^{\Psi}$. Another issue is the absence of an obvious candidate of a transporter system associated to ${\mathcal{F}}^{\Psi}$. To avoid the problems listed in Remark \[trouble\], we can try different strategies. For instance, instead of considering a single operation $\Psi$ acting on ${\mathcal{G}}$, we can consider a (suitable) family of operations $\{\Psi_i\}_{i \in {\mathbb{N}}}$ on ${\mathcal{G}}$. This will be specially useful when proving that certain properties hold after suitably increasing the power of $\Psi$. The situation is improved when we restrict our attention to the full subcategory ${\mathcal{L}}^{\bullet} \subseteq {\mathcal{L}}$, since, by [@JLL], unstable Adams operations are completely determined by its action on ${\mathcal{L}}^{\bullet}$. We can also restrict the morphism sets that we consider as fixed by imposing stronger invariance conditions. A family of operations ---------------------- Starting from the unstable Adams operation $\Psi$, we consider an specific family of operations which will satisfy our purposes. Set first $\Psi_0 = \Psi$, and let $\Psi_{i+1} = (\Psi_i)^p$, that is, the operation $\Psi_i$ iterated $p$ times. Consider the resulting family $\{\Psi_i\}_{i \in {\mathbb{N}}}$ fixed for the rest of this section, and note that, if an object or a morphism in ${\mathcal{L}}$ is fixed by $\Psi_i$ for some $i$, then it is fixed by $\Psi_j$ for all $j \geq i$. \[BI1\] By Corollary 4.2 [@JLL], we may assume that there exist a subset ${\mathcal{H}}\subseteq Ob({\mathcal{L}}^{\bullet})$ of representatives of the $S$-conjugacy classes in ${\mathcal{L}}^{\bullet}$ and a set $\widehat{{\mathcal{M}}} = \bigcup_{P,R \in {\mathcal{H}}} {\widehat{\mathcal{M}}}_{P,R}$, where each ${\widehat{\mathcal{M}}}_{P,R} \subseteq Iso_{{\mathcal{L}}}(P,R)$ is a set of representatives of the elements in $Rep_{{\mathcal{F}}}(P,R)$, and such that (i) $\Psi(P) = P$ for all $P \in {\mathcal{P}}$; and (ii) $\Psi(\varphi) = \varphi$ for all $\varphi \in {\widehat{\mathcal{M}}}$. Let us also fix some notation. For each $i$, set $$S_i \stackrel{def} = \{x \in S \mbox{ } \| \mbox{ } \Psi_i(x) = x\},$$ and more generally, for each subgroup $R \leq S$, set $R_i = R \cap S_i$. In particular, the notation $T_i$ means the subgroup of $T$ (the maximal torus) of fixed elements under $\Psi_i$ rather than the subgroup of $T$ of exponent $p^i$. There will not be place for confusion about such notation in this paper. For each $i$, $T_i \lneqq T_{i+1}$, and hence $T = \bigcup_{i \in {\mathbb{N}}} T_i$. As a consequence, we deduce the following. The following holds in ${\mathcal{L}}$. (i) Let $P \in Ob({\mathcal{L}})$. Then, there exists some $M_P$ such that, for all $i \geq M_P$, $P$ is $\Psi_i$-invariant. (ii) Let $\varphi \in Mor({\mathcal{L}})$. Then, there exists some $M_{\varphi}$ such that, for all $i \geq M_{\varphi}$, $\varphi$ is $\Psi_i$-invariant. Next we provide a tool to detect $\Psi_i$-invariant morphisms in ${\mathcal{L}}^{\bullet}$. Note that, for any morphism $\varphi \in Mor({\mathcal{L}})$, the following holds by Proposition 3.3 [@BLO3]. $$\Psi_i(\varphi) = \varphi \Longrightarrow \Psi_i(\varphi^{\bullet}) = \varphi^{\bullet}.$$ Our statement is proved by comparing morphisms in ${\mathcal{L}}^{\bullet}$ to the representatives fixed in ${\widehat{\mathcal{M}}}$, which we know to be $\Psi_i$-invariant (for all $i$) a priori. \[detectmorph\] Let $P, R$ be representatives fixed in \[BI1\], and let $Q \in {{P}^{S}}$ and $Q' \in {{R}^{S}}$. Then, a morphism $\varphi' \in Iso_{{\mathcal{L}}}(Q,Q')$ is $\Psi_i$-invariant if and only if for all $a \in N_S(P,Q)$ there exist $b \in N_S(R,Q')$ and a morphism $\varphi \in {\widehat{\mathcal{M}}}_{P,R}$ such that (i) $\varphi' = \delta(b) \circ \varphi \circ \delta(a^{-1})$; and (ii) $\delta(b^{-1} \cdot \Psi_i(b)) \circ \varphi = \varphi \circ \delta(a^{-1} \cdot \Psi_i(a))$. Note that condition (ii) above is equivalent to 1. $\delta(\Psi_i(b) \cdot b^{-1}) \circ \varphi' = \varphi' \circ \delta(\Psi_i(a) \cdot a^{-1})$. Suppose first that $\varphi'$ is $\Psi_i$-invariant. Choose $x \in N_S(P, Q)$ and $y \in N_S(R,Q')$, and set $\phi = \delta(y^{-1}) \circ \varphi' \circ \delta(x)$. Then, there exist $\varphi \in {\widehat{\mathcal{M}}}_{P,R}$ such that $[\rho(\varphi)] = [\rho(\phi)] \in Rep_{{\mathcal{F}}}(P,R)$ and $z \in R$ such that $\varphi = \delta(z) \circ \phi$. Let then $a = x \in N_S(P,Q)$ and $b = y \cdot z^{-1} \in N_S(R, Q')$. This way, condition (i) is satisfied, and we have to check that condition (ii) is also satisfied. Since both $\varphi$ and $\varphi'$ are $\Psi_i$-invariant, we may apply $\Psi_i$ to (i) to get the following equality $$\delta(b) \circ \varphi \circ \delta(a^{-1}) = \varphi' = \Psi_i(\varphi') = \delta(\Psi_i(b)) \circ \varphi \circ \delta(\Psi_i(a)^{-1}),$$ which is clearly equivalent to condition (ii) since morphisms in ${\mathcal{L}}$ are epimorphisms in the categorical sense. Suppose now that condition (i) and (ii) are satisfied for certain $a$, $b$ and $\varphi$. Write $\varphi = \delta(b^{-1}) \circ \varphi' \circ \delta(a)$ and apply $\Psi_i$ to this equality. Since $\varphi$ is $\Psi_i$-invariant, we get $$\delta(\Psi_i(b)^{-1}) \circ \Psi_i(\varphi') \circ \delta(\Psi_i(a)) = \delta(b^{-1}) \circ \varphi' \circ \delta(a).$$ Thus, after reordering the terms in this equation and using condition (ii’) above, we obtain $$\Psi_i(\varphi') \circ \delta(\Psi_i(a) \cdot a^{-1}) = \varphi' \circ \delta(\Psi_i(a) \cdot a^{-1}),$$ which implies that $\Psi_i(\varphi') = \varphi'$ since morphisms in ${\mathcal{L}}$ are epimorphisms in the categorical sense. A stronger invariance condition ------------------------------- Given an arbitrary $\Psi_i$-invariant object $P$ in ${\mathcal{L}}^{\bullet}$, there is no way a priori of relating $P_i$ to $P$, not to say of comparing $C_S(P_i)$ or $N_S(P_i)$ to $C_S(P)$ or $N_S(P)$ respectively. This turns out to be crucial if we want to study fixed points on ${\mathcal{G}}$ under the operation $\Psi_i$. This is the reason why we now introduce a stronger invariance condition for an object in ${\mathcal{L}}^{\bullet}$ to be $\Psi_i$-invariant. This is a condition on all objects in ${\mathcal{F}}^{\bullet}$. \[Kdeterm\] Let $K \leq S$ be a subgroup. We say that a subgroup $P \in Ob({\mathcal{F}}^{\bullet})$ is $K$-determined if $$(P \cap K)^{\bullet} = P.$$ For a $K$-determined subgroup $P \leq S$ we call the subgroup $P \cap K$ the $K$-root of $P$. Our interest lies on the case $K = S_i$, in which case $\Psi_i$-invariance is a consequence. Let $P \in Ob({\mathcal{F}}^{\bullet})$ be an $S_i$-determined subgroup for some $i$. Then, $P$ is $\Psi_i$-invariant. Note that, if $(P_i)^{\bullet} = P$, then $P = P_i \cdot T_P$, where $T_P$ is the maximal torus of $P$, since $P_i$ is a finite subgroup of $P$. Thus, by applying $\Psi_i$ to $P$, we get $$\Psi_i(P) = \Psi_i(P_i \cdot T_P) = P_i \cdot T_P = P,$$ since $\Psi_i(x) = x$ for all $x \in P_i$ and $\Psi_i(T_P) = T_P$ by definition of $\Psi_i$. We prove now that actually $S_i$-determined subgroups exist (for $i$ big enough). Let $P \in Ob({\mathcal{F}}^{\bullet})$. Then, there exists some $M_P \geq 0$ such that, for all $i \geq M_P$, $P$ is $S_i$-determined. Let $T_P$ be the maximal torus of $P$, and note that $P = \cup P_i$. Thus, there exists some $M$ such that, for all $i \geq M$, $R_i$ contains representatives of all the elements of the finite group $R/T_R$. Since $P^{\bullet} = P$ and $Aut_{{\mathcal{F}}}(T)$ is a finite group, it follows then that there must exist some $M_P \geq M$ such that, for all $i \geq M_P$, $(P_i)^{\bullet} = P$. We can then assume that all the objects fixed in Remark \[BI1\] are $S_i$-determined for all $i$, since there are only finitely many of them in the set ${\mathcal{H}}$. One must be careful at this point. Given $P, R$ $S_i$-determined subgroups, if $P_i$ and $R_i$ are ${\mathcal{F}}$-conjugate, then the properties of ${(\underline{\phantom{B}})^{\bullet}}$ imply that so are $P$ and $R$, but the converse is not so straightforward. \[rootconj\] There exists some $M_1 \geq 0$ such that, for all $P \in {\mathcal{H}}$ and all $i \geq M_1$, if $Q \in {{P}^{S}}$ is $S_i$-determined then $Q_i$ is $S$-conjugate to $P_i$. Since ${\mathcal{H}}$ contains finitely many $S$-conjugacy classes of subgroups and each $S$-conjugacy class contains finitely many $T$-conjugacy classes of subgroups, it is enough to prove the statement for a single $T$-conjugacy class, say ${{P}^{T}}$. Given such subgroup $P$, let $\pi = P/(P \cap T) \leq S/T$, and let $\widetilde{P} \leq S$ be the pull-back of $S \rightarrow S/T \leftarrow \pi$. Then, for any $Q \in {{P}^{T}}$, the following clearly holds: $Q \cap T = P \cap T$, $Q/(Q\cap T) = P/(P\cap T)$ and $Q \leq \widetilde{P}$. For any section $\sigma: \pi \to \widetilde{P}$ of the projection $\widetilde{P} \to \pi$, let $Q_{\sigma} = (P \cap T) \cdot {\langle \sigma(\pi) \rangle} \leq \widetilde{P}$. Given a random section $\sigma$, the subgroup $Q_{\sigma}$ will not in general be in the $T$-conjugacy class of $P$, but it is clear that for every $Q \in {{P}^{T}}$ there exists some $\sigma$ such that $Q = Q_{\sigma}$. Now, up to $T$-conjugacy, the set of sections $\sigma: \pi \to \widetilde{P}$ is in one to one correspondence with the cohomology group $H^1(\pi; T)$, which is easily proved to be finite by an standard transfer argument. Thus, we can fix representatives $\sigma_1, \ldots, \sigma_l$ of $T$-conjugacy classes of sections such that $Q_{\sigma_j} \in {{P}^{T}}$ for all $j$. For each such section, let $H_j = {\langle \sigma_j(\pi) \rangle} \leq \widetilde{P}$. It is clear then that there exists some $M_P$ such that, for all $i \geq M_P$, $H_1, \ldots, H_l \leq S_i$ and $Q_{\sigma_1}, \ldots, Q_{\sigma_l}$ are all $S_i$-determined. Let now $Q \in {{P}^{T}}$ be $S_i$-determined. In particular, this means that there exists a section $\sigma: \pi \to \widetilde{P} \cap S_i$ such that $Q = Q_{\sigma}$. Such a section is $T$-conjugate to some $\sigma_j$ in the list of representatives previously fixed, namely there exists some $t \in T$ such that $\sigma = c_t \circ \sigma_j$. Note that this implies that $H_{\sigma} \stackrel{def} = {\langle \sigma(\pi) \rangle} \in {{H_j}^{T}}$, and hence $Q \in {{Q_{\sigma_j}}^{T}}$. To finish the proof, note that $Q_i = (P \cap T_i) \cdot H_{\sigma}$ and $(Q_{\sigma_j})_i = (P \cap T_i) \cdot H_j$, and clearly $t \in T$ conjugates $(Q_{\sigma_j})_i$ to $Q_i$. We now prove some properties of $S_i$-determined subgroups. \[propcentral\] There exists some $M_2 \geq 0$ such that, for all $i \geq M_2$, if $P$ is $S_i$-determined, then $$C_S(P_i) = C_S(P).$$ Let $\mathfrak{X}$ be a set of representatives of the $S$-conjugacy classes in $Ob({\mathcal{F}}^{\bullet})$, and note that this is a finite set by Lemma 3.2 (a) in [@BLO3]. For any $P \in \mathfrak{X}$, consider the set $\{T_R \mbox{ } \| \mbox{ } R \in {{P}^{S}}\}$ of maximal tori of subgroups in ${{P}^{S}}$. This is a finite set, since, for any two $R, Q \in {{P}^{S}}$ and any $f \in Iso_{{\mathcal{F}}}(R,Q)$ the isomorphism $f_{\|T_R}: T_R \to T_Q$ has to be the restriction of an automorphism of $Aut_{{\mathcal{F}}}(T)$, by Lemma 2.4 (b) [@BLO3], and $Aut_{{\mathcal{F}}}(T)$ is a finite group. It is clear then that there exists some $M_P$ such that, for all $i \geq M_P$ and all $R \in {{P}^{S}}$, $$C_S((T_R)_i) = C_S(T_R).$$ Let now $i \geq M_P$ and let $R \in {{P}^{S}}$ be $S_i$-determined. We can then write $R = R_i \cdot T_R$ and $R_i = R_i \cdot (T_R)_i$, and it follows that $$C_S(R) = C_S(R_i) \cap C_S(T_R) = C_S(R_i) \cap C_S((T_R)_i) = C_S(R_i).$$ The proof is finished by taking $M_2 = \operatorname{max} \{M_P \mbox{ } \| \mbox{ } P \in \mathfrak{X}\}$. The following result is an easy calculation which is left to the reader. \[rootconjx\] Let $P,Q \leq S$ be $S_i$-determined subgroups such that $Q_i \in {{P_i}^{S}}$. Then, for all $x \in N_S(P_i,Q_i)$, $$x^{-1} \cdot \Psi_i(x) \in C_T(P).$$ Since, for any $H \leq K \leq S$ we have $C_K(H) = K \cap C_S(H)$, the following are immediate consequences of Proposition \[propcentral\]. \[corcentral1\] Let $i \geq M_2$ and let $P$ be $S_i$-determined. If $C_S(P) = Z(P)$, then $C_{S_i}(P_i) = Z(P_i)$. \[corcentral2\] There exists some $M_3 \geq 0$ such that, for all $i \geq M_3$, if $Q$ is $S_i$-determined and $C_S(Q) \gneqq Z(Q)$, then $C_{S_i}(Q_i) \gneqq Z(Q_i)$. As usual, since $Ob({\mathcal{F}}^{\bullet})$ contains finitely many $T$-conjugacy classes of subgroups $P$ such that $C_S(P) \gneqq Z(P)$, it is enough to prove the statement for a single $T$-conjugacy class of such subgroups. Fix such a subgroup $P$. We can assume that the statement holds for $P$, and let $z \in C_S(P) \setminus Z(P)$ be such that $z \in C_{S_i}(P_i) \setminus Z(P_i)$ (such an element exists by Proposition \[propcentral\]). Let now $Q \in {{P}^{T}}$ be $S_i$-determined, and let $x \in N_S(P_i, Q_i)$ (such an element exists by Lemma \[rootconj\]). Let also $z' = xzx^{-1} \in C_S(Q) \setminus Z(Q)$. If $C_T(P) \leq (Z(P) \cap T)$, then, by Lemma \[rootconj\], $$z \cdot (x^{-1} \Psi_i(x)) = (x^{-1} \Psi_i(x)) \cdot z,$$ which implies that $\Psi_i(z') = z' \in C_{S_i}(Q_i) \setminus Z(Q_i)$ by Lemma \[detectmorph\]. In this case, let $M_P = M_2$ as in Proposition \[propcentral\]. On the other hand, if $Z(P) \cap T \lneqq C_T(P)$, then we can take the element $z$ above to be in $C_T(P) \setminus (Z(P) \cap T)$. It is clear then that there exists some $M_P$ such that, for all $i \geq M_P$, $z \in C_{T_i}(P) \setminus (Z(P) \cap T_i)$, in which case $z' = z \in C_{T_i}(Q) \setminus (Z(Q) \cap T_i)$. The proof is finished then by taking $M_3$ to be the maximum of the $M_P$ among a finite set of representatives. We can also relate the normalizer of $P_i$ to the normalizer of $P$. \[propnormal\] There exists some $M_4 \geq 0$ such that, for all $i \geq M_4$, if $P$ is $S_i$-determined, then $$N_S(P_i) \leq N_S(P).$$ This is not obvious at all, since the properties of ${(\underline{\phantom{B}})^{\bullet}}$ only tell us that, for $x \in N_S(P_i)$, there exists a unique $f \in Aut_{{\mathcal{F}}}(P)$ extending the isomorphism $c_x \in Aut_{{\mathcal{F}}}(P_i)$. Again, it is enough to check the statement for a single $S$-conjugacy class of objects in ${\mathcal{F}}^{\bullet}$. Let then $P$ be a representative of such an $S$-conjugacy class, and consider the set $\{T_R \mbox{ } \| \mbox{ } R \in {{P}^{S}}\}$ (which is a finite set, as we have shown in the proof for Proposition \[propcentral\]). It follows then that there exists some $M_4$ such that, for all $i \geq M_4$ and all $R \in {{P}^{S}}$, if $g \in N_S((T_R)_i)$ then $g \in N_S(T_R)$. The proof is finished since, for $R \in {{P}^{S}}$ which is $S_i$-determined ($i \geq M_4$), there is an equality $R = R_i \cdot T_R$. Strongly fixed points ===================== Using the notion of $S_i$-determined subgroups we introduce the strongly fixed points of ${\mathcal{G}}$ under the action of $\Psi_i$, and prove their main properties. In particular this section contains the proof of Theorem \[ThA\]. For each $i$, consider the sets $$\label{generatorsets} \begin{array}{c} {\mathcal{H}}_i^{\bullet} \stackrel{def} = \{R \in Ob({\mathcal{L}}^{\bullet}) \mbox{ } \| \mbox{ } R \mbox{ is } S_i\mbox{-determined}\},\\ {\mathcal{H}}_i \stackrel{def} = \{R_i = R \cap S_i \mbox{ } \| \mbox{ } R \in {\mathcal{H}}_i^{\bullet}\},\\ \end{array}$$ and note that the functor ${(\underline{\phantom{B}})^{\bullet}}$ gives a one-to-one correspondence between these two sets. Let also $\widehat{{\mathcal{H}}}_i$ be the closure of ${\mathcal{H}}_i$ by overgroups in $S_i$. Finally, for each pair $P, R \in {\mathcal{H}}_i^{\bullet}$, consider the sets $$\begin{array}{c} Mor_{{\mathcal{L}}, i}(P,R) = \{\varphi \in Mor_{{\mathcal{L}}}(P, R) \mbox{ } \| \mbox{ } \Psi_i(\varphi) = \varphi\},\\ Hom_{{\mathcal{F}}, i}(P,R) = \{\rho(\varphi) \mbox{ } \| \mbox{ } \varphi \in Mor_{{\mathcal{L}},i}(P,R)\}. \end{array}$$ Recall from Lemma \[restrictmorph\] that, given a $\Psi_i$-invariant morphism $\varphi:P \to R$ in ${\mathcal{L}}$, the homomorphism $f = \rho(\varphi)$ restricts to a homomorphism $f_i: P_i \to R_i$. Thus, we can consider, for each pair $P_i, R_i \in {\mathcal{H}}_i$, the set $$A(P_i, R_i) = \{ f_i = res^P_{P_i}(f) \mbox{ } \| \mbox{ } f \in Hom_{{\mathcal{F}}, i}(P,R) \} \subseteq Hom_{{\mathcal{F}}}(P_i, R_i).$$ Again, the functor ${(\underline{\phantom{B}})^{\bullet}}$ provides a bijection from the above set to $Hom_{{\mathcal{F}}, i}(P,R)$. For each $i$, the $i$-th strongly fixed points fusion system is the fusion system ${\mathcal{F}}_i$ over $S_i$ whose morphisms are compositions of restrictions of morphisms in $\{A(P_i, R_i) \mbox{ } \| \mbox{ } P_i,R_i \in {\mathcal{H}}_i\}$. The category ${\mathcal{F}}_i$ is indeed a fusion system over $S_i$, as well as a fusion subsystem of ${\mathcal{F}}$. Let ${\mathcal{L}}_i^{\circ}$ be the category with object set ${\mathcal{H}}_i$ and whose morphism sets are spanned by the sets $Mor_{{\mathcal{L}},i}(P,R)$, after identifying the sets ${\mathcal{H}}_i$ and ${\mathcal{H}}_i^{\bullet}$ via ${(\underline{\phantom{B}})^{\bullet}}$. The category ${\mathcal{L}}_i^{\circ}$ is well-defined since, in fact, it can be thought as a subcategory of ${\mathcal{L}}$, although its actual definition will make more sense for the purposes of this paper. We want now to close ${\mathcal{L}}_i^{\circ}$ by overgroups, and one has to be careful at this step. Let $H, K \in \widehat{{\mathcal{H}}}_i$ be arbitrary subgroups, and let $P, R \in {\mathcal{H}}_i^{\bullet}$ be such that $H \leq P$, $K \leq R$. We say then that a morphism $\varphi \in Mor_{{\mathcal{L}},i}(P,R)$ restricts to a morphism $\varphi: H \to K$ if $f = \rho(\varphi):P \to R$ restricts to a homomorphism $f_{\|H}: H \to K$ in ${\mathcal{F}}$. We need a technical lemma before we define the closure of ${\mathcal{L}}_i^{\circ}$ by overgroups. \[bulletSideterm\] For any subgroup $H \leq S_i$, the subgroup $H^{\bullet} \leq S$ is $S_i$-determined. If, in addition, $H \in \widehat{{\mathcal{H}}_i}$, then $H^{\bullet}$ is ${\mathcal{F}}$-centric. To show that $H^{\bullet}$ is $S_i$-determined, we have to prove that $(H^{\bullet} \cap S_i)^{\bullet} = H^{\bullet}$. Since $H \leq H^{\bullet} \cap S_i \leq H^{\bullet}$, the equality follows by applying ${(\underline{\phantom{B}})^{\bullet}}$ to these inequalities. The centricity of $H^{\bullet}$ when $H \in \widehat{{\mathcal{H}}_i}$ follows by definition of the set $\widehat{{\mathcal{H}}}_i$ and by Proposition 2.7 [@BLO3]. As a consequence of this result, for any $H \in \widehat{{\mathcal{H}}}_i$, the subgroup $(H^{\bullet} \cap S_i) \in {\mathcal{H}}_i$. \[defigi\] For each $i$, the $i$-th strongly fixed points transporter system is the category ${\mathcal{L}}_i$ with object set $\widehat{{\mathcal{H}}}_i$ and with morphism sets $$Mor_{{\mathcal{L}}_i}(H, K) = \{\varphi \in Mor_{{\mathcal{L}},i}(H^{\bullet}, K^{\bullet}) \mbox{ } \| \mbox{ } \varphi \mbox{ restricts to a morphism } \varphi:H \to K\}.$$ Finally, the $i$-th strongly fixed points system is the triple ${\mathcal{G}}_i = (S_i, {\mathcal{F}}_i, {\mathcal{L}}_i)$. The composition rule in ${\mathcal{L}}_i$ is induced by the composition rule in ${\mathcal{L}}$, and hence is well-defined. ${\mathcal{L}}_i$ is called a transporter system since we will prove in this section that it actually has such structure. Properties of the strongly fixed points subsystems -------------------------------------------------- We now study the properties of each of the triples ${\mathcal{G}}_i$ defined above. At some point this will require increasing the degree of the initial operation $\Psi$ again, and also fix some more objects and morphisms in ${\mathcal{L}}$, apart from those already fixed in Remark \[BI1\]. First, we describe some basic properties of the triples ${\mathcal{G}}_i$, most of which are inherited from the properties of ${\mathcal{L}}$. \[morphi\] For all $i$ and for all $P_i, R_i \in {\mathcal{H}}_i$, there are equalities (i) $A(P_i, R_i) = Hom_{{\mathcal{F}}_i}(P_i, R_i)$; and (ii) $Mor_{{\mathcal{L}},i}(P,R) = Mor_{{\mathcal{L}}_i}(P_i, R_i)$. by definition of ${\mathcal{F}}_i$ and ${\mathcal{L}}_i$, it is enough to show only point (ii). The proof is done then by induction on the order of the subgroups $P_i, R_i \in {\mathcal{H}}_i$. First, we consider the case $P_i = R_i = S_i$. This case is obvious since in ${\mathcal{H}}_i$ the subgroup $S_i$ has no overgroups. Consider now a pair $P_i, R_i \lneqq S_i$. There is an obvious inclusion $Mor_{{\mathcal{L}},i}(P,R) \subseteq Mor_{{\mathcal{L}}_i}(P_i, R_i)$ by definition of ${\mathcal{L}}_i$. On the other hand, any morphism in $Mor_{{\mathcal{L}}_i}(P_i, R_i)$ is a composition of restrictions of $\Psi_i$-invariant morphisms in ${\mathcal{L}}$, by the induction hypothesis, and hence we have the equality. The category ${\mathcal{L}}_i$ also has associated a functor ${(\underline{\phantom{B}})^{\bullet}}_i$, induced by the original ${(\underline{\phantom{B}})^{\bullet}}$ in ${\mathcal{L}}$. Next we describe this functor in ${\mathcal{L}}_i$ and its main properties, most of which are identical to those of ${(\underline{\phantom{B}})^{\bullet}}$. Define first ${(\underline{\phantom{B}})^{\bullet}}_i$ on an object $H_i \in Ob({\mathcal{L}}_i)$ as $$(H_i)^{\bullet}_i \stackrel{def} = (H_i)^{\bullet} \cap S_i.$$ Recall that by definition, $Mor_{{\mathcal{L}}_i}(H_i, K_i) = \{ \omega \in Mor_{{\mathcal{L}},i}(H_i^{\bullet}, K_i^{\bullet}) \mbox{ } \| \mbox{ } \omega \mbox{ restricts to } \omega: H_i \to K_i\}$. Thus, on a morphism $\varphi \in Mor_{{\mathcal{L}}_i}(H_i, K_i)$, ${(\underline{\phantom{B}})^{\bullet}}_i$ is defined as the unique $\varphi \in Mor_{{\mathcal{L}}_i}(H_i^{\bullet} \cap S_i, K_i^{\bullet} \cap S_i)$ which restricts to $\varphi: H_i \to K_i$. Note that in particular ${(\underline{\phantom{B}})^{\bullet}}_i$ is the identity on ${\mathcal{H}}_i$ by construction. \[propertiesbulleti\] The following holds for ${(\underline{\phantom{B}})^{\bullet}}_i$. (i) For all $H_i \leq S_i$, $((H_i)^{\bullet}_i)^{\bullet}_i = (H_i)^{\bullet}_i$. (ii) If $H_i \leq K_i \leq S_i$, then $(H_i)^{\bullet}_i \leq (K_i)^{\bullet}_i$. (iii) Every morphism $\varphi \in Mor_{{\mathcal{L}}_i}(H_i,K_i)$ extends to a unique $(\varphi)^{\bullet}_i \in Mor_{{\mathcal{L}}_i}((H_i)^{\bullet}_i, (K_i)^{\bullet}_i)$ In particular, ${(\underline{\phantom{B}})^{\bullet}}_i$ is a functor from ${\mathcal{L}}_i$ to ${\mathcal{L}}_i^{\circ}$ which is left adjoint to the inclusion ${\mathcal{L}}_i^{\circ} \subseteq {\mathcal{L}}_i$. \(i) Let $H_i \leq S_i$ be any subgroup. By Lemma \[bulletSideterm\], $H \stackrel{def} = (H_i)^{\bullet}$ is $S_i$-determined, and hence $$\xymatrix@C=4mm{ (H_i)^{\bullet}_i \stackrel{def} = H \cap S_i \ar[rr]^{{(\underline{\phantom{B}})^{\bullet}}} & & (H \cap S_i)^{\bullet} = H \ar[rr]^{\underline{\phantom{A}} \cap S_i} & & H \cap S_i \stackrel{def} = ((H_i)^{\bullet}_i)^{\bullet}_i.\\ }$$ \(ii) This property follows from Lemma 3.2 (c) [@BLO3]. \(iii) Both the existence and uniqueness of $(\varphi)^{\bullet}_i$ hold by definition of ${\mathcal{L}}_i$. It follows now that ${(\underline{\phantom{B}})^{\bullet}}_i$ is a functor, and by Lemma \[bulletSideterm\], it sends objects and morphisms in ${\mathcal{L}}_i$ to objects and morphisms in ${\mathcal{L}}_i^{\circ}$. The adjointness property holds since the restriction map $$Mor_{{\mathcal{L}}_i}((H_i)^{\bullet}_i, P_i) \stackrel{res} \longrightarrow Mor_{{\mathcal{L}}_i}(H_i, P_i)$$ is a bijection for all $H_i \in \widehat{{\mathcal{H}}_i}$ and all $P_i \in {\mathcal{H}}_i$. \[inclequiv\] The inclusion ${\mathcal{L}}_i^{\circ} \subseteq {\mathcal{L}}_i$ induces a homotopy equivalence $$\|{\mathcal{L}}_i^{\circ}\| \simeq \|{\mathcal{L}}_i\|.$$ It is a consequence of Corollary 1 [@Quillen], since the inclusion ${\mathcal{L}}_i^{\circ} \subseteq {\mathcal{L}}_i$ has a right adjoint by Proposition \[propertiesbulleti\]. \[transpi\] There exists some $M_{\Psi} \geq 0$ such that, for all $i \geq M_{\Psi}$, ${\mathcal{L}}_i$ is a transporter system associated to ${\mathcal{F}}_i$. Clearly, each ${\mathcal{L}}_i$ is a nonempty finite category with $Ob({\mathcal{L}}_i) \subseteq Ob({\mathcal{F}}_i)$. The first step to prove the statement is to define the pair of functors $\varepsilon_i: {\mathcal{T}}_{Ob({\mathcal{L}}_i)}(S_i) \to {\mathcal{L}}_i$ and $\rho_i: {\mathcal{L}}_i \to {\mathcal{F}}_i$, but actually these two functors are naturally induced by ${\mathcal{L}}$. Indeed, the functor $\varepsilon: {\mathcal{T}}_{Ob({\mathcal{L}})}(S) \to {\mathcal{L}}$ restricts to a functor $\varepsilon_i$ as above. With respect to the functor $\rho_i$, the projection functor $\rho: {\mathcal{L}}\to {\mathcal{F}}$ naturally induces a functor $$\rho_i: {\mathcal{L}}_i \to {\mathcal{F}}_i$$ by the rule $\rho_i(\varphi: H \to H') = res^P_H(\rho(\varphi))$ for each morphism $\varphi \in Mor_{{\mathcal{L}}_i}(H, H')$ (Lemma \[restrictmorph\]). We next proceed to prove that all the axioms for transporter systems in definition \[defitransporter\] are satisfied (after considering a suitable power of $\Psi$). Note first that using the functor ${(\underline{\phantom{B}})^{\bullet}}_i$, it is enough to prove the axioms on the subset ${\mathcal{H}}_i \subseteq Ob({\mathcal{L}}_i)$. Recall by Proposition \[3.5OV\] that ${\mathcal{L}}$ satisfies the axioms of a transporter system. Axioms (A1), (A2), (B) and (C) hold by definition of ${\mathcal{G}}_i$ and by the same axioms on ${\mathcal{L}}$. Next we show that axiom (I) holds. By Proposition \[morphi\], there is an equality $Aut_{{\mathcal{L}}_i}(S_i) = Mor_{{\mathcal{L}},i}(S,S) \leq Aut_{{\mathcal{L}}}(S)$. Furthermore, by definition of all these groups, and because we have fixed representatives of the elements of $Out_{{\mathcal{F}}}(S)$ in ${\widehat{\mathcal{M}}}$, there is a group extension $$1 \to (\varepsilon_i)_{S_i,S_i}(S_i) \to Aut_{{\mathcal{L}}_i}(S_i) \to Out_{{\mathcal{F}}}(S) \to 1.$$ Thus, since $\{1\} \in Syl_p(Out_{{\mathcal{F}}}(S))$, the axiom follows. There is no need of checking that axiom (III) of transporter systems holds in this case, since ${\mathcal{L}}_i$ is a finite category. Axiom (II) will be proved by steps, since we need to discard finitely many of the first operations in $\{\Psi_i\}$. We recall here its statement. - Let $\varphi \in Iso_{{\mathcal{L}}_i}(P_i,Q_i)$, $P_i \lhd \widetilde{P}_i \leq S_i$ and $Q_i \lhd \widetilde{Q}_i \leq S_i$ be such that $\varphi \circ \varepsilon_i(\widetilde{P}_i) \circ \varphi^{-1} \leq \varepsilon_i(\widetilde{Q}_i)$. Then, there is some $\widetilde{\varphi} \in Mor_{{\mathcal{L}}_i}(\widetilde{P}_i, \widetilde{Q}_i)$ such that $\widetilde{\varphi} \circ \varepsilon_i(1) = \varepsilon_i(1) \circ \varphi$. The proof of axiom (II) is then organized as follows. First, we fix a finite list of representatives of all possible such extensions in ${\mathcal{L}}$ (up to conjugacy by an element in $S$). In the second step we prove the axiom for the representatives fixed in the set ${\mathcal{H}}$ of Remark \[BI1\], and finally in the third step we prove the general case. - Step 1. Representatives of the extensions. Let $P_i, Q_i \in {\mathcal{H}}_i$, $P_i \lhd \widetilde{P}_i$, $Q_i \lhd \widetilde{Q}_i$ and $\varphi \in Mor_{{\mathcal{L}}_i}(Pi, Q_i)$ as in the statement of axiom (II). By definition of ${\mathcal{L}}_i$ and ${\mathcal{H}}_i$, it is equivalent to consider in ${\mathcal{L}}$ the corresponding situation: $\varphi: P \to Q$ such that $\varphi \circ \varepsilon(\widetilde{P}) \circ \varphi^{-1} \leq \varepsilon(\widetilde{Q})$, where $P = (P_i)^{\bullet}$, $Q = (Q_i)^{\bullet}$, $\widetilde{P} = (\widetilde{P}_i)^{\bullet}$ and $\widetilde{Q} = (\widetilde{Q}_i)^{\bullet}$. Note that, by Lemma \[bulletSideterm\], the subgroups $P, Q, \widetilde{P}$ and $\widetilde{Q}$ are $S_i$-determined. We have then translated a situation in ${\mathcal{L}}_i$ to a situation in ${\mathcal{L}}$. We will keep this notation for the rest of the proof. Note first that if $P \in Ob({\mathcal{L}})$, then the quotient $N_S(P)/P$ is finite. Indeed, since $C_S(P) = Z(P)$, it follows that $N_S(P)/P \cong (N_S(P)/Z(P)) / (P/Z(P)) = Out_S(P) \leq Out_{{\mathcal{F}}}(P)$, and this group is finite by axiom (I) for saturated fusion systems (or by Proposition 2.3 [@BLO3]). As a consequence, if we fix $P, Q \in Ob({\mathcal{L}})$ and a morphism $\varphi \in Mor_{{\mathcal{L}}}(P,Q)$, then, up to conjugacy by elements in $S$, there are only finitely many morphisms $\widetilde{\varphi}: \widetilde{P} \to \widetilde{Q}$ such that $P \lhd \widetilde{P}$, $Q \lhd \widetilde{Q}$ and $\widetilde{\varphi}$ extends $\varphi$. Consider then the sets ${\mathcal{H}}$ and ${\widehat{\mathcal{M}}}$ fixed in Remark \[BI1\]. For each morphism $\varphi: P \to Q$ fixed in ${\widehat{\mathcal{M}}}$, we can fix representatives (up to $S$-conjugacy) of all possible extensions $\widetilde{\varphi}: \widetilde{P} \to \widetilde{Q}$. Let $\widetilde{{\mathcal{M}}}$ be s set of all such representatives: $$\widetilde{{\mathcal{M}}} \stackrel{def} = \{ \widetilde{\varphi}: \widetilde{P} \to \widetilde{Q} \mbox{ } \| \mbox{ } \varphi \in {\widehat{\mathcal{M}}}\}.$$ It is clear then that there exists some $M_{\Psi}$ such that, for all $i \geq M_{\Psi}$, the following holds: (i) each extension $\widetilde{\varphi}$ in the above set is $\Psi_i$-invariant; (ii) for each such $\widetilde{\varphi}$, the source subgroup, $\widetilde{P}$, is $S_i$-determined; and (iii) each $\widetilde{P}$ is $S_i$-conjugate to the corresponding representative of ${{\widetilde{P}}^{S}}$ fixed in ${\mathcal{H}}$. - Step 2. Both $P$ and $Q$ are in the set ${\mathcal{H}}$ fixed in \[BI1\]. In this case, there are $\varphi' \in {\widehat{\mathcal{M}}}$ and $x \in Q$ such that $\varphi = \delta(x) \circ \varphi'$. Furthermore, since both $\varphi$ and $\varphi'$ are $\Psi_i$-invariant, so is $\delta(x)$, and hence $x \in Q_i$. Let then $\widetilde{\varphi}' \in \widetilde{{\mathcal{M}}}$ be the extension of $\varphi'$ which sends $\widetilde{P}$ to $x \cdot \widetilde{Q} \cdot x^{-1}$, and let $$\widetilde{\varphi} = \delta(x) \circ \widetilde{\varphi}'.$$ It follows then that $\widetilde{\varphi}: \widetilde{P} \to \widetilde{Q}$ is an extension of $\varphi$, which in addition is $\Psi_i$-invariant since both $\delta(x)$ and $\widetilde{\varphi}'$ are. We just have to consider the corresponding morphism in ${\mathcal{L}}_i$ to prove that axiom (II) holds in this case, since $\widetilde{P}$ and $\widetilde{Q}$ are $S_i$-determined. - Step 3. One (or possibly both) of the subgroups $P, Q$ is not in ${\mathcal{H}}$. Since $\varphi$ is $\Psi_i$-invariant, it follows from Lemma \[detectmorph\] that there exist subgroups $R, R' \in {\mathcal{H}}$, a morphism $\varphi' \in {\widehat{\mathcal{M}}}_{R,R'}$, and elements $a \in N_S(R,P)$ and $b \in N_S(R',Q)$ such that (i) $\varphi = \delta(b) \circ \varphi' \circ \delta(a^{-1})$, and (ii) $\delta(b^{-1} \cdot \Psi_i(b)) \circ \varphi ' = \varphi' \circ \delta(a^{-1} \cdot \Psi_i(a))$. Let then $\widetilde{R} = a^{-1} \cdot \widetilde{P} \cdot a$ and $\widetilde{R}' = b^{-1} \cdot \widetilde{Q} \cdot b$, and let $\widetilde{\varphi}': \widetilde{R} \to \widetilde{R}'$ be the extension of $\varphi'$ fixed in $\widetilde{{\mathcal{M}}}$. Let also $$\widetilde{\varphi} \stackrel{def} = \delta(b) \circ \widetilde{\varphi}' \circ \delta(a^{-1}): \widetilde{P} \to \widetilde{Q}.$$ Since $\varphi$ is $\Psi_i$-invariant, and by Lemma 4.3 [@BLO3], it follows then that $\widetilde{\varphi}$ is also $\Psi_i$-invariant. Since $\widetilde{P}, \widetilde{Q}$ are $S_i$-determined, the proof is finished by considering then the morphism induced in ${\mathcal{L}}_i$ by $\widetilde{\varphi}$. By Corollary \[inclequiv\], the following statement is still true after replacing $Ob({\mathcal{L}}_i)$ by the subset ${\mathcal{H}}_i$. For each $i \geq M_{\Psi}$, ${\mathcal{F}}_i$ is $Ob({\mathcal{L}}_i)$-generated and $Ob({\mathcal{L}}_i)$-saturated. The $Ob({\mathcal{L}}_i)$-generation of ${\mathcal{F}}_i$ follows by definition of ${\mathcal{G}}_i$, and the $Ob({\mathcal{L}}_i)$-saturation of ${\mathcal{F}}_i$ follows directly by Proposition 3.6 [@OV]. Next we describe an interesting property of the family $\{{\mathcal{G}}_i\}$. Recall that, by Corollary \[inclequiv\], each inclusion ${\mathcal{L}}_i^{\circ} \subseteq {\mathcal{L}}_i$ induces a homotopy equivalence $\|{\mathcal{L}}_i^{\circ}\| \simeq \|{\mathcal{L}}_i\|$, and the striking point is that for each $i$ there is also a faithful functor $\Theta_i: {\mathcal{L}}_i^{\circ} \to {\mathcal{L}}_{i+1}^{\circ}$ defined by: $$\label{thetai} \xymatrix@R=1mm{ {\mathcal{L}}_i^{\circ} \ar[rr]^{\Theta_i} & & {\mathcal{L}}_{i+1}^{\circ}\\ P_i \ar@{\|->}[rr] & & (P_i)^{\bullet} \cap S_{i+1} = P_{i+1}\\ \varphi \ar@{\|->}[rr] & & \varphi.\\ }$$ It is easy to check that this is a well-defined functor: since $P_i \in {\mathcal{H}}_i$ is the (unique) $S_i$-root of the $S_i$-determined subgroup $P \in Ob({\mathcal{L}}^{\bullet})$, it follows by construction that $P_{i+1}$ is an element of the set ${\mathcal{H}}_{i+1}$, and by definition of ${\mathcal{L}}_i$ and ${\mathcal{L}}_{i+1}$, there is a natural inclusion of sets $$Mor_{{\mathcal{L}}_i}(P_i,R_i) \subseteq Mor_{{\mathcal{L}}_{i+1}}(P_{i+1}, Q_{i+1})$$ for all $P_i, Q_i \in {\mathcal{H}}_i$, which is a group monomorphism whenever $P_i = Q_i$. It follows then that $\Theta_i$ is faithful for all $i$. Note that in general the functor $\Theta_i$ does not induce a commutative square $$\xymatrix@R=1mm@C=1mm{ {\mathcal{L}}_i \ar[dd]_{\rho_i} \ar[rr]^{\Theta_i} & & {\mathcal{L}}_{i+1} \ar[dd]^{\rho_{i+1}}\\ & \times & \\ {\mathcal{F}}_i \ar[rr]_{incl_i} & & {\mathcal{F}}_{i+1}.\\ }$$ For instance, whenever $S$ has positive rank, we have $\rho_{i+1}(\Theta_i(S_i)) = (S_i)^{\bullet} \cap S_{i+1} = S_{i+1} \gneqq S_i = incl(\rho(S_i))$. This is not a great inconvenience, as we prove below. Let ${\mathcal{F}}_i^{{\mathcal{H}}_i} \subseteq {\mathcal{F}}_i$ be the full subcategory with object set ${\mathcal{H}}_i$, and let $\theta_i: {\mathcal{F}}_i^{{\mathcal{H}}_i} \to {\mathcal{F}}_{i+1}$ be the functor induced by $\Theta_i$. For all $i$, there is a natural transformation $\tau_i$ between the functors $incl_i$ and $\theta_i$. Set $\tau_i(P_i) = [incl_i(P_i) = P_i \hookrightarrow P_{i+1} = (P_i)^{\bullet} \cap S_i]$ for each $P_i \in {\mathcal{H}}_i$, and set also $$\xymatrix@R=1mm@C=1mm{ P_i \ar[rr]^{incl_i} \ar[dd]_{f_i} & & P_{i+1} \ar[dd]^{f_{i+1}} \\ & \tau_i(f_i) & \\ R_i \ar[rr]_{incl_i} & & R_{i+1}\\ }$$ for each $(f_i: P_i \to R_i) \in Mor({\mathcal{F}}_i^{{\mathcal{H}}_i})$, where $f_{i+1} = res^{(P_i)^{\bullet}}_{P_{i+1}}((f_i)^{\bullet})$. This is well-defined since $S_i$-roots are unique, and because of the properties of ${(\underline{\phantom{B}})^{\bullet}}$. In particular, it follows from Proposition 3.3 [@BLO3] that the above square is always commutative, and hence $\tau_i$ is a natural transformation. This way, we have a sequence of maps $$\xymatrix@R=5mm@C=1cm{ \ldots \ar[r] & \|{\mathcal{L}}_{i-1}^{\circ}\| \ar[r]^{\|\Theta_{i-1}\|} \ar[d]_{\simeq} & \|{\mathcal{L}}_i^{\circ}\| \ar[r]^{\|\Theta_i\|} \ar[d]_{\simeq} & \|{\mathcal{L}}_{i+1}^{\circ}\| \ar[r]^{\|\Theta_{i+1}\|} \ar[d]_{\simeq} & \ldots, \\ & \|{\mathcal{L}}_{i-1}\| & \|{\mathcal{L}}_i\| & \|{\mathcal{L}}_{i+1}\| & \\ }$$ and we can ask about the homotopy colimit of this sequence. Let $I$ be the poset of the natural numbers with inclusion, and let $\Theta: I \to \operatorname{Top}$ be the functor defined by $\Theta(i) = \|{\mathcal{L}}^{\circ}_i\|$ and $\Theta(i \to i+1) = \|\Theta_i\|$. \[hocolim1\] There is a homotopy equivalence $$(\operatorname{hocolim}_{\rightarrow I} \Theta)^{\wedge}_p \simeq B{\mathcal{G}}.$$ The statement follows since, as categories, ${\mathcal{L}}^{\bullet} = \bigcup_{i \in {\mathbb{N}}} {\mathcal{L}}_i^{\circ}$. We finally study the elements of a set ${\mathcal{H}}_i$ as objects in ${\mathcal{F}}_j$ for $j \geq i$. \[centricquasicentric\] Let $P_i \in {\mathcal{H}}_i$. Then, the following holds: (i) $P_i$ is ${\mathcal{F}}_i$-centric, (ii) $P_i$ is ${\mathcal{F}}_j$-quasicentric for all $j \geq i$, and (iii) $P_i$ is ${\mathcal{F}}$-quasicentric. Property (i) is a consequence of Proposition \[propcentral\]. Properties (ii) and (iii) are consequence of the Proposition below. Let $P \leq S$ be ${\mathcal{F}}$-quasicentric and $S_i$-determined for some $i$. Then, (i) $P_i$ is ${\mathcal{F}}$-quasicentric, and (ii) $P_i$ is ${\mathcal{F}}_j$-quasicentric for all $j \geq i$. The proof is done by steps. - Step 1. Let $H \in {{P_i}^{{\mathcal{F}}}}$ and $Q = (H)^{\bullet}$. Then there are equalities $$C_S(H) = C_S(Q).$$ Indeed, for $P_i$ the equality holds by Proposition \[propcentral\], since $P$ is $S_i$-determined. Also, since $P$ is $S_i$-determined, we can write $$\begin{array}{ccc} P = P_i \cdot T_P & \mbox{and} & P_i = P_i \cdot (T_P)_i. \end{array}$$ Let now $H \in {{P_i}^{{\mathcal{F}}}}$, $Q = (H)^{\bullet}$, and let $f \in Iso_{{\mathcal{F}}}(P_i,H)$. Using the infinitely $p$-divisibility property of $T_P$, we can write then $Q = f(P_i) \cdot T_Q = H \cdot T_Q$ and $H = f(P_i) \cdot f((T_P)_i) = H \cdot (T_Q)_i$. Thus, $$C_S(H) = C_S(H) \cap C_S((T_Q)_i) = C_S(H) \cap C_S(T_Q) = C_S(Q),$$ where the second equality holds by (the proof of) Proposition \[propcentral\], since we had previously fixed representatives of all the $S$-conjugacy classes in ${{P}^{{\mathcal{F}}}}$ in \[BI1\]. - Step 2. For each $H \in {{P_i}^{{\mathcal{F}}}}$, $H$ is ${\mathcal{F}}$-quasicentric. Let $C_{{\mathcal{F}}}(H)$ be the centralizer fusion system of $H$, and note that, in particular, $C_{{\mathcal{F}}}(H)$ is a fusion system over $C_S(H) = C_S(Q)$. Let also $f: R \to R'$ be a morphism in $C_{{\mathcal{F}}}(H)$. By definition of $C_{{\mathcal{F}}}(H)$, there is a morphism $\widetilde{f}: R \cdot H \to R' \cdot H$ in $C_{{\mathcal{F}}}(H)$ which extends $f$ and such that it restricts to the identity on $H$. By applying ${(\underline{\phantom{B}})^{\bullet}}$ to $\widetilde{f}$, we obtain a new morphim $(\widetilde{f})^{\bullet}$ which restricts to $f^{\bullet}: (R)^{\bullet} \to (R')^{\bullet}$ and to the identity on $(H)^{\bullet} = Q$. It follows then that $f^{\bullet}$ is a morphism in $C_{{\mathcal{F}}}(Q)$. On the other hand, there is an obvious inclusion of categories $C_{{\mathcal{F}}}(Q) \subseteq C_{{\mathcal{F}}}(H)$, which is in fact an equality by the above. Since $Q$ is ${\mathcal{F}}$-quasicentric by hypothesis, the proof of Step 2 is finished. - Step 3. For each $j \geq i$ and each $H \in {{P_i}^{{\mathcal{F}}_j}}$, $H$ is ${\mathcal{F}}_j$-quasicentric. This case follows by Step 1, together with the properties of the functor ${(\underline{\phantom{B}})^{\bullet}}$, since we can identify $C_{{\mathcal{F}}_j}(H)$ with a subcategory of $C_{{\mathcal{F}}}(H)$. Consequences of the existence of approximations by $p$-local finite groups -------------------------------------------------------------------------- We have skipped in the previous section the issue of the saturation of the fusion systems ${\mathcal{F}}_i$. This is a rather difficult question and we want to discuss it apart from the main results. In this section we will also study some consequences of the case when the triples ${\mathcal{G}}_i$ are $p$-local finite groups. Examples of this situation will be described in the following section. Recall that we have used Proposition 3.6 [@OV] to prove that for each $i$ the fusion system ${\mathcal{F}}_i$ is $Ob({\mathcal{L}}_i)$-generated and $Ob({\mathcal{L}}_i)$-saturated. Recall also that Proposition 3.6 [@OV] gives conditions for the fusion systems ${\mathcal{F}}_i$ to be saturated: each ${\mathcal{F}}_i$-centric subgroup $H \leq S_i$ not in $Ob({\mathcal{L}}_i)$ has to be ${\mathcal{F}}_i$-conjugate to some $K \leq S_i$ such that $$\label{condition} Out_{S_i}(K) \cap O_p(Out_{{\mathcal{F}}_i}(K)) \neq \{1\}.$$ The disadvantage of proving the saturation of ${\mathcal{F}}_i$ by means of this result lies obviously on the difficulty in checking the above condition, but the advantage of proving saturation using it is also great, since in particular this would mean that all ${\mathcal{F}}_i$-centric ${\mathcal{F}}_i$-radical subgroups are in $Ob({\mathcal{L}}_i)$. Indeed, note that if there was some ${\mathcal{F}}_i$-centric ${\mathcal{F}}_i$-radical not in $Ob({\mathcal{L}}_i)$, then the category ${\mathcal{L}}_i^{\circ}$ defined in the previous section could not be extended to a whole centric linking system associated to ${\mathcal{F}}_i$ (at least in an obvious way), and the functors $\Theta_i$ would not be valid any more. In order to check the condition above, we can consider the following two situations: (a) $H$ is not an $S_i$-root, that is, $H \lneqq (H)^{\bullet} \cap S_i$; or (b) $H$ is an $S_i$-root, that is, $H = (H)^{\bullet} \cap S_i$ but $(H)^{\bullet}$ is not ${\mathcal{F}}$-centric. The difficult case to study is (b), but we can prove rather easily that condition (\[condition\]) is always satisfied in case (a). \[nosiroot\] Let $H \leq S_i$ be an ${\mathcal{F}}_i$-centric subgroup not in $Ob({\mathcal{L}}_i)$ and such that $H \lneqq (H)^{\bullet} \cap S_i$. Then, $H$ satisfies condition (\[condition\]). Let $K \stackrel{def} = (H)^{\bullet} \cap S_i \leq S_i$. The functor ${(\underline{\phantom{B}})^{\bullet}}_i$ provides a natural inclusion $Aut_{{\mathcal{F}}_i}(H) \leq Aut_{{\mathcal{F}}_i}(K)$. Consider also the subgroup $A = \{c_x \in Aut_{{\mathcal{F}}_i}(H) \mbox{ } \| \mbox{ } x \in N_K(H)\}$. Via the above inclusion of automorphism groups in ${\mathcal{F}}_i$, we can see $A$ as $$A = Aut_{{\mathcal{F}}_i}(H) \cap Inn(K).$$ Since, by hypothesis, $H \lneqq K$, it follows that $H \lneqq N_K(H)$, and hence $Inn(H) \lneqq A$, since $H$ is ${\mathcal{F}}_i$-centric by hypothesis. The group $Aut_{{\mathcal{F}}_i}(H)$, seen as a subgroup of $Aut_{{\mathcal{F}}_i}(K)$, normalizes $Inn(K)$, and thus $A \lhd Aut_{{\mathcal{F}}_i}(H)$ and $$\{1\} \neq A/Inn(H) \leq O_p(Out_{{\mathcal{F}}_i}(H)).$$ Also, by definition of $A$, there is an inclusion $A/Inn(H) \leq Out_{S_i}(H)$ and this finishes the proof. It is still an open question whether condition (\[condition\]) is satisfied in case (b) in general. Let ${\mathcal{G}}$ be a $p$-local compact group, and let $\Psi$ be an unstable Adams operation acting on ${\mathcal{G}}$. We say that $\Psi$ approximates ${\mathcal{G}}$ by $p$-local finite groups if there exists some $M_{\Psi}'$ such that, for all $i \geq M_{\Psi}'$, condition (\[condition\]) holds for all $H \in Ob({\mathcal{F}}_i^c) \setminus Ob({\mathcal{L}}_i)$. We also say then that $\Psi$ induces an approximation of ${\mathcal{G}}$ by $p$-local finite groups. Let ${\mathcal{G}}$ be a $p$-local compact group such that, for all $P \in Ob({\mathcal{F}}^{\bullet}) \setminus Ob({\mathcal{L}}^{\bullet})$, $C_S(P) \gneqq Z(P)$. Then, any unstable Adams operation $\Psi$ induces an approximation of ${\mathcal{G}}$ by $p$-local finite groups. The Stable Elements Theorem --------------------------- When ${\mathcal{G}}$ is approximated by $p$-local finite groups, we can prove the Stable Elements Theorem (5.8 [@BLO2]) for ${\mathcal{G}}$. Such result holds, for instance, for the examples in the forthcoming section of this paper. \[inandout\] Let ${\mathcal{G}}$ be a $p$-local compact group, and let $\Psi$ be an unstable Adams operation that approximates ${\mathcal{G}}$ by $p$-local finite groups. Then, there are natural isomorphisms $$\begin{array}{ccc} H^{\ast}(BS; {\mathbb{F}}_p) \cong \varprojlim H^{\ast}(BS_i; {\mathbb{F}}_p) & \mbox{ and } & H^{\ast}(B{\mathcal{G}}; {\mathbb{F}}_p) \cong \varprojlim H^{\ast}(B{\mathcal{G}}_i; {\mathbb{F}}_p). \end{array}$$ Let $X$ be either $B{\mathcal{G}}$ or $BS$, and similarly let $X_i$ be either $B{\mathcal{G}}_i$ or $BS_i$, depending on which case we want to prove. Consider also the homotopy colimit spectral sequence for cohomology (XII.5.7 [@BK]): $$E^{r,s}_2 = \varprojlim \!\! \phantom{i}^rH^s(X_i;{\mathbb{F}}_p) \Longrightarrow H^{r+s}(X;{\mathbb{F}}_p).$$ We will see that, for $r \geq 1$, $E_2^{r,s} = \{0\}$, which, in particular, will imply the statement. For each $s$, let $H^s_i = H^s(X_i;{\mathbb{F}}_p)$, and let $F_i$ be the induced morphism in cohomology (in degree $s$) by the map $\|\Theta_i\|$. The cohomology ring $H^{\ast}(X_i;{\mathbb{F}}_p)$ is noetherian by Theorem 5.8 [@BLO2], and in particular $H^s_i$ is a finite ${\mathbb{F}}_p$-vector space for all $s$ and all $i$. Thus, the inverse system $\{H^s_i;F_i\}$ satisfies the Mittag-Leffler condition (3.5.6 [@Weibel]), and hence the higher limits $\varprojlim^rH^s_i$ vanish for all $r \geq 1$. This in turn implies that the differentials in the above spectral sequence are all trivial, and thus it collapses. \[stable\] (Stable Elements Theorem for $p$-local compact groups). Let ${\mathcal{G}}$ be a $p$-local compact group, and suppose that there exists $\Psi$, an unstable Adams operation on ${\mathcal{G}}$, that approximates ${\mathcal{G}}$ by $p$-local finite groups. Then, the natural map $$H^{\ast}(B{\mathcal{G}}; {\mathbb{F}}_p) \stackrel{\cong} \longrightarrow H^{\ast}({\mathcal{F}}) \stackrel{def} = \varprojlim_{\mathcal{O}({\mathcal{F}}^c)} H^{\ast}(\underline{\phantom{A}}; {\mathbb{F}}_p) \subseteq H^{\ast}(BS; {\mathbb{F}}_p)$$ is an isomorphism. Since each ${\mathcal{G}}_i$ is a $p$-local finite group (for $i$ big enough), we can apply the Stable Elements Theorem for $p$-local finite groups, Theorem 5.8 [@BLO2]: there is a natural isomorphism $$H^{\ast}(B{\mathcal{G}}_i; {\mathbb{F}}_p) \stackrel{\cong} \longrightarrow H^{\ast}({\mathcal{F}}_i) = \varprojlim_{\mathcal{O}({\mathcal{F}}_i^c)} H^{\ast}(\underline{\phantom{A}}; {\mathbb{F}}_p) \subseteq H^{\ast}(BS_i; {\mathbb{F}}_p)$$ Thus, by Proposition \[inandout\], there are natural isomorphisms $$H^{\ast}(B{\mathcal{G}}; {\mathbb{F}}_p) \cong \varprojlim H^{\ast}(B{\mathcal{G}}_i; {\mathbb{F}}_p) \cong \varprojlim H^{\ast}({\mathcal{F}}_i) \subseteq \varprojlim H^{\ast}(BS_i; {\mathbb{F}}_p) \cong H^{\ast}(BS;{\mathbb{F}}_p).$$ Furthermore, the functor ${(\underline{\phantom{B}})^{\bullet}}$ induces inclusions $\mathcal{O}({\mathcal{F}}_i^c) \subseteq \mathcal{O}({\mathcal{F}}_{i+1}^c)$ in a similar fashion as it induced the functors $\Theta_i$, and $\mathcal{O}({\mathcal{F}}^{\bullet c}) = \bigcup_{i \in {\mathbb{N}}} \mathcal{O}({\mathcal{F}}_i^c)$, from where it follows that $$\varprojlim_i H^{\ast}({\mathcal{F}}_i) \stackrel{def} = \varprojlim_i \varprojlim_{\mathcal{O}({\mathcal{F}}_i^c)} H^{\ast}(\underline{\phantom{A}}; {\mathbb{F}}_p) \cong \varprojlim_{\mathcal{O}({\mathcal{F}}^c)} H^{\ast}(\underline{\phantom{A}}; {\mathbb{F}}_p) \stackrel{def} = H^{\ast}({\mathcal{F}}).$$ A general proof (i.e. for all $p$-local compact groups) of the above result would lead to a proof of Theorem 6.3 [@BLO2] in the compact case, just by doing some minor modifications in the proof for the finite case. This in turn would allow us to reproduce (most of) the work in [@BCGLO2] for $p$-local compact groups. Suppose ${\mathcal{G}}$ is approximated by $p$-local finite groups. Then, by Proposition \[centricquasicentric\], together with Theorem B [@BCGLO1], we can define a zig-zag $$\xymatrix@R=4mm@C=1cm{ \ldots \ar[rd] & \|{\mathcal{L}}_{i-1}\| \ar[d]^{\simeq} \ar[rd] & \|{\mathcal{L}}_i\| \ar[d]^{\simeq} \ar[rd] & \|{\mathcal{L}}_{i+1}\| \ar[d]^{\simeq} \ar[rd] & \\ & \|{\mathcal{L}}_{i-1}^q\| & \|{\mathcal{L}}_i^q\| & \|{\mathcal{L}}_{i+1}^q\| & \ldots\\ }$$ where, for each $i$, ${\mathcal{L}}_i^q$ is the quasi-centric linking system associated to ${\mathcal{L}}_i$ (see [@BCGLO1]). This yields another homotopy colimit, which is easily seen to be equivalent to that in Theorem \[hocolim1\]. Examples of approximations by $p$-local finite groups ===================================================== We discuss now some examples of $p$-local compact groups which are approximated by $p$-local finite groups. The first example we consider is that of $p$-local compact groups of rank $1$, which will require rather descriptive arguments. The second example is that of $p$-local compact groups induced by the compact Lie groups $U(n)$. In this case, the particular action of $S/T$ over $T$ will be the key. $p$-local compact groups of rank $1$ ------------------------------------ The main goal of this section then is to prove the following. \[rank1\] Let ${\mathcal{G}}$ be a $p$-local compact group of rank $1$. Then, every unstable Adams operation $\Psi$ approximates ${\mathcal{G}}$ by $p$-local finite groups. To prove this result we will first study some technical properties of rank $1$ $p$-local compact groups, and then apply these properties to show that condition (\[condition\]) holds always. This process will imply again fixing some finite list of objects and morphisms in ${\mathcal{L}}$ and increasing the degree of $\Psi$ so that some properties hold. The approach here is rather exhaustive, and is not appropriate to study a more general situation. All the results achieved in the previous section are assumed to hold already. \[noclassification\] Possibly the main difficulty in this section is the absence of any kind of classification of rank $1$ $p$-local compact groups which we could use to reduce to a finite list of cases to study. In this sense, an attempt of a classification was made in §3 [@Gonza], but only with partial results which are of no use here. Namely, the author proved that every rank $1$ $p$-local compact group uniquely determines a connected rank $1$ $p$-local compact group which is in fact derived from either $S^1$, $SO(3)$ or $S^3$ (the last two only occurring for $p=2$), but there is still no reasonable notion equivalent to the group of components in classical Lie group theory. Note that the above list of connected $p$-local compact groups does not contain the Sullivan spheres. This is because of the notion of connectivity used, which was rather strict but lead to stronger results, such as Corollary 3.2.5 [@Gonza], which cannot be extended to weaker notions of connectivity. Roughly speaking, the condition for a morphism (in ${\mathcal{L}}$) to be $\Psi_i$-invariant is related to the existence of morphisms (in ${\mathcal{F}}$) sending elements of $T$ to elements outside $T$ (Lemma \[detectmorph\]). Define then $S_0 \leq S$ as the minimal strongly ${\mathcal{F}}$-closed subgroup of $S$ containing $T$. It is clear by definition of $S_0$ that such subgroup (if exists) is unique. Let ${\mathcal{G}}$ be a $p$-local compact group. Then, $S_0$ always exists. Furthermore, each element $x \in S_0$ is ${\mathcal{F}}$-subconjugate to $T$. To prove the existence of $S_0$, it is enough to consider the intersection of all the strongly ${\mathcal{F}}$-closed subgroups of $S$ containing $T$, since the intersection of two strongly ${\mathcal{F}}$-closed subgroups is again strongly ${\mathcal{F}}$-closed. To prove the second part of the statement, let $S_0' = \cup P_n$, where $P_0 = T$, and $P_{n+1}$ is the subgroup of $S$ generated by $P_n$ together with all the elements of $S$ which are ${\mathcal{F}}$-subconjugate to $P_n$. This is clearly an strongly ${\mathcal{F}}$-closed subgroup, hence $S_0 \leq S_0'$. On the other hand, if $S_0 \lneqq S_0'$, then there exists $x \in S_0' \setminus S_0$ and a morphism $f: {\langle x \rangle} \to T$ in ${\mathcal{F}}$ contradicting the fact that $S_0$ is strongly ${\mathcal{F}}$-closed. Next we describe the possible isomorphism types of $S_0$ in the rank $1$ case. The following criterion will be useful. Let ${\mathcal{G}}$ be a $p$-local compact group, and let $P \leq S$ be ${\mathcal{F}}$-subconjugate to $T$. Then, $$C_P(T) \stackrel{def} = C_S(T) \cap P = T \cap P.$$ This Lemma can be understood as follows. If $x \in S$ is ${\mathcal{F}}$-conjugate to an element in $T$, then either $x$ is already an element in $T$ or $x$ acts nontrivially on $T$. Let $f:P \to T$ be a morphism in ${\mathcal{F}}$. We can assume without loss of generality that $P = {\langle x \rangle}$ and that $P' = f(P)$ is fully ${\mathcal{F}}$-centralized, since it is a subgroup of $T$. This way we can apply axiom (II) for saturated fusion systems to $f$ to see that it extends to a morphism $\widetilde{f} \in Hom_{{\mathcal{F}}}(C_S(P) \cdot P, S)$. Suppose then that $x$ acts trivially on $T$. In particular, $T \leq C_S(P)$, and thus in particular $\widetilde{f}$ restricts to $\widetilde{f}: T \cdot P \to S$. The infinitely $p$-divisibility of $T$ and the hypothesis on $f$ imply then that $\widetilde{f}(T) = T$ and $\widetilde{f}(P) \leq T$ respectively, and hence $P \leq T$. The above result implies that the quotient $S_0/T$ can be identified with a subgroup of $Aut(T) = GL_r({{\mathbb{Z}}^\wedge_p})$, where $r$ is the rank of $T$. When $r=1$, $$Aut(T) \cong \left\{ \begin{array}{ll} {\mathbb{Z}}/2 \times {{\mathbb{Z}}^\wedge_2}, & p=2,\\ {\mathbb{Z}}/(p-1) \times {{\mathbb{Z}}^\wedge_p}, & p>2,\\ \end{array} \right.$$ and we can prove the following. \[isotype\] Let ${\mathcal{G}}$ be a rank $1$ $p$-local compact group. (i) If $p >2$, then $S_0 = T$. (ii) If $p=2$, then $S_0$ has the isomorphism type of either $T$, ${D_{2^{\infty}}}= \cup D_{2^n}$ or ${Q_{2^{\infty}}}= \cup Q_{2^n}$. The case $p>2$ is immediate, since $Aut(T)$ does not contain any finite $p$-subgroup, and hence $S_0/T$ has to be trivial. Suppose then the case $p=2$. In this case, $Aut(T)$ contains a finite $2$-subgroup isomorphic to ${\mathbb{Z}}/2$, and hence either $S_0/T = \{1\}$ or $S_0/T \cong {\mathbb{Z}}/2$. If $S_0/T = \{1\}$, then $S_0 = T$ and there is nothing to prove. Suppose otherwise that $S_0/T \cong {\mathbb{Z}}/2$. Then $S_0$ fits in an extension $T \to S_0 \to {\mathbb{Z}}/2$. By IV.4.1 [@MacLane] and II.3.8 [@Adem-Milgram], the group $$H^2({\mathbb{Z}}/2;T^{\tau}) \cong {\mathbb{Z}}/2$$ classifies all possible extensions $T \to S_0 \to S_0/T$ up to isomorphism. Here, the superindex on $T$ means that the coefficients are twisted by the action of ${\mathbb{Z}}/2$ on $T$. Thus, up to isomorphism, there are only two possible discrete $2$-toral groups of rank $1$ with the desired action on $T$ and such that $S_0/T \cong {\mathbb{Z}}/2$, and the proof is finished since both ${D_{2^{\infty}}}$ and ${Q_{2^{\infty}}}$ satisfy these conditions and are non-isomorphic. The proof of Theorem \[rank1\] will be done by cases, depending on the isomorphism type of $S_0$. As happened when proving that ${\mathcal{L}}_i$ is a transporter system associated to ${\mathcal{F}}_i$ (Theorem \[transpi\]), proving Theorem \[rank1\] will require fixing some finite list of objects and morphisms in ${\mathcal{F}}$ and considering operations $\Psi_i$ of degree high enough. \[BI3\] More specifically, we fix (i) a set ${\mathcal{P}}'$ of representatives of the $S$-conjugacy classes of non-${\mathcal{F}}$-centric objects in ${\mathcal{F}}^{\bullet}$; and (ii) for each pair $H, K \in {\mathcal{P}}'$ such that $K$ is fully ${\mathcal{F}}$-normalized, a set ${\mathcal{M}}_{H,K} \subseteq Hom_{{\mathcal{F}}}(H,K)$ of the classes in $Rep_{{\mathcal{F}}}(H,K)$. (iii) for each $f \in {\mathcal{M}}_{H,K}$ above, an “Alperin-like” decomposition (Theorem \[Alperin\]) $$\label{Alperinlike} \xymatrix@R=2mm@C=4mm{ & & & & & & & & & & & \\ & L_1 \ar[r]^{\gamma_1} & L_1 & & L_2 \ar[r]^{\gamma_2} & L_2 & & & & L_k \ar[r]^{\gamma_k} & L_k & \\ & & & & & & & & & & & \\ R_0 \ar[rrr]_{f_1} \ar[ruu] & & & R_1 \ar[luu] \ar[ruu] \ar[rrr]_{f_2} & & & R_2 \ar[luu] \ar[r] & \ldots \ar[r] & R_{k-1} \ar[ruu] \ar[rrr]_{f_k} & & & R_k, \ar[luu], \\ }$$ where $R_0 = H$, $R_k = K$, $L_j$ is ${\mathcal{F}}$-centric ${\mathcal{F}}$-radical and fully ${\mathcal{F}}$-normalized for $j = 1, \ldots, k$, and $$f_R = f_k \circ f_{k-1} \circ \ldots \circ f_2 \circ f_1.$$ (iv) for each $\gamma_j$ above, a lifting $\varphi_j$ in ${\mathcal{L}}$. This is clearly a finite list, and hence by Proposition \[finitesetinv\], there exists some $M_{\Psi}' \geq 0$ such that, for all $i \geq M_{\Psi}'$, all the subgroups in ${\mathcal{P}}'$ are $S_i$-determined and all the morphisms $\varphi_j$ are morphisms in ${\mathcal{L}}_i$. \[morphTid\] Let $H \in {\mathcal{P}}'$ be $S$-centric, and let $K \in {\mathcal{P}}' \cap {{H}^{{\mathcal{F}}}}$ be any non-$S$-centric object. Then, the set $Hom_{{\mathcal{F}}}(H,K)$ contains an element $f$, together with a decomposition as (\[Alperinlike\]), such that for all $j=1, \ldots, k$ $$f_j(C_T(R_{j-1})) \leq T.$$ Suppose first that $S_0 = T$. Since in this case $T$ is strongly ${\mathcal{F}}$-closed, the condition holds by axiom (C) for linking systems. Also if $S_0 \cong {Q_{2^{\infty}}}$ it is easy to see that $T_1 \leq T$ (the order $2$ subgroup of $T$) is strongly ${\mathcal{F}}$-closed (in fact it is ${\mathcal{F}}$-centra), and either $C_T(R) = T$ or $C_T(R) = T_1$. In both cases then the statement follows easily by axiom (C) of linking systems and the properties of $T$ and $T_1$. We are thus left to consider the case $S_0 \cong {D_{2^{\infty}}}$. Note that in this case every element in the quotient $S/S_0$ acts trivially on $T$. Also, $Z(S) \cap T = T_1$, but now this subgroup is not strongly ${\mathcal{F}}$-closed, and the subgroups $T_n$, $n \geq 2$, are all weakly ${\mathcal{F}}$-closed (this holds since the only elements of $S_0$ of order $2^n$ are all in $T$). Set for simplicity $L = L_j$, $\varphi = \varphi_j$ and $f = f_j$. If $C_T(L) \geq T_n$ for some $n \geq 2$, then the condition above holds directly by axiom (C) for linking systems, since $T_n$ is weakly ${\mathcal{F}}$-closed. We can assume thus that $C_T(L) = T_1$. Even more, if $L \cap S_0 = T_1$, then the condition above still holds since $S_0$ is strongly ${\mathcal{F}}$-closed. By inspection of $S_0$, this leaves only one case to deal with $$L_0 \stackrel{def} = L \cap S_0 = {\langle x, T_1 \rangle} = R_{j-1} \cap S_0 = R_j \cap S_0 \cong {\mathbb{Z}}/2 \times {\mathbb{Z}}/2$$ for some element $x$ which has order $2$. If we set $t_2$ for a generator of $T_2 \leq T$, then it is also easy to check that $t_2$ normalizes $L_0$, and in fact, since $L/L_0$ acts trivially on $T$, it also normalizes $L, R_{j-1}$ and $R_j$. Set also $t_1$ for the generator of $T_1$. The automorphism group of $L_0$ is isomorphic to $\Sigma_3$, generated by $c_{t_2}$ together with an automorphism $f_0$ of order $3$ which sends $t_1$ to $x$ and $x$ to $xt_1$. Note that the assumption that $S_0 \cong {D_{2^{\infty}}}$ implies that $Aut_{{\mathcal{F}}}(L_0) \cong \Sigma_3$. For the purposes of the proof we can now assume that $f$ restricts to $f_0$. Let then $\omega = f^{-1} \circ c_{t_2} \circ f^{-1} \circ c_{t_2}^{-1}$. It is easy to see that $\omega$ induces the identity on $L/L_0$, and by inspecting the automorphism group of $L_0$ it follows that $\omega_{\|L_0} = f_0$. Consider now $f' = \omega^{-1} \circ f$. By definition, $f'$ induces the same automorphism on $L/L_0$ as $f$, and the identity on $L_0$. To show that we can replace $f$ by $f'$ we have to show that the image of $R_{j-1}$ by $f$ and $f'$ are the same: $$\xymatrix{ R_{j-1} \ar[r]^{f} & R_j \ar[r]^{f} & R_j' \ar[r]^{c_{t_2}^{-1}} & R_j' \ar[r]^{f^{-1}} & R_j \ar[r]^{c_{t_2}} & R_j,\\ }$$ where $R_j' = f(R_j)$ is normalized by $t_2$ by the above arguments. We can assume then that, for each pair $H,K \in {\mathcal{P}}'$ (with $C_S(K)\geq Z(K)$) the set ${\mathcal{M}}_{H,K}$ fixed in Remark \[BI3\] contains at least a morphism $f$ satisfying Lemma \[morphTid\] above. (of Theorem \[rank1\]). Recall that, after Theorem \[transpi\] and by Proposition 3.6 [@OV], we only have to prove that there exists some $M_{\Psi}'$ such that, for all $i \geq M_{\Psi}'$, condition (\[condition\]) holds for all $H \in Ob({\mathcal{F}}_i^c) \setminus Ob({\mathcal{L}}_i)$. Actually we will prove the following: - there exists some $M_{\Psi}'$ such that, for all $i \geq M_{\Psi}'$, $Ob({\mathcal{F}}_i^c) = Ob({\mathcal{L}}_i)$. Using the functor ${(\underline{\phantom{B}})^{\bullet}}_i$, it is enough to prove that there exists such $M_{\Psi}$ such that, for all $S_i$-determined subgroups $R$, $R_i$ is ${\mathcal{F}}_i$-centric if and only if $R$ is ${\mathcal{F}}$-centric. Recall that Corollary \[corcentral1\] proves the “if” implication in the above claim. Furthermore, Corollary \[corcentral2\] says that if $R$ is not $S$-centric, then $R_i$ is not $S_i$-centric. The rest of the proof is then devoted to show that if $R_i \leq S_i$ is an $S_i$-root such that $R = (R_i)^{\bullet}$ is $S$-centric but not ${\mathcal{F}}$-centric, then $R_i$ is not ${\mathcal{F}}_i$-centric. We can also assume that $R_i$ is maximal in the sense that if $Q_i \leq S_i$ is such that $R_i \lneqq Q_i$, then either $Q_i$ is ${\mathcal{F}}_i$-centric or it is not an $S_i$-root. Let $H \in {{R}^{S}}$ be the representative of this $S$-conjugacy class fixed in ${\mathcal{P}}'$ (Remark \[BI3\]), and let $K \in {\mathcal{P}}' \cap {{R}^{{\mathcal{F}}}}$ be fully ${\mathcal{F}}$-normalized. Note that both $H_i$ and $K_i$ are not ${\mathcal{F}}_i$-centric by assumption (Remark \[BI3\]). Let $f \in {\mathcal{M}}_{H,K}$ be as in Lemma \[morphTid\], and let $$\xymatrix@R=2mm@C=4mm{ & & & & & & & & & & & \\ & L_1 \ar[r]^{\gamma_1} & L_1 & & L_2 \ar[r]^{\gamma_2} & L_2 & & & & L_k \ar[r]^{\gamma_k} & L_k & \\ & & & & & & & & & & & \\ R_0 \ar[rrr]_{f_1} \ar[ruu] & & & R_1 \ar[luu] \ar[ruu] \ar[rrr]_{f_2} & & & R_2 \ar[luu] \ar[r] & \ldots \ar[r] & R_{k-1} \ar[ruu] \ar[rrr]_{f_k} & & & R_k, \ar[luu], \\ }$$ be the decomposition (\[Alperinlike\]) fixed in Remark \[BI3\] for the morphism $f$, together with the liftings $\varphi_j \in Aut_{{\mathcal{L}}}(L_j)$. Let also $x \in N_S(H_i,R_i)$. By Lemma \[rootconjx\], $\Psi_i(x) x^{-1} \in C_T(R)$, or, equivalently, $\tau_0 = x^{-1} \Psi_i(x) \in C_T(H_i) = C_T(H)$. We can now apply axiom (C) to $\varphi_1$ and the element $\tau_0$. By hypothesis (Lemma \[morphTid\]), $f_1(C_T(H)) \leq T$, so in particular $f_1(\tau_0) = \tau_1$ for some $\tau_1 \in T$. Let then $t \in T$ be such that $$\tau_1 = t^{-1} \Psi_i(t),$$ and let $Q_1 = tR_1t^{-1}$, $L'_1 = t^{-1} L_1 t$ and $\varphi_1' = \delta(t) \circ \varphi_1 \circ \delta(x^{-1}) \in Aut_{{\mathcal{L}}}(L'_1)$. It follows from Lemma \[detectmorph\] that $\varphi_1'$ is $\Psi_i$-invariant, and $L_1'$ (or a certain proper subgroup) is ${\mathcal{F}}$-centric and $S_i$-determined. Proceeding inductively through the whole sequence $f_1, \ldots, f_k$, we see that ${\mathcal{F}}_i$ contains a morphism sending $R_i$ to a subgroup $Q_i$ which is not $S_i$-centric, and hence $R_i$ is not ${\mathcal{F}}_i$-centric. Since $p$-local compact groups of rank $1$ are approximated by $p$-local finite groups, we know (Theorem \[stable\]) that the Stable Elements Theorem hold for all of them. This result was used in [@BLO2] to prove Theorem 6.3, which states that, given a $p$-local finite group ${\mathcal{G}}$, a finite group $Q$ and a homomorphism $\rho: Q \to S$ such that $\rho(Q)$ is fully centralized in ${\mathcal{F}}$, there is a homotopy equivalence $$\|C_{{\mathcal{L}}}(\rho(Q))\|^{\wedge}_p \stackrel{\simeq} \longrightarrow Map(BQ, B{\mathcal{G}})_{B\rho},$$ where $C_{{\mathcal{L}}}(\rho(Q))$ is the centralizer linking system defined in appendix §A [@BLO2]. The proof for this result in [@BLO2] used an induction step on the order of $S$ and on the “size” of ${\mathcal{L}}$. However, since rank $0$ $p$-local compact groups are just $p$-local finite groups, we could use the same argument to prove the above statement for $p$-local compact groups of rank $1$, with some minor modifications. We have skipped it in this paper since it is not a general argument (it would only apply to $p$-local compact groups of rank $1$), and requires a rather long proof. The unitary groups $U(n)$ ------------------------- We prove now that the $p$-local compact groups induced by the compact Lie groups $U(n)$, $n \geq 1$, are approximated by $p$-local finite groups. As proved in Theorem 9.10 [@BLO3], every compact Lie group $G$ gives rise to a $p$-local compact group ${\mathcal{G}}$ such that $(BG)^{\wedge}_p \simeq B{\mathcal{G}}$. \[Un\] Let ${\mathcal{G}}(n)$ be the $p$-local compact group induced by the compact Lie group $U(n)$. Then every unstable Adams operation $\Psi$ approximates ${\mathcal{G}}(n)$ by $p$-local finite groups. The key point in proving this result is the particular isomorphism type of the Sylow subgroups of $U(n)$. Indeed, the Weyl group $W_n$ of a maximal torus of $U(n)$ is (isomorphic to) the symmetric group on $n$-letters, $\Sigma_n$. The action of $W_n$ on the maximal tori of $U(n)$ is easier to understand on the maximal torus of $U(n)$ formed by the diagonal matrices, $T$, where it acts by permuting the $n$ nontrivial entries of a diagonal matrix (see §3 [@Mimura-Toda] for further details). Furthermore, the following extension is split $$T \longrightarrow N_{U(n)}(T) \stackrel{\pi} \longrightarrow W_n.$$ Let us fix some notation. Let $\{t_k\}_{k \geq 0}$ be a basis for ${{\mathbb{Z}}/p^{\infty}}$, that is, each $t_k$ has order $p^k$, and $t_{k+1}^p = t_k$ for all $k$. Let also $T = ({{\mathbb{Z}}/p^{\infty}})^n$, with basis $\{(t_{k_1}^{(1)}, \ldots, t_{k_n}^{(n)})_{k_1, \ldots, k_n \geq 0}\}$. This way, the symmetric group $\Sigma_n$ acts on $T$ by permuting the superindexes. In addition, if $\Sigma \in Syl_p(\Sigma_n)$, then $S = T \rtimes \Sigma$ can be identified with the Sylow $p$-subgroup of the $p$-local compact group ${\mathcal{G}}(n)$. \[centralizerUn\] Let $P \leq S$. Then, $C_T(P)$ is a discrete $p$-subtorus of $T$. We proof that every element in $C_T(P)$ is infinitely $p$-divisible. Let $\pi = P/(P \cap T) \leq \Sigma \leq \Sigma_n$, and let $t \in C_T(P)$. Note that this means that $xtx^{-1} = t$ for all $x \in P$. In the basis that we have fixed above, $t = (\lambda_1 t_{k_1}^{(1)}, \ldots, \lambda_n t_{k_n}^{(n)})$, where the coefficients $\lambda_j$ are in $({\mathbb{Z}}/p)^\times$, and, for all $\sigma \in \pi$, if $\sigma(j) = l$, then $\lambda_j = \lambda_l$ and $k_j = k_l$. For each orbit of the action of $\pi$ in the set $\{1, \ldots, n\}$ let $j$ be a representative. Let also $t_j = \lambda_j t_{k_j}^{(j)}$ be the $j$-th coordinate of $t$, and let $u_j$ be a $p$-th root of $t_j$ in ${{\mathbb{Z}}/p^{\infty}}$. We can consider the element $t' \in T$ which, in the coordinate $l$, has the $p$-th root $u_j$ which corresponds to the orbit of $l$ in $\{1, \ldots, n\}$ under the action of $\pi$. This element is then easily seen to be a $p$-th root of $t$, as well as invariant under the action of $\pi$. Thus, $t' \in C_T(P)$, and this proves that every element in $C_T(P)$ is infinitely $p$-divisible. (of Theorem \[Un\]). We first prove the following statement: - Let $P \leq S$. There exists some $M'_P \geq 0$ such that, for all $i \geq M'_P$, if $R \in {{P}^{S}}$ is $S_i$-determined, then $R_i$ is $S_i$-conjugate to $P_i$. By Lemma \[rootconjx\], for all $y \in N_S(P_i,R_i)$ we have $y^{-1} \Psi_i(y) \in C_T(P_i)$. Also, since $S$ is $S_i$-determined, the subgroup $S_i$ contains representatives of all the elements in $\Sigma$, and hence we can assume that $y \in T$. Consider now the map $$\xymatrix@R=1mm{ T \ar[rr]^{\Psi_i^{\ast}} & & T\\ t \ar@{\|->}[rr] & & t^{-1} \Psi_i(t).\\ }$$ Since $T$ is abelian this is a group homomorphism for all $i$, and in fact it is epi by the infinitely $p$-divisibility property of $T$. The kernel of $\Psi_i^{\ast}$ is the subgroup of fixed elements of $T$ under $\Psi_i$. Also, this morphism sends each cyclic subgroup of $T$ to itself. It follows now by Lemma \[centralizerUn\] that $y \in N_S(P_i, R_i)$ has the form $y = t_1 t_2$, where $t_1 \in T_i = Ker(\Psi_i^{\ast})$ and $t_2 \in C_T(P_i)$, and hence $R_i$ is $S_i$-conjugate to $P_i$. In particular, since ${\mathcal{F}}^{\bullet}$ contains only finitely many $S$-conjugacy classes of non-${\mathcal{F}}$-centric objects, and using the above claim, it is clear that there exists some $M$ such that, for all $i \geq M$ and each $S_i$-determined subgroup $R$, $R$ is ${\mathcal{F}}$-centric if and only if $R_i$ is ${\mathcal{F}}_i$-centric. Hence, by Proposition 3.6 [@OV], it follows that for $i \geq M$ the fusion system ${\mathcal{F}}_i$ is saturated. The arguments to prove Theorem \[Un\] do not apply to any of the other families of compact connected Lie groups. 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--- abstract: 'On August 14, 2019, the LIGO and Virgo detectors observed GW190814, a gravitational-wave signal originating from the merger of a black hole with a compact object. GW190814’s compact-binary source is atypical both in its highly asymmetric masses and in its lower-mass component lying between the heaviest known neutron star and lightest known black hole in a compact-object binary. If formed through isolated binary evolution, the mass of the secondary is indicative of its mass at birth. We examine the formation of such systems through isolated binary evolution across a suite of assumptions encapsulating many physical uncertainties in massive-star binary evolution. We update how mass loss is implemented for the neutronization process during the collapse of the proto-compact object to eliminate artificial gaps in the mass spectrum at the transition between neutron stars and black holes. We find it challenging for population modeling to match the empirical rate of GW190814-like systems whilst simultaneously being consistent with the rates of other compact binary populations inferred by gravitational-wave observations. Nonetheless, the formation of GW190814-like systems at any measurable rate requires a supernova engine model that acts on longer timescales such that the proto-compact object can undergo substantial accretion immediately prior to explosion, hinting that if GW190814 is the result of massive-star binary evolution, the mass gap between neutron stars and black holes may be narrower or nonexistent.' author: - Michael Zevin - Mario Spera - Christopher P L Berry - Vicky Kalogera bibliography: - 'library.bib' title: Exploring the Lower Mass Gap and Unequal Mass Regime in Compact Binary Evolution --- [^1] Introduction {#sec:intro} ============ The third observing run of the Advanced LIGO–Virgo network [@aLIGO; @aVirgo] has already yielded unprecedented discoveries: the most massive system [@GW190425], and compact binaries with significantly asymmetric masses [@GW190412; @GW190814]. The most recently announced event, GW190814 [@GW190814], also had a compact object component that lies within the observational gap in masses between and , known as the lower mass gap [@Bailyn1998; @Ozel2010; @Farr2011; @Ozel2012]. Since no tidal signatures are measurable in the data and no electromagnetic or neutrino counterpart has been reported [see @GW190814 and references therein], the nature of the lighter object in the binary is uncertain. Nonetheless, this event establishes that compact objects do exist in binaries. The majority of non-recycled in the Galaxy have masses of ${\sim\,1.3\,\Msun}$ [@Ozel2012; @Kiziltan2013]. However, the maximum mass that a can achieve, $m_{\rm NS}^{\rm max}$, is currently uncertain. The Galactic millisecond pulsar J0740+6620 has a Shapiro-delay mass measurement of [$68\%$ credibility; @Cromartie2020]; this has been updated to when analyzed using a population-informed prior [@Farr2020]. The pulsar J1748$-$2021B has been estimated to have a mass of ($68\%$ confidence) assuming that the periastron precession is purely relativistic, but if there are contributions from the tidal or rotational distortion of the companion, this estimate would not be valid [@Freire2008a]. GW190425’s primary component had a mass greater than most Galactic , [$90\%$ credibility; @GW190425]; while high masses of $\lesssim\,2.5\,\Msun$ can be explained theoretically via stable accretion in low- and intermediate-mass systems [e.g., @Pfahl2003; @Lin2011; @Tauris2011], the explanation for a high-mass component in a system is open to debate [e.g., @Romero-Shaw2020; @Safarzadeh2020d; @Kruckow2020]. These observational constraints provide key insights about the ; although candidate for non-rotating can have maximum masses that extend as high as $\sim 3\,\Msun$ [@Rhoades1974; @Muller1996; @Kalogera1996a], population studies of known [@Alsing2018; @Farr2020], analysis of the tidal deformability of GW170817 [@GW170817_EoS; @Essick2020], modeling of the electromagnetic counterparts associated with GW170817 [@GW170817_MMA; @GW170817_kilonova; @Kasen2017; @Cowperthwaite2017; @Villar2017; @Margalit2017], and constraints from late-time observations of short gamma-ray bursts [@Schroeder2020] suggest a maximum mass of ${\lesssim\,2.7\,\Msun}$. The upper end of the lower mass gap is motivated by mass determinations in systems [e.g. @Bailyn1998; @Ozel2010; @Farr2011]. Though the lowest-mass candidate to date is between $2.6$–$6.1\,\Msun$ ($95\%$ confidence; @Thompson2019, see also the discussion in @vandenHeuvel2020), most observed in the Galaxy have masses $\gtrsim 5\,\Msun$. Selection effects may be affecting the observational sample [@Kreidberg2012], though it has been argued that such biases do not affect this broad picture [e.g., @Ozel2010]. A gap (or lack thereof) in the compact-object mass spectrum offers insights into the underlying mechanism responsible for their formation. In particular, if instability growth and launch of the proceed on rapid timescales ($\sim 10~\mathrm{ms}$ and $\sim 100~\mathrm{ms}$, respectively), stellar modeling and hydrodynamic simulations predict a dearth in remnant masses between $\sim 2$–$5\,\Msun$ [see @Fryer2012; @Belczynski2012a; @Muller2016]. Alternatively, if instabilities are delayed and develop over longer timescales ($\gtrsim 200~\mathrm{ms}$), accretion can occur on the proto- before the neutrino-driven explosion and this gap would be filled [@Fryer2012]. The mass distributions of and observed in our Galaxy provided initial evidence that this phenomenon proceeds on rapid timescales, and such prescriptions were therefore inherited by many rapid population studies examining the properties of and systems [e.g., @Dominik2012; @Belczynski2014; @Breivik2016a; @Rodriguez2016a; @Belczynski2016b; @Mapelli2018a; @Giacobbo2018b; @Kremer2019b; @Neijssel2019; @Spera2019; @Zevin2019b; @DiCarlo2020; @Rastello2020; @Banerjee2020]. However, recently discovered Galactic compact objects with mass estimates that extend inside the lower mass gap [e.g., @Thompson2019; @Wyrzykowski2020] and population fits to know Galactic [@Vigna-Gomez2018] raise tension with this interpretation; additional observations could further constrain physics [e.g., @Breivik2019]. GW190814 offers an unprecedented probe into the gap of compact object masses between and . The mass of the binary’s secondary component is ${m_2 = \SecondaryMassNinety}$ ($90\%$ credibility), making it the heaviest /lightest ever identified in a compact-object binary. The secondary of GW190814 has the potential to provide insights into the explosion mechanism, as it is a relatively clean probe of the compact object’s mass at birth. Additionally, the more massive primary component is ${m_1\,\PrimaryMassApprox}$, making this the most asymmetric compact binary discovered, with a mass ratio of ${q = m_2/m_1 \MassRatioApprox}$ [cf. @GW190412]. We investigate compact object formation in the lower mass gap and the formation of highly asymmetric compact binaries. We focus on the formation of GW190814-like systems using various standard assumptions for binary evolution, and if population models can produce such systems while being consistent with the empirical merger rates of different compact-binary populations. We do not make specific alterations to our population models to preferentially form GW190814-like systems, but instead explore how uncertain model parameters affect the formation rate and properties of such systems. In Sec. \[sec:populations\], we give an overview of our population models and present an updated prescription for determining remnant masses. We then discuss our model results in Sec. \[sec:results\], including the merger rates and formation pathways of GW190814-like systems. We explain implications for binary evolution physics in Sec. \[sec:conclusions\]. Throughout we assume Solar metallicity of ${Z_\odot=0.017}$ [@Grevesse1998] and Planck 2015 cosmological parameters of ${H_0 = 68~\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}}$, ${\Omega_\mathrm{m} = 0.31}$ and ${\Omega_{\Lambda} = 0.69}$ [@PlanckCollaboration2016]. ![image](Mzams_Mrem.png){width="\textwidth"} Population Models {#sec:populations} ================= We use the rapid binary population synthesis code `COSMIC` [@COSMIC] to examine the properties and rates of compact binaries.[^2] We investigate the impact of: initial binary properties, efficiency of evolution, survival in the phase, the determination of remnant masses, and natal kick prescriptions. We describe `COSMIC` and our model assumptions in Appendix \[app:pop\_models\], and our model variations are summarized in Table \[tab:table\]. Below we highlight updates to `COSMIC` that are pertinent to this work. Remnant Mass Prescription {#subsec:remnant_mass} ------------------------- We follow the remnant mass prescriptions from @Fryer2012 for determining the baryonic mass of the proto-compact object. The two prescriptions from @Fryer2012, Rapid and Delayed, are used to map the results of hydrodynamical simulations to rapid population synthesis. The two models differ by their assumed instability growth timescale (${\sim 10~\mathrm{ms}}$ and ${\sim 200~\mathrm{ms}}$ for Rapid and Delayed, respectively), with the Rapid model naturally leading to a low-mass gap. We make a small, but important, change when determining the final gravitational mass from the baryonic mass of the proto-compact object; further motivation and details are in Appendix \[app:remnant\_mass\]. Rather than assuming a fixed fractional mass loss of the total pre- mass for as in @Fryer2012, we cap the mass loss due to neutronization to $10\%$ of the maximum mass assumed for the iron core, as hydrodynamical simulations show that the mass loss from neutrino emission is $\sim 10\%$ of the iron-core mass rather than the total baryonic mass of the compact-object progenitor (C. Fryer 2020, private communication). Combining this criteria with the baryonic-to-gravitational mass prescription from @Lattimer1989 gives $$M_{\rm grav} = \begin{cases} \displaystyle \frac{20}{3} \left[(1 + 0.3 M_{\rm bar})^{1/2} - 1\right]& \Delta M\leq 0.1\,m_{\rm Fe}^{\rm max}\\ M_{\rm bar}-0.1\,m_{\rm Fe}^{\rm max}& \mathrm{otherwise}\\ \end{cases},$$ where $\Delta M = M_{\rm bar}-M_{\rm grav}$ and $m_{\rm Fe}^{\rm max}$ is the maximum possible mass assumed for the iron core, which we set to $5\,\Msun$. This upper limit on the mass loss when converting from baryonic to gravitational mass is slightly larger, though qualitatively similar to the procedure in [@Mandel2020a]. Whether the compact remnant is a or a can then be determined by comparing $M_{\rm grav}$ to $m_{\rm NS}^{\rm max}$. We show the mass–remnant mass relation for both our prescriptions in Fig. \[fig:Mzams\_Mrem\]. Merger Rates and Astrophysical Populations {#subsec:merger_rates} ------------------------------------------ Determining local merger rate densities from population synthesis can be used to directly compare population predictions with the empirical rates measured by LIGO–Virgo. We calculate local merger rates of different compact binary populations in a similar manner as [@Giacobbo2018b] and [@Spera2019], described in detail in Appendix \[app:merger\_rates\]. For compact binary coalescence class $i$, the local merger rate (i.e., the merger rate at redshifts ${z \leq z_{\rm loc}} = 0.01$) across all metallicity models $j$ is $$\mathcal{R}_{\mathrm{loc},\,i} = \frac{1}{t_\mathrm{l}(z_{\rm loc})} \int_{0}^{z_{\rm max}} \psi(z) \sum_{j}p(Z_j \| z) f_{\mathrm{loc},\,i}(z,Z_j) \frac{\mathrm{d} t_\mathrm{l}}{\mathrm{d}z} \mathrm{d}z,$$ where $t_\mathrm{l}$ is the lookback time, $\psi(z)$ is the star formation rate density, $p(Z_j \| z)$ is the likelihood of metallicity $Z_j$ at redshift $z$, $f_{\mathrm{loc},\,i}(z,Z_j)$ is the mass fraction of systems born at redshift $z$ with metallicity $Z_j$ that merge in the local universe, and $z_\mathrm{max} = 15$ is the maximum redshift we consider for binary formation. To get a representative astrophysical population of compact binary mergers, we combine the information from across metallicities for each population model. We assume that all systems merge at the median measured redshift of GW190814 (${z\,\EventRedshift}$).[^3] The delay time of each system thus provides its formation time and formation redshift. We then give a weight to each system based on the mass-weighted star formation weight (Eq. \[eq:sfr\]) at the formation redshift, the metallicity distribution (Eq. \[eq:met\]) at the formation redshift, the relative formation efficiency for the double compact object population in question in the particular metallicity model, and the relative number of systems formed in a particular metallicity model. Drawing a subset of systems based on these weights gives a representative sample at the merger redshift of GW190814. Results {#sec:results} ======= ![Total mass $M_\mathrm{tot} = m_1+m_2$ and mass ratio ${q =m_2/m_1}$ (${m_2 \leq m_1}$) distributions of astrophysical populations for models varying the efficiency $\alpha$ and remnant-mass prescription. For the other model variations, we use initial conditions from @Sana2012, the Bimodal natal-kick model, and the Pessimistic assumption for survival (see Appendix \[app:model\_assumptions\]). We include all systems with at least one component having ${m\,>\,3\,\Msun}$ in the plotted populations. Systems are drawn from metallicity models based on the star formation history and metallicity evolution outlined in Sec. \[subsec:merger\_rates\], assuming they merge at the redshift of GW190814. Densities are log-scaled, with contours containing $90\%$, $99\%$, and $99.9\%$ of systems. GW190814’s $M_\mathrm{tot}$ and $q$ are shown with a pink star; error bars are present, but are too small to be seen compared to the marker. []{data-label="fig:Mtot_q_credible"}](Mtot_q_credible.png){width="48.00000%"} We explore the properties and rates of compact binaries in our population models, focusing on how standard population modeling and variations of physical assumptions inherent to binary stellar evolution impact the formation rate of asymmetric-mass binaries and mergers with components residing in the lower mass gap, especially GW190814-like systems. Given the uncertainty in $m_{\rm NS}^{\rm max}$, we show combined distributions for all systems with at least one component having a mass ${> 3\,\Msun}$ unless otherwise specified. Probing the Low Mass Ratio Regime --------------------------------- ![Primary mass $m_1$ and secondary mass $m_2$ for the same models as in Fig. \[fig:Mtot\_q\_credible\]. Systems are colored by their metallicity. GW190814’s component masses are shown with a pink star; error bars are too small to be seen compared to the marker. []{data-label="fig:m1_m2_scatter"}](m1_m2_scatter.png){width="48.00000%"} We find that merger rates drop precipitously as component masses become more asymmetric, in agreement with many other population synthesis studies (e.g., @Dominik2012 [@Mapelli2018a; @Kruckow2018; @Neijssel2019], though see also @Eldridge2016 [@Eldridge2017]). Figure \[fig:Mtot\_q\_credible\] shows the distribution of mass ratio and total mass in a subset of our populations, with contour lines marking $90\%$, $99\%$, and $99.9\%$ of the population. Mass ratios are concentrated near unity for large total masses; since the pair instability process limits the maximum mass of [e.g., @Woosley2017; @Marchant2019; @Farmer2019], the degree of possible asymmetry decreases as a function of total mass. For systems with lower total mass, the mass ratio distribution extends to more asymmetric configurations, reaching down to ${q \lesssim 0.1}$. We do not see a strong difference in the distributions of total mass and mass ratio when varying the efficiency of ejection. However, for the Rapid mechanism we find fewer mass ratios near ${q \sim 0.4}$, and an island at lower mass ratios, whereas there is a continuum for the Delayed mechanism. This is a byproduct of the lower mass gap that is inherent to the Rapid prescription; since this gap extends from ${\sim 3}$–${6\,\Msun}$, a system with ${M_{\rm tot}=15\,\Msun}$ cannot have a mass ratio between ${\sim 0.2}$–${0.4}$. However, even with the Delayed mechanism we find ${\sim 90\%}$ of systems to have mass ratios of ${q\,\DelayedqTenPercentApprox}$. Regardless of our model assumptions, we find GW190814’s mass ratio and total mass to be an outlier, lying close to the contour for our populations. Populating the Lower Mass Gap ----------------------------- The impact of the lower mass gap is more apparent when examining systems’ component masses. Figure \[fig:m1\_m2\_scatter\] shows the primary and secondary masses of systems merging at the redshift of GW190814 for a subset of our populations. Systems are more sparse when moving away from equal mass. Although rare, we do find systems matching the GW190814’s component masses when using the Delayed prescription, which naturally populates the lower mass gap. However, it is impossible to form GW190814-like systems in our models using the Rapid prescription. ![Cumulative distribution function for the mass ratio $q$ and secondary mass $m_2$ for the same models presented in Fig. \[fig:Mtot\_q\_credible\]. Solid lines show the combined population for all systems containing at least one component with ${m\,>\,3\,\Msun}$, whereas dashed and dotted lines show the corresponding cumulative distribution functions for the (${m_1 > 3\,\Msun,\ m_2 > 3\,\Msun}$) and (${m_1 > 3\,\Msun,\ m_2 \leq 3\,\Msun}$) populations, respectively. The dotted pink line and shaded region show the median and $90\%$ credible interval for GW190814. []{data-label="fig:q_m2_cdf"}](q_m2_cdf.png){width="48.00000%"} In Fig. \[fig:q\_m2\_cdf\], we show cumulative distributions for the mass ratio and secondary mass in our populations. In the full population, we find that one in systems have a mass ratio similar to GW190814 or lower (${q\,\leq\,\MassRatioHigh}$). For systems with a secondary mass ${\leq\,3\,\Msun}$, of systems have a mass ratio of ${q\,\leq\,\MassRatioHigh}$, though this drops to when a more efficient is assumed, which is qualitatively similar to findings from other population synthesis work [e.g., @Giacobbo2018b]. For systems with a secondary mass ${>\,3\,\Msun}$, the mass ratio distribution deviates significantly when assuming the Rapid mechanism compared to the Delayed mechanism. We find of these systems have mass ratios of ${q\,\leq\,0.12}$ with a Delayed mechanism, whereas these systems are nonexistent when assuming a Rapid mechanism. The lower mass gap can be seen in the bottom panel of Fig. \[fig:q\_m2\_cdf\] as a plateau in the Rapid models as a function of $m_2$; the Delayed models, which populate the gap, have a more gradual buildup. For Delayed models, GW190814’s secondary mass lies at the percentile of the full population for ${\alpha=5}$, and drops to the percentile for ${\alpha=1}$. Compact Binary Merger Rates {#subsec:rate_predictions} --------------------------- Merger rates are a useful diagnostic for comparing predictions of population synthesis modeling to the empirical merger rate estimated by . Figure \[fig:model\_rates\] shows local merger rates of different compact binary populations for four variations of model assumptions. To compare with rates, we assume that compact objects with masses $\leq 3\,\Msun$ are , and those with masses $> 3\,\Msun$ are . For the four models examined, we find , , and merger rates to be consistent with the measured rate (bands for and , upper limit for ). We do not expect exact agreement in rates, since the results were calculated using mass distributions that are different from our populations. The single-event rate for GW190814-like systems is $1$–$23$Gpc$^{-3}$yr$^{-1}$ [90% credible level; @GW190814], and is shown with a pink band in Fig. \[fig:model\_rates\]. To compare our model predictions with the empirical rate, we choose two approximations for identifying GW190814-like systems in our models: a Narrow GW190814-like rate where we choose systems with ${0.06 \leq q \leq 0.16}$ and ${20 \leq M_{\rm tot}/\Msun\,\leq\,30}$ (pink stars) and a Broad GW190814-like rate where we choose systems with ${q \leq\,0.2}$ and ${M_{\rm tot}/\Msun \geq 20}$. In both cases, we find the local merger rate of GW190814-like systems to be over an order of magnitude lower than the empirical GW190814 rate. For example, in our model with a Delayed mechanism and $\alpha=1$, we find a local merger rate of for our Broad GW190814-like assumption and for our Narrow GW190814-like assumption; for a more efficient , the Broad and Narrow rates drop by a factor of and , respectively. Merger rates for our other model variations are presented in Table \[tab:table\]. ![Local merger rates for models where we vary the efficiency and remnant mass prescription. The blue and orange shaded regions mark the $90\%$ credible level for the current empirical [@GWTC1] and [@GW190425] merger rates from the , respectively, the green line marks the $90\%$ credible upper limit for the rate [@GWTC1], and the pink shaded region marks the $90\%$ credible single-event rate for GW190814-like systems [@GW190814]. Matching colored symbols mark the (blue triangles), (orange diamonds), and (green squares) merger rates from our models, assuming ${m_{\rm NS}^{\rm max} = 3\,\Msun}$. Pink symbols mark the merger rates from our models for systems with ${q \leq 0.2}$ and ${M_{\rm tot}/\Msun \geq 20}$ (crosses), and ${0.06\leq q \leq 0.16}$ and ${20 \leq M_{\rm tot}/\Msun \leq 30}$ (stars). []{data-label="fig:model_rates"}](model_rates.png){width="48.00000%"} Formation of GW190814-like systems {#subsec:formation_pathways} ---------------------------------- We identify two main channels for forming GW190814-like systems through isolated binary evolution in our population models. These can be broadly categorized as Channel A, where the primary (more massive) star at becomes the more massive component in the compact binary, and Channel B, where the primary star at becomes the lower mass object in the compact binary (either a or a ). Figure \[fig:formation\_channels\] shows evolutionary diagrams for examples from these general channels. In Channel A, the binary typically starts as a primary and secondary at . The binary evolves without interaction during the main sequence of the primary. At core helium burning, the primary overflows its Roche lobe. Depending on the mass ratio at the time of Roche lobe overflow, the mass transfer will either proceed stably or unstably, in the latter case triggering a phase [@Taam2000]. The primary then directly collapses into a $\sim 20\,\Msun$ . As the secondary crosses the Hertzsprung gap, it overflows its Roche lobe and proceeds through highly non-conservative mass transfer. Since the mass ratio between the donor star and the already-formed is close to unity, this phase of mass transfer typically proceeds stably. The naked helium-star does not proceed through another phase of mass transfer and becomes a following its . These systems generally have large orbital separations at double compact object formation, and thus to merge within a Hubble time the newly-formed needs to be kicked into a highly eccentric orbit, typically with a post- eccentricity of . In Channel B, the binary starts with a mass ratio of at . The primary fills its Roche lobe while on the main sequence. This phase of stable mass transfer donates a significant amount of material to the secondary, leading to a mass inversion where the lighter star at becomes the more massive star. The initially more massive star forms the lighter compact object before the secondary leaves the main sequence. Due to the large mass asymmetry between the already-formed and the now-much-more-massive secondary, when the secondary evolves into a giant and overflows its Roche lobe, mass transfer proceeds unstably and initiates a phase. The binary significantly hardens during the phase, and the second star directly collapses into a . At low metallicities, these channels operate with similar probability. For metallicities of ${Z\,\leq\,Z_\odot/30}$, we find systems in our simulated population with ${0.06\,\leq\,q\,\leq 0.16}$ and ${20\,\leq\,M_{\rm tot}/\Msun\,\leq\,30}$ lead to the primary star becoming the heavier compact object, whereas lead to the secondary star becoming the heavier compact object. At higher metallicities (${Z\,>\,Z_\odot/30}$) Channel A becomes dominant, with of systems proceeding through Channel A and through Channel B. As the metallicity increases to ${\gtrsim\,Z_\odot/8}$, GW190814-like systems no longer form since masses are suppressed due to line-driven winds. Since there are only a small number of GW190814-like systems in our models, the channels presented here and their relative likelihood only offer a broad picture of how systems with properties similar to GW190814 form. ![Evolutionary diagrams of two systems following the dominant pathways for forming GW190814-like systems in our models, as described in Sec. \[subsec:formation\_pathways\]. In Channel A, the more massive star at becomes the larger compact object, whereas in Channel B the system undergoes a mass inversion and the higher mass star at evolves into the lighter compact object. Stable mass transfer is denoted by an arrow from the donor to the accretor, and a phase is apparent in Channel B by the pink oval surrounding the two stars. []{data-label="fig:formation_channels"}](formation_channels.pdf){width="48.00000%"} Discussion and Conclusions {#sec:conclusions} ========================== GW190814 is a event which challenges compact binary population modeling, and places new constraints on the physics of massive-star evolution. We explore the formation of compact binary mergers with highly asymmetric masses and components residing in the lower mass gap. We find that systems with properties similar to GW190814 can form through isolated binary evolution; however, the predicted formation rates of such systems are an order of magnitude lower than the empirical single-event rate when considering models that match the observed rates for other compact binary populations. The mass of GW190814’s secondary lies in the dearth of compact objects between the heaviest and lightest , and if the result of isolated binary evolution, requires that instability growth and launch proceed on longer timescales than typically assumed. The most massive Galactic have low-mass stellar companions, allowing for stable mass accretion on significant timescales. Thus, the masses of these are not a direct probe of their masses at formation. GW190814’s lower mass component likely had minimal accretion after formation: its mass at merger is indicative of its birth mass. Even if there was a mass inversion in the system and the secondary component of GW190814 was the first-born compact object (Channel B in Sec. \[subsec:formation\_pathways\]), the amount of material it could feasibly accrete is limited by the evolutionary timescale of its massive companion (which goes on to form a $\gtrsim 20\,\Msun$ ). The amount accreted by a or at the Eddington limit is $$\Delta M_{\rm Edd} \approx 0.03 \left(\frac{R}{10~\mathrm{km}}\right) \left(\frac{t}{1~\mathrm{Myr}}\right) \Msun$$ where $R$ is the radius of the compact object [e.g., @Cameron1967]. Thus, even for accretion timescales of ${\mathcal{O}(1\,\mathrm{Myr})}$ and reasonable radii [@GW170817_EoS; @Riley2019; @Miller2019], the amount of mass that the lighter compact object could accrete is ${\approx\,0.03\,\Msun}$, far too low to bridge the gap between the canonical ${1.3\,\Msun}$ mass and the ${\SecondaryMassApprox}$ secondary of GW190814. While there is evidence of super-Eddington accretion in ultra-luminous X-ray sources [e.g., @Bachetti2014], it is unlikely that this could increase the mass of the GW190814’s lighter object significantly; the post-hydrogen exhaustion lifetime for the progenitor star of a ${\PrimaryMassApprox}$ is only ${\sim 0.5~\mathrm{Myr}}$, and mass transfer between the high-mass companion star and the already-formed compact object would almost assuredly be unstable due to the system’s large mass ratio. Thus, even at $10$ times the Eddington rate, GW190814’s lower-mass component would not accrete more than ${\mathcal{O}(0.1)\,\Msun}$ during the remaining lifetime of its stellar companion. As a clean probe of the natal mass of compact objects in the lower mass gap, GW190814 is an exquisite system for constraining the mechanisms that impact compact remnant masses. The secondary’s mass is inconsistent with instability growth and launch on rapid timescales (${t\,\sim\,10~\mathrm{ms}}$; see e.g., @Fryer2012 [@Muller2016]); if GW190814’s source formed from isolated binary evolution, it favors launch models where instabilities develop over longer timescale (${t \sim\,100~\mathrm{ms}}$). As we only consider two models for the mechanism, we cannot place a lower limit on the instability growth timescale. More detailed hydrodynamical simulations investigating instability growth, launch, and how they connect to compact remnant masses [e.g., @Ertl2020; @Patton2020] will be needed to determine a lower limit on the timescales necessary to produce systems with component masses in the lower mass gap, and whether there is a critical growth timescale that leads to a populated lower mass gap. Semi-analytic prescriptions and fitting formulae based on these detailed models can then be adopted by rapid population synthesis, either using deterministic or probabilistic [e.g., @Mandel2020a] approaches. The combination of a highly asymmetric binary with a low-mass secondary is predicted to be rare in most rapid binary population studies [e.g., @Dominik2012; @Giacobbo2018b; @Mapelli2018a; @Kruckow2018; @Neijssel2019; @Spera2019; @Mapelli2019; @Olejak2020], though some population modeling finds compact binaries with mass ratios of $\sim 10$:$1$ to have rates comparable to near-equal-mass mergers [@Eldridge2016; @Eldridge2017] and may be able to better match the mass asymmetry of GW190814. Even when we consider mechanisms that fill the lower mass gap, our predicted rate for GW190814-like systems is in tension with the empirical rate. There are many theoretical uncertainties in binary stellar evolution with complex correlations that strongly affect the rates and population properties of compact binary mergers [e.g., @Barrett2018], and we only choose a few to investigate. It is possible that variations in other uncertain physical prescriptions, such as the mass-transfer accretion rates, mass-transfer efficiency, the criteria for the onset of unstable mass transfer, and how each of these depend on the evolutionary stages of the stars involved, may help to alleviate the discrepancy between the merger rates of GW190814-like systems and other compact binary populations. Observations of compact binaries with unusual properties (such as GW190814) will be paramount in constraining uncertainties in this high-dimensional parameter space. This work focuses on the formation of systems with high mass ratios and component masses in the lower mass gap through canonical isolated binary evolution. Many other channels have been proposed for producing the compact binary mergers observed by LIGO–Virgo. Dynamical formation in dense stellar clusters such as globular clusters preferentially forms compact binaries with similar masses [e.g., @Sigurdsson1993a], and thus the formation rate of and other compact binaries with highly asymmetric masses are predicted to be rare [@Clausen2013; @Ye2020; @ArcaSedda2020]. The formation of compact binaries with highly asymmetric masses may be more prevalent in young star clusters [e.g., @DiCarlo2019; @Rastello2020; @Santoliquido2020], though [@Fragione2020a] finds the merger rate of systems in young massive and open clusters to be three orders of magnitude lower, similar to the predictions from old globular clusters. Other formation mechanisms have been explored for forming highly asymmetric compact binary mergers, such as hierarchical systems in the galactic field [e.g., @Silsbee2017; @Antonini2017a; @Fragione2019; @Safarzadeh2019c], hierarchical systems in galactic nuclei with a supermassive as the outer perturber [e.g., @Antonini2012; @Petrovich2017; @Hoang2018; @Stephan2019; @Fragione2019b], and in disks around supermassive in active galactic nuclei [e.g., @Yang2019b; @McKernan2019]. However, the rates and formation properties from these channels are highly uncertain. Nevertheless, a full picture of the landscape of compact binary mergers will require consideration of all these channels, and investigating how physical prescriptions (such as the connection between the underlying mechanism and compact remnant mass) jointly affect population properties, rates, and inferred branching ratios across these channels [e.g. @Vitale2017a; @Stevenson2017; @Zevin2017b; @Talbot2017; @ArcaSedda2020]. The identification of bona fide systems and other compact binary mergers with highly asymmetric masses will help to further constrain the relative contribution of various formation channels, and the underlying physical prescriptions inherent to these formation pathways. The authors thank Chris Fryer and Pablo Marchant for useful discussions. MZ acknowledges support from CIERA and Northwestern University. MS acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No. 794393. CPLB is supported by the CIERA Board of Visitors Professorship. VK is supported by a CIFAR G+EU Fellowship and Northwestern University. This work used computing resources at CIERA funded by NSF Grant No. PHY-1726951, and resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. Population Models {#app:pop_models} ================= `COSMIC` [@COSMIC] is based on the single-star fitting formulae from [@Hurley2000] and binary evolution prescriptions from [@Hurley2002]. Among many updates, `COSMIC` includes state-of-the-art physical prescriptions for stellar winds in massive stars [@Vink2001] and stripped stars [@Yoon2005; @Vink2005], multiple treatments for the onset [@Belczynski2008; @Claeys2014] and evolution [@Claeys2014] of unstable mass transfer, multiple prescriptions for natal kicks [@Hobbs2005; @Bray2016; @Giacobbo2020] with special treatment for electron capture [@Podsiadlowski2004] and ultra-stripped [@Tauris2015], as well as mass loss and orbital evolution from and pair instability [@Woosley2017; @Woosley2019; @Marchant2019]. Furthermore, `COSMIC` includes a number variations for how initial conditions are sampled [@Sana2012; @Moe2017], which can significantly affect the properties and rates of compact binary populations. Rather than simulating a predetermined number of systems, `COSMIC` runs populations specifically targeted for particular configurations of stellar types (such as or that merge within a Hubble time) until properties of the target population (such as their masses and orbital periods at formation) have converged [@COSMIC], thereby adequately exploring the tails of population distributions. Model Assumptions {#app:model_assumptions} ----------------- We investigate five uncertain aspects of binary evolution physics: 1. Initial conditions (primary mass, mass ratio, orbital period, and eccentricity) are sampled either independently using the best-fit values from @Sana2012 with a binary fraction of $0.7$, or using the correlated multidimensional distributions from @Moe2017. In the multidimensional sampling, the binary fraction is determined based on the probability that a system with a given primary mass is in a binary. 2. efficiency, which determines how easily the envelope is unbound from the system during a phase, is parameterized as in @Webbink1984 and @deKool1990. We vary the efficiency parameter $\alpha$, using either $\alpha=1$ or a higher value of $\alpha=5$ [@Giacobbo2019a; @Fragos2019], and use a variable prescription for the envelope binding energy factor $\lambda$ [@Claeys2014]. A higher efficiency will lead to wider post- binaries. 3. survival is chosen to either be an Optimistic or a Pessimistic scenario. In the Optimistic case, stars that overfill their Roche lobes on the Hertzsprung gap and proceed through unstable mass transfer are assumed to survive the phase, whereas in the Pessimistic case these systems are assumed to merge [cf. @Belczynski2008]. The Pessimistic scenario leads to significantly fewer compact binary mergers, particularly for . 4. Remnant masses are determined using the Rapid and Delayed prescriptions from @Fryer2012. The Rapid prescription yields a mass gap between and , whereas the Delayed prescription fills this gap (Fig. \[fig:Mzams\_Mrem\]). These prescriptions are updated as described in Sec. \[subsec:remnant\_mass\] and Sec. \[app:remnant\_mass\]. 5. natal kicks are determined in two ways. In the Bimodal prescription, iron core-collapse kicks are drawn from a Maxwellian distribution with a dispersion of $\sigma=265~\mathrm{km\,s}^{-1}$ [@Hobbs2005], whereas electron-capture and ultra-stripped are given weaker kicks drawn from a Maxwellian distribution with a dispersion of $\sigma=20~\mathrm{km\,s}^{-1}$ (e.g., @Podsiadlowski2004 [@VanDenHeuvel2007; @Tauris2015; @Beniamini2016], see @COSMIC for more details). The second kick prescription uses the scaling based on compact-object mass and mass loss in @Giacobbo2020. These represent only a few of the binary evolution parameters that can possibly affect the parameter distribution and merger rates of compact-binary populations. Besides the parameter variations described above, we anticipate that mass transfer conservation and the stellar-type specific criteria for the onset of unstable mass transfer will have the largest impact. In this study, we assume that mass transfer is limited to the thermal timescale of the accretor for stars and limited to the Eddington rate for compact objects, and that angular momentum is lost from the system as if the excess material is a wind from the accretor [@Hurley2002]. The onset of unstable mass transfer is determined as in @Belczynski2008 using critical mass ratios for a given stellar type: $q_{\rm crit}=3.0$ for H-rich stars ($k_\star = 1$–$6$), $q_{\rm crit}=1.7$ for helium main sequence stars ($k_\star = 7$), $q_{\rm crit}=3.5$ for evolved helium stars ($k_\star = 8,\,9$), and $q_{\rm crit}=0.628$ for compact objects ($k_\star \geq 10$). A full exploration of parameter space is reserved for future work. Remnant Mass Prescription {#app:remnant_mass} ------------------------- Here, we provide more details regarding the updated remnant mass prescription used in this study. To determine the mass of compact remnants, we follow the Rapid and Delayed prescriptions described in @Fryer2012. These allow for the results of hydrodynamical simulations exploring the timescale of instability growth and launch of the to be used directly in rapid population synthesis. Mass fallback is also accounted for in the determination of the baryonic mass of the proto-compact object. As in [@Giacobbo2020], we adjust the initial mass of the proto-compact object to be $1.1\,\Msun$ rather than $1.0\,\Msun$, as this better reproduces the typical masses of in the Galaxy. Following the determination of the baryonic mass of the remnant, the gravitational mass is calculated to account for neutronization in the collapsing core. In [@Fryer2012], the gravitational mass of the remnant is calculated differently for and . For , the gravitational mass is calculated according [@Lattimer1989] based on the neutrino observations of SN 1987A: $$M_{\rm grav} = \frac{20}{3} \left[(1 + 0.3 M_{\rm bar})^{1/2} - 1\right],$$ where $M_{\rm bar}$ is the pre-collapse baryonic mass calculated as in [@Fryer2012]. For the mass reduction is assumed to be a fixed percentage of the proto-compact object’s baryonic mass: $$M_{\rm grav} = 0.9 M_{\rm bar}.$$ This leads to an increasing amount of mass loss when converting from baryonic to gravitational mass as function of increasing mass. Since the true maximum mass is unknown and likely sensitive to other aspects of the proto-compact object such as rotation, the delineation between these two prescriptions is typically determined by an adjustable parameter for the maximum mass: $m_{\rm NS}^{\rm max}$. There are two issues with this simple prescription that affect the compact-object mass spectrum. First, the final mass of a remnant is a function of the total pre-collapse baryonic mass of the proto-compact object, though neutronization is instead occurring in the iron core of the proto-compact object. Even for massive and hot radiation-supported cores, the iron core mass is $\lesssim 5\,\Msun$ (C. Fryer 2020, private communication). Hydrodynamical simulations show that the mass loss from neutrino emission is $\sim 10\%$ of this core mass rather than the total baryonic mass of the progenitor (C. Fryer 2020, private communication). Second, using separate prescriptions for determining and gravitational mass leads to an artificial gap in the mass spectrum; this artificial gap is different than the lower mass gap and is apparent even when using the Delayed remnant mass prescription. For example, assuming $m_{\rm NS}^{\rm max}=2.5\,\Msun$ and $10\%$ mass loss when converting from baryonic to gravitational mass in , the most massive that can be formed is $2.5\,\Msun$ whereas the least massive that can be formed is $2.7\,\Msun$. With these in mind, we update how the final gravitational mass of a compact object is determined: $$M_{\rm grav} = \begin{cases} \displaystyle \frac{20}{3} \left[(1 + 0.3 M_{\rm bar})^{1/2} - 1\right]& \Delta M\leq 0.1\,m_{\rm Fe}^{\rm max}\\ M_{\rm bar}-0.1\,m_{\rm Fe}^{\rm max}& \mathrm{otherwise}\\ \end{cases},$$ where $\Delta M = M_{\rm bar}-M_{\rm grav}$ and $m_{\rm Fe}^{\rm max}$ is the maximum possible mass of the iron core, which we set to $5\,\Msun$. For $m_{\rm Fe}^{\rm max}=5\,\Msun$, the switchover in this conditional occurs at $\simeq\,3.1\,\Msun$. As shown in Figure \[fig:Mzams\_Mrem\], this update eliminates any artificial gaps in the mass spectrum between and when using the Delayed mechanism. Local Merger Rates {#app:merger_rates} ================== The mass fraction of binaries that are born at redshift $z$ and merge as compact binaries in the local universe is $$f_{\mathrm{loc},\,i}(z) = \frac{M_i(z;z_{\rm merge}<z_{\rm loc})}{M_{\rm samp}},$$ where $M_{\rm samp}$ is the total stellar mass sampled in the simulation, $i$ represents the class of compact binary merger (, , , etc.), $M_i$ is the stellar mass that leads to merger type $i$, and $z_{\rm loc}$ is the maximum redshift that we consider for local mergers, which we set to $z_{\rm loc}=0.01$. `COSMIC` accounts for the total mass sampled in $M_{\rm samp}$, incorporating both the binary fraction and the mass contribution from lower-mass stars that do not lead to compact binary formation. The number of mergers per unit volume that form in the redshift interval $[z,z+\Delta z]$ and merge in the local universe is thus $$\Delta \mathcal{N}_{\mathrm{loc},\,i}(z)= \psi(z) f_{\mathrm{loc},\,i}(z) \frac{\mathrm{d}t_\mathrm{l}}{\mathrm{d}z} \Delta z,$$ where $\psi(z)$ is the star formation rate density and $t_\mathrm{l}(z)$ is the lookback time at redshift $z$. We use @Madau2017a for the star formation rate density as a function of redshift, $$\label{eq:sfr} \psi(z) = 10^{-2} \frac{(1+z)^{2.6}}{1 + \left[(1+z)/3.2\right]^{6.2}}\,\Msun\,\mathrm{yr}^{-1}\,\mathrm{Mpc}^{-3}.$$ Integrating over all formation redshifts up to $z_{\rm max}=15$ and converting to the number of mergers per unit time gives us the local merger rate density $$\mathcal{R}_{\mathrm{loc},\,i} = \frac{1}{t_\mathrm{l}(z_{\rm loc})} \int_{0}^{z_{\rm max}} \psi(z) f_{\mathrm{loc},\,i}(z) \frac{\mathrm{d}t_\mathrm{l}}{\mathrm{d} z} \mathrm{d}z = \frac{1}{H_0 t_\mathrm{l}(z_{\rm loc})} \int_{0}^{z_{\rm max}} \frac{\psi(z) f_{\mathrm{loc},\,i}(z)}{(1+z) E(z)} \mathrm{d}z,$$ with $E(z) = \left[\Omega_{\rm rad}(1+z)^4 + \Omega_{\rm m}(1+z)^3 + \Omega_{\rm k}(1+z)^2 + \Omega_{\rm \Lambda} \right]^{1/2}$. In practice, we discretize this integral with $1000$ log-spaced redshift bins between $z_{\rm loc}$ and $z_{\rm max}$. Each population is run at a single metallicity and allows all binaries to evolve for the entire age of the Universe, allowing for rate calculations to be performed in post-processing. We simulate $16$ log-spaced metallicities between $Z_\odot/200$ and $2\,Z_\odot$ for each population model assumption. To account for metallicity evolution over cosmic time, we use the mean mass-weighted metallicity as a function of redshift in @Madau2017a, $$\label{eq:met} \log_{10}\left\langle Z/Z_\odot \right\rangle = 0.153 - 0.074 z^{1.34},$$ and assume a truncated log-normal distribution of metallicities at each redshift with a dispersion of 0.5 dex [@Bavera2020] that reflects over boundaries at $Z_{\rm min}=Z_\odot/200$ and $Z_{\rm max}=2\,Z_\odot$. The weights for each metallicity model $j$ at a given redshift $p(Z_j\|z)$ (which are normalized to unity to account for our discrete metallicity models, $\sum_j p(Z_j\|z) = 1$) are then folded into the rate calculation to give a local volumetric merger rate across all metallicity models: $$\mathcal{R}_{\mathrm{loc},\,i} \simeq \frac{1}{H_0 t_\mathrm{l}(z_{\rm loc})} \sum_k \frac{\psi(\bar{z}_k) \sum_{j}p(Z_j \| \bar{z}_k) f_{\mathrm{loc},\,i}(\bar{z}_k,Z_j)}{(1+\bar{z}_k) E(\bar{z}_k)} \Delta z_k,$$ where $\bar{z}_k$ is the midpoint (in log space) of the $k$-th redshift bin and $\Delta z_k$ is the size of the $k$-th redshift bin. Local merger rates for each population model we simulate are shown in Table \[tab:table\]. In addition to , , and rates for each model, we also give a Narrow rate for GW190814-like systems (defined as $0.06\,\leq\,q\,\leq\,0.16$, $20\,\leq\,M_{\rm tot}/\Msun\,\leq\,30$) and a Broad rate for GW190814-like systems (defined as $q\,\leq\,0.2$, $M_{\rm tot}/\Msun\,\geq\,20$). [^1]: zevin@u.northwestern.edu [^2]: [cosmic-popsynth.github.io](https://cosmic-popsynth.github.io/) (Version 3.3) [^3]: The low merger redshift of GW190814 makes these results a good proxy for local mergers in general.
--- abstract: 'Recognizing human activities from multi-channel time series data collected from wearable sensors is ever more practical. However, in real-world conditions, coherent activities and body movements could happen at the same time, like moving head during walking or sitting. A new problem, so-called “Coherent Human Activity Recognition (Co-HAR)”, is more complicated than normal multi-class classification tasks since signals of different movements are mixed and interfered with each other. On the other side, we consider such Co-HAR as a dense labelling problem that classify each sample on a time step with a label to provide high-fidelity and duration-varied support to applications. In this paper, a novel condition-aware deep architecture “Conditional-UNet” is developed to allow dense labeling for Co-HAR problem. We also contribute a first-of-its-kind Co-HAR dataset for head movement recognition under walk or sit condition for future research. Experiments on head gesture recognition show that our model achieve overall $2-3\%$ performance gain of F1 score over existing state-of-the-art deep methods, and more importantly, systematic and comprehensive improvements on real head gesture classes.' author: - bibliography: - 'main.bib' title: 'Conditional-UNet: A Condition-aware Deep Model for Coherent Human Activity Recognition From Wearables' --- =1 Introduction ============ With the rapid development and lower cost, wearable devices with embedded sensors are becoming more and more popular to be used for a plethora of applications, such as healthcare [@gray2007head], authentication [@parimi2018analysis], robotic control [@gray2007head], virtual/augmented reality [@zolkefly2018head; @hachaj2019head; @chen2019augmented] and e-learning [@deshmukh2018feedback]. Although there are many possible sensors in many real-world applications of wearables, people prefer to have limited devices with multiple functions, such as smart phone, virtual reality headset, smart glass, or wireless headphone, instead of wearing multiple devices at the same time. Another challenge is that body moves simultaneously during daily activities that generate complicated mixed signals for those limited devices mounted on the body. These multiple human activity and movements are interactively interfered with each other. For example, such a challenging task is to recognize head gestures during walk or sit conditions (Figure \[fig:example\]) using embedded accelerometer and gyroscope sensors in only one location, for example a headphone or Virtual Reality headset. The denoised signal collected under sitting situation (above red line) has a clearer pattern compared to signal under walking situation (the lower red line). This is a relatively hard task, because body parts except head during walking generates stronger inertia than the simultaneous head movements. Many previous works on head gestures [@gray2007head; @wu2017applying; @hachaj2019head] only focus on a controlled sitting situation, or they simply do not consider such coherent activities at all. In this work, a new problem, so-called “ Coherent Human Activity Recognition” (Co-HAR), is firstly proposed for such kind of tasks with interference movements. Recent development of deep learning shed a light on human activity recognition research, since deep learning allows to learn latent features using deep structures such as convolution layer, pooling layer and embedding layer [@goodfellow2016deep]. It usually requires much less or even no effort on feature extraction than pre-dated models before deep learning. In an end-to-end fashion, deep learning models have better generality which allow a model structure to be performed well in different data without domain-specific work that resulted in shorter development cycles. Deep learning could also be naively applied for Co-HAR. However, deep learning for Co-HAR problem has not been systematically explored and specially-designed. Beyond naively using deep methods, some critical technical challenges prevent current deep architectures to perform better, including: 1) single location of sensors has mutual impact of signals. As discussed, sensors are placed only in a headphone over a user’s head. It is impractical to ask a user to wear sensors all over body in the real-world scenario. It is also difficulty to exactly separating signals and reduce mutual impacts using existing approaches like a basic multi-label classification [@vaizman2018context; @mohamed2020multi]; 2) imbalanced domination of different activities could fade away signals of other activities. Sensors could have different sensitive levels to different body movements. The dominating movement might not be the most interesting one that we would like to examine. For example, in the head gesture problem (in Figure \[fig:example\]), sensors are placed in a headphone, walking generates strong signals in the forward inertia. However, head gestures are more critical for most applications like Virtual Reality. The current models might have limited power in such scenarios. 3) multi-label window problem for duration-varied activities. The time steps in one window may not always share the same ground truth label, and the duration of an activity always vary in different windows. Mixing of ground truth labels not only creates difficulty for underlying models but also reduces flexibility of usages due to a whole set of hyper-parameters to be considered, such as the best window length, sampling stride and window labelling strategy. By tackling these challenges for Co-HAR problem, we proposed a novel condition-aware deep structure, called “Conditional-UNet”, which could take multi-channel sensor data embedded only at one location of human body as input and hence explicitly capture conditional dependence within coherent labels. We propose a novel encoding module to model the conditional dependence which could reduce the mutual impact of coherent body movement and guide our model to learn better patterns in activities with imbalanced domination. Since it follows the dense labeling approach [@yao2018efficient], it avoids multi-label window problem, but it aims to classify multiple labels which is more challenging than previous dense labeling works. The major contributions of our work are summarized as follows: - We consider a challenging problem, so-called “Coherent Human Activity Recognition” (Co-HAR) which classify coherent activity labels beyond simple multi-label scenarios and use multi-channel time-series data from one wearable device only at one location of human body. - A novel condition-aware deep classification model, called “Conditional-UNet”, is developed to better densely classify coherent labels. Conditional-UNet explicitly models conditional dependence through a novel deep structure, including a new encoding module with specially-designed gradient-permitted sampling and embedding structure and a UNet-based decoding module. - The contribution of a new dataset for Co-HAR problem. To conduct experiments, we build an Ardino-based device to collect data and label head gestures and walk/sit condition. - Extensive experiments show that our proposed Conditional-UNet outperform existing state-of-the-art UNet model by $2-3\%$ of F1 score over head gesture label classification. Related Work ============ Feature Extraction based Methods. Pre-dated models before deep learning rely heavily on hand-crafted features (e.g., mean, variance, kurtosis, or other kinds of indexes) [@wu2017applying; @parimi2018analysis], motion (e.g., physical laws) [@zolkefly2018head], transform-based feature (e.g., wavelet [@chung2009realtime], fourier transform [@gamal2013hand]). Exacted features are then feed to classifiers such as Support Vector Machines [@zolkefly2018head], Boosting Tree [@wu2017applying] and Hidden Markov Model [@zolkefly2018head]. These approaches usually work well for a specific type of tasks while fails for other types of applications. Deep Learning based Dense Labeling. With the advancement of deep learning methods, the applications of deep learning to HAR using data from wearable sensors are relatively new. More and more works propose to utilize some kinds of deep learning methods [@yao2018efficient; @ignatov2018real; @ferrari2019human]. The success of deep-learning-based methods comes from their high expressiveness in learning underlying complex principles directly from the data in end-to-end fashion without handcrafted rules. Another most recent advancement by using deep learning is the Dense Labeling [@yao2018efficient] using a fully convolutional network [@long2015fully] to label each sample instead of a sliding window. It avoids the segmentation problem in most of conventional methods. Another work [@zhang2019human] achieves the same goal of dense labeling but utilizes another deeper structure called “UNet”. These works still assumes a single activity label rather than coherent activities. Multi-label classification A recent work [@varamin2018deep] for multiple overlapping labels output activity sets of multiple labels for each window using deep neural networks, but learned labels are not associated with each sample and there is not explicitly consideration of conditional dependence in different human activities. Another recent work [@mohamed2020multi] converts multiple labels into one label with all classes from different labels to solve the multi-label classification. There is no existing works considering coherent multiple labels which have conditional dependence within them. We point readers to more details in other survey papers in this area [@nweke2018deep; @chen2020deep]. In summary, deep learning methods, including the state-of-the-art UNet, are actively researched in exiting works with better performances than pre-dated conventional methods. However, to our best knowledge, there are no work considering conditional dependency in multiple dense labels beyond simple multi-label classification. Next, we would formally define our problem. Problem Definition ================== A set of sequences $\mathcal{D} = \{ (\pmb{X}^{(i)}, \pmb{Y}^{(i)}) \}_i^N, \forall \pmb{X}^{(i)} \in \mathbb{R}^{K \times T^{(i)}}, \pmb{Y}^{(i)} \in \mathbb{R}^{H \times T^{(i)}}$, which contains a multivariate sequence $\pmb{X}^{(i)}$ which have $K$ variables of sensors and the time length of each sequence is $T^{(i)}$. Here, each time step $t \in \{1, \dots, T^{(i)}\}$ is normally referred as a sample. Correspondingly, $\pmb{Y}^{(i)}$ is a multi-label sequence with $H$ labels. For each label $h \in \{1,\dots, H\}$, there are $C_h$ different numbers of classes for this label $h$, so an element $\pmb{Y}^{(i)}_{h, t} \in \{1, \dots, C_h\}$, where $\pmb{Y}^{(i)}_{h, t} = 1$ usually define for a null label (e.g. no hand gesture is performed). The sequence index $(i)$ is dropped in later parts whenever it is clear that we are referring to terms associated with a single sequence. We define our problem as follow: \[def:co-har\] is a multi-label classification problem with conditional dependency assumption within joint multiple coherent labels and has a goal to minimize the difference between a classifier’s predicted $H$-label sequence $\pmb{\hat{Y}}^{(i)}$ for $K$-channel sample sequence $\pmb{X}^{(i)}$ and the ground truth label $\pmb{Y}^{(i)}$ given multi-channel time-series sequences set $\mathcal{D}$. For example, in our head gesture task, a sample $\pmb{X}^{(i)}_t$ contain $K = 6$ variables of sensors including tri-axial acceleration and tri-axial gyroscopes. If a sampling rate is $1~Hz$, then there should be $T^{(i)} = 60$ total measures for a one-second window. Since we are interested in two types of labels, head gesture and walk/sit condition, so there are two label types $L = 2$. For walk label $y_1 \in \{1, 2\}$, $y_1 = 1$ if a human subject is sitting, and $y_0 = 2$ indicates a walk activity. Similarly, for head gesture label $y_2$, $y_2 = \{1, \dots 9\}$, and $y_2 = 1$ means no head gestures, and other numbers indicates other $8$ head gestures, and $C_1 = 9$. A key component of Co-HAR is that we keep the complete joint probability of all labels $p(Y_1^{(i)}, \dots, Y_H^{(i)})$ with conditional dependency, not simplified joint probability $p(Y_1^{(i)}) \times \dots \times p(Y_H^{(i)})$ that assumed independence of all labels in the basic multi-label classification. More details are in following methodology Section \[sec:framework\]. Methodology =========== In this section, condition-aware framework of Co-HAR and detailed modeling components of Conditional-UNet are introduced. Section \[sec:framework\] shows the condition-aware deep framework for Co-HAR problem and the formalization of its decomposed conditional losses. In Sections \[sec:sampling\], \[sec:embedding\], and \[sec:merging\], three key components of handle joint label conditions are introduced to capture the conditional dependence within coherent activity labels. Condition-aware deep framework {#sec:framework} ------------------------------ In this part, condition-aware deep framework is developed firstly to build up a probabilistic understanding of Co-HAR problem, and to formalize loss with an independent loss and a series of dependent losses. In general, the goal of Co-HAR in previous Co-HAR definition \[def:co-har\] is to learn a joint probability of multiple labels given multi-channel sensor data, noted as $p(Y_1, \dots, Y_H \| X)$. By an axiom of probability, joint probability of observing a sequence can be decomposed to a series of independence and dependence components in Equation \[eq:joint\_probability\]: $$\begin{split} &p(Y_1, \dots, Y_H \| X) = \\ &p_{\theta_1}(Y_1 \| X) p_{\theta_2}(Y_2\| Y_1, X) \dots p_{\theta_H}(Y_H \| P_{H-1}, \dots, Y_1, X) \end{split} \label{eq:joint_probability}$$ where $\theta_i$ are parameters of each probability function $p(\cdot)$. This is the complete conditional relationship for Co-HAR. By assuming conditional independence between all the labels, we can simplify it to be $p(Y_1, \dots, Y_H \| X) = p_{\theta_1}(Y_1 \| X) p_{\theta_2}(Y_2\| X) \dots p_{\theta_H}(Y_H \| X)$, which is a normal multi-label classification framework in many current works [@mohamed2020multi; @vaizman2018context]. However, in this work, we want to keep the conditional dependence since conditional independence assumption drop a lot of useful information. Next, the condition-aware loss function is introduced with its different components. Condition-aware multi-label dense classification loss: following the common approach of Maximum Likelihood Estimation (MLE) [@bishop2006pattern] and similar to dense labeling [@yao2018efficient], we get loss function by factorizing joint probability in Equation \[eq:joint\_probability\] to each temporal sample with each label, and get its logarithmic transformation as follows: $$\begin{split} \mathcal{L} = &log(p(Y_1, \dots, Y_H \| X)) = \sum_t^T \big( log(p_{\theta_1}(Y_{1,t} \| X)) + \\ & \dots + log(p_{\theta_H}(Y_{H,t} \| P_{H-1, t}, \dots, Y_{1,t}, X))\big) \end{split} \label{eq:log_probability}$$ where $log(p_{\theta_1}(Y_{1,t} \| X))$ is log-likelihood to observe different classes of label $1$ on time step $t$th sample. Furthermore, this log-likelihood with each class $m$ of a label is formulated as $\sum^{C_1}_m y_{1,t}^{m} log(p_{\theta_1}(Y_{1, t} = m \| X))$, where $y_{1,t}^m$ is observed frequency of class $m$ in all samples, and $p_{\theta_1}(Y_{1, t} = m \| X)$ is estimated likelihood of class $m$ got from deep model. The calculation of estimated likelihood is the same as deep model take sensor data as input and output estimated likelihood, noted as $\pmb{\hat{y}}_1 = f_{\theta_1}(X)$, where $\pmb{\hat{y}}_1$ is estimated logit vector with $m$th element on $t$th sample $\hat{y}_{1,t}^m$ as estimated probability of multi-label categorical distribution of label $1$. Until now, it is a normal MLE with more details in [@bishop2006pattern]. The difference of our condition-aware model start from label $2$. Instead of only taking $X$ as inputs for $Y_2$, our deep model takes conditional signals of $Y_1$ as input too, noted as $\pmb{\hat{y}}_2 = f_{\theta_2}(X, Y_1)$, or $\pmb{\hat{y}}_i = f_{\theta_i}(X, Y_1, \dots, Y_{i-1})$. It means that our condition-aware deep model should decode all previous labels as joint conditional dependence for the next label estimation. Before we show how conditional dependence is estimated, we summarize our condition-aware loss as follows: $$\begin{split} &\mathcal{L} = - \frac{1}{N} \big( \sum^T_t \sum^{C_1}_m y_{1,t}^m log(\hat{y}_{1,t}^m) \\ &+ \sum^T_t \sum^{C_2}_m y_{2,t}^m log(\hat{y}_{2,t}^m) + \dots + \sum^T_t \sum^{C_l}_m y_{H,t}^m log(\hat{y}_{H,t}^m) \big) \\ &\hat{y}_{1,t}^m = f_{\theta_1}(X), \hat{y}_{2,t}^m = f_{\theta_2}(Y_1, X), \\ &\dots, \hat{y}_{H,t}^m = f_{\theta_H}(Y_1,\dots,Y_{H-1}, X) \end{split} \label{eq:losses}$$ $$\operatorname{minimize}_{\Theta} {\mathcal{L}} \label{eq:minimize}$$ where $\Theta = \{\theta_1, \dots, \theta_H\}$ is the set of all deep model’s parameters that minimize negative log-likelihood loss $\mathcal{L}$ in Equation \[eq:losses\]. To model the joint label conditions within this condition-aware deep model, we create a chain of conditional deep models $f_{\theta_i}$, except the first $f_{\theta_1}$.* It follows the procedure illustrated in Figure \[fig:deep\_model\]: 1) Decoding module: uses a deep encoding module $f_{\theta_1}(X)$ to compute the logit vector of label $1$ for each sample $\pmb{y}_{1} = f_{\theta_1}(X)$. Here, the deep encoding module is UNet [@ronneberger2015u]. UNet is originally designed for image segmentation following the idea of fully convolutional network [@long2015fully]. It is a deep fully convolutional network, which internally contains multiple down-sampling convolutional layers and multiple up-sampling deconvolutional layers. It is recently used to boost performance in normal HAR task [@zhang2019human] as a more powerful alternative than basic fully convolutional network using in Dense Labeling [@yao2018efficient], CNN [@xu2018human], or SVM [@zhang2019human]. We utilize the same structure as [@zhang2019human] with more details; 2) Encoding module: uses a decoding module to convert logit vector $\pmb{y}_{1}$ to a conditional signal. It includes three sub-modules in this module: a) a generating sub-module $Generate(\cdot)$ to get a sampled class for the first label from the categorical distribution, $\hat{Y_1} = Generate(\pmb{y}_1)$. It is detailed in Section \[sec:sampling\]; b) an embedding sub-module $Embed(\cdot)$ (detailed in Section \[sec:embedding\]) is used to project $Y_1$ to a continuous embedding space with $Embed_{\phi_1}(\hat{Y_1})$, where $\phi_i$ is the set of parameters; c) a merging sub-module $Merge(\cdot)$ (detailed in Section \[sec:merging\]) to merge both $X$ and embedded signal as input into the next encoding module to get logit vector $\pmb{y}_{2}$ of label $Y_2$, $\pmb{y}_{2} = f_{\theta_2}(Merge(X, Embed(Cat(\pmb{y}_1))))$. For $Y_3$, the only difference is that it merges both $g_{\phi_1}(\hat{Y_1})$ and $g_{\phi_2}(\hat{Y_2})$ with $X$, $\pmb{y}_{3}$ of label $Y_3$, $\pmb{y}_{3} = f_{\theta_3}(Merge(X, Embed(Cat(\pmb{y}_1))), Embed(Cat(\pmb{y}_2))))$. we continue this chain of processes until it reaches the last conditional model for the last label $Y_H$. Here, all labels $Y_i$ are one-hot vectors; 3) Optimizing module: uses all logit vector $\hat{\pmb{y}}_i$ to calculate the multi-label dense classification loss $\mathcal{L}$, and minimize it through gradient back-propagation optimization techniques for deep neural network models, such as Adam [@goodfellow2016deep], or Stochastic-Gradient-Descent [@goodfellow2016deep]. Since we develop our code using PyTorch [@ketkar2017introduction], the Adam Optimizer is directly used to perform optimization and train our Conditional-UNet classification model. Module 1) and 3) are conventional works with more details in other works [@ronneberger2015u; @zhang2019human; @goodfellow2016deep], while Module 2) is our main structure to explicitly handle conditional dependence. We will now introduce different parts of this novel decoding module. Gradient-permitted generating sub-module {#sec:sampling} ---------------------------------------- This generating sub-module, noted as $Generate(\cdot)$ in Figure \[fig:deep\_model\], is the first step to incorporate conditional dependence information in Conditional-UNet. Its goal is to generate a sample of current label $\hat{Y}_i$ from estimated logit $\hat{y}_i$ of which each element is a probability to get $m$th class, so that sampled label can be used as a conditional input for the next decoding module. The keys here are both to allow gradient back-propagation and to better process conditional dependence signal. We proposed two variants for this sub-module as follows: ### Naive-Max trick which selects the maximum probability class $m$ in estimated logit vector $\hat{y}_i$ in Equation \[eq:naive\_max\]: $$\begin{split} \hat{Y}_i &= \operatorname{arg\,max}_m \hat{y}_i, \\ &\forall m \in \{1, \dots, C_h\}, \hat{y}_i \in \mathbb{R}^{C_h} \end{split} \label{eq:naive_max}$$ where $C_h$ is the number of classes in label $Y_i$. This Naive-Max trick simplifies a categorical distribution to focus only on its class with maximum probability, however, it does not capture the whole distribution information. For example in Figure \[fig:gumbel\_max\] (a), it can potentially learn a flatten distribution (differ from the true distribution in Figure \[fig:gumbel\_max\] (c)), whose class with the max probability does not differ a lot from other classes. In this case, the distinguishing power could be vanished because of the big variance in this approach. The Naive-Max does not block gradient flow, however, it is potentially unstable because the maximum class shift a lot during training process. If a maximum-probability class is changed in a followed training iteration, the gradient flowing path changes to the other class which has maximum probability in that training iteration. This is a huge instability disadvantage, while its advantage is its easy implementation. ### Gumbel-Max trick* which implements the true process of sampling a class $\hat{Y}_i$ from a categorical distribution using logit vector $\hat{y}_i$, noted as $\hat{Y}_{i} \sim Cat(\hat{y}_i)$. This sampling process captures the full distribution information because it can still generate classes which do not have the maximum probabilities. In this way, we approximate the full categorical distribution for each class, not only the one with maximum probability. Unfortunately, this operation of sampling from categorical distribution do not have gradients, so it prevents gradients flowing in back-propagation training. The work-around solution is to use re-parameterization trick, specifically we leverage a “Gumbel-Max” trick [@jang2016categorical] for categorical data of labels. For example in Figure \[fig:gumbel\_max\] (b), Gumbel-Max reduces the chance of flatten distribution like Naive-Max, while pushes the distribution to concentrate on one or a few classes and decrease the probabilities of other classes. It can get a distribution closer to the shape of true distribution (Figure \[fig:gumbel\_max\] (c)), not only capture the peak one. The Gumbel-Max trick is done in this way. Specifically, we create $\pmb{y}_{i, t}' = \tanh((\pmb{q}^{i,t} + \pmb{g}) / \tau)$, where $\tau$ is so-called “temperature” hyper-parameter. Each value $g_j$ in $\pmb{g}$ is an independent and identically distributed (i.i.d.) sample from standard Gumbel distribution [@jang2016categorical]. $\pmb{g}$ has the same dimension as $\pmb{y}_{i, t}$. We generate a one-hot representation $\pmb{y}_{i, t}$, whose $j$th element is one and all the others are zeros, where $j$ is got as the index of the maximum element in $\pmb{y}'_{i, t}$. Then, a OneHot operation is used to get the integer value of a class $Y_{i, t} = \operatorname{OneHot}(\pmb{y}_{i, t})$. In this way, gradients can be backpropagated through $\pmb{y}_{i, t}'$. The same approach is also used for all the labels except the last one $Y_H$. Notice that the larger $\tau$, the more uniformly regulated with more stable gradient flow are the sampled values. The typical approach is to decrease $\tau$ as training continues and we can adopt a decrease strategy similar to [@jang2016categorical] (Equations \[eq:decode\_cat\]). $$\begin{split} \pmb{y}_{i, t}' &= \tanh \big((\pmb{q}^{i,t}_v + \pmb{g}_v) / \tau \big) \\ \pmb{y}_{i, t} &= \arg\max \pmb{y}_{i, t}' \\ Y_{i, t} &= \operatorname{OneHot}(\pmb{y}_{i, t}) \end{split} \label{eq:decode_cat}$$ Class embedding sub-module {#sec:embedding} -------------------------- Generated class for each label and time $\hat{Y}_{i, t}$ is a categorical value. Inspired in Word2Vec [@rong2014word2vec] in Natural Language Processing, we convert categorical classes to a continuous space to be processed by the following neural networks. The embedded continuous value is the conditional signal we need for conditional dependence computing of the next label $Y_{i+1}$. To achieve this, an embedding weight table $\pmb{W}_i \in \mathbb{R}^{C_h \times E_i}$ contains all learnable embedding parameters, where $C_h$ is the number of classes in label $Y_i$, $E_i$ is a hyper-parameter of the dimension of continuous space, normally $E_i \ll C_h$. Here, we simply take $E_i = \frac{C_h}{2}$. Each label $Y_i$ has its own embedding table $\pmb{W}_i$. Embedding operation is $\Bar{\pmb{y}}_{i, t} = \pmb{W}_i Y_{i}$, where $\Bar{\pmb{y}}_{i, t}$ is projected continuous vector in a continuous space. Merging sub-module to capture joint conditions {#sec:merging} ---------------------------------------------- The embedded vector $\Bar{\pmb{y}}_{i, t}$ are concatenates with all previous embedded vector $\Bar{\pmb{y}}_{1, t}, \dots, \Bar{\pmb{y}}_{i-1, t}$ and the raw sensors $\pmb{X}_{,t}$. In this way, the next label $Y-i$’s joint conditions of all previous labels and sensors, noted as $p(Y_i \| X, Y_1, \dots, Y_{i-1})$ in MLE before, are captured in merged vector as input for the next decoding module $f_{\theta_i}$. The only exception is the last label’s embedding vector $\Bar{\pmb{y}}_{H, t}$, which has not concatenation operation, since we have reached the end. In our proposed Conditional-UNet, a natural question is that what is the best order to sequentially model joint label conditions. This is just another hyper-parameter to be tuned. If there are $H$ labels, there is potentially $(H-1)!$ orders. However, fortunately, there are normally not many labels in real-world applications (e.g. $2$ labels in our head gesture experiment, and there are $2! = 2$ orders to be tuned on). Experiments {#sec:experiments} =========== In order to demonstrate and verify the performance of the proposed Conditional-UNet for Co-HAR problem, we conduct experiments as follows: (i) collect a new dataset about head gesture under walk/sit situation through Arduino UNO and other hardwares; (ii) compare our method and its variants with state-of-the-art competing methods; (iii) a qualitative analysis to illustrate effectiveness of our proposed method. Device design and experiment settings {#sec:experiment_setting} ------------------------------------- Hardware design: To our best knowledge, there are no dataset that is collected for Co-HAR yet, especially sensor module locates only at one location of body, so that we can only retrieve mixed signals instead of signals from multiple locations. The types of labels should be conditional dependent and interactively impact each other, so that we need to classify multiple different labels for the same sample. With this in mind, we implement an Arduino UNO module with acceleration and gyroscope sensors located on a headphone (e.g. Figure \[fig:system\]). The data communication is done through a bluetooth HM-10 module which send data to an IOS iphone app. Also, a camera which is not shown in the figure simultanuously records a user’s head gesture and body movement video as the Arduino UNO collect sensor data. Briefly, there are the main software component for Arduino UNO. First, it callibrates the device for the initial few seconds. Then, it starts the read of sensor data. Then, it sends the data to a registered bluetooth transmit address to let bluetooth module send data to iphone through an simple IOS app to parse transmitted data and upload to a backend server. Notice that some basic manual cleaning of the data is done according our recorded videos (around $3-4$ minutes) like removing the starting and ending periods (about a few second duration). We then use synchronized cleaned videos (about $10$ videos for each combination of head gesture classes and walk/sit classes) to manually label head gesture label and walk/sit label. In general, we collect $9$ classes for head gesture label, namely left-roll, right-roll, head-right, head-left, right-lean, left-lean, head-up, head-down, and a null class of no-move (e.g. in Figure \[fig:example\]), and $2$ classes of walk/sit label, namely if a user is walking or sitting. The baud rate is set as $9600$ bits per second, and our data sample rate is chosen at $\frac{1}{12}Hz$. Under this setting, each ground truth head gestures contains about $20$ samples in a duration of about $1.6$ seconds. The over summary of our collected data is in Table \[tab:gesture\_stats\]. We can see that left roll and right roll takes a lot more time than other head gestures. Also, we can found that duration varies for each head gesture class too. Maximum duration could be $0.3$s more than the minimum duration, which is about $4$ more samples. The visualization of right-roll under both sit and walk condition in Figure \[fig:right\_roll\] also intuitively show such varied duration at different times and also the strong impact from body movement under walk condition. This is an indication that Co-HAR problem is more challenging than just sit without walk. [ \|m[.5in]{}\|m[.6in]{}\|m[.6in]{}\|m[.75in]{}\| ]{} Gesture & class number per sit video & class number per walk video & duration range of a gesture (second)\ head up & 9 & 9 & 1.5 - 1.8\ head down & 9 & 9 & 1.5 - 1.8\ head left & 9 & 10 & 1.5 - 1.8\ head right & 10 & 10 & 1.5 - 1.7\ left lean & 10 & 10 & 1.5 - 1.7\ right lean & 9 & 10 & 1.5 - 1.7\ left roll & 10 & 9 & 1.9 - 2.1\ right roll & 10 & 9 & 1.9 - 2.1\ no gesture & - & - & -\ \[tab:gesture\_stats\] Competing Methods and Conditional-UNet variants ----------------------------------------------- Two competing methods including a pre-dated conventional method, and a baseline deep UNet model are used. Two variants of our proposed Conditional-UNet model are also introduced here. ### Surpport Vector Machine (SVM) SVM is a conventional method widely used pre-dated deep learning methods. We use it as a naive baseline. The six sensor signals are used as raw feature inputs. Two different SVM models are trained separately for head gesture label and walk/sit label. ### UNet baseline (UNet) This is a baseline multi-label classification based on UNet, a state-of-the-art deep fully convolutional network for HAR [@zhang2019human], which only use one UNet decoding module to output both head gesture label and walk/sit label at the same time without any conditional dependence. ### Dense Head Conditioned on Dense Walk (DHcoDW) This model is a variant of our Conditional-UNet that first decoding modules model walk/sit label and an encoding module to encode conditional dependence of walk/sit condition, then sequentially, a second decoding module model head label. “Dense” means that both labels are classified for each sample (a.k.a. each time step). ### Dense Walk Conditioned on Dense Head (DWcoDH) This model is another variant of our Conditional-UNet that first decoding modules model head label and an encoding module to encode conditional dependence of head condition, then sequentially, a second decoding module model walk/sit label. Both labels are classified for each sample. Evaluation Metric ----------------- As a classification problem, we use common accuracy score and multi-label F1 score as evaluation metric to compare competing method with our proposed methods. Overall, multi-label F1 score consider both precision and recall in different classes, and is better than accuracy score. We also demonstrate confusion matrix to show the performance of precision and recall for each class of a label. Quantitative Analysis --------------------- Table \[tab:model\_performance\] contains the experiment results of accuracy and F1 scores by competing methods and variants of our proposed methods. The bolded values are the best one compared to other methods. We can see that SVM fail for head gesture label with a low F1 score of only $35.57\%$, while SVM perform better for walk/sit label with $66.71\%$ F1 score. It indicates that walk/sit has much stronger signals and is easier to be classified. UNet already perform very well as current state-of-the-art method with $90.46\%$ accuracy score and $84.60\%$ F1 score for head gesture label, and $94.94\%$ accuracy score and $94.15\%$ F1 score for walk/sit label. For head gesture label, DHcoDW variant perform about $2\%$ better on accuracy ($92.06\%$) and $3.2\%$ better on F1 ($87.83\%$). The performance on walk/sit label of DHcoDW is only a little worse than UNet ($1.94\%$ less on accuracy and $2.15\%$ less on F1). For DWcoDH variant, head gesture also improve about $1\%$ of accuracy and $2\%$ of F1 score, and walk/sit labels are almost equally as well as UNet baseline. This results show that DHcoDW variant utilize the conditional dependence of walk/sit to better classify head gestures with a little downgrading of walk/sit label. But, DWcoDH’s results show that conditional dependence of head gestures promote less performance gain because the stronger signals of walk/sit vanish signals of head movements. This is also a good illustration of advantages of our proposed Conditional-UNet for real-world applications, since head gesture label is more critical and interesting than walk/sit label in real-world applications. Walk/sit condition is more like a noise to be removed than a valuable label to be used. \[tab:model\_performance\] [ m[.4in]{} m[.4in]{} \| m[.4in]{} m[.4in]{} m[.4in]{} m[.4in]{} m[.4in]{} ]{} Labels & & SVM & UNet & DHcoDW & DWcoDH\ & Accuracy & 0.7516 & 0.9046 & 0.9206 & 0.9128\ & F1 & 0.3557 & 0.8460& 0.8783 & 0.8638\ & Accuracy & 0.6241 & 0.9494 & 0.9300 & 0.9426\ & F1 & 0.6671 & 0.9415 & 0.9201 & 0.9369\ Confusion matrix of different methods are also shown in Figure \[fig:confusion\_matrix\], which tell more details about different methods. All confusion matrix values are normalized by total number of ground truth in this class, or in another word by sum of each row (diagonal elements are true positive rate). The most important observation is that two variants of our proposed conditional-UNet achieve significantly gains on head gesture label, because naive UNet model mistakenly classify a large portion of head gesture as null class. Since null class and other head gestures are imbalanced, the accuracy score do not quite reflect the margin of improvement as shown by confusion matrix. By comparing DWcoHD and DHcoDW variants, we can see that DHcoDW achieve a higher true positive rate except head-right, left-lean, left-roll. If we compare the walk/sit label, it is found that DHcoDW variant achive more balance between walk class ($94\%$) and sit class ($93\%$), but both naive UNet and DWcoDH have low performances on walk class. This is another good demonstration that the conditional dependence design help to get more gains by learning challenging body movements during walk condition. Qualitative visualization ------------------------- We illustrate a few classification results here through visualization of raw sensor, ground truth, and classified classes for both head gesture label and walk/sit label in Figure \[fig:qualitative\_viz\]. Each column is for each methods, the first row is for head gesture label, the second row is for walk/sit label. Left Y axis of the first row plots are different classes of head gestures. Right Y axis of first and second row plots are for raw X-axis accelerometer values. The left Y axis of the second row plots are for walk/sit label. DHcoDW variant of Conditional-UNet outperform a good result (only one sample on the right is classified wrong for head gesture label). DWcoDH unfortunately classify wrong class for walk/sit label, while its classification for head gesture label are very good. In general, this error of one sample ($\frac{1}{12}$ second) could be minor issue for many real-world application. Conclusion ========== We proposed a Coherent Human Activity Recognition problem here with a novel condition-aware deep model of Conditional-UNet to model the joint probability of multiple labels in Co-HAR with explicitly structures to handle conditional dependence in a sequential manner. The conducted experiments show that the proposed method outperforms the pre-dated SVM and state-of-the-art UNet deep model by $3\%$ in F1 score. Moreover, it gets significant gains for different head gestures with a little sacrification of walk/sit label performance. The experiments show that our proposed Conditional-UNet successfully capture the conditional dependence as expected. In this work, a Co-HAR dataset is also contributed to research communities.
--- abstract: 'We investigate the continuous optimal transport problem in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider regularization of the problem with so-called Young’s functions, which forces the optimal transport plan to be a function in the corresponding Orlicz space rather than a Radon measure. We derive the predual problem and show strong duality and existence of primal solutions to the regularized problem. Existence of (pre-)dual solutions will be shown for the special case of $L^p$ regularization for $p\geq2$. Then we derive four algorithms to solve the dual problem of the quadratically regularized problem: A cyclic projection method, a dual gradient decent, a simple fixed point method, and Nesterov’s accelerated gradient, all of which have a very low cost per iteration.' author: - \| Dirk Lorenz\ Institute of Analysis and Algebra\ TU Braunschweig\ 38092 Braunschweig, Germany\ `d.lorenz@tu-braunschweig.de`\ Hinrich Mahler\ Institute of Analysis and Algebra\ TU Braunschweig\ 38092 Braunschweig, Germany\ `h.mahler@tu-braunschweig.de`\ bibliography: - 'literature.bib' title: 'Orlicz-space regularization for optimal transport and algorithms for quadratic regularization' --- Introduction ============ We consider the optimal transport problem in the following form: For compact sets $Ω_1,\,Ω_2\subset ℝ^n$, measures $\mu_{1},\mu_{2}$ on $\Omega_{1},\Omega_{2}$, respectively, with the same total mass and a real-valued cost function $c:\Omega_{1}\times\Omega_{2}\to ℝ$ we want to solve $$\inf_{\pi}\int_{\Omega_{1}\times\Omega_{2}}c\dpi$$ where the infimum is taken over all measures on $\Omega_{1}\times \Omega_{2}$ which have $\mu_{1}$ and $\mu_{2}$ as their first and second marginals, respectively (see [@peyre2019computational; @villani2003topics]). Since optimal plans $\pi$ tend to be singular measures (even for marginals with smooth densities [@villani2008optimal; @Santambrogio]), regularization of the problem have become more important, most prominently entropic regularization [@carlier2017convergence; @cuturi2013sinkhorn; @Benamou:2015; @cuturi2016smoothed; @clason:2019] which ensures that optimal plans have densities. It has been shown in [@clason:2019] that the analysis of entropically regularized optimal transport problems naturally takes place in the function space $L\log L$ (also called Zygmund space[@Bennett:1988]) and that optimal plans for entropic regularization are always in $L\log L(\Omega_{1}\times \Omega_{2})$ and exist if and only if the marginals are in the spaces $L\log L(\Omega_{i})$. These spaces are an example of so-called Orlicz spaces [@Rao:1988] and hence, we consider regularization in these spaces in this paper. Another motivation to study a more general regularization comes from the fact that regularization with the $L^{2}$-norm have been shown to be beneficial in some applications, see [@roberts2017gini; @blondel2017smooth; @dessein2018regularized; @Lorenz:2019]. To simplify notation, denote $Ω:= Ω_1 \times Ω_2$. The regularized problems we consider are $$\tag{P}\label{eq:reg_kantorovich} \inf_{\substack{\pi\geq 0,\\{(P_{i})_\#}{π} = {μ_i},\,i=1,2}} \int_{\Omega} c \dpi + γ \int_{\Omega} Φ\compos π \commath$$where $γ>0$ and the infimum is taken over all positive measure, which have densities with respect to the Lebesgue measure and the constraints ${(P_{i})_\#}{π} = {μ_i}$ state that $\pi$ should have the marginals $\mu_{i}$. The two main cases we will consider in this paper are $\Phi(t) = t\log t$ (entropic regualarization) and $\Phi(t) = \nicefrac{t^2}2$ (quadratic regularization), however, many results in this paper hold for more general functions $\Phi$. The rest of the paper is organized as follows. First, we introduce the so-called Young’s functions and Orlicz spaces in \[sec:yf\_os\]. Moreover, a slight generalization of Young’s functions is defined. In \[sec:existence\] we give a strong duality result for , which allows us to prove existence of primal solutions. Existence of solutions for the (pre-)dual problem will be discussed for the special case $Φ(t) = \nicefrac{(t_+)^p}{p}$ for $p\geq 2$. Due to limited space, we omit some proofs in this section and the proofs will be published in an extended version. Finally, we consider numerical methods for the quadratic regularization in \[sec:numerics\]. Here, we will work with the duality result presented in \[sec:existence\] and concentrate on algorithms with very low cost per iteration. Young’s Functions & Orlicz Spaces {#sec:yf_os} ================================= We briefly introduce some notions about Young’s functions and Orlicz spaces. For a more detailed introduction, see [@Bennett:1988; @Rao:1988]. \[def:yf\] 1. Let $ϕ : [0, ∞) →{} [0, ∞]$ be increasing and lower semicontinuous, with $ϕ(0) = 0$. Suppose that $ϕ$ is neither identically zero nor identically infinite on $(0,∞)$. Then the function $Φ$, defined by $$Φ(t) := \int_0^t ϕ(s) \ds\commath$$ is said to be a Young’s function. 2. A Young’s function $Φ$ is said to have the $\Delta_2$-property near infinity if $Φ(t)<∞$ for all $t$ and $$∃C>0,\,t_0\geq 0:\,∀t\geq t_0:\,Φ(2t) \leq C Φ(t).$$ By definition, Young’s functions are convex and for a Young’s function $\Phi$ it holds that the complementary Young’s function $\Psi(x) := \int_{0}^{x}(\Phi')^{-1}(s)\ds$ is also a Young’s function. Indeed, the complementary Young’s function is related to the convex conjugate $Φ^$. The negative entropy regularization uses the regularization functional $\int_{\Omega}\pi\log\pi$ and the function $t\mapsto t\log t$ is not a Young’s function. Hence, we introduce a slight generalization. \[def:qyfs\] Let $Φ$ be a Young’s function and $t_{0}\geq 0$. Let $\check{Φ}$ be a convex, lower semicontinuous function bounded from below with $$\check{Φ}(t) = Φ(t)-Φ(t_{0})\qquad\forall t\geq t_{0}$$ and $\check{Φ}(t)\leq 0$ for all $t<t_{0}$. Then $\check{Φ}$ is said to be a quasi-Young’s function* induced by $Φ$. The function $\check{Φ}(t) = t\log_+ t$ is a quasi-Young’s function induced by the Young’s function $Φ(t) = t\log t$, with $t_{0} = 1$. It holds $Φ(t) = \max\{0,\check{Φ}(t)\}$. Let $Φ$ be a Young’s function and $Ω⊂ℝ^n$. Define the Luxemburg norm of a measurable function $f: Ω → ℝ$ as $$\lux{f} := \inf \set{γ\geq0 }{ \int_{\Omega} Φ\compos\of{\frac{{\left\|{f}\right\|}}{γ}}\dleb \leq 1}\commath[.]$$ Then the space $$\Lorl(Ω) := \set{f:Ω→ℝ\,\text{measurable}}{\lux f < ∞}$$ of measurable functions on $Ω$ with finite Luxemburg norm is called the Orlicz space of $Φ$. One can verify that the definitions of $\lux{\blank}$ and $\Lorl$ are essentially independent of whether $Φ$ is a Young’s function or just a quasi-Young’s function. To simplify notation, $\lux[\tilde{Φ}]{\blank}$ and $\Lorl[\tilde{Φ}]$ will therefore also be used for quasi-Young’s function[s]{} $\tilde{Φ}$. Note that for a quasi-Young’s function $\tilde{Φ}$ induced by a Young’s function $Φ$, it holds that $\Lorl[\tilde{Φ}] = \Lorl$, while in general $\lux[\tilde{Φ}]{\blank}$ and $\lux{\blank}$ are equivalent but not equal. It is well known that for a quasi-Young’s function $\Phi$ with $\lim_{t→∞}\nicefrac{Φ(t)}{t} = ∞$ and bounded domain $\Omega$ it holds that $\Lorl(Ω) ⊂ \L 1(Ω)$. Moreover for complementary Young’s function[s]{} $Φ$ and $Ψ$ which are proper, locally integrable and have the $\Delta_2$-property near infinity it holds that $(\Lorl{})^$ is canonically isometrically isomorphic to $(\Lorl[Ψ],\lux[Ψ]{\blank})$ (see, e.g. [@diening:2011]). Let $Φ(t) = t \log t$ and $\tilde{Φ}(t) = t \log_+ t$. The space of measurable functions $f$ with $\int_{\Omega} \tilde{Φ}\compos {\left\|{f}\right\|}\dleb<∞$ is called $\LlogL{}$. Because $\Lorl = \Lorl[\tilde{Φ}]$, the space of measurable functions $g$ with $\int_{\Omega} {Φ}\compos{\left\|{g}\right\|}\dleb<∞$ is equal to $\LlogL$ as well. The complementary Young’s function $\tilde{Ψ}$ of $\tilde{Φ}$ is given by $$\tilde{Ψ}(t) = \begin{cases} t,&t\leq 1\\ \e^{t-1},&\text{else.} \end{cases}$$ Hence, the dual space of $\LlogL{}$ is given by the space of measurable functions $h$ that satisfy $\int_{\Omega} \tilde{Ψ}\compos{\left\|{h}\right\|}\dleb<∞$, which is called ${L_{\exp}}{}$. Similar to [@clason:2019 Lemma 2.11] one can show that the marginals of a transport plan $π\in\Lorl$ are also in $\Lorl$: \[thm:proj\_contraction\] If $π\in\Lorl(Ω)$ for a quasi-Young’s function $Φ$, then ${(P_{i})_\#}π\in\Lorl(Ω_i)$ for $i=1,2$ with $$\lux{{(P_{i})_\#}π} \leq \max\of{1, {\left\|{Ω_{3-i}}\right\|}} \lux{π}\commath[.]$$ Conversely, for the direct product of two marginals to lie in $\Lorl(Ω)$, some assumptions have to be made. \[thm:tensor\_feasible\] Let $Ω_1, Ω_2⊂ℝ^n $ be bounded and $Φ$ be a quasi-Young’s function satisfying either $$\label{cond:tensor_feasible:1} Φ(xy) \leq C Φ(x)Φ(y)$$ for some $C>0$ or $$\label{cond:tensor_feasible:2} \begin{split} Φ(xy) &\leq C_1 xΦ(y) + C_2 Φ(x)y\\ \frac{Φ(t)}{t} & \xrightarrow[t\to\infty]{} ∞ \end{split}$$ for some $C_1,C_2 \geq 0$. If $π = μ_1\dprod μ_2$ and $μ_i\in\Lorl(Ω_i)$ for $i=1,2$, where $(μ_1\dprod μ_2)(x_1,x_2):=μ_1(x_1)μ_2(x_2)$, then $π\in\Lorl(Ω)$. 1. For $Φ(t) = \frac{t^p}{p}$, $p>1$, \[cond:tensor\_feasible:2,cond:tensor\_feasible:1\] both hold trivially. 2. For $Φ(t) = t\log t$, \[cond:tensor\_feasible:2\] holds, since $\log(xy) = \log(x) + \log(y)$. Existence of Solutions {#sec:existence} ====================== In this section, strong duality will be shown for the regularized mass transport using Fenchel duality in the spaces $\radon(Ω)$ and $\CC(Ω)$. Here, the general framework as outlined in [@Attouch:2006 Chap. 9] or [@Ekeland:1999 Sec. III.4], is used. The result will then be used to study the question of existence of solutions for both the primal and the dual problem. \[thm:str:dual\] Let $Φ$ be a quasi-Young’s function and let $$\tilde{Φ}(t) := \begin{cases} ∞,&t<0,\\ Φ(t),&\text{else}. \end{cases}$$ If $\tilde{Φ}^\compos\of{\nicefrac{-c}{γ}}$ is integrable, then the predual problem to is $$\tag{P}\label{eq:dual_reg_kantorovich} \sup_{\substack{α_i\in\CC(Ω_i),\\i=1,2}} \int_{\Omega_1} α_1 \dμ_1 + \int_{\Omega_2}α_2\dμ_2 - γ\int_{\Omega} \tilde{Φ}^\compos\of{\frac{{α_1}\dplus{α_2}-c}{γ}} \dleb\,,$$ where $(α_1\oplus α_2)(x_1,x_2):=α_1(x_1) + α_2(x_2)$, and strong duality holds. Furthermore, if the supremum is finite, posseses a minimizer. This proof follows the outline of the proof in [[@clason:2019 Theorem 3.1]]{}. First note that $\radon(\Omega)$ is the dual space of $\CC(\Omega)$ for compact $\Omega$. Furthermore Slater’s condition is fulfilled with ${α_1},{α_2} = 0$ so that strong duality holds and (assuming a finiteness of the supremum) the primal possesses a minimizer. Additionally, the integrand of the last integral in is normal, so that it can be conjugated pointwise [@rockafellar:1968 Theorem 2]. Carrying out the conjugation one obtains $$\begin{aligned} \sup_{\substack{α_i\in\CC(Ω_i),\\i=1,2}} &\int_{\Omega_1} {α_1} \dμ_1 + \int_{\Omega_2}{α_2}\dμ_2 - γ\int_{\Omega} \tilde{Φ}^\compos\of{\frac{α_1\dplus α_2 - c}{γ}} \dleb\\ &\begin{multlined} =\sup_{\substack{α_i\in\CC(Ω_i),\\i=1,2}} \int_{\Omega_1} {α_1} \dμ_1 + \int_{\Omega_2}{α_2}\dμ_2 + \int_{\Omega} \of{\inf_{π} \of{c - α_1\dplus α_2} π + γ\tilde{Φ}\compos π }\dleb \end{multlined}\\ &\begin{multlined} =\sup_{\substack{α_i\in\CC(Ω_i),\\i=1,2}} \inf_{π} \int_{\Omega} cπ + γ \tilde{Φ}\compos π \dleb + \int_{\Omega_1} {α_1}\d ({μ_1}-{(P_{1})_\#}π)\\ + \int_{\Omega_2} {α_2}\d({μ_2}-{(P_{2})_\#}π) \end{multlined}\\ &=\inf_{\substack{π,\\{(P_{i})_\#}{π} = {μ_i},\,i=1,2}} \int_{\Omega} c \dpi + γ \int_{\Omega} \tilde{Φ}\compos π \dleb\\ &=\inf_{\substack{0\leqπ,\\{(P_{i})_\#}{π} = {μ_i},\,i=1,2}} \int_{\Omega} c \dpi + γ \int_{\Omega} Φ(π) \dleb\commath \end{aligned}$$ which is . 1. Using $Φ(t) = t\log t$, one obtaines the result for $\LlogL{}$ as stated in [@clason:2019 Theorem 3.1] 2. Using $Φ(t) = \nicefrac{t^2}2$, one obtains the result for $\L2$ as stated in [@Lorenz:2019]. \[thm:str:dual\] does not* claim that the supremum is attained, i.e. the predual problem admits a solution. Moreover, the solutions of cannot be unique since one can add and subtract constants to $α_1$ and $α_2$, respectively, without changing the functional value. Existence Result for the Primal Problem --------------------------------------- The duality result can now be used to address the question of existence of a solution to . \[thm:dual:prop\_solution\] Problem admits a minimizer $\bar{π}$ if and only if $μ_i \in\Lorl(Ω_i)$ for $i=1,2$ and $$\label{cond:prod_of_fcts} μ_1\dprod μ_2 \in\Lorl(Ω)\quad \forall μ_i \in\Lorl(Ω_i),\,i=1,2 \commath[.]$$ In this case, $\bar{π} \in\Lorl(Ω)$. Moreover, the minimizer is unique, if $Φ$ is strictly convex. The proof given in [@clason:2019 Theorem 3.3] for $Φ(t) = t \log t$ holds for arbitrary $Φ$. That is, the necessity of the condition $μ_i \in\Lorl(Ω_i)$, $i=1,2$ only relies on \[thm:proj\_contraction\]. For sufficiency, it is noted that for $μ_i \in\Lorl(Ω_i)$, $i=1,2$ it holds that $π=μ_1\dtimes μ_2\in\Lorl(Ω)$, which is ensured by \[cond:prod\_of\_fcts\]. Thus, the infimum in is finite and weak duality shows that the supremum in is finite as well. Existence of a solution for now follows from \[thm:str:dual\]. If strict convexity holds for $Φ$, it directly implies uniqueness. For example, \[cond:prod\_of\_fcts\] is satisfied when $Φ$ satisfies either \[cond:tensor\_feasible:1\] or \[cond:tensor\_feasible:2\] since in those cases \[thm:tensor\_feasible\] holds. Existence Result for the Predual Problem ---------------------------------------- The question of existence of solutions to the predual problem proves to be more difficult for general Young’s functions. There are results that shows existence for the predual problem in the entropic case [@clason:2019] and in the quadratic case [@Lorenz:2019], but their proofs are quite different in nature. Here, we only treat Young’s functions of the type $Φ(t)= \nicefrac{t^p}{p}$ for $p>1$, i.e, only regularization in $L^{p}$. Note that in this case ${Φ}^(t) = \nicefrac{(\pos{t})^q}{q}$, where $\nicefrac 1 p + \nicefrac 1 q = 1$ and the predual is actually the dual. \[asspt:dual\] Let $Ω_1$ and $Ω_2$ to be compact comains, let the cost function $c$ be continuous and fulfill $c \geq c^{\dagger} > -∞$. Furthermore, the marginals ${μ_i}\in\Lp(Ω_i)$ satisfy ${μ_i}\geq δ > 0$ a.e. for $i=1,2$ and finally assume that $\int_{\Omega_1} {μ_1} =\int_{\Omega_2} {μ_2} = 1$. It can not be expected for to have continuous solutions ${α_1}$, ${α_2}$. However, observe that the objective function of is also well defined for functions ${α_i}\in\L{1}(Ω_i)$, $i=1,2$, with $\pos{({α_1}\oplus{α_2} - c)}/γ \in \L{q}(Ω)$. This gives rise to the following variant of the dual problem, for which existence of minimizers can be shown: $$\tag{P$^\dagger$}\label{eq:dual_aux} \begin{multlined} \min\left\{ Λ({α_1},{α_2}) := \frac{1}{q} {\\|{\pos{({α_1}\oplus {α_2} -c)}}\\|}_q^q \vspace{-1em} - γ^{q-1}\int_{\Omega_1} {α_1} \dμ_1 - γ^{q-1}\int_{\Omega_2}{α_2}\dμ_2 \right.\\ \left.\vphantom{\int_{\Omega_2}{α_2}}\middle\|\, α_i\in\L{1}(Ω_i),\,i=1,2,\,\pos{({α_1}\oplus{α_2} - c)}/γ \in \L{q}(Ω) \right\} \end{multlined}$$ The strategy is now as follows. 1. First, show that admits a solution $({\bar{α}_1},{\bar{α}_2}) \in \L{1}(Ω_1) \times \L{1}(Ω_2)$. 2. Then, prove that ${\bar{α}_1}$ and ${\bar{α}_2}$ possess higher regularity, namely that they are functions in $\L{q}(Ω_i)$. The objective function is extended to allow to deal with weakly-$$ converging sequences. To that end, define $$G:\L{q}(Ω)\ni w \mapsto \int_{\Omega} \of{\frac{1}{q} (\pos{w})^q - w μ }\dleb \inℝ \commath$$ where $μ := γ^{q-1}({μ_1}\otimes {μ_2})$. Then, thanks to the normalization of ${μ_1}$ and ${μ_2}$, $$Λ({α_1},{α_2}) = G({α_1}\oplus {α_2} - c) - \int_{\Omega} c μ \dleb\quad ∀{α_1},{α_2} \in\L{q}\commath[.]$$ Of course, $G$ is also well defined as a functional on the feasible set of and this functional will be denoted by the same symbol to ease notation. In order to extend $G$ to the space of Radon measures, consider for a given measure $w \in\radon(Ω)$, the Hahn-Jordan decomposition $w=\pos w + \negprt w$ and assume $\pos w \in\Lq(Ω)$. Then, set $$G(w) := \int_{\Omega} \frac{1}{q} (\pos{w})^q \dleb - \int_{\Omega} μ \dw\commath[.]$$ With slight abuse of notation, this mapping will be denoted by $G$, too. If $w\acl$, then $\pos w \in\L1(Ω)$ and $\pos w(x) = \max\sset{0, w(x)}$ $\leb$-a.e. in $Ω$. Hence, both functionals denoted by $G$ conincide on $\Lq(Ω)$, which justifies this notation. The following auxiliary results are generalizations of the corresponding results in [@Lorenz:2019] and can be proven with little effort. \[thm:dual:boundedness\] Let \[asspt:dual\] hold and suppose that a sequence $\seq w n⊂ \L q (Ω)$ fulfills $$G(w_n) \leq C < ∞\quad ∀ n \inℕ$$ for some $C>0$. Then, the sequences $\pos{\seq w n}$ and $\negprt{\seq w n}$ are bounded in $\L q(Ω)$ and $\L 1 (Ω)$, respectively. The assertion w.r.t. $\pos{\seq w n}$ can be proven by the same argument used in [@Lorenz:2019 Lemma 2.6]. The second one can be seen by making use of $μ\geq δ$ with $δ$ from \[asspt:dual\], which yields the estimate $$\begin{aligned} C \geq G(w_n) & = \frac{1}{q}\int_{\Omega} \pos{(w_n)}^q \dleb - \int_{\Omega} \pos{(w_n)} μ\dleb + \int_{\Omega} \negprt{(w_n)} μ\dleb\\ &\geq \frac{1}{q} {\\|{\pos{(w_n)}}\\|}_q^q - {\\|{μ}\\|}_p{\\|{\pos{(w_n)}}\\|}_q + γ^{q-1}δ^2 {\\|{\negprt{(w_n)}}\\|}_1\\ &\geq - {\\|{μ}\\|}_p{\\|{\pos{(w_n)}}\\|}_q + γ^{q-1}δ^2 {\\|{\negprt{(w_n)}}\\|}_1\commath[.] \end{aligned}$$ Since ${\\|{\pos{(w_n)}}\\|}_q$ is already known to be bounded, the second assertion holds. \[thm:dual:wsconv\] Let \[asspt:dual\] hold and a sequence $\seq w n⊂ \L q (Ω)$ be given such that $w_n {\xrightharpoonup}\bar{w}$ in ${{\mathcal{M}}}(Ω)$ and $G(w_n) \leq C < ∞$ for all $n\inℕ$. Then it holds that $\pos{\bar w}\in\L q (Ω)$ and $$G(\bar{w}) \leq \liminf_{n→∞} G(w_n).$$ \[thm:dual-l1-solultion\] Let \[asspt:dual\] hold. Then, admits a solution $({\bar{α}_1},{\bar{α}_2})\in\L 1 (Ω_1)\times \L 1 (Ω_2)$. In [@Lorenz:2019 Proposition 2.10] the statement is proven for $p=2$ via the classical direct method of the calculus of variations using only [@Lorenz:2019 Lemmas 2.8 & 2.9] and \[thm:dual:boundedness,thm:dual:wsconv\], where [@Lorenz:2019 Lemmas 2.8 & 2.9] are rather technical results holding independently of the choice of $Φ$. Hence, the proof also holds for $p\geq 2$. The next results states that $α_i$, $i=1,2$ are indeed functions in $\Lq(Ω_i)$. \[thm:dual:lq\] Let \[asspt:dual\] hold and let $p\geq 2$. Then every optimal solution $({\bar{α}_1}, {\bar{α}_2})$ from \[thm:dual-l1-solultion\] satisfies ${\bar{α}_i}\in\Lq(Ω_i)$, $i=1,2$. Numerical Methods for Quadratic Regularization {#sec:numerics} ============================================== In this section we turn to numerical methods and focus only on the case of quadratic regularization. For the special case of the negative entropy, $Φ(t) = t\log t$, there is the celebrated Sinkhorn method [@cuturi2013sinkhorn; @Sinkhorn:1966; @Sinkhorn:1967a] which can be interpreted as an alternating projection method [@Benamou:2015]. For the case of quadratic regularization, i.e. $\Phi(t) = t^{2}/2$ [@Lorenz:2019] proposed a Gauß-Seidel method (which is similar to the Sinkhorn method) and a semismooth Newton method (which is similar to the Sinkhorn-Newton method from [@brauer2017sinkhorn] for entropic regularization). Both methods converge reasonably well, but the iterations become expensive for large scale problems. In [@blondel2017smooth] used the standard solver L-BFGS method to solve the dual problems which also works good for medium scale problems, but is not straightforward to parallelize. Here we focus on methods that come with very low cost per iteration and which allow for simple parallelization. We switch to the discrete case and slightly change notation. The marginals are two non-negative vectors $\mu\in{{\mathbb{R}}}^{N}$ and $\nu\in{{\mathbb{R}}}^{M}$ with $\sum_{i}\mu_{i} = \sum_{j}\nu_{j}$ and the cost is $c\in{{\mathbb{R}}}^{N\times M}$. We denote by ${\mathbf{1}}$ the vector of all ones (of appropriate size). A feasible transport plan is now a matrix $\pi\in{{\mathbb{R}}}^{N\times M}$ with $\pi{\mathbf{1}}= \mu$ (matching row-sums) and $\pi^{T}{\mathbf{1}}= \nu$ (matching colum sums). The quadratically regularized optimal transport problem is then, for some $\gamma>0$ $$\label{eq:qrot-discrete} \min_{\substack {\pi\geq 0\\\pi{\mathbf{1}}=\mu\\\pi^{T}{\mathbf{1}}= \nu}}\sum_{ij}c_{ij}\pi_{ij} + \tfrac\gamma2{\\|{\pi}\\|}_{2}^{2}.$$ The starting point for our algorithms for the quadratically regularized problem is the optimality system: $\pi$ is optimal if and only if there are two vectors $\alpha\in{{\mathbb{R}}}^{N}$ and $\beta\in{{\mathbb{R}}}^{M}$ such that $$\begin{aligned} \pi & = (\alpha\oplus\beta - c)_{+}/\gamma\\ \sum_{j}\pi_{ij} & = \mu_{i},\quad \sum_{i}\pi_{ij} = \nu_{j}\end{aligned}$$ where we used the notation $\oplus$ to denote the outer sum, i.e. $\alpha\oplus\beta\in{{\mathbb{R}}}^{N\times M}$ with $(\alpha\oplus\beta)_{ij} = \alpha_{i}+\beta_{j}$. Note the similarity to entropic regularization: There one can show that a plan $\pi$ is optimal if it is of the form $\pi = \exp(\tfrac{\alpha\oplus\beta-c}\gamma)$ and has correct row and column sums. An alternative formulation of the optimality system is: $\pi = (\rho+\alpha\oplus\beta-c)/\gamma$ is optimal if $$\begin{aligned} \rho & = (\alpha\oplus\beta-c)_{-}\\ \sum_{j}(\rho_{ij} + \alpha_{i}+\beta_{j}-c_{ij}) & = \gamma\mu_{i}\\ \sum_{i}(\rho_{ij} + \alpha_{i}+\beta_{j}-c_{ij}) & = \gamma\nu_{j}\\\end{aligned}$$ This leads us to a very simple algorithm: Initialize $\alpha$ and $\beta$ and cyclically solve the first equation above for $\rho$, the second for $\alpha$, and the third for $\beta$. This algorithm is described as Algorithm \[alg:qrot\_cp\]. Note that we can interpret Algorithm \[alg:qrot\_cp\] as a cyclic projection method: The quadratically regularized optimal transport problem is equivalent to minimizing ${\\|{-\tfrac{c}{\gamma}-\pi}\\|}_{2}^{2}$ over the constraints $\pi\geq0$, $\pi{\mathbf{1}}=\mu$, and $\pi^{T}{\mathbf{1}}=\nu$, i.e. the solution is the projection of $-c/\gamma$ onto the set defined by these three constraints. Algorithm \[alg:qrot\_cp\] does implicitly project $\pi$ cyclically onto these three constraints (without actually forming $\pi$ during the iteration). While iterative cyclic projections are guaranteed to find a feasible point, it is not guaranteed that the iteration converges to the projection in general [@bauschke1997method]. However, in this case the fixed points $\alpha^{}$, $\beta^{}$ of the algorithm are indeed solutions of , since the resulting $\pi = (\alpha^{}\oplus\beta^{}-c)_{+}/\gamma$ has the correct form and marginals. Initialize: $\alpha^{0}=0\in{{\mathbb{R}}}^{N}$, $\beta^{0}=0\in{{\mathbb{R}}}^{M}$, set $n=0$ $\rho^{n+1}_{ij} = (\alpha_{i}^{n} + \beta_{j}^{n} - c_{ij})_{-}$ $\alpha^{n+1}_{i} = \tfrac{\gamma}M\Big(\mu_{i} - \tfrac1\gamma\sum_{j}\rho_{ij}^{n+1}+\beta_{j}^{n}-c_{ij}\Big)$ $\beta^{n+1}_{j} = \tfrac{\gamma}N\Big(\nu_{j} - \tfrac1\gamma\sum_{i}\rho_{ij}^{n+1}+\alpha_{i}^{n+1}-c_{ij}\Big)$ $n \gets n+1$ Output $\pi = (\alpha^{n}\oplus\beta^{n}-c)/\gamma$ Another natural choice for an algorithm is the gradient method on the dual problem of , namely on $$\min_{\alpha,\beta}\left\{F(\alpha,\beta) := \tfrac12{\\|{(\alpha\oplus\beta-c)_{+}}\\|}_{2}^{2} -\gamma{\langle{\alpha},{\mu}\rangle} - \gamma{\langle{\beta},{\nu}\rangle}\right\}.$$ The gradients with respect to $\alpha$ and $\beta$ are $$\nabla_{\alpha}F(\alpha,\beta) = (\sum_{j}(\alpha_{i}+\beta_{j}-c_{ij})_{+}-\gamma\mu_{i}),\quad \nabla_{\beta}F(\alpha,\beta) = (\sum_{i}(\alpha_{i}+\beta_{j}-c_{ij})_{+}-\gamma\nu_{j}),$$ respectively. With the help of the plans $\pi = (\alpha\oplus\beta-c)_+/\gamma$ one can express the gradients as $\nabla_{\alpha}F = \gamma(\pi{\mathbf{1}}-\mu)$ and $\nabla_{\beta}F = \gamma(\pi^{T}{\mathbf{1}}-\nu)$, respectively. A natural stepsize that leads to good performance is $\tau = 1/(M+N)$. This amounts to Algorithm \[alg:qrot\_gd\]. Initialize: $\alpha^{0}\in{{\mathbb{R}}}^{N}$, $\beta^{0}\in{{\mathbb{R}}}^{M}$, stepsize $\tau=1/(M+N)$, set $n=0$ $\pi^{n} = (\alpha^{n} \oplus \beta^{n} - c)_{+}/\gamma$ $\alpha^{n+1} = \alpha^{n} -\tau\gamma(\pi^{n}{\mathbf{1}}-\mu)$ $\beta^{n+1} = \beta^{n} -\tau\gamma\big((\pi^{n})^{T}{\mathbf{1}}-\nu\big)$ $n \gets n+1$ Algorithm \[alg:qrot\_fp\] below is another algorithm which works with extremely low cost per iteration. It can be derived as follows: The gradients are differentiable almost everywhere and the Hessian of $F$ is $$G(\alpha,\beta) = \begin{pmatrix} \operatorname{diag}(\sigma{\mathbf{1}}) & \sigma\\ \sigma^{T} & \operatorname{diag}(\sigma^{T}{\mathbf{1}}) \end{pmatrix},\quad\text{with}\quad \sigma_{ij} = \begin{cases} 1 & \alpha_{i} + \beta_{j}-c_{ij}\geq 0\\ 0 & \text{otherwise.} \end{cases}$$ The (semismooth) Newton method from [@Lorenz:2019] performs updates of the form $$\begin{pmatrix} \alpha^{n+1}\\\beta^{n+1} \end{pmatrix} = \begin{pmatrix} \alpha^{n}\\\beta^{n} \end{pmatrix} - G(\alpha^{n},\beta^{n})^{-1} \begin{pmatrix} \nabla_{\alpha}F(\alpha^{n},\beta^{n})\\\nabla_{\beta}F(\alpha^{n},\beta^{n}) \end{pmatrix}.$$ To reduce the computation, we can omit the inversion of $G$, by replacing it with the simpler matrix $$M = \begin{pmatrix} M(I + \tfrac1N{\mathbf{1}})& 0\\ 0 & N(I + \tfrac1M{\mathbf{1}}) \end{pmatrix},\quad\text{with}\quad M^{-1} = \begin{pmatrix} \tfrac1M(I - \tfrac1{2N}{\mathbf{1}}) & 0\\ 0 & \tfrac1N(I - \tfrac1{2M}{\mathbf{1}}) \end{pmatrix}.$$ where $I$ denotes the identity matrix and ${\mathbf{1}}$ denotes the matrix of all ones (of appropriate sizes). Fixed points of Algorithm \[alg:qrot\_fp\] are optimal solutions of the quadratically regularized optimal transport problem . The vectors $\alpha$, $\beta$ are fixed points if and only if $f$ and $g$ are zero. But this means that $\pi{\mathbf{1}}= \mu$ and $\pi^{T}{\mathbf{1}}= \nu$ which is, by definition of $\pi$ in the algorithm, the optimality condition. This shows that fixed points are optimal. Note that Algorithm \[alg:qrot\_fp\] is very similar to the dual gradient descent in Algorithm \[alg:qrot\_gd\] (it mainly differs in the stepsizes and the subtraction of the mean values). Initialize: $\alpha^{0}\in{{\mathbb{R}}}^{N}$, $\beta^{0}\in{{\mathbb{R}}}^{M}$, set $n=0$ $\pi^{n}_{ij} = (\alpha_{i}^{n} + \beta_{j}^{n} - c_{ij})_{+}/\gamma$ $f_{i}^{n} = -\gamma(\sum_{j}\pi_{ij}^{n} - \mu_{i})$ $\alpha^{n+1}_{i} = \alpha_{i}^{n} + \tfrac1M\left(f_{i}^{n} - \tfrac{\sum_{i}f_{i}^{n}}{2N}\right)$ $g_{j}^{n} = -\gamma(\sum_{i}\pi_{ij}^{n} - \nu_{j})$ $\beta_{j}^{n+1} = \beta_{j}^{n} + \tfrac1N\left( g_{j}^{n} - \tfrac{\sum_{j}g_{j}^{n}}{2M}\right)$ $n \gets n+1$ As a final algorithm we tested Nesterov’s accelerated gradient descent of the dual as stated in Algorithm \[alg:qrot\_ng\]. We used the same stepsize as for Algorithm \[alg:qrot\_gd\]. Initialize: $\alpha^{0}=\alpha^{-1}\in{{\mathbb{R}}}^{N}$, $\beta^{0}=\beta^{-1}\in{{\mathbb{R}}}^{M}$, stepsize $\tau=1/(M+N)$, set $n=0$ $\bar\alpha^{n} = \alpha^{n} + \sigma_{n}(\alpha^{n}-\alpha^{n-1})$, $\bar\beta^{n} = \beta^{n} + \sigma_{n}(\beta^{n}-\beta^{n-1})$ with $\sigma_{n} = n/(n+3)$ $\pi^{n} = (\bar\alpha^{n} \oplus \bar\beta^{n} - c)_{+}/\gamma$ $\alpha^{n+1} = \bar\alpha^{n} -\tau\gamma(\pi^{n}{\mathbf{1}}-\mu)$ $\beta^{n+1} = \bar\beta^{n} -\tau\gamma\big((\pi^{n})^{T}{\mathbf{1}}-\nu\big)$ $n \gets n+1$ Although the pseudo-code for all algorithms explicitly forms the outer sums $\alpha\oplus\beta$ at some points, this is not needed in implementations. In all cases we only need row- and colum-sums of these larger quantities of size $N\times M$ and these can be computed in parallel. Figure \[fig:example\_1d\_quad\] shows results for simple one-dimensionals marginals and quadratic cost function $c(x,y) = \|x-y\|^{2}$ and in Figure \[fig:example\_1d\_abs\] we used the absolute value $c(x,y) = \|x-y\|$. In all examples the cyclic projection (Algorithm \[alg:qrot\_cp\]) and the fixed-point iteration (Algorithm \[alg:qrot\_fp\]) perform good (Algorithm \[alg:qrot\_fp\] always slightly ahead) while dual gradient descent (Algorithm \[alg:qrot\_gd\]) is always significantly slower. Nesterov’s gradient descent (Algorithm \[alg:qrot\_ng\]) oscillates heavily, takes longer to reduce the error in the beginning but keeps reducing the error faster than the other methods. [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|^{2}$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_quad"}](code/example_1d_1.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|^{2}$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_quad"}](code/example_1d_2.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|^{2}$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_quad"}](code/example_1d_3.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|^{2}$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_quad"}](code/example_1d_4.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_abs"}](code/example_1d_5.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_abs"}](code/example_1d_6.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_abs"}](code/example_1d_7.pdf "fig:"){width="\textwidth"} [0.45]{} ![Results for one-dimensional examples with $c(x,y) = \|x-y\|$. Each subfigure on the left: Marginals and optimal plan. Each subfigure on the right: maximal violation of constraints over iteration count.[]{data-label="fig:example_1d_abs"}](code/example_1d_8.pdf "fig:"){width="\textwidth"}
--- abstract: 'It was recently shown in [@tHMRR19] that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach space are not essentially incomparable.' address: - 'S. ter Horst, School of Mathematical and Statistical Sciences, North-West University, Research Focus Area: Pure and Applied Analytics, Private Bag X6001, Potchefstroom 2520, South Africa and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)' - 'M. Messerschmidt, Department of Mathematics and Applied Mathematics; University of Pretoria; Private bag X20 Hatfield; 0028 Pretoria; South Africa' - 'A.C.M. Ran, Department of Mathematics, Faculty of Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The Netherlands and Research Focus Area: Pure and Applied Analytics, North-West University, Potchefstroom, South Africa' author: - 'S. ter Horst' - 'M. Messerschmidt' - 'A.C.M. Ran' title: Equivalence after extension and Schur coupling for relatively regular operators --- [^1] Introduction {#S:Intro} ============ Equivalence After Extension (EAE) and Schur Coupling (SC) are two relations on (bounded linear) Banach space operators that originated in the study of integral operators [@BGK84], along with the relation Matricial Coupling (MC), and which have since found many other applications, cf., [@CP16; @CP17; @CDS14; @ET17; @S17; @GKR17; @tHMRRW18] for a few recent references. The applications of these operator relations often rely on the fact that EAE, MC and SC coincide, in the specific context of the application, and it is important that one can easily and explicitly move between the three operator relations. This led to the question, posed by H. Bart and V.É. Tsekanovskii in [@BT94], whether these three operator relations might coincide at the level of general Banach space operators. In fact, by then it was known that EAE and MC coincide [@BGK84; @BT92a] and in [@BT94], see also [@BT92b], it was proved that EAE and MC are implied by SC. All these implications are obtained by explicit constructions, see Section 2 in [@tHR13] for an overview. Various attempts to prove that EAE (or MC) implies SC followed and several positive results in special cases were obtained [@BGKR05; @tHR13; @T14; @tHMRRW18]. However, in the recent paper [@tHMRR19] an explicit counterexample showing that EAE need not imply SC was obtained from the characterization of EAE and SC on Banach spaces that are essentially incomparable. This reiterated the observation from [@tHMR15] that the Banach space geometries of the underlying spaces play an important role. The counterexample of [@tHMRR19] involves Fredholm operators, which motivated us into a further investigation of EAE and SC for this class of operators, and more generally for the class of relatively regular operators, without the essential incomparability assumption. In order to describe the content of this paper in some more detail, we introduce some notation and terminology. Throughout, $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ will be two Banach space operators. Here, for Banach spaces ${{\mathcal V}}$ and ${{\mathcal W}}$, with ${{\mathcal B}}({{\mathcal V}},{{\mathcal W}})$ we indicate the Banach space of bounded linear operators mapping ${{\mathcal V}}$ into ${{\mathcal W}}$, abbreviated to ${{\mathcal B}}({{\mathcal V}})$ in case ${{\mathcal V}}={{\mathcal W}}$. In the sequel, the term operator will always mean bounded linear operator, and invertibility of an operator will mean the operator has a bounded inverse. We say that the operators $U$ and $V$ are [equivalent after extension (EAE)]{} if there exist Banach spaces ${{\mathcal X}}_0$ and ${{\mathcal Y}}_0$ and invertible operators $E\in{{\mathcal B}}({{\mathcal Y}}\oplus{{\mathcal Y}}_0, {{\mathcal X}}\oplus{{\mathcal X}}_0)$ and $F\in{{\mathcal B}}({{\mathcal X}}\oplus{{\mathcal X}}_0,{{\mathcal Y}}\oplus{{\mathcal Y}}_0)$ so that $$\label{EAE} {\ensuremath{\left[\begin{array}{cc}U&0\\0&I_{{{\mathcal X}}_0}\end{array} \right]}}=E{\ensuremath{\left[\begin{array}{cc}V&0\\0& I_{{{\mathcal Y}}_0}\end{array} \right]}}F.$$ In other words, $U\oplus I_{{{\mathcal X}}_0}$ and $V\oplus I_{{{\mathcal Y}}_0}$ are equivalent. The operators $U$ and $V$ are called [Schur coupled (SC)]{} if there exists an operator matrix $M={\left[\begin{smallmatrix} A&B\\C&D\end{smallmatrix}\right]}\in{{\mathcal B}}({{\mathcal X}}\oplus{{\mathcal Y}},{{\mathcal X}}\oplus{{\mathcal Y}})$ so that $A$ and $D$ are invertible and $$U=A-BD^{-1}C {\quad\mbox{and}\quad}V=D-CA^{-1}B.$$ Thus, the Schur complements of $M$ with respect to $D$ and $A$ are the operators $U$ and $V$, respectively. In this paper we investigate the notions of EAE and SC, and in particular the question whether EAE of $U$ and $V$ implies SC of $U$ and $V$, for the class of [relatively regular operators]{}, that is, operators which have complemented kernels and ranges. In particular, their ranges must be closed. The terminology relatively regular goes back to Atkinson [@A53], but this class of operators has also figured under the monikers generalized Fredholm operators [@C74] and generalized invertible operators. More generally, assume that ${\textup{Ker\,}}U$ and ${\textup{Ker\,}}V$ are complementable and so are the closures of ${\textup{Ran\,}}U$ and ${\textup{Ran\,}}V$. Equivalently, ${{\mathcal X}}$ and ${{\mathcal Y}}$ admit decompositions ${{\mathcal X}}_1\oplus{{\mathcal X}}_2={{\mathcal X}}={{\mathcal X}}_1'\oplus {{\mathcal X}}_2'$ and ${{\mathcal Y}}_1\oplus{{\mathcal Y}}_2={{\mathcal Y}}={{\mathcal Y}}_1'\oplus {{\mathcal Y}}_2'$ such that with respect to these decompositions $U$ and $V$ take the form $$\begin{aligned} U={\ensuremath{\left[\begin{array}{cc}U'&0\\ 0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\ {{\mathcal X}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1'\\ {{\mathcal X}}_2'\end{array} \right]}} &{\quad\mbox{and}\quad}V={\ensuremath{\left[\begin{array}{cc}V'&0\\ 0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\ {{\mathcal Y}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1'\\ {{\mathcal Y}}_2'\end{array} \right]}},\notag\\ \mbox{with}\quad{{\mathcal X}}_2={\textup{Ker\,}}U,\quad {{\mathcal X}}_1'=\overline{{\textup{Ran\,}}U},&\quad {{\mathcal Y}}_2={\textup{Ker\,}}V,\quad {{\mathcal Y}}_1'=\overline{{\textup{Ran\,}}V}.\label{UVdec}\end{aligned}$$ Operators $U$ and $V$ that admit a decomposition as above will be called [complementable]{}. Thus $U$ and $V$ are relatively regular if they are complementable and have closed range. A [Fredholm operator]{} is a relatively regular operator with finite dimensional kernel and cokernel, while a [left (respectively, right) Atkinson]{} operator is a relatively regular operator with finite dimensional kernel (respectively, cokernel). Here, with some abuse of terminology we refer to a complement of the range of a relatively regular operator $T$ as its [cokernel]{}, denoted as ${\textup{Coker\,}}T$. Note that all left (respectively, right) Atkinson operators are left (respectively, right) Fredholm operators, but that the converse need not hold, since left and right Fredholm operators need not be relatively regular; see [@A04 Chapter 7] for more details. In Section \[S:EAE-SR\], for given complementable operators $U$ and $V$ that are EAE we characterise the invertible operators $E$ and $F$, of a special form identified in [@tHR13], cf., below, that establish the EAE of $U$ and $V$. Specifying to the case of relatively regular operators, we recover the result from [@BT92a] that $U$ and $V$ are EAE if and only if the kernels of $U$ and $V$ are isomorphic and the cokernels of $U$ and $V$ are isomorphic. However, more importantly, it enables us to identify a Banach space operator problem which is in some way equivalent to the question whether the EAE operators $U$ and $V$ are also SC. We view this observation as the main contribution of our paper. The Banach space operator problem is presented in Section \[S:EquivBSOP\], as Problem \[P:BSOP\], and in that section we also explain the connection with the question whether given relatively regular operators $U$ and $V$ that are EAE are also SC. Some analysis of Problem \[P:BSOP\] is conducted in Section \[S:AnalyseBSOP\], and the implications of the obtained results are translated into the context of EAE and SC in Section \[S:EAEvsSC\]. Again, Banach space geometry plays an important role. In particular, in Section \[S:BSP\] we further analyse a Banach space property (which we already encountered in [@tHMRR19]) that plays an important role in the analysis of Section \[S:AnalyseBSOP\], and which turns out to be equivalent to the existence of Fredholm operators of a certain index. Although this analysis in Section \[S:AnalyseBSOP\] does not provide a complete answer to Problem \[P:BSOP\], it shows some of the intricacies that occur when considering the question whether EAE implies SC beyond the known cases, e.g., Hilbert spaces or essentially incomparable Banach spaces, even when restricting to relatively regular operators. We conclude this introduction with some comments on notation and terminology and recall some facts from Banach space (operator) theory. Throughout this paragraph let ${{\mathcal G}}$ and ${{\mathcal H}}$ be Banach spaces. An operator $T\in{{\mathcal B}}({{\mathcal G}},{{\mathcal H}})$ is called [inessential]{} whenever $I_{{\mathcal G}}-ST$ is Fredholm for any $S\in{{\mathcal B}}({{\mathcal H}},{{\mathcal G}})$, or, equivalently, $I_{{\mathcal H}}-TS$ is Fredholm for any $S\in{{\mathcal B}}({{\mathcal H}},{{\mathcal G}})$; in fact, in this case the operators $I_{{\mathcal G}}-ST$ and $I_{{\mathcal H}}-TS$ are Fredholm with index 0. The class of inessential operators contains all compact, strictly singular and strictly co-singular operators and is closed under left and right multiplication, that is, if $T\in{{\mathcal B}}({{\mathcal G}},{{\mathcal H}})$ is inessential, then so is $STR$ for all $S\in{{\mathcal B}}({{\mathcal H}},{{\mathcal L}})$ and $R\in{{\mathcal B}}({{\mathcal K}},{{\mathcal G}})$, with ${{\mathcal L}}$ and ${{\mathcal K}}$ arbitrary Banach spaces. See Sections 6.1 and 6.2 of [@A04] for further details. The Banach spaces ${{\mathcal G}}$ and ${{\mathcal H}}$ are called [essentially incomparable]{} in case all operators in ${{\mathcal B}}({{\mathcal G}},{{\mathcal H}})$ are inessential, or, equivalently, all operators in ${{\mathcal B}}({{\mathcal H}},{{\mathcal G}})$ are inessential. In case ${{\mathcal G}}$ and ${{\mathcal H}}$ are essentially incomparable, they are also [projection incomparable]{} which means that there is no infinite dimensional complemented subspace of ${{\mathcal G}}$ that is isomorphic to a complemented subspace of ${{\mathcal H}}$; see Section 7.5 in [@A04] for further discussion and many examples. Equivalence after extension for relatively regular operators {#S:EAE-SR} ============================================================ Recall from [@tHR13], see also [@tHMRRW18 Lemma 2.1], that if $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ are EAE, then it is always possible to select the operators $E$ and $F$ in with ${{\mathcal X}}_0={{\mathcal Y}}$, ${{\mathcal Y}}_0={{\mathcal X}}$ and so that $E$ and $F$ have the form $$\label{EFform1} \begin{aligned} F={\ensuremath{\left[\begin{array}{cc}F_{11}& I_{{\mathcal Y}}\\ I+F_{22}F_{11} & F_{22}\end{array} \right]}},&\quad E={\ensuremath{\left[\begin{array}{cc}E_{11} & U\\ E_{21} & -F_{11}\end{array} \right]}}\\ F^{-1}={\ensuremath{\left[\begin{array}{cc}-F_{22}& I_{{\mathcal X}}\\ I+F_{11}F_{22} & -F_{11}\end{array} \right]}},&\quad E^{-1}={\ensuremath{\left[\begin{array}{cc}{{\widehat{E}}}_{11} & V\\ {{\widehat{E}}}_{21} & F_{22}\end{array} \right]}}. \end{aligned}$$ In Theorem \[T:EAEcomplement\] below, for complementable operators $U$ and $V$ that are EAE we completely characterize the invertible operators $E$ and $F$ of the form that establish the EAE of $U$ and $V$, and in Theorem \[T:EAErelativelyreg\] below we specialize this result to the case where $U$ and $V$ are relatively regular. By Proposition 1 in [@BT92a], a necessary condition for complementable operators $U$ and $V$ as in to be EAE is that ${{\mathcal X}}_2$ and ${{\mathcal Y}}_2$ are isomorphic and ${{\mathcal X}}_2'$ and ${{\mathcal Y}}_2'$ are isomorphic, that is, there exist operators $$\label{E'F'} E'\in{{\mathcal B}}({{\mathcal X}}_2,{{\mathcal Y}}_2) {\quad\mbox{and}\quad}F'\in{{\mathcal B}}({{\mathcal X}}_2',{{\mathcal Y}}_2'),\quad\mbox{both invertible}.$$ While for complementable operators this condition need not be sufficient, by Example 6 in [@BT92a], it is sufficient if one restricts to relatively regular operators [@BT92a Theorem 2]. We recover this result in Theorem \[T:EAErelativelyreg\] below. Assuming we have invertible operators $E'$ and $F'$ as in , we consider operators $E$ and $F$ as in that are of the following form $$\label{EFform2} \begin{aligned} F&={\ensuremath{\left[\begin{array}{cc\|cc} Y_1 & 0 & I_{{{\mathcal Y}}_1} & 0 \\ Y_2 & E' & 0 & I_{{{\mathcal Y}}_2} \\\hline I+Y_3Y_1 & 0 & Y_3 & 0 \\ Y_4Y_1-E'^{-1}Y_2 & 0 & Y_4 & -E'^{-1}\end{array} \right]}} : {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\{{\mathcal X}}_2\\ \hline {{\mathcal Y}}_1\\{{\mathcal Y}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\{{\mathcal Y}}_2\\\hline {{\mathcal X}}_1\\{{\mathcal X}}_2\end{array} \right]}},\\ E&={\ensuremath{\left[\begin{array}{cc\|cc} X_1 & X_2 & U' & 0 \\ 0 & F'^{-1} & 0 & 0 \\\hline X_5 &X_3 & -Y_1 & 0 \\ X_6 & X_4 & -Y_2 & -E' \end{array} \right]}} : {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1'\\{{\mathcal Y}}_2'\\ \hline {{\mathcal X}}_1\\{{\mathcal X}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1'\\{{\mathcal X}}_2'\\ \hline {{\mathcal Y}}_1\\{{\mathcal Y}}_2\end{array} \right]}}. \end{aligned}$$ One easily verifies that an operator $F$ of this form is invertible, with inverse given by $$\label{Finvform} F^{-1}={\ensuremath{\left[\begin{array}{cc\|cc} -Y_3 & 0 & I_{{{\mathcal X}}_1} & 0 \\ -Y_4 & E'^{-1} & 0 & I_{{{\mathcal X}}_2} \\ \hline I_{{{\mathcal Y}}_1}+Y_1Y_3 & 0 & -Y_1 & 0 \\ Y_2Y_3+E'Y_4 & 0 & -Y_2 & -E'\end{array} \right]}} : {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\{{\mathcal Y}}_2\\ \hline{{\mathcal X}}_1\\{{\mathcal X}}_2\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\{{\mathcal X}}_2\\ \hline{{\mathcal Y}}_1\\{{\mathcal Y}}_2\end{array} \right]}},$$ while the invertibility of $F'$ and $E'$ imply that $E$ is invertible if and only if the block operator $$\label{Eblock} {\ensuremath{\left[\begin{array}{cc}X_1 & U'\\ X_5 & -Y_1\end{array} \right]}}$$ is invertible. The motivation of considering $E$ and $F$ in this special form becomes clear from the following result. \[T:EAEcomplement\] Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be complementable Banach space operators as in . Assume $U$ and $V$ are EAE with the operators $E$ and $F$ that establish the EAE of the form . Then $E$ and $F$ are of the form with $E'$ and $F'$ invertible operators as in and such that $$\label{EAEcondition1} \begin{aligned} &{\ensuremath{\left[\begin{array}{cc}X_1 & U'\\ X_5 & -Y_1\end{array} \right]}}{\ensuremath{\left[\begin{array}{cc}V' & 0 \\ 0 & I_{{{\mathcal X}}_1}\end{array} \right]}}= {\ensuremath{\left[\begin{array}{cc}U' & 0 \\ 0 & I_{{{\mathcal Y}}_1}\end{array} \right]}}{\ensuremath{\left[\begin{array}{cc}-Y_3 & I_{{{\mathcal X}}_1} \\ I_{{{\mathcal Y}}_1} +Y_1 Y_3 & - Y_1\end{array} \right]}}\\[2mm] &\qquad \qquad \mbox{and}\quad Y_4=F'^{-1}(X_6V' -Y_2Y_3). \end{aligned}$$ Conversely, if there exist invertible operators $E'$ and $F'$ as in and operators $X_1, X_5, Y_1, Y_3$ such that the block operator is invertible and satisfies the first identity in , then $U$ and $V$ are EAE and the EAE of $U$ and $V$ is established by $E$ and $F$ given by , where the operators $X_2,X_3,X_4, X_6,Y_1,Y_2,Y_3$ can be chosen arbitrarily and $Y_4$ is given by the second identity in . The relation between $U'$ and $V'$ in the first line of and with the operator invertible means that $U'\in{{\mathcal B}}({{\mathcal X}}_1,{{\mathcal X}}_1')$ and $V'\in{{\mathcal B}}({{\mathcal Y}}_1,{{\mathcal Y}}_1')$ are EAE in the ‘non-square’ sense of [@BGK84], i.e., with possibly ${{\mathcal X}}_1\neq{{\mathcal X}}_1'$ and ${{\mathcal Y}}_1\neq{{\mathcal Y}}_1'$. The relations EAE and MC were originally studied for Banach space operators $U\in{{\mathcal B}}({{\mathcal X}},{{\mathcal X}}')$ and $Y\in{{\mathcal B}}({{\mathcal Y}},{{\mathcal Y}}')$, cf., [@BGK84], however, for SC to make sense it is required that ${{\mathcal X}}\simeq{{\mathcal X}}'$ and ${{\mathcal Y}}\simeq{{\mathcal Y}}'$, so that one usually assumes ${{\mathcal X}}={{\mathcal X}}'$ and ${{\mathcal Y}}={{\mathcal Y}}'$. Part of the first claim, under the assumption that $U$ and $V$ are relatively regular, was already given in the proof of [@tHR13 Proposition 3.2], but we will repeat the arguments here for completeness. Assume $U$ and $V$ as in are EAE with $E$ and $F$ of the form . Note that it is clear that the right upper corner of $F$ is as in , and that once it is established that $F_{11}$ and $F_{22}$ are as in , then the left lower corner of $F$ in as well as the formula for ${F^{-1}}$ in also follow. Furthermore, it then also follows that the right upper and right lower corner of $E$ are as in . The fact that $E$ and $F$ are invertible operators satisfying implies that $F$ maps ${{\mathcal X}}_2={\textup{Ker\,}}U$ onto ${{\mathcal Y}}_2={\textup{Ker\,}}V$ and $E^{-1}$ maps any complement of ${\textup{Im\,}}V$ onto any complement of ${\textup{Im\,}}U$, in particular, $E^{-1}$ maps ${{\mathcal Y}}_2'$ onto ${{\mathcal X}}_2'$. This establishes the zero entries in the second block column of $F$ and in the second block row of $E$ in , as well as the fact that the remaining entries, $F'^{-1}\in{{\mathcal B}}({{\mathcal Y}}_2,{{\mathcal X}}_2)$ and $E'\in{{\mathcal B}}({{\mathcal X}}_2',{{\mathcal Y}}_2')$ are invertible. Write $F_{22}={\left[\begin{smallmatrix} Y_3 & Y_5\\ Y_4 & Y_6\end{smallmatrix}\right]}$, compatible with the decomposition in . The fact that the left lower corner in $F$ is given by $I+F_{22}F_{11}$ and has zero entries in the right upper and right lower corner yields $$Y_5F'=0 {\quad\mbox{and}\quad}I+ Y_6F'=0.$$ Since $F'$ is invertible, we then obtain that $Y_5=0$ and $Y_6=-F'^{-1}$. Hence $F$ is of the form . In passing, we also showed that $E$ is as in . The fact that $-E'$ and $F'^{-1}$ are invertible and appear in $E$ in a block column and block row, respectively, that otherwise only contains zero-operators implies that $E$ is invertible if and only if the $2 \times 2$ block operator matrix obtained by removing the rows and columns that contain $-E'$ and $F'^{-1}$ is invertible. To see this, note that this operator matrix appears after taking the Schur complement of $E$ with respect to $-E'$ and then the Schur complement with respect to $F'^{-1}$. The resulting $2 \times 2$ block operator matrix is as in , so that we obtain that $E$ is invertible if and only if the operator in is invertible. Now, writing out the EAE relation with $E$ and $F$ as in , it follows directly that all that remains to be verified is . Thus we have proved the first claim of Theorem \[T:EAEcomplement\]. For the converse direction, assume that $X_1, X_5, Y_1, Y_3$ are operators such that the block operator is invertible, and the first identity in holds. One then easily verifies that with $E$ and $F$ as in , with $X_2,X_3,X_4, X_6,Y_1,Y_2,Y_3$ arbitrary and $Y_4$ given by , the EAE relation holds. When specialized to the case that $U$ and $V$ are relatively regular, i.e., $U'$ and $V'$ invertible, we obtain the following result. The necessary and sufficient conditions for EAE of two relatively regular operators were also given in Theorem 2 in [@BT92a], however, only with a sketch of the proof. The characterization of the operators $E$ and $F$ that establish the EAE relation appears to be new. \[T:EAErelativelyreg\] Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be relatively regular Banach space operators as in , i.e., with $U'$ and $V'$ invertible. Then $U$ and $V$ are EAE if and only if there exist invertible operators $E'$ and $F'$ as in . Moreover, in that case, the operators $E$ and $F$ of the form that establish the EAE of $U$ and $V$ are precisely the operators of the form where $E'$ and $F'$ are arbitrary invertible operators as in , the operators $Y_1,Y_2,Y_3,X_2,X_3,X_4, X_6$ can be chosen arbitrarily, and the operators $X_1$ $X_5$ and $Y_4$ are given by $$\label{X1X5Y4} X_1= -U' Y_3 V'^{-1},\quad X_5= (I + Y_1Y_3)V'^{-1},\quad Y_4=E'^{-1}(X_6V' -Y_2Y_3).$$ According to Proposition 1 in [@BT92a], when $U$ and $V$ are EAE, invertible operators $E'$ and $F'$ as in exist. For the converse direction, assume we have invertible operators $E'$ and $F'$ as in . Using the converse implication in Theorem \[T:EAEcomplement\], it suffices to find operators $X_1, X_5, Y_1, Y_3$ such that the block operator is invertible and the first identity in holds. Since $U'$ and $V'$ are invertible, we can choose $Y_1$ and $Y_3$ freely, and define $X_1$ and $X_5$ via the identity $$\label{EblockFact} {\ensuremath{\left[\begin{array}{cc}X_1 & U'\\ X_5 & -Y_1\end{array} \right]}}= {\ensuremath{\left[\begin{array}{cc}U' & 0 \\ 0 & I\end{array} \right]}}{\ensuremath{\left[\begin{array}{cc}-Y_3 & I \\ I +Y_1 Y_3 & - Y_1\end{array} \right]}}{\ensuremath{\left[\begin{array}{cc}V'^{-1} & 0 \\ 0 & I\end{array} \right]}},$$ which implies that first identity in holds, and also yields the formulas in . Since $E'$ and $F'$ are invertible as well as the middle factor in the factorization of the block operator , it follows that the block operator is invertible. Hence, by Theorem \[T:EAEcomplement\] we obtain that $U$ and $V$ are EAE. Furthermore, the second part of Theorem \[T:EAEcomplement\] also tells us that the operators $E$ and $F$ of the form are as claimed, since the relation is equivalent to the identities when $U'$ and $V'$ are invertible. Recall that $U$ and $V$ are called [strongly equivalent after extension (SEAE)]{} in case $U$ and $V$ are EAE in such a way that the operators $E$ and $F$ that establish the EAE relation have the left lower corner (from ${{\mathcal X}}_0$ to ${{\mathcal Y}}$) and right upper corner (from ${{\mathcal Y}}$ to ${{\mathcal X}}_0$) invertible, respectively. By Theorem 2 in [@BT92b], see also Theorem 2.4 in [@tHR13], $U$ and $V$ are SEAE if and only if they are SC. It is not directly clear that if $U$ and $V$ are SEAE, then there are also $E$ and $F$ that establish the SEAE and are of the form . We next show that this is the case. \[L:SEAEspecialform\] Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be SEAE. Then the SEAE can be established with $E$ and $F$ of the form . Let $E$ and $F$ be invertible operators that establish the SEAE of $U$ and $V$. Decompose $E$ and $F$ as block operator matrices $E=[E_{ij}]_{i,j=1,2}$ and $F=[F_{ij}]_{i,j=1,2}$ compatible with the EAE identity . Then $F_{12}$ and $E_{21}$ are invertible. In the transfer to the special form , by Lemma 4.1 in [@tHR13] the new $(1,2)$-entry of $F$ becomes $I_{{\mathcal Y}}$ and the new $(2,1)$-entry of $E$ becomes $F_{12}E_{21}$, both of which are invertible. It then follows as in the proofs of Corollary 4.2 in [@tHR13] and Lemma 2.1 in [@tHMRRW18] that the new operators that establish the SEAE of $U$ and $V$ are also of the form . An equivalent Banach space operator problem {#S:EquivBSOP} =========================================== In this section we introduce the following Banach space operator problem which is related to the question whether EAE and SC coincide for relatively regular operators, in a way described below. \[P:BSOP\] Consider Banach spaces ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ so that $$\label{BanProps} {{\mathcal V}}\oplus {{\mathcal Z}}_1 \simeq {{\mathcal V}}\oplus {{\mathcal Z}}_2,\quad {{\mathcal W}}\oplus {{\mathcal Z}}_1 \simeq {{\mathcal W}}\oplus {{\mathcal Z}}_2.$$ Find operators $$\label{Ops} \begin{aligned} &A_{12}\in{{\mathcal B}}({{\mathcal Z}}_1,{{\mathcal V}}),\quad A_{21}\in{{\mathcal B}}({{\mathcal V}},{{\mathcal Z}}_2),\quad A_{22}\in{{\mathcal B}}({{\mathcal Z}}_1,{{\mathcal Z}}_2),\\ &\qquad\qquad B_1\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}}),\quad B_2\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}}), \end{aligned}$$ so that the $2 \times 2$ block operator $$\label{T} T={\ensuremath{\left[\begin{array}{cc}I_{{\mathcal V}}-B_1B_2 & A_{12} \\ A_{21} & A_{22}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_1\end{array} \right]}}\to{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_2\end{array} \right]}}$$ is invertible. If such an invertible operator $T$ exists, then we say that Problem \[P:BSOP\] associated with the spaces ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$, ${{\mathcal Z}}_2$ is solvable, or simply that Problem \[P:BSOP\] is solvable in case there should not be any unclarity about the spaces. Note that by assumption invertible operators between ${{\mathcal V}}\oplus{{\mathcal Z}}_1$ and ${{\mathcal V}}\oplus{{\mathcal Z}}_2$ exist, the question is whether there exists one of the form . We will now explain the connection with EAE and SC. Assume $U$ and $V$ are relatively regular operators as in . Assume $U$ and $V$ are EAE. We want to know when they are also SC, or, equivalently, SEAE. According to Theorem \[T:EAErelativelyreg\], all operators of the form that establish the EAE relation of $U$ and $V$ are given by with $Y_1$, $Y_2$, $Y_3$, $X_2$, $X_3$, $X_4$, $X_6$ arbitrary operators (acting between the spaces indicated in ) and $X_1$, $X_5$ and $Y_4$ are given by . By Lemma \[L:SEAEspecialform\], the question whether $U$ and $V$ are also SEAE now reduces to the question whether we can find $E$ and $F$ of the form with $E_{21}$ invertible. Now set $$A_{12}=X_3,\quad A_{21}=X_6 V',\quad A_{22}=X_4,\quad B_1=Y_1,\quad B_2= Y_3,$$ so that $$\label{VWZ1Z2def} {{\mathcal V}}={{\mathcal Y}}_1,\quad {{\mathcal W}}={{\mathcal X}}_1,\quad {{\mathcal Z}}_1={{\mathcal Y}}_2, \quad {{\mathcal Z}}_2={{\mathcal Y}}_2'.$$ It is then clear from Theorem \[T:EAErelativelyreg\] that $E_{21}$ and $T$ in are related through $$E_{21}{\left[\begin{smallmatrix} V'& 0 \\ 0& I\end{smallmatrix}\right]}= T,$$ and thus $E_{21}$ is invertible if and only if $T$ is invertible. Since in the choices for $E$ and $F$ we are free to choose $X_3$, $X_4$, $X_6$, $Y_1$ and $Y_3$, to see that the problem whether $U$ and $V$ are SEAE reduces to the above Banach space operator problem, it remains to show that for our choice of the spaces ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ condition is satisfied. Note that $U$ and $V$ being relatively regular, i.e., $U'$ and $V'$ invertible, yields ${{\mathcal X}}_1 \simeq {{\mathcal X}}_1'$ and ${{\mathcal Y}}_1 \simeq {{\mathcal Y}}_1'$, while the assumption that $U$ and $V$ are EAE gives ${{\mathcal X}}_2\simeq {{\mathcal Y}}_2$ and ${{\mathcal X}}_2'\simeq {{\mathcal Y}}_2'$. This implies that $${\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\{{\mathcal Z}}_1\end{array} \right]}}={\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\{{\mathcal Y}}_2\end{array} \right]}}={{\mathcal Y}}={\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1'\\{{\mathcal Y}}_2'\end{array} \right]}}\simeq {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\{{\mathcal Y}}_2'\end{array} \right]}}={\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_2\end{array} \right]}}$$ and $${\ensuremath{\left[\begin{array}{c}{{\mathcal W}}\\{{\mathcal Z}}_1\end{array} \right]}}={\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\ {{\mathcal Y}}_2\end{array} \right]}}\simeq{\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\ {{\mathcal X}}_2\end{array} \right]}}={{\mathcal X}}={\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1'\\{{\mathcal X}}_2'\end{array} \right]}}\simeq {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\ {{\mathcal Y}}_2'\end{array} \right]}}={\ensuremath{\left[\begin{array}{c}{{\mathcal W}}\\ {{\mathcal Z}}_2\end{array} \right]}}.$$ Hence condition is indeed satisfied. We summarise the above discussion in the following proposition. Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be relatively regular operator as in that are EAE. Then ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ defined in satisfy . Moreover, $U$ and $V$ are SEAE if and only if there exists an invertible operator matrix $T$ as in with operator entries as in . The above shows how for any relatively regular EAE operators $U$ and $V$ a version of Problem can be set up in such a way that solvability of this problem coincides with $U$ and $V$ being SEAE. It is also the case that with any version of Problem \[P:BSOP\] we can associate relatively regular EAE operators $U$ and $V$ so that Problem \[P:BSOP\] is the derived problem for these operators $U$ and $V$. Specifically, for Banach spaces ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ satisfying , let $U'\in{{\mathcal B}}({{\mathcal W}})$ and $V'\in{{\mathcal B}}({{\mathcal V}})$ be invertible operators (in particular $U'=I_{{\mathcal W}}$ and $V'=I_{{\mathcal V}}$ is a possibility), set ${{\mathcal X}}={{\mathcal W}}\oplus{{\mathcal Z}}_1$ and ${{\mathcal Y}}={{\mathcal V}}\oplus{{\mathcal Z}}_1$, and define $$\label{UCcounter} \begin{aligned} U&={\ensuremath{\left[\begin{array}{cc}U'&0\\0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal W}}\\{{\mathcal Z}}_1\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal W}}\\{{\mathcal Z}}_2\end{array} \right]}}\simeq {\ensuremath{\left[\begin{array}{c}{{\mathcal W}}\\{{\mathcal Z}}_1\end{array} \right]}},\\ V&={\ensuremath{\left[\begin{array}{cc}V'&0\\0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\{{\mathcal Z}}_1\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\{{\mathcal Z}}_2\end{array} \right]}}\simeq {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\{{\mathcal Z}}_1\end{array} \right]}}. \end{aligned}$$ Clearly $U$ and $V$ are EAE, by Theorem \[T:EAErelativelyreg\]. If there exist operators $A_{12}$, $A_{21}$, $A_{22}$, $B_1$ and $B_2$ as in such that $T$ in is invertible, then $U$ and $V$ are SEAE, independently of the choice of $U'$ and $V'$, and if there are no such operators that make $T$ invertible, then $U$ and $V$ are not SEAE, again independently of the choice of $U'$ and $V'$. The value of this observation is that it provides a way to construct operators that are EAE but not SC. \[C:EAEnotSC\] Let ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ be Banach spaces satisfying so that the associated Problem \[P:BSOP\] is not solvable. Then for any invertible operators $U'\in{{\mathcal B}}({{\mathcal W}})$ and $V'\in{{\mathcal B}}({{\mathcal V}})$, the operators $U$ and $V$ in are EAE but not SEAE. A Banach space property {#S:BSP} ======================= In this section we further analyse a Banach space property considered in [@tHMRR19], which we will also encounter in the following sections. Given Banach space ${{\mathcal Z}}$ and an integer $k>0$, we say that ${{\mathcal Z}}$ is [stable under finite dimensional quotients of dimension $k$]{}, abbreviated to ${{\mathcal Z}}$ is [SUFDQ$_k$]{}, if for any finite dimensional subspace ${{\mathcal F}}\subset {{\mathcal Z}}$ with $\dim ({{\mathcal F}}) =k$ we have ${{\mathcal Z}}\simeq {{\mathcal Z}}/{{\mathcal F}}$. Furthermore, we say that ${{\mathcal Z}}$ is [stable under finite dimensional sums of dimension $k$]{}, abbreviated to ${{\mathcal Z}}$ is [SUFDS$_k$]{}, if ${{\mathcal Z}}\simeq {{\mathcal Z}}\oplus {{\mathcal F}}$ for any finite dimensional Banach space ${{\mathcal F}}$ with $\dim ({{\mathcal F}})=k$. Note that if ${{\mathcal Z}}$ is SUFDQ$_k$ (respectively, SUFDS$_k$) then ${{\mathcal Z}}$ is also SUFDQ$_{nk}$ (respectively SUFDS$_{nk}$) for all integers $n>0$. In particular, if ${{\mathcal Z}}$ is SUFDQ$_1$ (respectively, SUFDS$_1$), then ${{\mathcal Z}}$ is SUFDQ$_k$ (respectively, SUFDS$_k$) for all integers $k>0$. Therefore, we shall abbreviate SUFDQ$_1$ and SUFDS$_1$ by SUFDQ and SUFDS, respectively. The Banach space properties SUFDQ and SUFDS appeared in [@tHMRR19], and it was shown there that SUFDQ implies SUFDS [@tHMRR19 Lemma 3.3], as well as that the Banach space sum of two spaces that are SUFDQ is also SUFDQ [@tHMRR19 Proposition 3.2]. As also observed in [@tHMRR19], all primary Banach spaces are SUFDQ, hence they are SUFDQ$_k$ for all integers $k>0$. \[P:BanPropEquiv\] For all integer $k>0$ and Banach spaces ${{\mathcal Z}}$ the following are equivalent: - ${{\mathcal Z}}$ is SUFDQ$_k$; - ${{\mathcal Z}}$ is SUFDS$_k$; - in ${{\mathcal B}}({{\mathcal Z}})$ there exist Fredholm operators of index $k$; - in ${{\mathcal B}}({{\mathcal Z}})$ there exist Fredholm operators of index $-k$. The implication (i) $\Rightarrow$ (ii) follows by the same argument as in the proof of Lemma 3.3 in [@tHMRR19], if one restricts the finite dimensional Banach spaces to those having dimension $k$. Next we prove (ii) $\Rightarrow$ (i). Let ${{\mathcal F}}\subset {{\mathcal Z}}$ with $\dim ({{\mathcal F}})=k$ and let ${{\mathcal W}}$ be a complement of ${{\mathcal F}}$ in ${{\mathcal Z}}$, so that ${{\mathcal W}}\simeq {{\mathcal Z}}/{{\mathcal F}}$. Since ${{\mathcal Z}}$ is SUFDS$_k$, we have $${{\mathcal W}}\oplus {{\mathcal F}}={{\mathcal Z}}\simeq {{\mathcal Z}}\oplus{{\mathcal F}}.$$ By Lemma 3.4 in [@tHMRR19], which applies since $\dim ({{\mathcal F}})<\infty$, we have ${{\mathcal Z}}/{{\mathcal F}}\simeq {{\mathcal W}}\simeq {{\mathcal Z}}$. Thus (i) holds. For the equivalence of (ii) and (iii), first assume (iii) holds and let ${{\mathcal F}}$ be a Banach space of dimension $k$. Let $F\in{{\mathcal B}}({{\mathcal Z}})$ be a Fredholm operator with index $k$. Without loss of generality $F$ is surjective, so that $\dim ({\textup{Ker\,}}F)=k$. Now let $\Xi\in{{\mathcal B}}({{\mathcal Z}},{{\mathcal F}})$ be any operator that maps ${\textup{Ker\,}}F$ onto ${{\mathcal F}}$ and is zero on a complement of ${\textup{Ker\,}}F$ in ${{\mathcal Z}}$. Then ${\left[\begin{smallmatrix} F\\\Xi\end{smallmatrix}\right]}$ is an invertible operator in ${{\mathcal B}}({{\mathcal Z}},{{\mathcal Z}}\oplus{{\mathcal F}})$, hence ${{\mathcal Z}}\simeq{{\mathcal Z}}\oplus {{\mathcal F}}$. Thus (iii) implies (ii). Conversely, assume (ii) holds. Let ${{\mathcal F}}$ be any Banach space of dimension $k$. By assumption ${{\mathcal Z}}\simeq{{\mathcal Z}}\oplus{{\mathcal F}}$. Now let $T$ be an isomorphism between ${{\mathcal Z}}$ and ${{\mathcal Z}}\oplus{{\mathcal F}}$ and decompose $T$ as $T={\left[\begin{smallmatrix} F\\\Xi\end{smallmatrix}\right]}$ with $F\in {{\mathcal B}}({{\mathcal Z}},{{\mathcal Z}})$ and $\Xi\in {{\mathcal B}}({{\mathcal Z}},{{\mathcal F}})$. In particular, $T$ is Fredholm with index 0. Since $\Xi$ is a finite rank operator, $T_0:={\left[\begin{smallmatrix} F\\ 0\end{smallmatrix}\right]}=T-{\left[\begin{smallmatrix} 0\\\Xi\end{smallmatrix}\right]}$ is also Fredholm with index 0. Since $\dim ({\textup{Ker\,}}T_0)=\dim ({\textup{Ker\,}}F)$ and $\dim ({\textup{Coker\,}}T_0) =\dim ({\textup{Coker\,}}F) +k$, it follows that $F$ is Fredholm with index $k$. Similarly one proves that (ii) and (iv) are equivalent. For the implication (ii) $\Rightarrow$ (iv), to see that there exists a Fredholm operator in ${{\mathcal B}}({{\mathcal Z}})$ with index $-k$ a similar argument applies based on an invertible operator in ${{\mathcal B}}({{\mathcal Z}}\oplus{{\mathcal F}},{{\mathcal Z}})$. Conversely, assuming there exists a Fredholm operator with index $-k$ in ${{\mathcal B}}({{\mathcal Z}})$, one can modify the argument in the proof of the implication (iii) $\Rightarrow$ (ii) to construct an invertible operator from ${{\mathcal Z}}\oplus {{\mathcal F}}$ to ${{\mathcal Z}}$. We conclude this section with a few additional observations and comments. \[L:BSPsum\] Let ${{\mathcal Z}}$ be SUFDQ$_k$. Then ${{\mathcal W}}\oplus {{\mathcal Z}}$ is SUFDQ$_k$ for any Banach space ${{\mathcal W}}$. This result is straightforward from the fact that ${{\mathcal Z}}$ has property (ii) (or (iii)) in Proposition \[P:BanPropEquiv\]. \[L:BSPfindimcomp\] Let ${{\mathcal Z}}$ be SUFDQ$_k$ and let ${{\mathcal F}}\subset{{\mathcal Z}}$ be finite dimensional. Then ${{\mathcal Z}}/{{\mathcal F}}$ is SUFDQ$_k$. Let ${{\mathcal Z}}_1$ be a complement of ${{\mathcal F}}$ in ${{\mathcal Z}}$. It suffices to prove that ${{\mathcal Z}}_1$ is SUFDQ$_k$. Let $T\in{{\mathcal B}}({{\mathcal Z}})$ be a Fredholm operator of index $k$ and write $T={\left[\begin{smallmatrix} T_{11}&T_{12}\\ T_{21}& T_{22}\end{smallmatrix}\right]}$ with respect to the decomposition ${{\mathcal Z}}={{\mathcal Z}}_1\oplus{{\mathcal F}}$. Then $T_1:={\left[\begin{smallmatrix} T_{11}&0\\0&0\end{smallmatrix}\right]}=T-{\left[\begin{smallmatrix} 0&T_{12}\\ T_{21}& T_{22}\end{smallmatrix}\right]}$ is Fredholm with index $k$ because ${\left[\begin{smallmatrix} 0&T_{12}\\ T_{21}& T_{22}\end{smallmatrix}\right]}$ is finite rank. We have ${\textup{Ker\,}}T_1 = {\textup{Ker\,}}T_{11} \oplus {{\mathcal F}}$ and ${\textup{Coker\,}}T_1 = {\textup{Coker\,}}T_{11} \oplus {{\mathcal F}}$. Thus ${\textup{Index}}(T_{11})={\textup{Index}}(T_{1})=k$ and $T_{11}\in{{\mathcal B}}({{\mathcal Z}}_1)$. Hence ${{\mathcal Z}}_1$ is SUFDQ$_k$. \[C:BSPkTo-k\] For any integer $k$ and Banach space ${{\mathcal Z}}$, ${{\mathcal B}}({{\mathcal Z}})$ contains Fredholm operators of index $k$ if and only if ${{\mathcal B}}({{\mathcal Z}})$ contains Fredholm operators of index $-k$. \[C:BSPlargerk\] Let ${{\mathcal Z}}$ be a Banach space that has a complemented subspace which is SUFDQ$_k$. Then ${{\mathcal B}}({{\mathcal Z}})$ contains Fredholm operators of index $nk$ for all $n\in{{\mathbb Z}}$. In particular, if ${{\mathcal Z}}$ contains a complemented copy of a primary Banach space, then ${{\mathcal B}}({{\mathcal Z}})$ contains Fredholm operators of all indices. \[R:BSexamples\] In [@GM97], W.T. Gowers and B. Maurey constructed a Banach space ${{\mathcal Z}}$ which is isomorphic to its subspaces of even codimension, but not to those of odd codimension. In other words, ${{\mathcal Z}}$ is SUFDQ$_2$ but not SUFDQ, so that on ${{\mathcal Z}}$ Fredholm operators exist, but only of even index. Furthermore, there also exist Banach spaces where all Fredholm operators have index 0, i.e., so that there is no $k>0$ so that the Banach spaces is SUFDQ$_k$. Examples of such spaces are Banach spaces with few operators and very few operators. A Banach space ${{\mathcal Z}}$ has [few operators]{} if all operators in ${{\mathcal B}}({{\mathcal Z}})$ are of the form ${\lambda}I_{{\mathcal Z}}+S$ with ${\lambda}\in{{\mathbb C}}$ and $S$ strictly singular, and ${{\mathcal Z}}$ has [very few operators]{} if all operators in ${{\mathcal B}}({{\mathcal Z}})$ are of the form ${\lambda}I_{{\mathcal Z}}+K$ with ${\lambda}\in{{\mathbb C}}$ and $K$ compact. All hereditary indecomposable Banach spaces have few operators [@GM93], and example of a Banach space with very few operators was first constructed by Argyros and Haydon [@AH11]. Analysis of Problem \[P:BSOP\] {#S:AnalyseBSOP} ============================== In this section we analyse Problem \[P:BSOP\]. Throughout, let ${{\mathcal V}}$, ${{\mathcal W}}$, ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ be Banach spaces satisfying . We begin with some general results and a few corollaries, after which we consider some special cases in Subsections \[SubS:Z1Z2findim\] and \[SubS:EssInc\]. \[L:Z1Z2iso\] Assume ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_2$. Then Problem \[P:BSOP\] is solvable. Taking $A_{22}\in{{\mathcal B}}({{\mathcal Z}}_1,{{\mathcal Z}}_2)$ invertible and setting $A_{12}$, $A_{21}$, $B_1$ and $B_2$ equal to zero-operators yields $T$ in invertible. \[C:VorWfindim\] Assume $\dim({{\mathcal V}})<\infty$ or $\dim({{\mathcal W}})<\infty$. Then ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_2$ and hence Problem \[P:BSOP\] is solvable. Apply Lemma 3.4 of [@tHMRR19] to the first isomorphism in in case $\dim({{\mathcal V}})<\infty$ and to the second isomorphism in in case $\dim({{\mathcal W}})<\infty$. As indicated, by assumption invertible operators from ${{\mathcal V}}\oplus{{\mathcal Z}}_1$ to ${{\mathcal V}}\oplus{{\mathcal Z}}_2$ exist, the challenge is to find one with the left upper corner as in . Hence we are interested in the question which operators are contained in the set $${{\mathfrak K}}:=\{I_{{\mathcal V}}-B_1B_2\colon B_1\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}}),\, B_2\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}})\}.$$ Note that the set ${{\mathfrak K}}$ only depends on ${{\mathcal V}}$ and ${{\mathcal W}}$. Apart from zero operators, we can take $B_1$ and $B_2$ to be finite rank operators, in which case $I-B_1B_2$ is Fredholm with index 0. \[L:FredK\] Assume ${{\mathcal V}}$ and ${{\mathcal W}}$ are infinite dimensional. Then for any Fredholm operator $F$ in ${{\mathcal B}}({{\mathcal V}})$ with index 0 there exists an invertible operator $G\in{{\mathcal B}}({{\mathcal V}})$ so that $GF\in{{\mathfrak K}}$. Since ${{\mathcal V}}$ and ${{\mathcal W}}$ are infinite dimensional all finite rank operators in ${{\mathcal B}}({{\mathcal V}})$ can be factored as $B_1B_2$ with $B_1\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}})$ and $B_2\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}})$. Let $F$ in ${{\mathcal B}}({{\mathcal V}})$ be Fredholm with index 0. By Theorem XI.5.3 in [@GGK90], we have $F=H-K$ for operators $H,K\in{{\mathcal B}}({{\mathcal V}})$ with $H$ invertible and $K$ finite rank. Thus $H^{-1}F=I_{{\mathcal V}}- H^{-1}K$. Since $H^{-1}K$ is also finite rank, it follows that $H^{-1}F\in{{\mathfrak K}}$. Hence we can take $G=H^{-1}$. It can happen that ${{\mathfrak K}}$ contains only Fredholm operators of index 0. \[L:fKonlyFredholm\] The set ${{\mathfrak K}}$ contains only Fredholm operators (of index 0) if and only if ${{\mathcal V}}$ and ${{\mathcal W}}$ are essentially incomparable Banach spaces. This is directly clear from the definition of essentially incomparable. In Lemma \[L:fKonlyFredholm\], the parenthesised phrase can be included or removed without changing the validity of the statement. In case ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$, Problem \[P:BSOP\] is solvable for any ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ so that hold. This case can easily be characterized. \[L:K=B(V)\] It holds that ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$ if and only if ${{\mathcal V}}$ is isomorphic to a complemented subspace of ${{\mathcal W}}$. Furthermore, ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$ holds if and only if $0\in{{\mathfrak K}}$. Note that the condition $0\in{{\mathfrak K}}$ is equivalent to the existence of a left invertible operator from ${{\mathcal V}}$ into ${{\mathcal W}}$, whose range provides a complemented subspace of ${{\mathcal W}}$ which is isomorphic to ${{\mathcal V}}$. Conversely, if ${{\mathcal V}}_0\subset {{\mathcal W}}$ is isomorphic to ${{\mathcal V}}$ and complemented in ${{\mathcal W}}$, say with complement ${{\mathcal W}}_0$ and isomorphism $R_0\in{{\mathcal B}}({{\mathcal V}},{{\mathcal V}}_0)$, then $R={\left[\begin{smallmatrix} R_0\\ 0\end{smallmatrix}\right]}\in{{\mathcal B}}({{\mathcal V}},{{\mathcal V}}_0\oplus{{\mathcal W}}_0)={{\mathcal B}}({{\mathcal V}},{{\mathcal W}})$ is left invertible, so that $0\in{{\mathfrak K}}$. Now assume $R\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}})$ is left invertible, with left inverse $R^+\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}})$. Let $X\in{{\mathcal B}}({{\mathcal V}})$ be arbitrary. Then take $B_1=(I_{{\mathcal V}}-X)R^+$ and $B_2=R$, and it follows that $X=I-B_1B_2\in{{\mathfrak K}}$. Hence ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$. Conversely, it is clear that ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$ implies that $0\in{{\mathcal B}}({{\mathcal V}})$. \[C:IsoTocompSubs\] Assume ${{\mathcal V}}$ is isomorphic to a complemented subspace of ${{\mathcal W}}$. Then Problem \[P:BSOP\] is solvable. \[C:V=W,Z1Z2findim\] Assume ${{\mathcal V}}\oplus {{\mathcal Z}}_1\simeq {{\mathcal W}}\oplus {{\mathcal Z}}_1$ or ${{\mathcal V}}\oplus {{\mathcal Z}}_2\simeq {{\mathcal W}}\oplus {{\mathcal Z}}_2$ (note that one implies the other via ). Then Problem is solvable in case ${{\mathcal Z}}_1$ or ${{\mathcal Z}}_2$ is finite dimensional. Without loss of generality we may assume $\dim ({{\mathcal Z}}_1)<\infty$. Then Lemma 3.4 in [@tHMRR19] implies ${{\mathcal V}}\simeq{{\mathcal W}}$. By Lemma \[L:K=B(V)\] we then find that ${{\mathfrak K}}={{\mathcal B}}({{\mathcal V}})$, so that for $T$ we can take any invertible operator from ${{\mathcal V}}\oplus {{\mathcal Z}}_1$ to ${{\mathcal V}}\oplus {{\mathcal Z}}_2$. The case where ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional {#SubS:Z1Z2findim} ------------------------------------------------------------------------------- We start with a characterization of for the case that ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional. \[L:BanPropChar-findim\] Assume ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional. Then holds if and only if ${{\mathcal V}}$ and ${{\mathcal W}}$ are SUFDQ$_k$ for $k:=\|\dim({{\mathcal Z}}_2)-\dim({{\mathcal Z}}_1)\|$ or $k=0$. In case $k=0$ we have ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_2$ and hence holds. Thus it remains to consider the case $k>0$. We prove the claim for ${{\mathcal V}}$ using the first isomorphism in ; the claim for ${{\mathcal W}}$ follows by an analogous argument. Assume holds and let $S\in{{\mathcal B}}({{\mathcal V}}\oplus{{\mathcal Z}}_1,{{\mathcal V}}\oplus{{\mathcal Z}}_2)$ be invertible. Decompose $S$ as $$\label{Sdec} S={\ensuremath{\left[\begin{array}{cc}S_{11}&S_{12}\\ S_{21}&S_{22}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_1\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_2\end{array} \right]}}.$$ Since $S$ is invertible, $S$ is Fredholm with index 0. Write $$S={{\widetilde{S}}} + R \quad\mbox{with}\quad {\widetilde{S}}={\ensuremath{\left[\begin{array}{cc}S_{11}&0\\ 0&0\end{array} \right]}},\ R={\ensuremath{\left[\begin{array}{cc}0&S_{12}\\ S_{21}&S_{22}\end{array} \right]}}.$$ Since $R$ is finite rank, also ${{\widetilde{S}}}$ is Fredholm with index 0. Therefore the dimension of ${\textup{Ker\,}}S_{11}$ is equal to $\dim ({\textup{Ker\,}}{{\widetilde{S}}})-\dim ({{\mathcal Z}}_1)$ and the dimension of ${\textup{Coker\,}}S_{11}$ is equal to $\dim ({\textup{Coker\,}}{{\widetilde{S}}})-\dim ({{\mathcal Z}}_2)$. From this we conclude that $S_{11}\in{{\mathcal B}}({{\mathcal V}})$ is Fredholm with index $\dim ({{\mathcal Z}}_2) - \dim ({{\mathcal Z}}_1)$. Hence ${{\mathcal V}}$ is SUFDQ$_k$ by Proposition \[P:BanPropEquiv\]. Conversely, assume ${{\mathcal V}}$ is SUFDQ$_k$. We have to consider two cases. First assume $\dim({{\mathcal Z}}_2)\geq \dim({{\mathcal Z}}_1)$, so that $k=\dim({{\mathcal Z}}_2)-\dim({{\mathcal Z}}_1)$. Let $F\in{{\mathcal B}}({{\mathcal V}})$ be a Fredholm operator of index $k$. Without loss of generality we may assume $F$ is surjective. To obtain an invertible operator $S$ as in , set $S_{11}=F$, $S_{12}=0$, let $S_{22}$ be any linear injection from ${{\mathcal Z}}_1$ into ${{\mathcal Z}}_2$. Then ${\textup{Ker\,}}F$ and ${\textup{Coker\,}}S_{22}$ both have dimension $k$. Finally, take for $S_{21}$ an operator in ${{\mathcal B}}({{\mathcal V}},{{\mathcal Z}}_2)$ with kernel equal to a complement of ${\textup{Ker\,}}F$ while $S_{21}$ maps ${\textup{Ker\,}}F$ isomorpically onto ${\textup{Coker\,}}S_{22}$. It is easy to see that $S$ constructed in this way is invertible. Hence ${{\mathcal V}}\oplus{{\mathcal Z}}_1\simeq{{\mathcal V}}\oplus{{\mathcal Z}}_2$. In case $\dim({{\mathcal Z}}_2)<\dim({{\mathcal Z}}_1)$, a similar construction based on an injective Fredholm operator with index $-k$ provides an invertible operator $S$ from ${{\mathcal V}}\oplus{{\mathcal Z}}_1$ to ${{\mathcal V}}\oplus{{\mathcal Z}}_2$. The above lemma implies in particular that ${{\mathcal B}}({{\mathcal V}})$ contains Fredholm operators of index $\pm k$. To solve Problem \[P:BSOP\] we need to know if ${{\mathfrak K}}\subset {{\mathcal B}}({{\mathcal V}})$ contains Fredholm operators of index $k$ or $-k$. \[P:BSOPchar-findimZs\] Assume ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional. Then Problem \[P:BSOP\] is solvable if and only if ${{\mathfrak K}}$ contains a Fredholm operator of index $l:=\dim ({{\mathcal Z}}_2) - \dim ({{\mathcal Z}}_1)$. In order to prove Proposition \[P:BSOPchar-findimZs\] we first prove the following lemma, which shows that for $l\neq 0$ without loss of generality the Fredholm operator in Proposition \[P:BSOPchar-findimZs\] is injective (if $l< 0$) or surjective (if $l> 0$). \[L:SurInjFredholm\] Assume ${{\mathfrak K}}$ contains a Fredholm operator of index $l\neq 0$. Then ${{\mathfrak K}}$ contains a Fredholm operator of index $l$ which is injective (if $l<0$) or surjective (if $l> 0$). Moreover, if $l< 0$, then ${{\mathcal V}}$ and ${{\mathcal W}}$ are SUFDQ$_k$ for $k:=\|l\|$. Let $X=I_{{\mathcal V}}-B_1B_2\in{{\mathfrak K}}$, with $B_1\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}})$ and $B_2\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}})$, be Fredholm with index $l$. Since $X\in{{\mathcal B}}({{\mathcal V}})$, we know ${{\mathcal V}}$ is SUFDQ$_k$ for $k=\|l\|$. Since $X$ is Schur coupled to $Y:=I_{{\mathcal W}}-B_2B_1 \in{{\mathcal B}}({{\mathcal W}})$, via the operator matrix ${\left[\begin{smallmatrix} I_{{\mathcal V}}&B_1\\B_2&I_{{\mathcal W}}\end{smallmatrix}\right]}$, it follows [@BGKR05 p. 211] that $Y$ is Fredholm with index $l$, hence also ${{\mathcal W}}$ is SUFDQ$_k$. This proves the last claim of the proposition. Next we construct a Fredholm operator ${\widetilde{X}}\in{{\mathfrak K}}$ of index $l$ which is injective or surjective. There exists a finite rank operator $R\in{{\mathcal B}}({{\mathcal V}})$, say $R$ has rank $r$, so that ${\widetilde{X}}:=X-R$ is injective (if $l\leq 0$) or surjective (if $l\geq 0$). In that case ${\widetilde{X}}$ is also Fredholm with index $l$. Determine an integer $n>0$ so that $r\leq n k$ and note that ${{\mathcal W}}$ is also SUFDQ$_{nk}$ and hence SUFDS$_{nk}$. Let ${{\mathcal Z}}\subset{{\mathcal V}}$ be a subspace of dimension $nk$ that contains the range of $R$. Then ${{\mathcal W}}\simeq {{\mathcal W}}\oplus {{\mathcal Z}}$. Let $L\in{{\mathcal B}}({{\mathcal W}}\oplus{{\mathcal Z}},{{\mathcal W}})$ be an isomorphism. Since ${{\mathcal Z}}$ is complemented in ${{\mathcal V}}$, the embedding $\tau_{{\mathcal Z}}\in{{\mathcal B}}({{\mathcal Z}},{{\mathcal V}})$ of ${{\mathcal Z}}$ into ${{\mathcal V}}$, along an arbitrary complement of ${{\mathcal Z}}$, has a left inverse $\pi_{{\mathcal Z}}\in{{\mathcal B}}({{\mathcal V}},{{\mathcal Z}})$. Now set $${\widetilde{B}}_1={\ensuremath{\left[\begin{array}{cc}B_1&\tau_{{\mathcal Z}}\end{array} \right]}}L^{-1}\in{{\mathcal B}}({{\mathcal W}},{{\mathcal V}}) {\quad\mbox{and}\quad}{\widetilde{B}}_2=L{\ensuremath{\left[\begin{array}{c}B_2\\ \pi_{{\mathcal Z}}R\end{array} \right]}}\in{{\mathcal B}}({{\mathcal V}},{{\mathcal W}}).$$ We then obtain that $${\widetilde{X}}=I_{{\mathcal V}}-B_1B_2-R=I_{{\mathcal V}}-{\widetilde{B}}_1{\widetilde{B}}_2\in{{\mathfrak K}}.\qedhere$$ In case $l=0$, note that $I_{{\mathcal V}}\in{{\mathfrak K}}$, hence ${{\mathfrak K}}$ contains a Fredholm operator index $l$. On the other hand, for $l=0$ we have ${{\mathcal Z}}_1\simeq{{\mathcal Z}}_2$, so that Problem \[P:BSOP\] is solvable by Lemma \[L:Z1Z2iso\]. In the remainder of the proof assume $l\neq 0$. Assume Problem \[P:BSOP\] is solvable and $T$ as in is invertible. Then the left upper corner of $T$ is in ${{\mathfrak K}}$ and reasoning as in the first paragraph of the proof of Lemma \[L:BanPropChar-findim\] this operator is Fredholm with index $l$. Conversely, assume ${{\mathfrak K}}$ contains a Fredholm operator $F$ with index $l$. According to Lemma \[L:SurInjFredholm\], we may assume $F$ is surjective if $l\geq 0$, or injective if $l\leq 0$. Assume $l\leq 0$, so that $F$ is injective and the range of $F$ has codimension $-l$. Let ${{\mathcal V}}'$ be any complement of the range of $F$. Decompose ${{\mathcal Z}}_1={{\mathcal Z}}_1'\oplus{{\mathcal Z}}_1''$ with ${{\mathcal Z}}_1'\simeq {{\mathcal V}}'$ and ${{\mathcal Z}}_1''\simeq{{\mathcal Z}}_2$. Let $A'_{12}\in{{\mathcal B}}({{\mathcal Z}}_1',{{\mathcal V}})$ be a left invertible operator with range ${{\mathcal V}}'$ and $A'_{22}\in{{\mathcal B}}({{\mathcal Z}}_1'',{{\mathcal Z}}_2)$ an isomorphism. Then Problem \[P:BSOP\] is solvable via the invertible operator $$T={\ensuremath{\left[\begin{array}{ccc}F& A'_{12}&0\\ 0&0& A'_{22}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_1' \\ {{\mathcal Z}}_1''\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}\\ {{\mathcal Z}}_2\end{array} \right]}}.$$ In case $l\geq 0$ a similar construction provides an invertible $T$ as in with $A_{12}=0$. Assume ${{\mathcal V}}$ and ${{\mathcal W}}$ are such that ${{\mathfrak K}}$ contains a Fredholm operator of index $l$. Then holds and Problem \[P:BSOP\] is solvable for all finite dimensional Banach spaces ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ so that $\dim ({{\mathcal Z}}_2) -\dim({{\mathcal Z}}_1)=n l$ for a positive integer $n$. The next corollary follows directly by combining Proposition \[P:BSOPchar-findimZs\] with Lemma \[L:fKonlyFredholm\]. \[C:EssIncFinDimZs\] Assume ${{\mathcal V}}$ and ${{\mathcal W}}$ are essentially incomparable and ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional. Then Problem \[P:BSOP\] is solvable if and only if $\dim({{\mathcal Z}}_1)=\dim({{\mathcal Z}}_2)$. We now employ Proposition \[P:BSOPchar-findimZs\] to obtain a solution criterion that is independent of the set ${{\mathfrak K}}$. \[P:BSOPfindimiso\] Assume ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional. Problem \[P:BSOP\] is solvable in case ${{\mathcal V}}$ has a finite codimensional subspace that is isomorphic to a complemented subspace of ${{\mathcal W}}$ or in case ${{\mathcal W}}$ has a finite codimensional subspace that is isomorphic to a complemented subspace of ${{\mathcal V}}$. In case ${{\mathcal Z}}_1\simeq{{\mathcal Z}}_2$ we are done. So assume $k:=\|\dim ({{\mathcal Z}}_2) - \dim ({{\mathcal Z}}_1)\|>0$. By Lemma \[L:BanPropChar-findim\], ${{\mathcal V}}$ and ${{\mathcal W}}$ are SUFDQ$_k$. Assume ${{\mathcal V}}={{\mathcal V}}_1\oplus{{\mathcal V}}_2$ and ${{\mathcal W}}={{\mathcal W}}_1\oplus{{\mathcal W}}_2$ with ${{\mathcal V}}_1\simeq {{\mathcal W}}_1$. If $\dim({{\mathcal V}}_2)<\infty$, then ${{\mathcal V}}_1$ is SUFDQ$_k$ by applying Lemma \[L:BSPfindimcomp\] to the first isomorphism in , and hence ${{\mathcal W}}_1$ is SUFDQ$_k$. In case $\dim({{\mathcal W}}_2)<\infty$, we get that ${{\mathcal V}}_1$ and ${{\mathcal W}}_1$ are SUFDQ$_k$ by applying the same argument to the second isomorphism in . Under either of the conditions in the proposition we thus get that ${{\mathcal V}}_1$ and ${{\mathcal W}}_1$ are SUFDQ$_k$ and it suffices to prove that Problem \[P:BSOP\] is solvable from this fact. Now let $T_1\in{{\mathcal B}}({{\mathcal V}}_1)$ be a Fredholm operator of index $l:=\dim ({{\mathcal Z}}_2) - \dim ({{\mathcal Z}}_1)$, which exists by Proposition \[P:BanPropEquiv\]. Let $S_1\in{{\mathcal B}}({{\mathcal V}}_1,{{\mathcal W}}_1)$ be invertible. Now set $$\begin{aligned} B_1&={\ensuremath{\left[\begin{array}{cc}S_1^{-1}&0\\ 0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal W}}_1\\{{\mathcal W}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_1\\{{\mathcal V}}_2\end{array} \right]}},\\ B_2&={\ensuremath{\left[\begin{array}{cc}S_1 (T_1-I_{{{\mathcal V}}_1})&0\\0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_1\\{{\mathcal V}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal W}}_1\\{{\mathcal W}}_2\end{array} \right]}}.\end{aligned}$$ It then follows that $$I_{{{\mathcal V}}}-B_1 B_2 ={\ensuremath{\left[\begin{array}{cc}T_1&0\\ 0& I_{{{\mathcal V}}_2}\end{array} \right]}}\in{{\mathfrak K}},$$ and thus $I_{{{\mathcal V}}}-B_1 B_2$ is Fredholm with index $l$. Hence Problem \[P:BSOP\] is solvable by Proposition \[P:BSOPchar-findimZs\]. In Proposition \[P:BSOPfindimiso\] one cannot remove the condition that one of the subspaces of ${{\mathcal V}}$ and ${{\mathcal W}}$ has finite codimension, as shown in the next example. \[E:EAEnotSC\] Consider ${{\mathcal V}}={{\mathcal V}}_0\oplus \ell^p$ and ${{\mathcal W}}={{\mathcal V}}_0\oplus \ell^q$ with $1<p\neq q<\infty$ and where ${{\mathcal V}}_0$ is an infinite dimensional Banach space with few operators, see Remark \[R:BSexamples\], in particular ${{\mathcal V}}_0$ is SUFDQ$_k$ for no $k>0$. Since $\ell^p$ and $\ell^q$ are prime, they are SUFDQ, and thus ${{\mathcal V}}$ and ${{\mathcal W}}$ are SUFDQ, by Lemma \[L:BSPsum\]. Hence for all finite dimensional ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ we have . Although ${{\mathcal V}}$ and ${{\mathcal W}}$ are not essentially incomparable, we claim that still all Fredholm operators in ${{\mathfrak K}}$ have index $0$. We claim that ${{\mathcal V}}_0$ and $\ell^p$ are projection incomparable. Indeed, assume this is not the case and ${{\mathcal K}}$ is an infinite dimensional complemented subspace of $\ell^p$ which is isomorphic to a complemented subspace of ${{\mathcal V}}_0$. Since $\ell^p$ is prime, $\ell^p$ and ${{\mathcal K}}$ are isomorphic, so that ${{\mathcal V}}_0$ contains a complemented copy of $\ell^p$. But then ${{\mathcal V}}_0$ would be SUFDQ. The contradiction proves our claim. Likewise ${{\mathcal V}}_0$ and $\ell^q$ are projection incomparable. In fact, by Theorem 7.97 in [@A04], since all operators on ${{\mathcal V}}_0$ are either Fredholm with index 0 or inessential (in fact strongly singular), ${{\mathcal V}}_0$ and $\ell^p$ are essentially incomparable, and so are ${{\mathcal V}}_0$ and $\ell^q$. From the above it follows that $B_1\in {{\mathcal B}}({{\mathcal V}}_0\oplus\ell^q,{{\mathcal V}}_0\oplus\ell^p)$ and $B_2\in {{\mathcal B}}({{\mathcal V}}_0\oplus\ell^p,{{\mathcal V}}_0\oplus\ell^q)$ have the form $$\begin{aligned} B_1&={\ensuremath{\left[\begin{array}{cc}{\lambda}_1 I_{{{\mathcal V}}_0} -S_1 & B_{12}^{(1)}\\ B_{21}^{(1)} & B_{22}^{(1)} \end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_0\\ \ell^q\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_0\\ \ell^p\end{array} \right]}},\\ B_2&={\ensuremath{\left[\begin{array}{cc}{\lambda}_2 I_{{{\mathcal V}}_0} -S_2 & B_{12}^{(2)}\\ B_{21}^{(2)} & B_{22}^{(2)} \end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_0\\ \ell^p\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal V}}_0\\ \ell^q\end{array} \right]}} \end{aligned}$$ with ${\lambda}_1,{\lambda}_2\in{{\mathbb C}}$, $S_1$, $S_2$ strictly singular (hence inessential) and all $B_{ij}^{(1)}$ and $B_{ij}^{(2)}$ inessential. Since the class of inessential operators is closed under left and right multiplication with any bounded operator, it follows that $$I-B_1B_2={\ensuremath{\left[\begin{array}{cc}(1-{\lambda}_1{\lambda}_2)I_{{{\mathcal V}}_0} &0 \\ 0& I_{\ell^p}\end{array} \right]}} -B$$ where $B$ is an inessential operator in ${{\mathcal B}}({{\mathcal V}}_0\oplus\ell^p)$. In particular, $I-B_1B_2$ is Fredholm with index 0 in case ${\lambda}_1{\lambda}_2\neq 0$ and not Fredholm in case ${\lambda}_1{\lambda}_2=0$ (for otherwise ${\left[\begin{smallmatrix} 0&0\\0&I_{\ell^p}\end{smallmatrix}\right]}$ would be Fredholm). Hence, in this case also all Fredholm operators in ${{\mathfrak K}}$ have index 0, so that Problem \[P:BSOP\] with ${{\mathcal V}}$ and ${{\mathcal W}}$ as in this example and ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ finite dimensional is only solvable in case ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_2$. Essentially incomparable Banach spaces {#SubS:EssInc} -------------------------------------- In Lemma \[L:fKonlyFredholm\] and Corollary \[C:EssIncFinDimZs\] above we already encountered the notion of essentially incomparable Banach spaces. Next we consider some other cases of Problem \[P:BSOP\] where essential incomparability plays a role. All the results rely on the following general lemma. \[L:EssIncImplic\] Let $S_1,{{\mathcal S}}_2,{{\mathcal T}}_1,{{\mathcal T}}_2$ be Banach spaces so that ${{\mathcal S}}_1\oplus{{\mathcal T}}_1 \simeq {{\mathcal S}}_2\oplus{{\mathcal T}}_2$ and so that ${{\mathcal T}}_1$ and ${{\mathcal T}}_2$ are essentially incomparable. Then there is a subspace of ${{\mathcal T}}_1$ with finite codimension which is isomorphic to a complemented subspace of ${{\mathcal S}}_2$. In particular, if ${{\mathcal T}}_1$ or ${{\mathcal S}}_2$ is SUFDQ, then ${{\mathcal S}}_2$ contains a complemented copy of ${{\mathcal T}}_1$. Let $T$ be an isomorphism from ${{\mathcal S}}_1\oplus{{\mathcal T}}_1$ to ${{\mathcal S}}_2\oplus{{\mathcal T}}_2$ and decompose $T$ and $T^{-1}$ as $$T={\ensuremath{\left[\begin{array}{cc}T_{11}&T_{12}\\T_{21}&T_{22}\end{array} \right]}}{\ensuremath{\left[\begin{array}{c}{{\mathcal S}}_1\\ {{\mathcal T}}_1\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\mathcal S}}_2\\ {{\mathcal T}}_2\end{array} \right]}},\ T^{-1}={\ensuremath{\left[\begin{array}{cc}S_{11}&S_{12}\\S_{21}&S_{22}\end{array} \right]}}{\ensuremath{\left[\begin{array}{c}{{\mathcal S}}_2\\ {{\mathcal T}}_2\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\mathcal S}}_1\\ {{\mathcal T}}_1\end{array} \right]}}.$$ Since ${{\mathcal T}}_1$ and ${{\mathcal T}}_2$ are essentially incomparable, $T_{22}$ and $S_{22}$ are inessential operators. Writing out $T^{-1}T=I$ in terms of the decompositions of $T$ and $T^{-1}$ we find that $I_{{{\mathcal T}}_1}=S_{21}T_{12}+S_{22}T_{22}$. Hence $S_{21}T_{12}=I_{{{\mathcal T}}_1}-S_{22}T_{22}$, which implies $S_{21}T_{12}$ is Fredholm with index 0, because $T_{22}$ and $S_{22}$ are inessential. This in turn implies that $T_{12}$ is left Atkinson, that is, $T_{12}$ has finite dimensional kernel and complemented range, cf., Definition 7.1 and Theorem 7.2 in [@A04] (noting that we can replace compact operators in Definition 7.1 by inessential operators). Likewise, $S_{21}$ is right Atkinson, but we do not need this fact in the current proof. Let ${{\mathcal T}}_1'\subset{{\mathcal T}}_1$ be any complement of ${\textup{Ker\,}}T_{12}$, so that ${{\mathcal T}}_1'$ has finite codimension. Then $T_{12}$ restricted to ${{\mathcal T}}_1'$ is an isomorphism from ${{\mathcal T}}_1'$ onto the range of $T_{12}$, which is a complemented subspace of ${{\mathcal S}}_2$. This proves the first claim. If ${{\mathcal T}}_1$ is SUFDQ, then ${{\mathcal T}}_1$ is isomorphic to ${{\mathcal T}}_1'$ and hence to a complemented subspace of ${{\mathcal S}}_2$. Assume ${{\mathcal S}}_2$ is SUFDQ. Note that $${{\mathcal T}}_1\simeq {{\mathcal T}}_1'\oplus {\textup{Ker\,}}T_{12}\simeq T_{12}({{\mathcal T}}_1')\oplus {\textup{Ker\,}}T_{12}.$$ Let $S_2'$ be a complement of $T_{12}({{\mathcal T}}_1')$ in ${{\mathcal S}}_2$, so that ${{\mathcal S}}_2=T_{12}({{\mathcal T}}_1')\oplus{{\mathcal S}}_2'$. Since $\dim({\textup{Ker\,}}T_{12})< \infty$, we have $${{\mathcal S}}_2\simeq {{\mathcal S}}_2\oplus{\textup{Ker\,}}T_{12} =T_{12}({{\mathcal T}}_1')\oplus{{\mathcal S}}_2' \oplus{\textup{Ker\,}}T_{12} \simeq T_{12}({{\mathcal T}}_1')\oplus{\textup{Ker\,}}T_{12}\oplus{{\mathcal S}}_2' \simeq {{\mathcal T}}_1\oplus{{\mathcal S}}_2'.$$ Hence also in this case ${{\mathcal S}}_2$ contains a complemented copy of ${{\mathcal T}}_1$. In Corollary \[C:EssIncFinDimZs\] we concluded that in case ${{\mathcal V}}$ and ${{\mathcal W}}$ are essentially incomparable and ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are finite dimensional, them Problem \[P:BSOP\] is only solvable in case $\dim({{\mathcal Z}}_1)=\dim({{\mathcal Z}}_2)$. We now consider another case with ${{\mathcal V}}$ and ${{\mathcal W}}$ essentially incomparable, where ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are infinite dimensional. \[L:VW-EssInc\] Assume ${{\mathcal V}}\oplus {{\mathcal Z}}_1\simeq {{\mathcal W}}\oplus {{\mathcal Z}}_1$ or ${{\mathcal V}}\oplus {{\mathcal Z}}_2\simeq {{\mathcal W}}\oplus {{\mathcal Z}}_2$ (note that one implies the other via ). Then ${{\mathcal Z}}_1\simeq{{\mathcal Z}}_2$ and Problem \[P:BSOP\] is solvable if ${{\mathcal V}}$ and ${{\mathcal W}}$ are essentially incomparable and one of the following conditions is satisfied - ${{\mathcal V}}\simeq {{\mathcal V}}\oplus{{\mathcal V}}$ and either ${{\mathcal V}}$ is SUFDQ or both ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are SUFDQ; - ${{\mathcal W}}\simeq {{\mathcal W}}\oplus{{\mathcal W}}$ and either ${{\mathcal W}}$ is SUFDQ or both ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are SUFDQ. Assume ${{\mathcal V}}$ and ${{\mathcal W}}$ are essentially incomparable and condition (i) holds; the proof for (ii) goes analogously. For $i=1,2$, apply Lemma \[L:EssIncImplic\] with ${{\mathcal S}}_1={{\mathcal S}}_2={{\mathcal Z}}_i$, ${{\mathcal T}}_1={{\mathcal V}}$ and ${{\mathcal T}}_2={{\mathcal W}}$. We find that ${{\mathcal Z}}_i\simeq{{\mathcal V}}\oplus{{\mathcal Z}}_i'$ for some Banach space ${{\mathcal Z}}_i'$. Since also ${{\mathcal V}}\simeq {{\mathcal V}}\oplus{{\mathcal V}}$, we have $${{\mathcal Z}}_i\simeq {{\mathcal V}}\oplus{{\mathcal Z}}_i' \simeq {{\mathcal V}}\oplus{{\mathcal V}}\oplus{{\mathcal Z}}_i' \simeq {{\mathcal V}}\oplus{{\mathcal Z}}_i.$$ Using the first isomorphism in we obtain that ${{\mathcal Z}}_1\simeq {{\mathcal V}}\oplus{{\mathcal Z}}_1\simeq {{\mathcal V}}\oplus{{\mathcal Z}}_2 \simeq {{\mathcal Z}}_2$, which proves our claim. Next we consider the case where ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are essentially incomparable. \[L:Z1Z2EssInc\] Assume ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are essentially incomparable and ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ are SUFDQ. Then Problem \[P:BSOP\] is solvable in case ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_1\oplus {{\mathcal Z}}_1$ and ${{\mathcal Z}}_2\simeq {{\mathcal Z}}_2\oplus {{\mathcal Z}}_2$. Applying Lemma \[L:EssIncImplic\] to it follows that ${{\mathcal V}}$ contains complemented copies of ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$ and likewise ${{\mathcal W}}$ contains complemented copies of ${{\mathcal Z}}_1$ and ${{\mathcal Z}}_2$. Hence for $i=1,2$ we have ${{\mathcal V}}\simeq{{\mathcal V}}_i\oplus{{\mathcal Z}}_i$ and ${{\mathcal W}}\simeq{{\mathcal W}}_i\oplus {{\mathcal Z}}_i$ for Banach spaces ${{\mathcal V}}_i$ and ${{\mathcal W}}_i$. In particular, we find that $${{\mathcal V}}_1\oplus{{\mathcal Z}}_1\simeq {{\mathcal V}}_2\oplus{{\mathcal Z}}_2.$$ Again applying Lemma \[L:EssIncImplic\] it follows that ${{\mathcal V}}_1$ contains a complemented copy of ${{\mathcal Z}}_2$ and ${{\mathcal V}}_2$ contains a complemented copy of ${{\mathcal Z}}_2$. In both cases we obtain that ${{\mathcal V}}\simeq{{\widetilde{{{\mathcal V}}}}}\oplus{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2$ for a Banach space ${{\widetilde{{{\mathcal V}}}}}$ and by similar arguments ${{\mathcal W}}\simeq{{\widetilde{{{\mathcal W}}}}}\oplus{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2$ for a Banach space ${{\widetilde{{{\mathcal W}}}}}$. By assumption we have ${{\mathcal Z}}_1\simeq {{\mathcal Z}}_1\oplus {{\mathcal Z}}_1$ and ${{\mathcal Z}}_2\simeq {{\mathcal Z}}_2\oplus {{\mathcal Z}}_2$. Thus $${{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\oplus {{\mathcal Z}}_1 \simeq {{\mathcal Z}}_1\oplus {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2 \simeq {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2 \simeq {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2 \oplus{{\mathcal Z}}_2.$$ Now let $$S={\ensuremath{\left[\begin{array}{cc}S_{11}& S_{12}\\ S_{21} & S_{22}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\\ {{\mathcal Z}}_1\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\\ {{\mathcal Z}}_2\end{array} \right]}}$$ be invertible. Subject to the identifications ${{\mathcal V}}\simeq{{\widetilde{{{\mathcal V}}}}}\oplus{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2$ and ${{\mathcal W}}\simeq{{\widetilde{{{\mathcal W}}}}}\oplus{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2$ we obtain an invertible operator $$T={\ensuremath{\left[\begin{array}{ccc}I_{{{\widetilde{{{\mathcal V}}}}}}&0&0\\ 0&S_{11}& S_{12}\\ 0&S_{21}&S_{22}\end{array} \right]}} :{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\\{{\mathcal Z}}_1\end{array} \right]}}\to{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\\{{\mathcal Z}}_2\end{array} \right]}}$$ which is as in via $$\begin{aligned} &B_1={\ensuremath{\left[\begin{array}{cc}0&0\\0& I_{{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal W}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}},\\ &B_2={\ensuremath{\left[\begin{array}{cc}0&0\\0& I_{{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2}-S_{11}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}} \to {\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal W}}}}}\\ {{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}},\\ & A_{12}={\ensuremath{\left[\begin{array}{c}0\\S_{12}\end{array} \right]}}:{{\mathcal Z}}_1\to{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}},\ A_{21}={\ensuremath{\left[\begin{array}{cc}0&S_{21}\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\widetilde{{{\mathcal V}}}}}\\{{\mathcal Z}}_1\oplus{{\mathcal Z}}_2\end{array} \right]}}\to{{\mathcal Z}}_2,\\ &A_{22}=S_{22}.\qedhere\end{aligned}$$ In Lemmas \[L:VW-EssInc\] and \[L:Z1Z2EssInc\] we encountered Banach spaces ${{\mathcal Z}}$ with the property ${{\mathcal Z}}\simeq{{\mathcal Z}}\oplus {{\mathcal Z}}$. This property also appears in the Pe[ł]{}czyński decomposition technique for the Banach space Schroeder-Bernstein problem, cf., Theorem 2.2.3 in [@AK16]. All separable Banach spaces with countable primary unconditional bases have this property [@K99], see also [@FR05] for further examples. Prime spaces that have infinite dimensional complemented subspaces with infinite codimension are isomorphic to their squares. However, there are prime spaces whose infinite dimensional complemented subspaces are all finite codimensional [@M03], so these prime spaces cannot be isomorphic to their squares. Other examples are a reflexive Banach space not isomorphic to its square [@F72] and a Banach space that is isomorphic to its cube, but not to its square [@G96]. New cases where EAE and SC coincide, or not {#S:EAEvsSC} =========================================== In this section we translate some of the results from the analysis of Problem \[P:BSOP\] to the question in which cases EAE implies SC for relatively regular operators. Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be relatively regular operators as in . For the reader’s convenience we recall the setting and translation to Problem \[P:BSOP\] here. Hence $U$ and $V$ decompose as $$U={\ensuremath{\left[\begin{array}{cc}U'&0\\ 0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1\\ {{\mathcal X}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal X}}_1'\\ {{\mathcal X}}_2'\end{array} \right]}} {\quad\mbox{and}\quad}V={\ensuremath{\left[\begin{array}{cc}V'&0\\ 0&0\end{array} \right]}}:{\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1\\ {{\mathcal Y}}_2\end{array} \right]}}\to {\ensuremath{\left[\begin{array}{c}{{\mathcal Y}}_1'\\ {{\mathcal Y}}_2'\end{array} \right]}},\\$$ where $U'$ and $V'$ are invertible and $${{\mathcal X}}_2={\textup{Ker\,}}U,\quad {{\mathcal X}}_1'={\textup{Ran\,}}U,\quad {{\mathcal Y}}_2={\textup{Ker\,}}V,\quad {{\mathcal Y}}_1'={\textup{Ran\,}}V.$$ Although not uniquely determined, we will use the notation $${{\mathcal X}}_2'={\textup{Coker\,}}U {\quad\mbox{and}\quad}{{\mathcal Y}}_2'={\textup{Coker\,}}V.$$ Since $U$ and $V$ are relatively regular, we have $${{\mathcal X}}_1\simeq {{\mathcal X}}_1' {\quad\mbox{and}\quad}{{\mathcal Y}}_1\simeq {{\mathcal Y}}_1'.$$ Throughout this section we will assume $U$ and $V$ are EAE, which yields $${{\mathcal X}}_2\simeq{{\mathcal Y}}_2 {\quad\mbox{and}\quad}{{\mathcal X}}_2'\simeq{{\mathcal Y}}_2'.$$ Hence in that case, up to isomorphy we only have the spaces $${{\mathcal V}}:={\textup{Ran\,}}U,\quad {{\mathcal W}}:={\textup{Ran\,}}V,\quad {{\mathcal Z}}_1:={\textup{Ker\,}}U\simeq{\textup{Ker\,}}V,\quad {{\mathcal Z}}_2:={\textup{Coker\,}}U \simeq{\textup{Coker\,}}V,$$ and this also provides the translation to Problem \[P:BSOP\], noting that we may interchange the roles of $U$ and $V$, and hence ${{\mathcal V}}$ and ${{\mathcal W}}$. Using the above we translate some results of Section \[S:AnalyseBSOP\] to the current setting. \[P:Implic\] Let $U\in{{\mathcal B}}({{\mathcal X}})$ and $V\in{{\mathcal B}}({{\mathcal Y}})$ be relatively regular operators as in which are EAE. Then $U$ and $V$ are SC in the following cases: - ${\textup{Ker\,}}U\simeq {\textup{Coker\,}}U$ (Lemma \[L:Z1Z2iso\]); - $U$ or $V$ finite rank (Corollary \[C:VorWfindim\]); - ${\textup{Ran\,}}U$ isomorphic to a complemented subspace of ${\textup{Ran\,}}V$ or ${\textup{Ran\,}}V$ isomorphic to a complemented subspace of ${\textup{Ran\,}}U$ (Corollary \[C:IsoTocompSubs\]); - ${{\mathcal X}}\simeq{{\mathcal Y}}$, $U$ and $V$ Atkinson operators (Corollary \[C:V=W,Z1Z2findim\]); - ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ finite dimensional, ${\textup{Ran\,}}U$ has a finite codimensional subspace which is isomorphic to a complemented subspace of ${\textup{Ran\,}}V$ or ${\textup{Ran\,}}V$ has a finite codimensional subspace which is isomorphic to a complemented subspace of ${\textup{Ran\,}}U$ (Proposition \[P:BSOPfindimiso\]); - ${\textup{Ran\,}}U$ and ${\textup{Ran\,}}V$ essentially incomparable, ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ finite dimensional with the same dimension (Corollary \[C:EssIncFinDimZs\]); - ${\textup{Ran\,}}U$ and ${\textup{Ran\,}}V$ essentially incomparable, ${\textup{Ran\,}}U\simeq {\textup{Ran\,}}U\oplus{\textup{Ran\,}}U$ and either ${\textup{Ran\,}}U$ is SUFDQ or both ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ are SUFDQ or ${\textup{Ran\,}}V\simeq {\textup{Ran\,}}V\oplus{\textup{Ran\,}}V$ and either ${\textup{Ran\,}}V$ is SUFDQ or both ${\textup{Ker\,}}V$ and ${\textup{Coker\,}}V$ are SUFDQ (Lemma \[L:VW-EssInc\]); - ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ essentially incomparable, ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ SUFDQ, ${\textup{Ker\,}}U\simeq {\textup{Ker\,}}U \oplus {\textup{Ker\,}}U$ and ${\textup{Coker\,}}U\simeq {\textup{Coker\,}}U \oplus {\textup{Coker\,}}U$ (Lemma \[L:Z1Z2EssInc\]). We point out here that case (i) was obtained in [@tHR13 Theorem 3.3], while case (iv) can easily be derived from the main result in [@tHMRR19] (noting that with ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ finite dimensional we have that ${{\mathcal X}}$ and ${{\mathcal Y}}$ are essentially incomparable if and only if ${\textup{Ran\,}}U$ and ${\textup{Ran\,}}V$ are essentially incomparable). Case (iv) is an improvement of Proposition 5.1 (1) where in addition to $U$ and $V$ Atkinson it was also asked that ${{\mathcal X}}$ and ${{\mathcal Y}}$ are SUFDQ. All other cases discussed in Proposition \[P:Implic\] do not seem to be covered in the literature, to the best of our knowledge. Corollary \[C:EssIncFinDimZs\] also provides a way to construct an example that shows EAE does not imply SC, with ${{\mathcal X}}$ and ${{\mathcal Y}}$ essentially incomparable and ${\textup{Ker\,}}U$ and ${\textup{Coker\,}}U$ finite dimensional, but this, and much more, was already covered in [@tHMRR19]. The main result of [@tHMRR19], as well as [@tHMRR19 Proposition 2.4], may lead one to hope that for relatively regular operators the case with ${{\mathcal X}}$ and ${{\mathcal Y}}$ essentially incomparable is the only case where EAE and SC do not coincide. Although maybe somewhat contrived, the following example shows this is not the case. Following the construction preceding Corollary \[C:EAEnotSC\] we can use Example \[E:EAEnotSC\] to obtain a new example which shows EAE need not imply SC. In Example \[E:EAEnotSC\] we take $${{\mathcal V}}={{\mathcal V}}_0\oplus \ell^p {\quad\mbox{and}\quad}{{\mathcal W}}={{\mathcal V}}_0\oplus \ell^q$$ with ${{\mathcal V}}_0$ a Banach space with few operators and $p\neq q$. Since $\ell^q$ and $\ell^p$ are SUFDQ, so are ${{\mathcal V}}$ and ${{\mathcal W}}$, by Lemma \[L:BSPsum\], hence ${{\mathcal V}}\oplus{{\mathcal Z}}\sim {{\mathcal V}}$ and ${{\mathcal W}}\oplus{{\mathcal Z}}\sim {{\mathcal W}}$ for any finite dimensional Banach space ${{\mathcal Z}}$, hence we may take ${{\mathcal X}}={{\mathcal W}}$ and ${{\mathcal Y}}={{\mathcal V}}$. We then set $$U={\ensuremath{\left[\begin{array}{cc}I_{{{\mathcal V}}_0}&0\\0&S_q\end{array} \right]}} {\quad\mbox{and}\quad}V={\ensuremath{\left[\begin{array}{cc}I_{{{\mathcal V}}_0}&0\\0&S_p\end{array} \right]}},$$ with $S_q$ and $S_p$ the forward shifts on $\ell^q$ and $\ell^p$, respectively. Translating back to Problem \[P:BSOP\] we obtain ${{\mathcal Z}}_1 =\{0\}$ and ${{\mathcal Z}}_2\simeq {{\mathbb C}}$, so that Problem \[P:BSOP\] is not solvable by the analysis in Example \[E:EAEnotSC\], hence $U$ and $V$ are not SC. On the other hand, $U$ and $V$ are Fredholm and the dimensions of their kernels and cokernels coincide, so that $U$ and $V$ are EAE by Proposition 1 in [@BT92a]. #### Declarations: $ $ #### Conflict of interest The authors declare that they have no conflict of interest. #### Funding This work is based on research supported in part by the National Research Foundation of South Africa (NRF) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard. #### Availability of data and material Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. #### Code availability Not applicable. [99]{} P. Aiena, [Fredholm and local spectral theory, with applications to multipliers]{}, Kluwer Academic Publishers, Dordrecht, 2004. F. 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--- abstract: 'We construct for all $ k\in \mathbb{N} $ a $ k $-edge-connected digraph $ D $ with $ s,t\in V(D) $ such that there are no edge-disjoint $ s \rightarrow t $ and $ t\rightarrow s $ paths. We use in our construction “self-similar” graphs which technique could be useful in other problems as well.' author: - 'Attila Joó [^1]' date: 2015 title: 'Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair' --- This is the peer reviewed version of the following article: [@joo2016highly], which has been published in final form at <http://dx.doi.org/10.1002/jgt.22046>. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Introduction ============ Basic notions ------------- In this paper by “path” we mean a finite, simple, directed path. Sometimes we define a path of a digraph $ D=(V,A) $ by a finite sequence $ v_0,\dots,v_n $ of vertices of $ D $. If there are more than one edges from $ v_i $ to $ v_{i+1} $ for some $ i<n $, then it is not specified which edge is used by the path, so we use this kind of definition only if it does not matter. An $ u\rightarrow v $ path is a path with initial vertex $ u $ and terminal vertex $ v $. Its length is the number of its edges. We call a digraph $ D $ connected if for all $ u,v\in V(D) $ there is a $ u\rightarrow v $ path in $ D $. For $ U\subseteq V $ let $ \mathsf{span}_D(U) $ be the set of those edges of $ D $ whose heads and tails are contained in $U $ and let $ D[U]= (U,\mathsf{span}_D(U)) $. If it is clear what digraph we talk about, then we omit the subscripts.\ Background and Motivation ------------------------- R. Aharoni and C. Thomassen proved by a construction the following theorem that shows that several theorems about edge-connectivity properties of finite graphs and digraphs become “very” false in the infinite case. For all $ k\in \mathbb{N}$ there is an infinite graph $ G=(V,E) $ and $ s,t\in V $ such that $ E $ has a $ k $-edge-connected orientation but for each path $ P $ between $ s $ and $ t $ the graph $ G=(V,E\setminus E(P)) $ is not connected. In this article we would like to introduce a similar result. If $ D $ is a $ k $-edge-connected finite digraph, then for all $ s_1,t_1,\dots ,s_k,t_k\in V(D) $ there are pairwise edge-disjoint paths $P_1,\dots ,P_k $ such that $ P_i $ is an $ s_i\rightarrow t_i $ path. This fact is implied by the following Theorem of W. Mader as well as the (strong form of) Edmonds’ Branching theorem (see [@frank2011connections] p. $ 349 $ Theorem $ 10.2.1 $). Let $ D=(V,A) $ be a $ k+1 $-edge-connected, finite digraph and $ s,t\in V $. Then there is an $ s\rightarrow t $ path $ P $ such that $ (V,A\setminus A(P)) $ is $ k $-edge-connected. We will show that in the infinite case there is no $ k\in \mathbb{N} $ such that $ k $-edge-connectivity guarantees even the existence of edge-disjoint $ s_1 \rightarrow t_1 $ and $ s_2 \rightarrow t_2 $ paths for all $ s_1, t_1, s_2, t_2 $ vertices. Not even in the special case where the two ordered vertex pair is the reverse of each other. Main result =========== \[result\] For all $ k\in \mathbb{N} $ there exists a $ k $-edge-connected digraph without back and forth edge-disjoint paths between a certain vertex pair. Let $ k\geq2 $ be fixed, $ I=\{ 0,\dots,2k-1 \}$, $ I_e=\{ i\in I: i\text{ is even } \} $, $ I_o=I\setminus I_e $. Denote by $ I^{} $ the set of finite sequences from $ I $. Let the vertex set $ V $ of the digraph is the union of the disjoint sets $ \{ s_\mu : \mu\in I^{} \} $ ( we mean $ s_\mu=s_\nu $ iff $ \mu=\nu $) and $ \{ t_\mu : \mu\in I^{} \} $ ( $ t_\mu=t_\nu $ iff $ \mu=\nu $). If $ \mu $ is the empty sequence we write simply $ s,t $ and we denote the concatenation of sequences by writing them successively. For $ \nu\in I^{} $ let denote the set $ \{ r_{\nu\mu}: r\in \{ s,t \},\ \mu\in I^{} \}\subseteq V $ by $ V_\nu $. The edge-set $ A $ of the digraph consists of the following edges. For all $ \mu \in I^{} $ there are $ k $ edges in both directions between the two elements of the following pairs: $\{ s_\mu, t_{\mu1} \},\ \{ s_{\mu i}, t_{\mu (i+2)}\}\ (i=0,\dots,2k-3),\ \{ s_{\mu(2k-2)}, t_\mu \} $. Simple directed edges are $ (s_\mu, t_{\mu0} ), (t_{\mu i}, s_{\mu (i+1)} )_{i\in I_e}, (s_{\mu i},t_{\mu (i+1)})_{i\in I_o\setminus \{ 2k-1 \}},$ $(s_{\mu(2k-1)},t_\mu)\ $ for all $ \mu\in I^{} $. Finally $ D{\stackrel{\text{def}}{=}}(V,A) $ (see figure \[önhas k=3\]). \(s) at (-6,0) [s]{}; (t\_1) at (-3.5,0) [$t_1$]{}; (s\_1) at (-2.5,0) [$s_1$]{}; (t\_3) at (-0.5,0) [$t_3$]{}; (s\_3) at (0.5,0) [$s_3$]{}; (t\_5) at (2.5,0) [$t_5$]{}; (s\_5) at (3.5,0) [$s_5$]{}; (t\_0) at (-3.5,3) [$t_0$]{}; (s\_0) at (-2.5,3) [$s_0$]{}; (t\_2) at (-0.5,3) [$t_2$]{}; (s\_2) at (0.5,3) [$s_2$]{}; (t\_4) at (2.5,3) [$t_4$]{}; (s\_4) at (3.5,3) [$s_4$]{}; (t) at (6,3) [t]{}; (s) edge\[<->,thick\] (t\_1); (s\_1) edge\[<->,thick\] (t\_3); (s\_3) edge\[<->,thick\] (t\_5); (s\_0) edge\[<->,thick\] (t\_2); (s\_2) edge\[<->,thick\] (t\_4); (s\_4) edge\[<->,thick\] (t); (s) edge\[->\] (t\_0); (t\_0) edge\[->\] (s\_1); (s\_1) edge\[->\] (t\_2); (t\_2) edge\[->\] (s\_3); (s\_3) edge\[->\] (t\_4); (t\_4) edge\[->\] (s\_5); (s\_5) edge\[->\] (t); (-3,0) node circle (1); (0,0) node circle (1); (3,0) node circle (1); (-3,3) node circle (1); (0,3) node circle (1); (3,3) node circle (1); at (-0.25,1.20) [$\Huge{V_3}$]{}; at (3.35,1.15) [$\Huge{V_5}$]{}; at (-3.25,1.20) [$\Huge{V_1}$]{}; at (-3.25,4.20) [$\Huge{V_0}$]{}; at (0.25,4.15) [$\Huge{V_2}$]{}; at (3.25,4.15) [$\Huge{V_4}$]{}; One can avoid using parallel edges (without losing the desired properties of the digraph) by dividing each of these edges with one-one new vertex and drawing between them $ k (k-1) $-many new directed edges, one-one for each ordered pair. One can also achieve $ k $-connectivity instead of $ k $-edge-connectivity by using some similarly easy modification. \[isomorph\] For $ \nu\in I^{} $ the function $ f_\nu:V\rightarrow V_\nu,\ f_\nu(r_\mu){\stackrel{\text{def}}{=}}r_{\nu\mu}\ (r\in \{ s,t \}) $ is an isomorphism between $ D $ and $ D[V_\nu] $. It is a direct consequence of the definition of the edges since the number of edges from $ r_\mu $ to $ r'_{\mu'} $ are the same as from $ r_{\nu \mu} $ to $ r'_{\nu \mu'} $ for all $ r,r'\in \{ s,t \},\ \nu,\mu,\mu'\in I^{} $. \[Dv\] Denote by $ D_v $ the digraph that we obtain from $ D $ by contracting for all $ i\in I $ the set $ V_i $ to a vertex $ v_i $. Then $ D_v $ is $ k $-edge-connected. In the vertex-sequence $ s,v_1, v_3,\dots,v_{2k-1} $ there are $ k $ edges in both directions between the neighboring vertices such as in the sequence $ v_0,v_2,\dots,v_{2k-2},t $. Finally there are in both directions at least $ k $ edges between the vertex sets of the sequences above. \ For $ u\neq v $ we denote by $ \lambda(u,v) $ the local edge-connectivity from $ u $ to $ v $ in $ D $ (i.e. $\lambda(u,v)= \min \{ \left\|A'\right\|: A'\subseteq A,\text{ there is no path from } u\text{ to }v\text{ in }(V,A\setminus A') \} $) and let $ \lambda \{ u,v \}{\stackrel{\text{def}}{=}}\min \{ \lambda(u,v), \lambda(v,u) \} $. \[D connected\] $ D $ is connected. We will show that $ \lambda\{ s,r_{\mu} \}\geq 1 $ for all $ r\in \{ s,t \},\ \mu\in I^{} $. We will use induction on length of $ \mu $ (which is denoted by $ \left\|\mu\right\| $). Consider first the $ \left\|\mu\right\|=0,1 $ cases directly. The path $ s,t_0,s_1,t_2,s_3,\dots, t_{2k-2},s_{2k-1},t $ shows that $ \lambda(s,t)\geq 1 $. Using the isomorphism $ f_i $ (see Proposition \[isomorph\]) we may fix an $ s_i \rightarrow t_i $ path $ P_{s_i,t_i} $ in $ D[V_i] $ for all $ i\in I $. The path $$\begin{aligned} & t,P_{s_{2k-2},t_{2k-2}},\dots, P_{s_{2k-2j},t_{2k-2j}},\dots, P_{s_{0},t_{0}},P_{s_{1},t_{1}},s\\ \end{aligned}$$ justifies that $ \lambda(t,s)\geq 1 $ (thus $ \lambda\{ s, t \}\geq 1 $). Then we may fix a $ t_i \rightarrow s_i $ path $ P_{t_i,s_i} $ in $ D[V_i] \ (i\in I) $. The paths $$\begin{aligned} & s,P_{t_1,s_1},P_{t_3,s_3},\dots,P_{t_{2j+1},s_{2j+1}},\dots, P_{t_{2k-1},s_{2k-1}}\\ & P_{s_{2k-1},t_{2k-1}}, P_{s_{2k-3},t_{2k-3}},\dots,P_{s_{2k-1-2j},t_{2k-1-2j}},\dots, P_{s_1,t_1},s \end{aligned}$$ certify that $ \lambda \{ s,r_i \}\geq 1 $ if $ r\in \{ s,t \},\ i\in I_o $. The paths $$\begin{aligned} & t,P_{s_{2k-2},t_{2k-2}},P_{s_{2k-4},t_{2k-4}},\dots,P_{s_{2k-2-2j},t_{2k-2-2j}}\dots, P_{s_{0},t_{0}}\\ & P_{t_{0},s_{0}}, P_{t_{2},s_{2}},\dots,P_{t_{2j},s_{2j}},\dots,P_{t_{2k-2},s_{2k-2}},t \end{aligned}$$ certify that $ \lambda \{ t,r_i \}\geq 1 $ if $ r\in \{ s,t \}\geq 1,\ i\in I_e $ and thus (by $ \lambda\{ s, t \}\geq 1 $ and by transitivity) $ \lambda \{ s,r_i \}\geq 1 $ if $ r\in \{ s,t \},\ i\in I_e $. Hence the cases $\mu \in I$ with $\|\mu \| \leq 1$ are settled. Let be $ l\geq 1 $ and suppose $ \lambda\{ s,r_\mu \}\geq 1 $ if $r\in \{ s,t \},\ \mu\in I^{},\ \left\|\mu\right\|\leq l $. Let $ \nu=\mu i $, where $ i\in I $ and $ \left\|\mu\right\| =l$. By the induction hypothesis we have $ \lambda\{ s,s_\mu \}\geq 1 $. By the induction hypothesis for $ l=1 $ we have $ \lambda\{ s, r_i \}\geq 1 $ and so $ \lambda\{ s_\mu, r_{\mu i } \}\geq 1 $ by the isomorphism $ f_\mu $. Combining these, we get $ \lambda \{ s,r_{\mu i} \}\geq 1 $. \[k-edge-connected\] $ D $ is $ k $-edge-connected. Let $ k>l\geq 1 $. \[oroklod\] Let $ \mu \in I^{*} $ arbitrary. If we delete at most $ l $ edges of the digraph $ D[V_\mu] $ in such a way that its subgraphs $ D[V_{\mu i}]\ (i\in I) $ remain connected after the deletion, then $ D[V_\mu] $ also remains connected after the deletion. Because the isomorphism $ f_{\mu} $ it is enough to deal with the case where $\mu $ is the empty sequence. Denote by $ D' $ the digraph that we have after the deletion. Let $ D'_v $ be the digraph that we get from $ D' $ by contracting the sets $ V_i $ to a vertex $ v_i $ for all $ i\in I $. The digraphs $ D'[V_i]\ (i\in I) $ are connected by assumption, thus $ D' $ is connected iff $ D'_v $ is connected. The digraph $ D'_v $ arises by deleting at most $ l<k $ edges of the $ k $-edge-connected digraph $ D_v $ (see Proposition \[Dv\]) hence it is connected. \ We will prove that if $ D $ is $ l $-edge-connected, then it is also $ l+1 $ edge-connected. This is enough since we have already proved $ 1 $-connectivity of $ D $ in Proposition \[D connected\]. Assume that $ D $ is $ l $-edge-connected. Let $ C\subseteq A,\ \left\|C\right\|=l $ arbitrary and $ D'{\stackrel{\text{def}}{=}}(V,A\setminus C) $. By the definition of $ l+1 $-edge connectivity we need to show that $ D' $ is connected. Suppose for contradiction that it is not. Since the connectivity of the subgraphs $ D'[V_i]\ (i\in I) $ implies the connectivity of $ D' $ (by Proposition \[oroklod\]) there is an $ i_0\in I $ such that $ D'[V_{i_0}] $ is not connected. Since the connectivity of the subgraphs $ D'[V_{i_0 i}]\ (i\in I) $ implies the connectivity of $ D'[V_{i_0}] $ there is an $ i_1\in I $ such that $ D'[V_{i_0 i_1}] $ is not connected$ \dots $ By recursion we obtain an infinite sequence $ (i_n)_{n\in \mathbb{N}} $ such that the digraphs $ D'[V_{i_0\dots i_n}]\ (n\in \mathbb{N}) $ are all disconnected. Note that the digraphs $ D[V_{i_0\dots i_n}]\ (n\in \mathbb{N}) $ are $ l $-connected because $ D $ is $ l $-connected by assumption and they are isomorphic to it, hence necessarily $ C \subseteq \mathsf{span}(V_{i_0\dots i_n}) $ for all $ n\in \mathbb{N} $. But then $$C\subseteq \bigcap_{n=0}^{\infty} \mathsf{span}(V_{i_0 \dots i_n})=\mathsf{span}\left( \bigcap_{n=0}^{\infty} V_{i_0\dots i_n}\right)=\mathsf{span}(\varnothing)=\varnothing$$ which is a contradiction since $ \left\| C\right\|= l\geq 1 $. \[no pair-connected\] There are no edge-disjoint back and forth paths between $ s $ and $ t $ in $ D $. Suppose, seeking a contradiction, that there are. Let $P_{s,t} $ be an $ s \rightarrow t $ path and $ P_{t,s} $ be a $ t \rightarrow s$ path such that they are edge-disjoint and have a minimal sum of lengths among these path pairs. For $ u,v\in V $ call a set $ U\subseteq V $ an $ uv $-cut iff $ u\in U $ and $ v\notin U $. The set $ \{ t \}\cup \bigcup \{ V_i: i\in I_e \} $ is a $ ts $-cut and its outgoing edges are $ \{ (t_i,s_{i+1}) \}_{i\in I_e} $. Let $i_0\in I_e $ be the maximal index such that $ P_{t,s} $ uses the edge $ (t_{i_0}s_{i_0+1}) $. Then an initial segment of $ P_{t,s} $ is necessarily of the form $ t,P_{s_{2k-2},t_{2k-2}},P_{s_{2k-4},t_{2k-4}},\dots,P_{s_{i_0},t_{i_0}},s_{i_0+1} $ where $ P_{s_i,t_i} $ is an $ s_i\rightarrow t_i $ path in $ D[V_i] $. The set $T{\stackrel{\text{def}}{=}}\{ t \}\cup \bigcup \{ V_i : i_0\leq i\in I \} $ is also a $ ts $-cut and all the tails of its outgoing edges are in $ \{ t_{i_0},t_{i_0+1} \} $. $ P_{t,s} $ has already used the edge $ (t_{i_0},s_{i_0+1}) $ so it may not use another edge with tail $ t_{i_0} $ hence $ P_{t,s} $ leave $ T $ using an edge with tail $ t_{i_0+1} $. But then $ P_{t,s} $ contains an $s_{i_0+1}\rightarrow t_{i_0+1} $ subpath $ P_{s_{i_0+1},t_{i_0+1}} $ in $ D[V_{i_0+1}] $.\ $S{\stackrel{\text{def}}{=}}\{ s \}\cup \bigcup \{V_i: i_0+1\geq i\in I \} $ is an $ st $-cut and all the tails of its outgoing edges are in $ \{ s_{i_0},s_{i_0+1} \} $. Therefore $ P_{s,t} $ has an initial segment in $ D[S] $ that terminates in this set. We know that $ P_{s,t} $ does not use the edge $ (t_{i_0},s_{i_0+1}) $ because $ P_{t,s} $ has already used it. Therefore there is an $ m\in \{ i_0,i_0+1 \} $ such that $ P_{s,t} $ has a $ t_m\rightarrow s_m $ subpath $ P_{t_m,s_m} $ in $ D[V_m] $. But then the paths $ P_{t_m, s_m} $ and $ P_{s_m,t_m} $ are proper subpaths of $ P_{s,t} $ and $ P_{t,s} $ respectively. By Proposition \[isomorph\] $ f_m $ is an isomorphism between $ D $ and $D[V_m] $ and thus the inverse-images of the paths $ P_{t_m, s_m} $ and $ P_{s_m,t_m} $ are edge-disjoint back and forth paths between $ s $ and $ t $ with strictly less sum of lengths than the added length of paths $ P_{s,t}$ and $P_{t,s} $, which contradicts with the choice of $ P_{s,t} $ and $ P_{t,s} $. \ [1]{} Infinite, highly connected digraphs with no two arc-disjoint spanning trees. , 1 (1989), 71–74. , vol. 38. OUP Oxford, 2011. Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair. , 1 (2017), 51–55. On a property of n-edge-connected digraphs. , 4 (1981), 385–386. [^1]: MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary. Email: [joapaat@cs.elte.hu]{}
--- abstract: 'We experimentally explore the underlying pseudo-classical phase space structure of the quantum delta kicked accelerator. This was achieved by exposing a Bose-Einstein condensate to the spatially corrugated potential created by pulses of an off-resonant standing light wave. For the first time quantum accelerator modes were realized in such a system. By utilizing the narrow momentum distribution of the condensate we were able to observe the discrete momentum state structure of a quantum accelerator mode and also to directly measure the size of the structures in the phase space.' author: - 'G. Behinaein, V. Ramareddy, P. Ahmadi, and G.S. Summy' title: Exploring the phase space of the quantum delta kicked accelerator --- For more than a century the study of chaotic phenomena has been recognized as being crucial to developing a fuller understanding of nature. One aspect of this study which was missing until relatively recently was experimental scrutiny of quantum systems which in the classical limit exhibit chaotic behavior. Theoretical work on this subject had largely concentrated on the investigation of idealized systems such as the quantum delta-kicked rotor (QDKR) which were already well known from extensive work in the classical regime [@Lichtenberg]. The experimental study of this system gained new impetus through its realization using laser cooled atoms exposed to a corrugated potential from a pulsed off-resonant standing light wave [@Moore]. This system has subsequently led to many discoveries in the field of quantum chaos including observation of quantum resonances [@Moore; @Ryu], dynamical localization [@Moor2; @Ringot] and quantum diffusion [@Ammann; @Duffy2004]. In the theoretical description of the QDKR the effective value of Planck’s constant scales with the time between the pulses [@Oskay2000; @Sadgrove]. Therefore, to achieve classical correspondence in which $\hbar \rightarrow 0$, the time between pulses needs to be close to zero. However, for technical reasons this value can not be made arbitrarily small in experiments. Recently, Fishman, Guarneri, and Rebuzzini (FGR) [@Fishman] have shown that this difficulty can be circumvented for kicking periods close to a quantum resonance time. These resonance times [@Oskay; @darcy3] occur at integer multiples of the Talbot time, a time interval during which plane waves with certain equally spaced momenta in the kicking direction can acquire a phase factor which is an integer multiple of $2\pi$. This is analogous to the Talbot effect in optics [@berry]. Using this method, FGR have studied the quantum delta-kicked accelerator (QDKA) which can be created by taking the QDKR and adding a linear potential along the direction of the standing wave. They showed that the effective value of Planck’s constant scales with the deviation of the pulse separation time from certain non-zero resonance times, thus making it feasible to make the effective Planck’s constant very small. Therefore, if the time between pulses is chosen close to these resonance times, a pseudo-classical approach can be adopted to study the system. Perhaps the simplest way of experimentally realizing the QDKA is by applying the pulsed standing wave in the direction of gravity [@Oberthaler; @Godun]. This experiment has led to the discovery of quantum accelerator modes (QAMs) near the resonance times. One of the most important characteristics of QAMs is that they are comprised of atoms which show a linear momentum growth with pulse number in a freely falling frame [@Oberthaler]. It has been shown that QAMs are quantum nondissipative counterparts of mode locking [@Buchleitner]. They have also been suggested for use in the preparation of well defined initial conditions for quantum chaos experiments . FGR attributed the QAMs to the existence of stability islands in the pseudo-classical phase space. These studies have shown that this underlying phase space has a complex structure which is highly sensitive to the experimental parameters. However, the broad momentum distribution of the laser cooled atoms which have been used so far to study this system have prevented the examination of the local structures in the phase space. In this letter we report on the realization of QAMs using a Bose-Einstein condensate (BEC) of rubidium 87 atoms and the exploration of the pseudo-classical phase space structure of the QDKA. Figure\[momenutm\_pulsenumebr\] shows experimentally observed momentum distributions as a function of the number of standing wave pulses applied to a BEC. This figure demonstrates that the QAM gains momentum linearly as the number of pulses increases. Note that this is the first time that it has been possible to determine that the QAM is made up of several distinct momentum states as originally postulated in Ref.[@Godun]. This quantization of momentum is observable because the initial momentum uncertainty of the condensate was much smaller than two photon recoils. The Hamiltonian of the delta kicked accelerator for an atom with mass $m$ can be written as, $H = p^2/2m + m g' x + {U_{{\rm max}}\over 2}[1+\cos (G x)] \sum_{n} \delta (t - n T)$, where $p$ is the atomic momentum along the standing wave, $g'$ is the component of the gravitational acceleration in the direction of the standing wave, $G=2k$ the grating vector where $k$ is the light wave vector, $n$ the pulse number, $T$ the pulse period, and $U_{{\rm max}}$ is the well depth of the standing wave due to the light shift. The net effect of the time dependent potential is to distribute the condensate into different momentum states via a diffractive process. The population of these momentum states is determined by $\|J_{n'}(\phi_d)\|^2$, where $J_{n'}(\phi_d)$ is an $n'$th order Bessel function of the first kind with argument $\phi_d = U_{{\rm max}}\triangle t/(2 \hbar)$, the phase modulation depth [@Godun]. It is useful to consider this system for pulse periods close to integer multiples of the half Talbot time, $T_{\ell} = \ell \times 2\pi m / \hbar G^2 (= \ell \times 33.3 \mu s$ for Rb87 atoms). With this restriction the system can be described by the classical map [@Fishman], $$\begin{aligned} \theta_{n+1} &=& \theta_n + {\rm {sgn}} (\epsilon) J_n \nonumber \\ J_{n+1} &=& J_n - K \sin (\theta_{n+1}) + {\rm{sgn}}(\epsilon) \tau \eta,\label{map}\end{aligned}$$ where $\epsilon = 2 \pi \ell(T/T_{\ell} - 1)$ is a small number, $\ell$ is any positive integer number, and $K = \|\epsilon\| \phi_d$. The dimensionless $J$ and $\theta$ parameters are defined as, $$\begin{aligned} \theta &=& G~ x ~{\rm {mod}} (2 \pi) \nonumber \\ J_n &=& I_n + {\rm{sgn}}(\epsilon) [\pi \ell + \tau (\beta + n \eta + \eta /2)] \label{dynamic}\end{aligned}$$ where $p/(\hbar G) = I /\|\epsilon\| + \beta$, $\beta$ is the fractional momentum, $\tau = \hbar T G^2 /m$, and $\eta = m g' T/(\hbar G)$. Figure \[unitcell\] shows a typical phase space portrait for the map of Eq.(\[map\]) with $\phi_d = 1.4$, $T=29.5 \mu$s and $\epsilon = -0.72$. Perhaps the most important feature of this plot is the existence of a stable fixed point surrounded by an island of stability. If the size of these islands is large enough to capture a significant fraction of the wavepacket they give rise to observable accelerator modes. According to this model the momentum gain of an atom in a period $\mathfrak{p}$ accelerator mode after $n$ kicks is given by, $$q = n \Big [{\eta \tau \over \epsilon} + {2 \pi {\mathfrak{m}} \over {\mathfrak{p}} \| \epsilon \|} \Big],$$ where ${\mathfrak{m}}$ is an integer and $\mathfrak{(p,m)}$ specifies a particular accelerator mode [@Fishman; @Buchleitner; @Schlunk]. The initial conditions of atoms that can be accelerated is another important property of the accelerator modes. The limited range of these conditions is a consequence of the fact that the stability islands do not cover the whole unit cell of the phase space. Furthermore, the initial momentum required for an accelerator mode to emerge is periodic. This can be seen by using Eq.(\[dynamic\]) and the fact that $J$ has a periodicity of $2 \pi$. This is equivalent to a momentum periodicity of $\Delta p = 2 \pi \hbar G /\tau$. Observing this phase space structure requires that the atomic momentum distribution be narrower than $2 \pi \hbar G /\tau$. This implies a temperature of $\approx 450$ nK for Rb87 atoms exposed to a pulsed standing wave of 390 nm wavelength and pulse period close to the Talbot time ($\ell = 2$). To this extent, a BEC is an ideal candidate [@Deng]. The momentum width of a condensate is shown with two white lines in Fig.\[unitcell\]. Note that this width is smaller than the momentum extent of the island, indicating that the momentum resolution of the experiment is more than sufficient to clearly detect and identify a stability island from the chaotic background. However, for experiments utilizing cold thermal atomic samples, the momentum distribution is significantly wider than $\hbar G$. Although this wide momentum distribution makes it relatively easy to observe the accelerator modes, there is no direct way of examining the structure of the phase space. In order to use BEC to explore the phase space of the kicked accelerator we used the experimental setup described in detail by Ref.[@ahmadi_2]. A standard six-beam MOT was used to trap about $50 \times 10^6$ atoms which were loaded into the trapping potential created by a focused $\mathrm{CO_2}$ laser beam. In order to optimize the loading efficiency, the waist of the beam was chosen to be 100 $\mu$m [@ahmadi]. Typically $4\times 10^6$ atoms were trapped at the focus of the $\mathrm{CO_2}$ laser beam. Subsequently, one of the lenses through which the beam passed was moved 17 mm in 1 s so as to reduce the beam waist to 12 $\mu$m and compress the trap for efficient evaporative cooling [@Weiss]. The power in the beam was then reduced to 100 mW in 4 s to create a pure condensate with $\sim 50000$ atoms in the $5S_{1/2}F=1,m_F=0$ state. The $\mathrm{CO_2}$ laser was then turned off and after a variable time interval the kicking potential was turned on. Varying this time allowed the BEC to fall under the influence of gravity, thus changing the momentum of the condensate at the commencement of the kicks. The light for the kicks was obtained by passing the light from the MOT laser through a 40 MHz acousto-optic modulator (AOM). This light was 6.7 GHz red detuned from the $5S_{1/2}F=1 \rightarrow 5P_{3/2}F=2$ transition of Rb87 and propagated at $41^{\circ}$ relative to the vertical direction. The beam size was 1 mm, such that $\phi_d$ did not change appreciably while the BEC was interacting with the series of kicks. The phase modulation depth, $\phi_d$, was inferred by comparing the population of the first and zeroth order momentum states after one pulse. The temporal profile of the standing light wave was controlled by periodically switching the AOM on in order to create a sequence of pulses each $250$ns in length. The momentum distribution was measured using a time of flight method. That is, the condensate expanded for a controlled time interval, typically 9 ms, and was then destructively imaged using an absorptive technique. To observe the pseudo-classical phase space structure of the QAMs, a series of data were taken for pulse periods near both Talbot and half-Talbot times. Figure\[initial1\] shows a typical data set taken at (a) $T=61 \mu$s, (b) $T=72.2 \mu$s, (c) $T=28.5 \mu$s, and (d) $T=37.1 \mu$s pulse separations for different values of the BEC’s initial momentum. (a) and (b) occur on either side of the Talbot time at $T = 66.6 \mu$s while (c) and (d) occur on either side of the half-Talbot time at $T = 33.3 \mu$s. The data in Figure\[initial1\] was created by horizontally stacking 60 time-of-flight images of the condensate, each for a different initial momentum. These data confirm the periodicity of the QAMs with momentum. Furthermore, the data of Fig.\[initial1\] provides a direct way to validate the theoretical prediction of the island size. To do so, the data of Fig.\[initial1\] was summed along the initial momentum axis. $\Delta p_{{\rm island}}$ was then determined by measuring when the accumulated signal of the QAM had dropped to 1/$e$ of its maximum value. The theoretical values were inferred by plotting the map of Eq.(\[map\]) for the corresponding experimental values of $K$. The experimental and theoretical values for the momentum extent of the islands are given in Fig.\[islandsize\], near (a) half-Talbot and (b) Talbot times. The circle and asterisk signs are the experimental and theoretical values for $\Delta J_{{\rm island}}$ (as defined in Figs.\[unitcell\] and \[initial1\]). It can be seen that the experimental values are very close to the theoretical predictions. Note that for values of $K > 2$ at half-Talbot time, the stability island elongates in $J$ and becomes narrow in $\theta$. This behavior reduces the effective overlap between the BEC’s wavefunction and the stability island and consequently the QAMs were not visible in Fig.\[islandsize\](b) for higher values of $K$. Figure\[initial1\] also shows that there can be little overlap between the initial conditions that will populate a QAM at two different values of $T$. This behavior particularly affects what happens in experiments in which the momentum distribution is measured as the pulse period is scanned across a resonance time. Unlike the experiments with cold atomic samples where the QAMs on both sides of a resonance could be populated [@Oberthaler], in the case of the condensate, only the QAMs which have significant overlap with the condensate wavefunction will be observable. This can be seen in Fig.\[period\_scane\], where we performed a scan of pulse period across the Talbot time for two different initial momenta. The initial momentum for Fig.\[period\_scane\](a) was set to $1.2 \hbar G$ such that the QAMs were efficiently loaded at pulse periods near $T=72 \mu$s, whereas in Fig.\[period\_scane\](b), the initial momentum was set at $1.5 \hbar G$ to mainly populate the QAMs around pulse periods near $T=61 \mu$s. As can be seen, the QAM with indices $\mathfrak{(p,m)}=$(1,0) appear at pulse periods greater than the Talbot time in Fig.\[period\_scane\](a), whereas in Fig.\[period\_scane\](b) the (1,0) QAM mostly appears at pulse periods smaller than the Talbot time. Note that this is the first time that it has been possible to selectively populate an island at a particular position in phase space. In conclusion our experiments have demonstrated the feasibility of observing quantum accelerator modes using a BEC. Using a BEC we were able to examine the underlying pseudo-classical phase space structure of the quantum delta kicked accelerator. 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--- abstract: 'Photocurrent in solids is an important phenomenon with many applications including the solar cells. In conventional photoconductors, the electrons and holes created by light irradiation are separated by the external electric field, resulting in a current flowing into electrodes. Shift current in noncentrosymmetric systems is distinct from this conventional photocurrent in the sense that no external electric field is needed, and, more remarkably, is driven by the Berry phase inherent to the Bloch wavefunction. It is analogous to the polarization current in the ground state but is a d.c. current continuously supported by the nonequilibrium steady state under the pumping by light. Here we show theoretically, by employing Keldysh-Floquet formalism applied to a simple one-dimensional model, that the local photo excitation can induce the shift current which is independent of the position and width of the excited region and also the length of the system. This feature is in stark contrast to the conventional photocurrent, which is suppressed when the sample is excited locally at the middle and increases towards the electrodes. This finding reveals the unconventional nature of shift current and will pave a way to design a highly efficient photovoltaic effect in solids.' address: - ' $^1$Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, JAPAN ' - ' $^2$RIKEN Center for Emergent Matter Sciences (CEMS), Wako, Saitama, 351-0198, JAPAN ' author: - 'Hiroaki Ishizuka$^1$ and Naoto Nagaosa$^{1,2}$' title: ' Local photo-excitation of shift current in noncentrosymmetric systems ' --- [Keywords]{}: photovolatic effect, shift current, nonequilibrium Green’s function method. Introduction ============ Photo-carrier generation and the associated photocurrent is a subject of intensive studies in condensed matter physics and its applications. For example, the excitation spectrum of photocurrent together with the optical absorption spectrum offers a powerful tool to investigate the electronic states in bulk solids. Namely, the absorption can occur both by the excitons, i.e., the bound states of electron and hole, and electron-hole continuum, while the photocurrent can be carried only by the latter, as the exciton is a charge neutral object and cannot contribute to the current. By increasing the external electric field for the photocurrent, the excitons split into electrons and holes and begins to contribute to the photocurrent. In solar cells, the external electric field is generated by the p-n junction, where the electrons and holes feel the potential gradient in the opposite directions and are separated [@Nelson2003; @Wuerfel2009]. In contrast to the conventional view of the photocurrent discussed above, the shift current in noncentrosymmetric systems has been attracting intensive interest [@Fridkin2001; @Choi2009; @Young2012_1; @Young2012_2; @Grinberg2013; @Nie2015; @Shi2015; @deQuilettes2015; @Morimoto2016]. In this phenomenon, since the system itself has a fixed “direction”, the light irradiation alone can produce the d.c. current without the external electric field. One might think that the internal electric field plays a similar role to that of the external one. However, the situation is much more subtle. Due to the periodicity of the crystal potential, there is no macroscopic potential gradient. Microscopically, the band structure has certain symmetry between $\vec{k}$ and $-\vec{k}$ due to the time-reversal symmetry $T$ ( $\vec{k}$ is the crystal momentum ). Namely, $T$-symmetry gives the relation $\varepsilon_{\uparrow} (\vec{k}) = \varepsilon_{\downarrow}(-\vec{k})$ with $\uparrow(\downarrow)$ represents the up (down) spin. Therefore, it is highly nontrivial whether the excitation of electrons and holes contributes to the shift current. The lack of inversion symmetry $I$, however, opens the possibility of the Berry phase of Bloch wavefunction to be finite. (More precisely, one can choose the gauge of the Bloch wavefunction where the Berry phase connection is zero in the presence of both $T$ and $I$ symmetries, i.e., no Berry phase contribution to material properties.) It have been shown that, the shift current comes from the Berry connection of the Bloch wavefunction $\|\vec{k}\,n\rangle$ for conduction ($n=c$) and valence $(n=v$) bands [@Kral2000; @Sipe2000; @Morimoto2016]. An important observation here is that the Berry connection has the meaning of the intracell coordinate in the sense that the position $\vec{r}_n$ of the wavepacket made from the Bloch wavefunctions shifts proportionally by the Berry connection. Indeed, it is related to the polarization in ferroelectric materials [@Resta1992; @King-Smith1993]. This scenario for the ground state finds its generalization to the nonequilibrium steady state, where the continuous pump from the valence to conduction band occurs by the photo-excitation. In this case, the [difference]{} of the Berry phase between the conduction and valence bands corresponds to the shift of the electron position induced by the transition, and contribute to the d.c. photocurrent. This is the picture of shift current, and hence is different from the transport current supported by the free carriers. It is more quantum mechanical origin, i.e., cannot be attributed to the motion of individual carriers, and is expected to be nonlocal in nature. Therefore, the optical response of noncentrosymmetric photovoltaic systems to local photo-excitation is nontrivial. To study the shift current due to the local photo-excitation, in this paper, we consider a noncentrosymmetric non-interacting fermion chain as a basic model that exhibits shift current. As we ignore the Coulomb interaction of electrons, our model has no internal bias due to the ferroelectric polarization, therefore, no photocurrent from the conventional mechanism similar to photoconductors. We study the model by a nonequilibrium Green’s function technique used to study nonequilibrium phenomena in semiconductors [@Haug2008] and in organic solids [@Wei2006; @Wei2007; @Wei2007_2]; the details of the method are elaborated in appendices. By applying the method, we numerically study the shift current induced by the local photo-excitation using the fermion chain up to $N=1200$ sites. Although there are no potential gradient, we find that a finite photocurrent appears in this model. Furthermore, we find that the locally excited shift current in the fermion chain behaves in a distinct manner from the photocurrent in photoconductors and in p-n junctions, as shown in Fig. \[fig:bdchain\]. In particular, we find that the magnitude of shift current does not depend on the position of the photoexcitation. This is a contrasting feature to the conventional case, in which the current is suppressed for the photoexcitation at the center of the chain. This is an interesting feature that experimentally distinguishes the shift current from the conventional photocurrent. Furthermore, this is an advantageous feature for application to highly efficient optoelectronic devices. We also find that the magnitude of photocurrent does not change by increasing the width of the bonds excited. As presented in Sec. \[sec:results:lightwidth\], we discuss that this is likely to be a consequence of high carrier density excited by light. The remainder of the paper is organized as follows. In Sec. \[sec:model\], we introduce the model we consider in this paper. In Sec. \[sec:results\], we present the result of theoretical calculations on the light-position dependence of the shift current in a fermion chain. Sections \[sec:discussion\] and \[sec:summary\] are devoted to discussions and summary, respectively. The nonequilibrium Green’s function method for the Floquet theory and its numerical implementation are elaborated in \[sec:method\] and \[sec:computation\], respectively. The detail of the quasiclassical theory for carrier concentration and the shift current is explained in \[sec:carrier\_density\]. Model and Method {#sec:model} ================ ![Schematic figure of the model. (a) an $N$ site fermion chain with bond alternation and staggered potential. The chain is coupled to the leads at each end. The incident light excites $l_A$ sites of the chain at $i_A$. (b) Floquet bands of the fermion chain in the bulk limit ($N\to\infty$). See the text for details.[]{data-label="fig:model"}](fig01.pdf){width="\linewidth"} To study the effect of local photo-excitation in noncentrosymmetric systems, we consider Rice-Mele model [@Rice1982] coupled to leads (electrode) at each end and with monochromatic incident light \[Fig. \[fig:model\](a)\]; the light is taken into account as the time-dependent electric field. The Hamiltonian for the chain is given by $$\begin{aligned} H= H_0 + H_A,\label{eq:H}\end{aligned}$$ where the first term is the Rice-Mele Hamiltonian, $$\begin{aligned} H_0&=& -t\sum_i c_{i+1}^\dagger c_i + c_{i}^\dagger c_{i+1} - \frac{B}2\sum_i (-1)^i\left(c_{i+1}^\dagger c_i + c_{i}^\dagger c_{i+1} \right)\nonumber\\ &&\qquad+ \frac{d}2\sum_i (-1)^i c_i^\dagger c_i, \label{eq:H0}\end{aligned}$$ and the second term is the interaction with the incident light, $$\begin{aligned} H_A&=& -itA\sin(\Omega \tau) \sum_i{}^\prime c_{i+1}^\dagger c_i - c_{i}^\dagger c_{i+1} \nonumber\\ && + \frac{iAB}2\sin(\Omega \tau)\sum_i{}^\prime (-1)^i\left(c_{i+1}^\dagger c_i - c_{i}^ \dagger c_{i+1}\right). \label{eq:HA}\end{aligned}$$ Here, $c_i$ ($c_i^\dagger$) is the annihilation (creation) operator of the electron at the $i$th site; $t$ ($B$) is the uniform (staggered) hopping between the nearest-neighbor sites, and $d$ is the staggered potential. In Eq. (\[eq:HA\]), $\Omega$ is the frequency of the light and $A$ is the amplitude of vector field. Here, we only consider the leading order in $A$. The sum $\sum^\prime$ is over consecutive $l_A$ sites centered at $i_A$ \[see Fig. \[fig:model\](a)\]. In periodically driven systems, there are no eigenstates as in the time-independent Hamiltonians. However, according to the Floquet theory [@Floquet1883; @Hanggi2005], one can define similar eigenstates for the periodically driven Hamiltonian called Floquet states. In the Floquet theory, in addition to the bands that consists of the eigenstate of time-independent Hamiltonian, we consider the side bands of the original band with the energy shift of $-\kappa\Omega$ ($\kappa\in\mathbb Z$) \[Fig. \[fig:model\](b)\]. When the chain is exposed to light, it gives the mixing term between the side bands with different $\kappa$; the leading order contribution from the light comes in as the mixing between two neighboring subbands (subbands with the indices $\kappa$ and $\kappa+1$). Therefore, to take into account of the leading order contributions from the light, in the subsequent calculations, we focus on the bands with indices $\kappa=0$ and $-1$ bands and ignore all others (\[sec:computation\]). To study the nonequilibrium transport phenomena in the Rice-Mele model, the photocurrent from the local excitation is studied using a nonequilibrium Green’s function theory [@Meir1992; @Rammer2007; @Haug2008]. We generalized the method to the case of periodically driven systems and computed the shift current numerically. The general formalism and further details on the application of the method to the model in Eq. (\[eq:H\]) are elaborated in \[sec:method\]. Results: Shift Current in Noncentrosymmetric Crystals {#sec:results} ===================================================== Light-Position Dependence of Shift Current ------------------------------------------ ![The result of the photocurrent in a $N=1200$ site fermion chain with bond alternation ($B=1$) and staggered potential ($d=1$). The figure shows the dependence of the photocurrent on the location of incident light ($x_A=i_A/N$) for different widths of light ($l_A$).[]{data-label="fig:bdchain"}](fig02.pdf){width="0.65\linewidth"} In this section, we study the shift current induced by the local excitation by light. Figure \[fig:bdchain\] shows the light-position dependence of the shift current by the local excitation at $x_A=i_A/N$ for $N=1200$. The three different lines correspond to different $l_A$, the number of bonds we shine the light onto \[see fig. \[fig:model\](a)\]. The results show that the shift current by the local excitation is almost independent of $x_A$. Besides, the shift current is symmetric with respect to $x_A$, i.e., $J(x_A)\simeq J(1-x_A)$. For later convenience, we here define the “symmetric” and “asymmetric” components of the photocurrent by $$J_s(x_A)\equiv\frac{J(x_A)+J(1-x_A)}2$$ and $$J_{as}(x_A)\equiv\frac{J(x_A)-J(1-x_A)}2,$$ respectively. By calculating $J_s$ and $J_{as}$ from the data in Fig. \[fig:bdchain\], we indeed find that $J_{as}\lesssim10^{-8}$, which is below the numerical precision. Hence, our results show that the symmetric part of the photocurrent is dominant in the Rice-Mele chain. This is a contrasting feature from what is expected in the conventional mechanism in p-n junctions, by which the photocurrent is borne by charged quasi-particles. In p-n junctions, the photocurrent is essentially an interfacial phenomenon. At the interface (or surface), a potential gradient occurs due to the nonuniform structure. Due to this potential gradient, when an electron-hole pair is created by light, the electron and hole drift toward opposite directions, resulting in a net charge current. In the setup shown in Fig. \[fig:model\](a), the junction of the chain and the lead plays the role of the interface. Therefore, in this mechanism, $J$ is expected to increase as we approach the edges. Besides, if the carriers are generated close to the edge, we expect that the excited carriers can escape to the electrodes easily, before being scattered back into the valence band. Hence, the current is expected to be larger at the ends of the chain than in the center. Another important feature is the symmetry. Although there is no potential gradient in the bulk, in principle, a finite photocurrent can appear from the conventional mechanism when the position of the light is sufficiently close to one of the edges. This is a mechanism similar to p-n junctions where the potential gradient appears due to the interface. If this mechanism is dominant, for a centrosymmetric chain (or if the effect of broken inversion symmetry is negligible), the photocurrent from the local excitation is expected to be antisymmetric. On the contrary, if the photocurrent is induced by the bulk effect, e.g., anomalous photovoltaic effect, then the $x_A$ dependence of the photocurrent is expected to be independent of the position of light for a sufficiently long chain, with possibly some effect of the surface near the edges, i.e., they are (almost) symmetric. Therefore, we expect the light-position dependence of the locally excited photocurrent reflects the microscopic mechanism of the photocurrent; $J_{as}$ represents the contribution of the edges, e.g. the photocurrent by the conventional mechanism at the edge, while $J_{s}$ reflects the bulk effect such as the anomalous photovoltaic effects or shift current. These observations imply that the symmetric feature of the photocurrent with respect to $x_A$ reflects the unconventional photocurrent in the noncentrosymmetric chains, namely, the shift current. Therefore, in this paper, we call the symmetric photocurrent “shift current”. Our result in Fig. \[fig:bdchain\] indicates that shift current is the dominant source of the photocurrent in Rice-Mele chain. To compare the result of Rice-Mele model with a model with no shift current, we also performed the calculation of $x_A$ dependence for a symmetric chain with $t=1$, $B=1.2$, and $d=0$ (not shown). In this model, however, we find no observable photocurrent within the numerical precision (only small $J_{as}$ that is below our numerical precision), consistent with the above discussion. ![Light-position dependence of the photocurrent in the fermion chain with both the bond alternation ($B=1$) and the staggered potential ($d=1$). Each line shows the result for different length $N$.[]{data-label="fig:chain_len"}](fig04.pdf){width="0.65\linewidth"} In the last, we discuss the effect of finite size effect. Fig. \[fig:chain\_len\] shows the result of the asymmetric chain for $l_A=39$ with $N= 600$, $900$, and $1200$. The results for the different $N$ show similar magnitude of the shift current. They are also symmetric with respect to $x_A$. From this observation, we conclude that our data for $N=1200$ reflects $N\to\infty$ behavior in the ballistic limit. Light Width Dependence of Shift Current {#sec:results:lightwidth} --------------------------------------- ![$\Omega$ dependence of the photocurrent in an noncentrosymmetric chain with $t=B=d=1$. The light is shined at the center of the chain ($x_A=i_A/N=0.5$). (a) photocurrent by local excitation with different widths of light $l_A=19$, $39$, and $59$. (b) photocurrent by the local excitation with different length $N$.[]{data-label="fig:omega_dep"}](fig06.pdf){width="\linewidth"} Another point to be noted in Fig. \[fig:bdchain\] is the $l_A$ dependence of the shift current. In Fig. \[fig:bdchain\], the induced shift current appears to be roughly the same for all $l_A$. Naively, the photocurrent is expected to be proportional to the amount of the photon energy absorbed by the chain, therefore, proportional to $l_A$. Our results, however, show that there is nearly no dependence of the induced current on $l_A$. In our calculation, such behavior was observed for all range of $\Omega$. Figure \[fig:omega\_dep\] shows the result of shift current excited by the local excitation at $x_A=0.5$. Figure \[fig:omega\_dep\](a) is the result for different $l_A$ in $N=1200$ noncentrosymmetric chain. A finite photocurrent is observed above $\Omega\sim2.6$ and form a triangular shape with a peak at $\Omega\sim3.2$. The lower limit $\Omega\sim2.6$ roughly corresponds to the band gap $\sqrt{\Delta^2+4B^2}=\sqrt5\sim2.24$, while the peak corresponds to an $\Omega$ which a subband of the conduction band perfectly overlap with that of the valence band, $\frac12(\sqrt{\Delta^2+16t^2}+\sqrt{\Delta^2+4B^2})=\frac12(\sqrt5+\sqrt{17})\sim3.17$. The results show almost no $l_A$ dependence for all range of $\Omega\in [2.5:4.0]$, where the shift current is observed. Figure \[fig:omega\_dep\](b) shows the $N$ dependence of the shift current with $l_A=39$ and $A=0.2$ at $x_A=0.5$. For most values of $\Omega$, the results of shift current show almost no dependencies to $N$. On the other hand, some finite size effect is seen at $\Omega\sim3.1$. The $N$ sensitive behavior around $\Omega\sim3.1$ is likely to come from the sparse density of states in the center of the band. In the Floquet theory, the effect of light primarily appears around the crossing of two bands with different Floquet indices $\kappa$. In our model with $t=B=d=1$, it is around the crossing of the conduction band with $\kappa=0$ and valence band with $\kappa=-1$ (see Fig. \[fig:model\]); the two bands touch at their edge for $\Omega\sim2.24$, and crosses at the center of the bands for $\Omega\sim3.17$. As the density of states is relatively sparse at the center of the band than at the edges, the discreteness of the energy levels due to the finite length become more evident at the band center. However, the $N$ dependence at $\Omega\sim3.1$ also appears to be converged for the chains we used here ($N=1200$). The result indicates that, for a sufficiently long chain, there seems to be almost no $l_A$ dependence. Furthermore, from the $\Omega$ dependence, it seems the $l_A$ independent behavior has less to do with the microscopic structure of the wave functions, e.g., Berry connection. According to the Floquet theory, as the effects of light have much smaller energy scale than other parameters, the effect of light on the electrons appears primarily around the band crossing point, where $\varepsilon_v(k)+\Omega=\varepsilon_c(k)$ [@Oka2009]. Here, $\varepsilon_v(k)$ is the energy eigenstate of the valence band with wavenumber $k$, and the $\varepsilon_c(k)$ being that of conduction band \[see Fig. \[fig:model\](b)\]. As the crossing point changes by changing $\Omega$, if the behavior reflects the microscopic structure of the electronic states, e.g., Berry phase, then it is natural to expect $\Omega$ dependence. Therefore, it is unlikely that the $l_A$-independent behavior is related to the mechanism of the shift current. ![Schematic figure of charge transfer between the conduction and valence bands of a system irradiated with light and coupled to a lead. We assume that the lead is in its thermodynamical equilibruium state.[]{data-label="fig:quasi_classic"}](fig08.pdf){width="0.5\linewidth"} From this observation, the $l_A$-independent behavior is likely to come from the fact that a large number of carriers are excited in the conduction band. To show this, we here consider a quasi-classical model schematically shown in Fig. \[fig:quasi\_classic\]. The model consists of valence band, conduction band, and the lead(s) as shown in Fig. \[fig:quasi\_classic\]. The carriers are transferred between the valence and conduction bands by transition rate $W_{vc}$ and between the conduction band and the lead by hybridization $\Gamma$. Further details on the model and the calculation are given in \[sec:carrier\_density\]. From the general formalism of the Green’s function theory, the charge current induced in the model is proportional to the product of $\Gamma$ and the change of electron distribution function from the equilibrium distribution, as shown in Eq. \[eq:J\]. Therefore, we here focus on the carrier density in the conduction (valence) band, $\delta n_c$ ($\delta n_v$), assuming that the current is proportional to $\delta n_c$ and $\delta n_v$. In our case, $W_{vc}$ comes from the irradiation of light. A perturbation theory based on Fermi’s golden rule shows that this term is proportional to $l_A$. In the steady state ($\delta\dot{n}_{c,v}=0$), the calculation based on this simple model gives $$\delta n_c = \frac{W_{vc} \rho_v}{\Gamma+W_{vc}(\rho_c+\rho_v)}.$$ Here, we used charge neutrality condition, $$\rho_c\delta n_c = \rho_v\delta n_v.$$ Similarly, we get $$\delta n_v = \frac{W_{vc} \rho_c}{\Gamma+W_{vc}(\rho_c+\rho_v)}.$$ Therefore, for $W_{vc}\ll \Gamma$, the carrier densities reads $\delta n_{c,v}\propto W_{vc}\propto l_A$; the number of carriers and the photocurrent is proportional to $l_A$. On the other hand, if $W_{vc}\gg\Gamma$, $\delta n_c$ and $\delta n_v$ become (almost) independent of $W_{vc}$, i.e., it is independent of the width of excitation. ![$l_A$ dependence of (a) the concentration of the excited electrons in the conduction band and (b) photocurrent in a noncentrosymmetric chain ($t=B=d=1$). The light is shined onto $l_A=39$ sites at the center of the chain ($x_A=0.5$. Different lines corresponds to the different length of the chain $N$.[]{data-label="fig:carrier_dens"}](fig07.pdf){width="\linewidth"} From the numerical calculations, it is difficult to directly give an estimate of $W_{vc}$. However, we can investigate to which limit the system is close to by calculating the density of excited carriers. If the system is in $W_{vc}\gg\Gamma$ limit, there exists a large density of carriers. On the other hand, the carrier density is about $\delta n_c\sim W_{vc} \rho_v/\Gamma$ for $W_{vc}\ll \Gamma$; the carrier density is much smaller than the density of state. In Fig. \[fig:carrier\_dens\], we calculated the carrier density in the conduction band with respect to the number of states, $\delta n_c/\rho_c$. The number of carriers is calculated by $$\begin{aligned} \delta n_c = \frac1{2\pi N} \Im\left\{{\rm Tr}\int_0^\infty d\omega\, G^<_{00}(\omega)\right\},\end{aligned}$$ and the number of the number of states by $$\begin{aligned} \rho_c = \frac1{2\pi N} \Im\left\{{\rm Tr}\int_0^\infty d\omega\, \left[ G^A_{00}(\omega)- G^R_{00}(\omega)\right]\right\}.\end{aligned}$$ Here, $G^<_{00}(\omega)$ is the $N\times N$ matrix of lesser Green’s function and $G^{R,A}_{00}(\omega)$ is the retarded and the advanced Green’s function with Floquet index $\kappa=0$. The $l_A$ dependence of $\delta n_c$ is plotted in Fig. \[fig:carrier\_dens\](a). Here, the calculation were done for $l_A=39$ and $x_A=0.5$ with $A=0.2$. The three curves were for the different length of chain, $N=600,900,1200$. For the noncentrosymmetric chain, the result shows a rapid increase of the carrier density below $l_A=20$ and a saturation to $\delta n_c/\rho_c\sim0.25$ above that. In coincidence with the saturation of the carrier density, the photocurrent also shows saturation above $l_A=20$. These features are consistent with the above mechanism for the $l_A$-independent behavior of the shift current. Current-Voltage Characteristic ------------------------------ In this section, we study the current-voltage characteristic of the chains with shift currents by applying the external bias to the leads on the both sides. In our calculation, however, we find no current dependence on the external voltage within the numerical precision, if the bias is smaller than the charge gap (not shown). This is a natural consequence as the chain have no density of states at the Fermi level. As the carriers have higher energy than the bias, the effuluence of carriers to the leads is not affected by the change of electron distribution in the leads, which only takes place close to the Fermi level. Discussion {#sec:discussion} ========== Our results presented in Fig. \[fig:bdchain\] show that a large shift current can be induced in a ferroelectric chain by local excitation. In the clean limit of a fermion chain, our result implies that the photocurrent in bulk photovoltaic materials can be robustly transported through the material for a long distance without decaying. This sheds light on to the potential realization of a highly efficient photovoltaic device using ferroelectric insulators [@Choi2009; @Young2012_1; @Young2012_2; @Grinberg2013; @Nie2015; @Shi2015; @deQuilettes2015]. An important point related to the robustness of the current is the effect of impurities. In experiments, the impurities often suppress electric current. In one-dimensional systems, the fermions are susceptible to Anderson localization. If the localization occurs, the current is likely to be suppressed at the scale of the localization length, i.e., no d.c. current is transported for the chains much longer than the localization length. Therefore, we expect the shift current to decay by the length scale of localization when a light is shined onto only a part of the chain. In addition to the localization, another potential cause of the dissipation is the inelastic scattering by phonons. The inelastic scattering may contribute to the decay of shift current, limiting the distance the current is transmitted. Study on the effect of electron-phonon scattering to the shift current is left for future works. Conclusion {#sec:summary} ========== In this paper, we theoretically studied the shift currents from local excitation by light. Considering a noncentrosymmetric insulating fermion chain, we studied the dependence of shift current on the position of excitation. By applying a non-equilibrium Green’s function theory in \[sec:model\], we show that the local excitation in the noncentrosymmetric chain induces photocurrent that is almost independent of the position of excitation and is symmetric to the location of light ($x_A$), as shown in Fig. \[fig:bdchain\]. This is in contrast to the conventional mechanism, where the photocurrent is antisymmetric and maximized at the edge of the device, where the carriers can easily escape to the coupled leads. This difference in the position dependence of the photocurrent provides an experimental method to distinguish the unconventional photocurrent from that of the conventional ones, such as in the semiconductor junctions. Furthermore, this feature is advantageous for highly efficient optoelectronic devices. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank M. Kawasaki, M. Nakamura, M. Sotome, Y. Tokura, and M. Ueda for fruitful discussions. This work was supported by JSPS Grant-in-Aid for Scientific Research (No. 16H06717, No. 24224009, No. 26103006, and No. 26287088), from MEXT, Japan, and ImPACT Program of Council for Science, Technology and Innovation (Cabinet office, Government of Japan). Nonequilibrium Green’s Methods {#sec:method} ============================== To study the transport phenomena of the model in Eq. \[eq:H\] coupled to the leads, we use a nonequilibrium Green’s function method for time-independent Hamiltonians [@Meir1992; @Haug2008]. In this section, we summarize the basic formulation we use to study the photocurrent. The details of the numerical implementation are given in \[sec:computation\]. Keldysh-Floquet Theory for Systems Coupled to Leads --------------------------------------------------- For our calculation, we use an extension of non-equilibrium Green’s function approach used widely to study non-equilibrium phenomena [@Meir1992; @Haug2008]; the theory is extended to Floquet space in order to study periodically driven systems [@Hanggi2005; @Haug2008]. In this section, we elaborate the general formalism and the derivation of Dyson’s equation which we use to calculate the Green’s function. We also elaborate the application to the model we study in the main text. In the Floquet theory, the Green’s functions have additional indices that corresponds to the Floquet states [@Floquet1883; @Eliasson1992], $$\begin{aligned} G_{\kappa\kappa^\prime}(\omega) &=& G_{\kappa-\kappa^\prime}(\omega+\frac{\kappa+\kappa^\prime}2\Omega),\label{eq:keldysh-wigner}\end{aligned}$$ where $$\begin{aligned} \hspace{-5mm}G_n(\omega)= \frac1T\int_0^T dt_a \int dt_r G(t_a+\frac{t_r}2,t_a-\frac{t_r}2)\,e^{{\rm i} (n\Omega t_a+ \omega t_r)}\end{aligned}$$ is the Wigner representation of the Green’s function, $$\begin{aligned} G(t,t^\prime)=G(t+T,t^\prime+T),\end{aligned}$$ with $T=2\pi/\Omega$ being the periodicity of the Hamiltonian in the time direction. Using this notation, the general form of Dyson’s equation for the free particle systems is generally given by $$\begin{aligned} (\omega+\kappa\Omega)\,&&G_{\kappa,\kappa'}(\omega) - \sum_{\kappa^{\prime\prime}}{\cal H}_{\kappa,\kappa''}^0\, G_{\kappa'',\kappa'}(\omega) = 1.\label{eq:eom_keldysh}\end{aligned}$$ Here, $$\begin{aligned} \mathcal H_{\kappa,\kappa'}^0=\frac1T\int_0^T dt H^0(t)\,e^{\mathrm i (\kappa-\kappa') \Omega t}\end{aligned}$$ is the Fourier transform of the single particle Hamiltonian. As the Hamiltonian, we here consider a free fermion system coupled to free fermion leads, $$\begin{aligned} H(t)&=& H^0(\{c_i\},\{c_i^\dagger\},t) + \sum_{k,\alpha} \epsilon_{k\alpha} d_{k\alpha}^\dagger d_{k\alpha} + \sum_{k,\alpha,i} V_{k\alpha,i} d_{k\alpha}^\dagger c_i + {\rm h.c.}.\end{aligned}$$ Here, $c_i$ ($c_i^\dagger$) is the annihilation (creation) operator for the $i$th state in the system and $d_{k\alpha}$ ($d_{k\alpha}^\dagger$) is the annihilation (creation) operator for $k$th eigenstate in the $\alpha$th lead. $\epsilon_{k\alpha}$ is the eigenenergy for the $k$ th state in the $\alpha$th lead, and $V_{k\alpha,i}$ is the hybridization between the $k\alpha$th state and $i$th state in the system. For this type of models, the leads can be treated as a self-energy. The explicit form of self-energy can be deduced as follows. From Eq. \[eq:eom\_keldysh\], we get the relation $$\begin{aligned} \left[G_{\kappa,\kappa'}\right]_{k\alpha,i}(\omega) = \sum_j \left[g_{\kappa,\kappa}\right]_{k\alpha,k\alpha}(\omega) V_{k\alpha,j} \left[G_{\kappa,\kappa'}\right]_{j,i}(\omega), \label{eq:dyson2}\end{aligned}$$ where $g_{\kappa,\kappa'}(\omega)$ is the one-particle Green’s function for $V_{k\alpha,i}=0$. By substituting Eq. \[eq:dyson2\] to Eq. \[eq:eom\_keldysh\], the Dyson equation for $\left[G_{\kappa,\kappa'}\right]_{i,j}(\omega)$ reads $$\begin{aligned} (\omega+&&\kappa\Omega)\,G_{\kappa,\kappa'}(\omega) - \sum_{\kappa^{\prime\prime}}{\cal H}_{\kappa,\kappa''}^0\, G_{\kappa'',\kappa'}(\omega) - \sum_{\kappa^{\prime\prime}} \Sigma'_{\kappa,\kappa''}(\omega)\,G_{\kappa'',\kappa'}(\omega) = 1,\nonumber\\\label{eq:dyson_ren}\end{aligned}$$ with $$\begin{aligned} \left[\Sigma'_{\kappa,\kappa}\right]_{i,j}(\omega) = \sum_{k\alpha}V_{i,k\alpha} \left[g_{\kappa,\kappa}\right]_{k\alpha,k\alpha}(\omega) V_{k\alpha,j}. \label{eq:S_imp}\end{aligned}$$ Not that, in Eq. (\[eq:dyson\_ren\]), the Hilbert space is reduced to the degrees of freedoms in the chain, not the direct sum of chain and the leads. In this paper, we particularly consider the case in which the leads are coupled only to the ends of the chain. From Eq. (\[eq:S\_imp\]), the self-energy that comes from the leads, $\Sigma'_{\kappa,\kappa}(\omega)$, reads $$\begin{aligned} \left[ \Sigma'_{\kappa,\kappa}(\omega) \right]_{i,j}&=& \delta_{i,0}\delta_{j,0}i\Gamma_R(\omega)\left( \begin{array}{cc} -\frac12 & 2f_L(\omega+\kappa\Omega)-1 \\ 0 & \frac12 \end{array} \right)\nonumber\\ &&\quad+\delta_{i,N-1}\delta_{j,N-1}i\Gamma_L(\omega)\left( \begin{array}{cc} -\frac12 & 2f_R(\omega+\kappa\Omega)-1 \\ 0 & \frac12 \end{array} \right), \label{eq:Sigma}\end{aligned}$$ where $\alpha=L$ and $\alpha=R$ corresponds to left and right leads, respectively, and $$\begin{aligned} \Gamma_\alpha = 2\pi \sum_k \|V_{k\alpha,i_\alpha}\|^2 \delta(\omega-\epsilon_{k\alpha}).\end{aligned}$$ Here, $f_R(\omega)$ \[$f_L(\omega)$\] is the distribution function of electrons in the right (left) lead, and $i_\alpha$ is the site the lead is coupled to: $n_L=0$ and $n_R=N-1$, where $N$ is the number of sites in the fermion chain. In our calculation, we employed the wide-band approximation, namely, assuming $\Gamma_\alpha(\omega)$ to be a constant $$\Gamma_\alpha(\omega)=\Gamma_\alpha=\Gamma.$$ Calculation of the Current -------------------------- By using a similar procedure we used to derive our Dyson’s equation, the charge current that flows into/out of the system can also be calculated from the Green’s function of the chain [@Meir1992]. In our setup, the current between the $\alpha$th lead and the system is given by $$\begin{aligned} J_\alpha&=& i q_e \sum_{k,i} \int \frac{d\omega}{2\pi} \left\{ V_{k\alpha,i} \left[G^<_{\kappa,\kappa}\right]_{i,k\alpha}(\omega) - V_{i,k\alpha}\left[G^<_{\kappa,\kappa}\right]_{k\alpha,i}(\omega)\right\}, \label{eq:Jorig}\end{aligned}$$ where $V_{i,k\alpha}=(V_{k\alpha,i})^\ast$. Here, $\kappa\in\mathbb Z$ is an arbitrary number. Following a similar procedure in Ref. [@Meir1992] and by substituting $\left[G^<_{\kappa,\kappa}\right]_{i,k\alpha}(\omega)$ by Eq. \[eq:dyson2\], Eq. (\[eq:Jorig\]) reads $$\begin{aligned} J_\alpha= i q_e \int \frac{d\omega}{2\pi}\; {\rm tr}&&\left[ f_\alpha(\omega) \hat\Gamma_\alpha(\omega)\left\{\hat{G}^r_{\kappa\kappa}(\omega)-\hat{G}^a_{\kappa\kappa}(\omega)\right\}+\hat\Gamma_\alpha(\omega)\hat{G}^<_{\kappa\kappa}(\omega) \right].\nonumber\\ \label{eq:current}\end{aligned}$$ Here, $\hat\Gamma_\alpha(\omega)$ is a square matrix with indices that corresponds to each state of the system, $$\begin{aligned} \left[\hat\Gamma_\alpha\right]_{i,j}(\omega) = 2\pi \sum_{k\alpha} V_{j,k\alpha} V_{k\alpha,i} \delta(\omega-\epsilon_{k\alpha}),\end{aligned}$$ and $\hat{G}^<_{\kappa\kappa'}(\omega)$ \[$\hat{G}^r_{\kappa\kappa'}(\omega)$, $\hat{G}^a_{\kappa\kappa'}(\omega)$\] is the matrix for lesser (retarded, advanced) Green’s functions with $\kappa$ and $\kappa'$ indices for the Floquet states. In the case of the model in Eq. (\[eq:H\]), $\hat\Gamma_\alpha(\omega)$ reads $$\begin{aligned} \left[\hat\Gamma_\alpha\right]_{i,j}(\omega) = \delta_{i,i_\alpha}\delta_{j,i_\alpha}\Gamma_\alpha(\omega)\end{aligned}$$ for $\alpha=L, R$. Hence, Eq. (\[eq:current\]) become $$\begin{aligned} J_\alpha= i q_e \int \frac{d\omega}{2\pi}\; && f_\alpha(\omega) \Gamma_\alpha(\omega) \left\{\left[G^r_{\kappa\kappa}\right]_{i_\alpha,i_\alpha}(\omega)-\left[G^a_{\kappa\kappa}\right]_{i_\alpha,i_\alpha}(\omega)\right\}\nonumber\\ &&\qquad+i q_e \int \frac{d\omega}{2\pi}\Gamma_\alpha(\omega)\left[G^<_{\kappa\kappa}\right]_{i_\alpha,i_\alpha}(\omega). \label{eq:current2}\end{aligned}$$ Using these results, the current that flows through the system is given by [@Meir1992] $$\begin{aligned} J&=&J_R-J_L.\end{aligned}$$ In the wide band approximation, Eq. (\[eq:current2\]) reads $$\begin{aligned} J_a&=& -i\Gamma_a\int\frac{d\omega}{2\pi}\left\{G_{\kappa\kappa}^<(\omega)+f_a(\omega)\left[G^R_{\kappa\kappa}(\omega)-G^A_{\kappa\kappa}(\omega)\right]\right\}_{i_a,i_a},\\ &=& -i\Gamma\int\frac{d\omega}{2\pi}\left\{G_{\kappa\kappa}^<(\omega)+f_a(\omega)\left[G^R_{\kappa\kappa}(\omega)-G^A_{\kappa\kappa}(\omega)\right]\right\}_{i_a,i_a}. \label{eq:J}\end{aligned}$$ In the second equation, we set $\Gamma_\alpha=\Gamma$. Numerical Calculations {#sec:computation} ====================== For numerical calculation using the method explained in \[sec:method\], we introduce some approximation to implement the theory onto the computer. In addition, to accelerate the calculation, we implemented a matrix decomposition method. In this section, we illustrate the details on the numerical implementation of the nonequilibrium Green’s function formalism. Truncation of the Dyson Equation -------------------------------- The Green’s function for our model in Eq. (\[eq:H\]) can be calculated from Eq. (\[eq:dyson\_ren\]), by inverting the matrix $$(\omega+\kappa\Omega)\hat1-{\cal H}^0-\Sigma',$$ where ${\cal H}^0$ is a matrix of ${\cal H}^0_{\kappa\kappa'}$ given by $$\begin{aligned} {\cal H}^0=\left( \begin{array}{cccc} \ddots & \vdots & \vdots & \\ \cdots & {\cal H}^0_{\kappa\;\kappa} & {\cal H}^0_{\kappa\;\kappa+1} & \cdots \\ \cdots & {\cal H}^0_{\kappa+1\;\kappa} & {\cal H}^0_{\kappa+1\;\kappa+1} & \cdots \\ & \vdots & \vdots & \ddots \end{array} \right),\end{aligned}$$ and $\Sigma'$ is defined in the same manner with $\Sigma'_{\kappa\kappa'}$; $\hat1$ is the unit matrix. This matrix, however, is an infinite dimension matrix due to the Floquet indices. In our calculation, as we are interested in the leading order effect by the external light, we truncate the Floquet indices and leave the states that are close to the Fermi level, i.e., $\kappa=0$ and $-1$. Physically, this corresponds to considering only the leading order of scattering between the electrons in the valence band and that in the conduction band induced by light. One point to be noted is that, due to this approximation, the periodicity of the Green’s function with respect to $\omega$ is violated. To reduce the error that arises from the above approximation, for the calculation of shift current using Eq. (\[eq:J\]), we used $G_{00}(\omega)$ for $\omega\ge0$ and $G_{-1-1}(\omega)$ for $\omega<0$. Eigendecomposition of $G^{-1}(\omega)$ -------------------------------------- For the calculation of the Green’s functions in long chains, an efficient algorithm based on the eigendecomposition of matrices is used. Here, we briefly review the algorithm we used in this calculation. Suppose we have a $\omega$-independent Hamiltonian $H$ and self-energy $\Sigma$. Then the inverse of the Green’s function, $\omega\hat1-H-\Sigma$, is decomposed as $$\begin{aligned} \omega\hat1-H-\Sigma = R(\omega-\Lambda)L^T,\end{aligned}$$ where $R=(\vec\epsilon_1,\vec\epsilon_2,\cdots)$ and $L=(\vec{e}_1,\vec{e}_2,\cdots)$ are the $4N\times4N$ matrices of eigenvectors ($\epsilon_i$ and $e_i$ are right and left eigenvectors), and $\Lambda$ is a $4N\times4N$ diagonal matrix with eigenvalues as its diagonal components. Using this decomposition, the Green’s function is given by $$\begin{aligned} G(\omega)= R(\omega-\Lambda)^{-1}L^T.\end{aligned}$$ Therefore, if the Hamiltonian and the self-energies are $\omega$ independent, then the arbitrary element of the Green’s function can be calculated efficiently. In our formalism, in general, the self-energies are $\omega$ dependent. However, within the wide-band approximation and at zero temperature, the $\omega$ dependence of the self-energy appears as a step function, as shown in Eq. (\[eq:Sigma\]). Therefore, for a time-independent model, the above method is applicable by splitting the $\omega$ into two sectors, above and below the chemical potential. In the Floquet formalism, the number of sectors is given by $N_s+1$, where $N_s$ is the number of subbands. For the numerical integration in the calculation of the shift current using Eq. (\[eq:J\]), we find that the cutoff of $\Lambda_\omega^\pm=\pm4$ and discreteness of $\delta\omega=10^{-3}$ gives sufficient convergence. ![The finite size effect of photocurrent calculated with $l_A=39$ on a model with $B=1$ and $d=1$. The transverse axis is the location of incident light ($x_A=i_A/N$).[]{data-label="fig:size_eff"}](fig09.pdf){width="0.65\linewidth"} In Fig. \[fig:size\_eff\], we show an example of the calculation using the approximation introduced above. The figure shows the results of the light-position dependence of shift current for $l_A=39$ with different length of chain (the results for larger sizes are given in Fig. \[fig:chain\_len\]). The result for $N=100$ shows deviation from the results for larger $N$ with an arch-like structure. The results for $N\ge200$ show almost the same magnitude of the photocurrent and it is nearly independent of $x_A$. The oscillation of the result around $x_A=0.5$, on the other hand, remains robust for larger sizes. Presumably, this is related to the competition between the typical distance between the states in a band and the broadening of the pole of Green’s function due to the leads. In general, the typical distance between the states is proportional to $1/N$; the numerical calculation converges when the typical distance become smaller than that of the broadening of the poles. In our setup, on the other hand, we only couple the edge sites to a lead. Hence, the poles of the Green’s function become sharper with increasing size deep inside the bulk. Due to the competition between the two energy scales, the finite size effect around $x_A$ persists up to larger sizes. In the main text, we mostly use $N=1200$ for calculation. Quasiclassical Analysis on the Carrier Density {#sec:carrier_density} ============================================== This section elaborates the quasiclassical method discussed in Sec. \[sec:results:lightwidth\]. In the Green’s function theory developed in \[sec:method\], the current given in Eq. (\[eq:J\]) is proportional to the change of the charge density from its “equilibrium” distribution, $f_\alpha(\omega)\rho_{i_\alpha}(\omega)$, where $\rho_{i_\alpha}(\omega)$ is the local density of states at site $i_\alpha$. Therefore, for the analysis on the $l_A$-independence, we focus on the carrier density in the conduction band. We consider a simple model that consists of a valence band, a conduction band, and the lead(s) as shown in Fig. \[fig:quasi\_classic\]. The carriers are transferred between the valence and conduction bands by $W_{vc}$ and between the conduction band and the lead by $\Gamma$. The carrier concentration of the conduction band is given by solving a differential equation: $$\begin{aligned} \rho_c \delta \dot{n}_c&=&-W_{vc}\rho_v\delta n_v\rho_c\delta n_c + W_{vc}\rho_v(1-\delta n_v)\rho_c(1-\delta n_c) - \Gamma \rho_c \delta n_c. \label{eq:phenom1}\end{aligned}$$ Here, $\rho_{c}$ ($\rho_{v}$) is the density of states for conduction (valence) band, $\delta n_c$ ($\delta n_v$) is the concentration of the electrons (holes) in the conduction (valence) band. The first and the second term represents the transition of electrons between the conduction and valence bands (we discuss the detail in the following paragraphs), and the last term represents the effluence of carriers to the lead(s). The form of the last term can be guessed intuitively, or formally derived from the Keldysh theory, which the current that flows out of the chain is given by the product of hybridization and the modulation of electron density from the equilibrium distribution. To give a qualitative understanding, we estimate $W_{vc}$ using a perturbative approach. From the Fermi’s golden rule, for a perturbative Hamiltonian $H'(t)=H'(0)\sin(\Omega t)$, the transition amplitude from initial state $i$ to the final state $j$ reads $$\begin{aligned} c_f(t)&=& \frac{i}{2\hbar} \langle f \left\| H^\prime(0)\right\| i\rangle \left[ \frac{\exp\left\{it(\omega_{fi}+\Omega)\right\}-1}{\omega_{fi}+\Omega} - \frac{\exp\left\{it(\omega_{fi}-\Omega)\right\}-1}{\omega_{fi}-\Omega} \right],\nonumber\\\end{aligned}$$ where $\omega_{fi}\equiv\varepsilon_f-\varepsilon_i$ with $\varepsilon_i$ being the eigenenergy of $i$ th state. Here, we assumed the initial state $c_i(t)\sim 1$ and $c_f(t)\sim 0$ ($f\ne i$). In the $t\to \infty$ limit, the inside of the braces in this equation has two peaks in its absolute value, at $\omega=\pm \Omega$. We take the perturbative Hamiltonian to be the irradiation by light for electrons between sites $l_L$ and $l_R$ ($l_A=l_R-l_L$): $$\begin{aligned} H_A(t) = -iA\sin(\Omega t) \sum_{n=l_L}^{l_R-1} \left(t-(-1)^n\frac{B}2\right)c_{n+1}^\dagger c_n - \left(t-(-1)^n\frac{B}2\right)c_n^\dagger c_{n+1}.\nonumber\\ \label{eq:pert_fermi}\end{aligned}$$ By the Fourier transformation of $c_n$’s, $H_A$ reads $$\begin{aligned} H_A(t) &=& -iA\sin(\Omega t) \sum_{k,k'} I_{\frac{l_R-l_L-1}2}(k-k')\nonumber\\ &&\times\left[ \left\{\left(t-\frac{B}2\right)e^{ik+i(k-k')l_L}-\left(t+\frac{B}2\right)e^{-ik'+i(k-k')(l_L+1)}\right\}c_{Bk}^\dagger c_{Ak'} \right.\nonumber\\ &&\left.- \left\{\left(t-\frac{B}2\right)e^{-ik'+i(k-k')l_L}-\left(t+\frac{B}2\right)e^{ik+i(k-k')(l_L+1)}\right\}c_{Ak}^\dagger c_{Bk'} \right],\nonumber\\ \label{eq:pert_fermi2}\end{aligned}$$ with $$\begin{aligned} I_n(dk) = \left\{ \begin{array}{ll} \frac{n+1}N & \qquad(dk=0)\\ \frac{1-e^{i2(n+1)dk}}{N(1-e^{i2dk})} & \qquad(dk\ne0) \end{array} \right..\end{aligned}$$ Here, $c_{Ak}$ ($c_{Bk}$) are the annihilation operators for electrons on the even (odd) sites with wave number $k$. For simplicity, we assumed $l_L$ to be an even number site and $l_R$ to be odd. For a given $l_L$ and $l_R$, $I_n(dk)$ is given by a periodic function with periodicity $\pi$ (which corresponds to the size of the Brillouin zone), and with a peak of hight $(n+1)/N$ and the full width at half maximum $\sim \frac\pi{(n+1)}$. As the eigenstates for the bulk limit with wave number $k$ is given by a superposition of $c_{Bk}$ and $c_{Ak}$, the change in the wave number while exciting the electrons from the valence state to the conduction state is limited by the width of $I_n(dk)$. In the $t\to \infty$ limit, the transition from the valence band electron with momentum $k$ to the conduction band electron with $k'$ occurs only between the $k$ and $k'$ with energies precisely separated by $\Omega$. The transition amplitude for a state in the valence band with $k$ to that in a conduction band with $k'$ is given by $\sim A^2\|I_n(k-k')\|^2\sim (l_A+1)^2/4N^2$ ($l_A=l_R-l_L$). Due to the constraint from energy conservation, the transition occurs only between the state close to the crossing between $\varepsilon_v+\Omega$ and $\varepsilon_c$ \[see Fig. \[fig:model\](b)\]; the width of the wave number the transition takes place is given by $\sim \pi/n+1=2\pi/l_A+1$. Finally, the density of electron states per a width of $k$ is given by $N/2\pi$. From these facts, $W_{vc}$ is estimated to be $$\begin{aligned} W_{vc}\sim A^2\left(\frac{l_A+1}{2N}\right)^2\frac\pi{l_A+1}\frac{N}{2\pi} = \frac{l_A+1}{8N}A^2 \sim\frac{l_AA^2}{N},\end{aligned}$$ linearly proportional to $l_A$ and $A^2$. In the steady state ($\delta\dot{n}_c=0$), the solution of Eq. (\[eq:phenom1\]) is given by $$\delta n_c = \frac{W_{vc} \rho_v}{\Gamma+W_{vc}(\rho_c+\rho_v)}.$$ Here, we used charge neutrality condition, $$\rho_c\delta n_c = \rho_v\delta n_v.$$ Similarly, we get $$\delta n_v = \frac{W_{vc} \rho_c}{\Gamma+W_{vc}(\rho_c+\rho_v)}.$$ [99]{} Nelson J 2003 [The Physics of Solar Cells]{} (Imperical College Press, London, 1ed). Würfel P, Würfel U 2009 [Physics of Solar Cells: From Basic Principles to Advanced Concepts]{} (Wiley-VCH, Weinheim, 2ed). Fridkin V M 2001 Crystallogr. Rep. (Transl. Kristallografiya) [46]{} 654. Choi T, Lee S, Choi Y, Kiryukhin V, and Cheong S-W 2009 Science [324]{} 63. Young S M and Rappe A M 2012 Phys. Rev. Lett. [109]{} 116601. Young S M, Zheng F, and Rappe A M 2012 Phys. Rev. Lett. [109]{} 236601. Grinberg I, Vincent West D, Torres M, Gou G, Stein D M, Wu L, Chen G, Gallo E M, Akbashev A R, Davies P K, Spanier J E, and Rappe A M 2013 Nature [503]{} 509. 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Xie, L. M. Mei, J. Berakdar, Y. J. Yan, Org. Electron. [8]{}, 487 (2007). J. H. Wei, S. J. Xie, L. M. Mei, Y. J. Yan, Appl. Phys. Lett. [91]{}, 022115 (2007). Rice M J and Mele E J 1982 Phys. Rev. Lett. [49]{} 1455. Floquet G 1983 Ann. de l’Ecole Norm. Sup. [12]{} 47. Kohler S, Lehmann J, and Hanggi P 2005 Phys. Rep. [406]{} 379. Meir Y and Wingreen N 1992 Phys. Rev. Lett. [68]{} 2512. Rammer J 2007 [Quantum Field Theory of Non-equilibrium States]{} (Cambridge University Press, Cambridge, 1ed). Thouless D J 1983 Phys. Rev. B [27]{} 6083. Niu Q and Thouless D J 1984 J. Phys. A Math. Gen. [17]{} 2453. Eliasson L H 1992 Commun. Math Phys. [146]{} 447. Oka T and Aoki H 2009 Phys. Rev. B [79]{} 081406. Lines M E and Glass A M 1977 [Principles and applications of ferroelectrics and related materials]{} (Clarendon Press, Oxford).
--- bibliography: - 'bibliography.bib' - 'Lukas\_Bibliography.bib' --- \#1 \#1[\#1\|]{} \#1\#2[\#1\|.\#2]{} \#1\#2[\|\#2\#1\|]{} \#1\#2\#3[\#1\|\#2\|\#3]{} Introduction ============ Control is a key component in turning science into technology[@Glaser2015], [@Acin2018]. Broadly and colloquially speaking, control looks at providing the user / experimenter with external parameters to steer a given dynamical system to her liking [^1] rather than simply observing its internal dynamics. Control is in this sense ubiquitous to modern technology. In this colloquial sense, quantum control is transferring that idea to quantum systems and thus contains both hard- and software of many kinds. The ubiquity of control has given rise to the field of control theory. This is a field of applied mathematics that looks at how to choose said external parameters in order to drive the dynamical system to ones liking. It has spawned ideas of open-loop control, i.e., the pre-determination of controls given the laws of nature (that were a key ingredient to, e.g., the Apollo program) as well as closed-loop-control, interleaving of observation and adjustment as we know it in our daily lives from thermostats. This type of optimal control theory takes the hardware setup as a given, however ideally, these are developed in tandem. The mathematical procedures of open-loop-control typically involve optimizing a cost function, hence the name optimal control. The application of optimal control is not an entirely new idea. Pioneering applications were primarily chemistry, such as the laser control of chemical reactions and magnetic resonance. By now, quantum optimal control is also applied to a large spectrum of modern quantum technologies (Quantum 2.0) [@Acin2018]. This implies a certain tradition of fragmentation - quantum optimal control researchers tend to be in departments of mathematics, chemistry, computer science, and physics and follow their specific idiosyncrasies [@Glaser2015]. Modern efforts have gone very far in overcoming this fragmentation which is fruitful in learning from each other and respecting the different goals – quantum control of complex reactions does for example deal with large Hilbert spaces whereas control in quantum computing aims at sufficiently low errors in order to meet error correction thresholds. In this series of lectures, we would like to introduce the audience to quantum [optimal control]{}. The first lecture will cover basic ideas and principles of optimal control with the goal of demystifying its jargon. The second lecture will describe computational tools (for computations both on paper and in a computer) for its implementation as well as their conceptual background. The third chapter will go through a series of popular examples from different applications of quantum technology. These are lectures notes. Other than a textbook, it makes a significant difference to attend the lectures it goes with rather than use it to self-study. Other than a review, it is not complete but rather serves to introduce clarify the concepts of the field. This also means that the choice of references is certainly not complete, rather, it is the subjective choice of what the authors find most suitable and got inspired by. Elementary optimal control ========================== We start with classical examples of control, which lay a lot of foundations for quantum systems. Optimal control of a classical harmonic oscillator \[CH:SHO\] ------------------------------------------------------------- In order to understand the basic concept and structure of quantum optimal control, let us start with a simple classical example: control of the harmonic oscillator. The equation of motion of a harmonic oscillator driven by force $F(t)=mf(t)$ where $m$ is the mass and eigenfrequency $\Omega$ is given by $$\ddot{x}+\Omega^{2}x(t)=f(t).$$ Its general solution is parameterized through the Green’s function $$G(\tau)=\frac{\theta(\tau)}{\Omega}\sin\left(\Omega\tau\right)$$ (where $\theta$ is the Heaviside function) as $$x(t)=x(0)\cos\Omega t+\frac{\dot{x}(0)}{\Omega}\sin\Omega t+\int_{0}^{t}dt^{\prime}\ \frac{\sin\Omega\left(t-t^{\prime}\right)}{\Omega}f\left(t^{\prime}\right)$$ Readers not familiar with Green’s functions can easily verify that this expression does indeed solve the equation of motion of the driven oscillator. From this we get the velocity $$\dot{x}(t)=\dot{x}(0)\cos\Omega t-\Omega x(0)\sin\Omega t+\int_{0}^{t}dt^{\prime}\cos\Omega(t-t^{\prime})f\left(t^{\prime}\right)$$ Thus, imposing target values $x(T)$ and $\dot{x}(T)$ we find the conditions $$\begin{aligned} x(T)-x(0)\cos\Omega T - \frac{\dot{x}(0)}{\Omega}\sin\Omega t& =\int_{0}^{T}dt^{\prime}\ \frac{\sin\Omega\left(T-t^{\prime}\right)}{\Omega}f\left(t^{\prime}\right)\\ \dot{x}(T)-\dot{x}\left(0\right)\cos\Omega T +\Omega x(0)\sin\Omega t& =\int_{0}^{T}dt^{\prime}\,\cos\Omega(T-t^{\prime})f(t^{\prime}).\end{aligned}$$ These equations allow a few observations that have analogies all over quantum control: Firstly, the control $f(t)$ is needed to push the system away from its natural dynamics (the terms on the left) – it is redirecting the natural drift of the system. Secondly, there are two constraints for a function given through an integral – so we can expect many solutions. As an example, let’s look at the case that we move a particle by a fixed distance $x(0)=0$ and $x(T)=X$ from rest to rest $\dot{x}(T)=\dot{x}(0)=0$. We thus need to satisfy $$\int_{0}^{T}dt^{~\prime}\sin\left[\Omega\left(t-t^{\prime}\right)\right]f\left(t^{\prime}\right)=\Omega X\quad\int_{0}^{T}dt^{\prime}\cos\left[\Omega(t-t^{\prime})\right]f\left(t^{\prime}\right)=0$$ and we can easily show that this fixes low Fourier components of $f\left(t^{\prime}\right)$ but leaves higher ones open. The situation changes, if we impose, e.g., an energetic constraint to the control. This typically leads to constraints of the form $$\int_0^T dt\ f^2(t)=\int_{-\infty}^\infty d\omega f(\omega)f^\ast(\omega)\le A$$ where A is the imposed maximum and we have used symmetry properties of the Fourier transform of real-valued functions. Thus, the sum of Fourier components needs to be bounded and if the constraint is too close, there may not even be any solution. This is an example showing that constraints clearly influence the number of accessible solutions and their potential performance, which is commonly seen in practice. Optimal control for a classical system -------------------------------------- The previous section hinged on having a closed-form Green’s function solution of the equation of motion, which is not always available. This follows chapter 2.3 of Bryson and Ho [@bryson1975applied]. Suppose we have a dynamical system that can be controlled by a control parameter $u$ that enters a dynamic equation for the state variable $x$ in the form $$\dot{x}=f\left[x(t),u(t),t\right]\quad0\le t\le T\label{eq:Classic_EOM}$$ with a given $x(0)$. Both $x$ and $u$ can be single variables or vectors of variables. We wish to optimize a cost function at the end of the process $J\left[x(T),T\right].$ We recall classical Lagrangian mechanics and introduce a [Lagrange multiplier]{} function $\lambda$ and can thus state based on the constrained calculus of variations that we need to find a stationary point of $$\bar{J}=J\left[x(T),T\right]+\int_{0}^{T}dt\ \lambda^{T}(t)\left(f\left[x(t),u(t),t\right]-\dot{x}\right)$$ where we have allowed for the complex of coupled equations and thus vector-valued Lagrange multipliers. The introduction of the Lagrange multiplier allows for the optimization of $J$, while satisfying the equation of motion (\[eq:Classic\_EOM\]) at specified times. As such, this means $\lambda$ has to be time-dependent as well. We introduce the associated Hamilton’s function (which has a similar mathematical origin in the calculus of variations as the Hamiltonians of mechanics yet a very different physical motivation) $$H\left[x(t),u(t),\lambda(t),t\right]=\lambda^{T}(t)f\left[x(t),u(t),t\right] \label{eq:hamiltonian}$$ and rewrite our constrained cost function by integrating the last term by parts $$\bar{J}=J\left[x(T),T\right]+\lambda^{T}(T)x(T)-\lambda^{T}(0)x(0)+\int_{0}^{T}dt\ \left\{ H\left[x(t),u(t),\lambda(t),t\right]+\dot{\lambda}^{T}x(t)\right\} .$$ Now let’s consider the variation in $\bar{J}$ based on variations in $u(t)$ recalling that the times as well as the initial state variable are given. We find $$\delta\bar{J}=\left.\left(\frac{\partial J}{\partial x}-\lambda^{T}\right)\delta x\right\|_{t=T}+\left.\lambda^{T}\delta x\right\|_{t=0}+\int_{0}^{T}dt\ \left[\left(\frac{\partial H}{\partial x}+\dot{\lambda}^{T}\right)\delta x+\frac{\partial H}{\partial u}\delta u\right].$$ Note that in general we choose the variation at the beginning to be $\delta x(0)=0$, since we know the exact initial state of the dynamics. Now the variations of $x$ and $u$ are not independent, they are linked by the equation of motion. Were we not to work with the Lagrange multiplier, we would need to tediously solve the equation of motion for different control functions and then work out how these variations are related. Fortunately, the Lagrange multiplier method allows us to circumvent that problem. Our goal is for $\delta\bar{J}$ to vanish to first order. Choosing a specific Lagrange multiplier to realize this, we finally arrive at $$\dot{\lambda}^{T}=-\frac{\partial H}{\partial x}=-\lambda^{T}\frac{\partial f}{\partial x}\quad\lambda^{T}(t_{f})=\frac{\partial J}{\partial x(t_f)}.\label{eq:classical_back-propagation}$$ These are the Euler-Lagrange equations pertaining to the system. That being satisfied, we are left with the total variation $$\delta\bar{J}=\underset{=0}{\underbrace{\lambda^{T}(0)\delta x(0)}}+\int_{0}^{T}dt\ \frac{\partial H}{\partial u}\delta u$$ For an extremum to be reached under any variation of the control, we need $$\frac{\partial H}{\partial u}=\lambda^{T}\frac{\partial f}{\partial u}=0\quad0\le t\le T.\label{eq:classical_gradient}$$ We have shown the ingredients to what can be formalized as the [Pontryiagin Maximum Principle]{} (PMP). More pragmatically, these equations give us a recipe on how to solve the thus formulated optimal control problem by a coupled gradient search: From a suitable initial guess for $u(t)$ 1. Solve the equation of motion eq. (\[eq:Classic\_EOM\]) to find $x(t)$ using the initial value $x(0)$ that is part of the control problem 2. Find the Lagrange multiplier by solving eq. (\[eq:classical\_back-propagation\]). Note that there is a definite value given at the end time $T$, i.e., we have a final value problem – that is solved like an initial value problem but propagating backwards in time. This back-propagation is typical when we consider this cost functional. 3. With these, compute the effective gradient in eq. (\[eq:classical\_gradient\]) and update the values of $u$ following the direction of the gradient. Adjust the step size as needed. Iterating these three steps will get us to a local solution, depending on the initial conditions, if the control landscape admits one. ### Example: Driven harmonic oscillator Let us get back to formulating these steps for the optimal control problem of the driven harmonic oscillator described above in section \[CH:SHO\]. We identify the control as the dimensionless force $u\equiv f$ and write the equation of motion as a coupled system $$\begin{aligned} \frac{dx}{dt} & =\dot{x}\\ \frac{d\dot{x}}{dt} & =-\Omega^{2}x+u\\ x(0) & =0\quad\dot{x}(0)=0\end{aligned}$$ In order to have a differentiable performance index that forces the particle to end at $a$ at time $T$ and in rest we can write $$J=\Omega^{2}\left(x-a\right)^{2}+\dot{x}^{2}. \label{eq:sho_index}$$ This leads us to Hamilton’s function following the prescription of \[eq:hamiltonian\] $$H=\lambda_{1}\dot{x}+\lambda_{2}\left(u-\Omega^{2}x\right).$$ So the Euler-Lagrange equations \[eq:classical\_back-propagation\] describing the Lagrange Multiplier $$\dot{\lambda}_{1}=\lambda_{2}\Omega^{2}\quad\dot{\lambda}_{2}=-\lambda_{1} \label{eq:sho_adjoint}$$ which remarkably describes a free harmonic oscillator. It is such interpretations that lead to the Lagrange multiplier to be called the [adjoint system. ]{}The final conditions from eq. (\[eq:classical\_back-propagation\]) are $$\lambda_{1}(T)=2\Omega^{2}(x(T)-a)\quad\lambda_{2}(T)=2\dot{x}(T) \label{eq:sho_final}$$ which are of course both zero if the final conditions are met (thus, for the optimal solution, the adjoint system vanishes at $T$). The gradient flow for the control is given by eq. (\[eq:classical\_gradient\]) $$\frac{\partial H}{\partial u}=\lambda_{2}.$$ Again, iterating these equations will give us a suitable control. We could guess as a first control that $u_{0}(t)=\Omega^{2}a$ (which is the force needed to keep the particle at rest at the final position, so at least a motivated guess) thus leading to the equation of motion $$\ddot{x}_{0}+\Omega^{2}(x_{0}-a)=0$$ with the solution$x_{0}(t)=a\left(1-\cos\Omega t\right)$ and thus $\dot{x}_{0}=a\Omega\sin\Omega T.$ This clearly does not solve the control problem, we have from eq. (\[eq:sho\_index\]) $J=\Omega^{2}a^{2}$. In fact, the final conditions eq. \[eq:sho\_final\] for the adjoint system are $\lambda_{1}(T)=-2\Omega^{2}a\cos\Omega T$ and $\lambda_{2}(T)=2a\Omega\sin\Omega T$ leading us, by solving eq. (\[eq:sho\_adjoint\]) $$\begin{aligned} \lambda_{1} & =-2\Omega^{2}a\cos\Omega T\cos\left[\Omega\left(t-T\right)\right]+2a\Omega^{2}\sin\Omega T\sin\left[\Omega\left(t-T\right)\right]\\ & =-2a\Omega^{2}\cos\Omega t\end{aligned}$$ and $\lambda_{2}=-2\sin\Omega t$ . This means that the gradient suggests introducing a resonant drive – as we have seen from the exact solution above. For further treatment of the classical Harmonic oscillator, see [@andresen2011optimal]. Gradient-based optimal quantum control with the GRAPE algorithm --------------------------------------------------------------- These principles can be transferred to the control of quantum systems in a straightforward way. This is easily illustrated with the [GRadient Ascent Pulse Engineering (GRAPE)]{} algorithm [@GRAPE]. ### State-to-state control We start with a simple state preparation problem. Suppose WLOG that our system is described by a Hamiltonian $$\hat{H}(t)=\hat{H}_{0}+\sum_{i=1}^{n}u_{i}(t)\hat{H}_{i}.$$ We call the time-independent part of the Hamiltonian $\hat{H}_{0}$ the [drift]{}, the fields $u_{i}$ are the controls and $\hat{H}_{i}$ are the control Hamiltonians. In atomic physics, say, $\hat{H}_{0}$ describes the energy level structure of the atom, $u_{i}$ are laser or microwave fields and $\hat{H}_{i}$ are dipole operators describing the different field modes including polarization. Our task is now to start at an initial state $\|\psi_{0}\rangle$ at time $t=0$ and find controls such that we reach state $\|\psi_{1}\rangle$ at time $t=T.$ As in quantum physics the global phase is meaningless, this corresponds to maximizing the overlap $J=\left\|\left\langle \psi_{1}\|\psi(T)\right\rangle \right\|^{2}$. The dynamics of our system is, of course, subject to the Schrödinger equation $$i\hbar\partial_{t}\|\psi(t)\rangle=\hat{H}(t)\|\psi(t)\rangle.$$ Mathematically we got ourselves a system of the exact same structure as the previous one. We give its derivation in the form of Ref. [@GRAPE]. Many practical generators for $u_{i}$ such as standard arbitrary wave form generators (AWGs or Arbs) used in superconducting qubits represent[^2] the pulse in a piecewise constant fashion , so it is natural [^3] to represent the $u_{i}(t)$ in that same way: We chop the total time into $N$ intervals of length $\delta t=T/N$ and write $$u_{i}(t)=u_{i}(j)\quad{\rm for}\quad(j-1)\delta t\le t<j\delta t.$$ This allows us to write down the formal solution of the Schrödinger equation as $$\hat{U}(T)=\hat{U}_{N}\hat{U}_{N-1}\cdots\hat{U}_{2}\hat{U}_{1}$$ with $$\hat{U}_{k}=\exp\left(-\frac{i}{\hbar}\delta_{t}\left(\hat{H}_{o}+\sum_{i}u_{i}(j)\hat{H}_{i}\right)\right)\label{eq:PWC-Uk}$$ which we can introduce into the performance index as $$J=\left\|\left\langle \psi_{1}\|\hat{U}(T)\psi_{0}\right\rangle \right\|^{2}=\left\|\left\langle \psi_{1}\|\hat{U}_{N}\cdots\hat{U}_{1}\psi_{0}\right\rangle \right\|^{2}.$$ We are at liberty to move some of the factors into the adjoint state, giving us $$J=\left\|\left\langle U_{m+1}^{\dagger}\cdots U_{N}^{\dagger}\psi_{1}\|\hat{U}_{m}\cdots\hat{U}_{1}\psi_{0}\right\rangle \right\|^{2}$$ or $J=\left\|\left\langle \lambda_{m}\|\rho_{m}\right\rangle \right\|^{2}$ with $\|\rho_{m}\rangle=\hat{U}_{m}\cdots\hat{U}_{1}\|\psi_{0}\rangle$ $\|\lambda_{m}\rangle=\hat{U}_{N}\cdots\hat{U}_{m+1}\|\psi_{1}\rangle$. Here, the partially propagated state $\|\rho_m(t)\rangle$ is overlapped with the partially back-propagated [adjoint state]{} $\|\lambda_m(t)\rangle$ – both states are overlapped at time $t_m$. We thus sweep the time at which we calculate the overlap based on the actual pulse that we apply. Now the final ingredient we need is the derivative of an exponential proven in Theorem 4.5 of [@Hall00] (see also [@Hall03]) $$\left.\frac{d}{dt}\right\|_{t=0}e^{X+tY}=e^{X}\left\{ Y-\frac{\left[X,Y\right]}{2!}+\frac{\left[X,\left[X,Y\right]\right]}{3!}-\dots\right\} \label{eq:Deriv-of-time-slice}$$ Both of these together allow us to determine all the gradients needed to compute an update at any time step as shown in the left column of figure \[fig:update\]. We can rewrite this as $$\left.\frac{d}{dt}\right\|_{t=0}e^{X+tY}=e^{X}\int_{0}^{1}d\tau\ e^{\tau X}Ye^{-\tau X}$$ by simple power counting. This allows us to analytically compute the derivative of the propagator across one time step by identifying $\hat{X}=\hat{H}(t)$ (the Hamiltonian including the current values of the control) and $\hat{Y}=\hat{H}_{i}$ , one of the control Hamiltonians. In order to simplify the right-hand side, we define $\hat{U}_{k}(j)=\hat{U}_{k}^{j}$ (taking the exponential here simply means to stretch time and study the integral on the right $$\begin{aligned} \int_{0}^{1}d\tau\ &\hat{U}_{k}(j)\hat{H}_{i}\hat{U}_{k}^{\dagger}(j)\\ & =\int d\tau\ \left(1-i\tau\delta_{t}\hat{H}-\tau^{2}\delta_{t}^{2}\hat{H}^{2}+\dots\right)\hat{H}_{i}\left(1+i\tau\delta_{t}\hat{H}-\tau^{2}\delta_{t}^{2}\hat{H}^{2}+\dots\right)\\ & =\int d\tau\ \left(\hat{H}_{i}-i\tau\delta_{t}\left[\hat{H},\hat{H}_{i}\right]+\dots\right)\\ &\simeq\hat{H}_{i}\end{aligned}$$ where we assume that the time steps chosen are so small that the integral over the commutator can be neglected[^4]. A self-contained derivation is presented later in \[subsec:The-gradient-of-a-matrix-exponential\]. Restoring all the units leads us to the closed gradient formula $$\frac{\partial J}{\partial u_{i}(j)}=-i\delta_{t}\left\langle \lambda_{j}\left\|\hat{H}_{i}\right\|\rho_{j}\right\rangle \label{eq:grape_gradient_state}$$ meaning that we can expect, with an appropriate value of $\epsilon$ compute a gradient-based update $$u_{i}(j)\mapsto u_{i}(j)+\epsilon\frac{\partial J}{\partial u_{i}(j)}\label{eq:grape_update_state}$$ This allows us to extremalize $J$ hence to find controls that best approximate the final state with the following algorithm. Starting from an initial guess for the controls: 1. Compute the propagated initial state $\|\rho_{m}\rangle=\hat{U}_{m}\cdots\hat{U}_{1}\|\psi_{0}\rangle$ for all $m\le N$ by iterative matrix multiplication. 2. Compute the back-propagated final state $\|\lambda_{m}\rangle=\hat{U}_{N}\cdots\hat{U}_{m+1}\|\psi_{1}\rangle$ by iterative matrix multiplication 3. Compute the gradient of the performance index and update the controls following eqs. (\[eq:grape\_gradient\_state\]), (\[eq:grape\_update\_state\]) 4. Iterate until the value of $J$ is satisfactory or the updates are below a threshold There are a lot of practical improvements that were found beyond this which we will describe below. One must not underestimate the importance of this analytical derivation of a gradient. Whenever a gradient is available, it greatly improves the convergence of a search specifically when going from a rough initial guess that can often be obtained by solving an approximate version of the problem at hand to a solution that has the very high precision generally demanded by quantum technologies. If a gradient is available, its analytical and exact derivation is also paramount – numerical gradients are very hard to control numerically as they involve a small difference between two potentially large numbers. In pioneering, pre-GRAPE work [@Niskanen03] this was rather obvious – even with large computational effort, only few parameters could be optimized. ### An alternative, direct derivation An alternative derivation of the variational approach to quantum optimal control is as follows: Let us again look at the state transfer task. We shall construct a functional, $J$, to be maximized, and utilize Lagrange multipliers to enforce both the intial condition and the equation of motion. We shall parameterize our control fields, $u\left(t\right)$ using a vector of scalar real parameters $\vec{\alpha}$. Our aim is to maximize the overlap of the goal state $\ket{\psi_{\textrm{goal}}}$ and the state at final time $T$, $\ket{\psi\left(T\right)}$, $$J_{\textrm{goal}}=\left\|\braket{\psi\left(T\right)}{\psi_{\textrm{goal}}}\right\|^{2}.$$ We need to impose an initial condition, utilizing a Lagrange multiplier $\lambda_{\textrm{init}}$ $$J_{\textrm{init}}=\lambda_{\textrm{init}}\left(\left\|\braket{\psi\left(0\right)}{\psi_{\textrm{init}}}\right\|^{2}-1\right).$$ Next, we must guarantee the Schrödinger equation, $\left(i\hbar\partial_{t}-H\left(\bar{\alpha},t\right)\right)\ket{\psi\left(t\right)}=0$ is upheld at all times. To do that, at each point in time, $t$, we must multiply the equation of motion by the Lagrange multiplier $\bra{\chi\left(t\right)}$, and we must add the contributions for all points in time: $$J_{\textrm{e.o.m}}=\int_{o}^{T}\Braket{\chi\left(t\right)}{i\hbar\partial_{t}-H\left(\bar{\alpha},t\right)}{\psi\left(t\right)}$$ Note that in $J_{\textrm{e.o.m}}$, $\bra{\chi\left(t\right)}$ can be interpreted as a conjugate state, propagating backwards in time, as the term can be rewritten as $\braket{\left(-i\hbar\partial_{t}-H\left(\bar{\alpha},t\right)\right)\chi\left(t\right)}{\psi\left(t\right)}$. The functional to be minimized is then $$J=J_{\textrm{init}}+J_{\textrm{e.o.m}}+J_{\textrm{goal}}$$ We then proceed in the standard variational approach, taking the gradient of this functional with respect to $\bar{\alpha}$ and requiring $$\partial_{\bar{\alpha}}J=0.$$ ### Synthesis of unitary gates We will now go to the topic of finding controls that best approximate a quantum gate. This can be viewed as a generalization of the state preparation problem to rotating a full basis of the Hilbert space into a desired new basis. This first begs the question of how to find an appropriate performance index. It can be accomplished by starting with a distance measure between the desired and the actual final unitary $\left\Vert \hat{U}_{{\rm target}}-\hat{U}(T)\right\Vert $. The most common choice is based on the 2-norm $$\begin{aligned} \left\Vert \hat{U}_{{\rm target}}-\hat{U}(T)\right\Vert _{2}^{2} & ={\rm Tr}\left[\left(\hat{U}_{{\rm target}}^{\dagger}-\hat{U}^{\dagger}(T)\right)\left(\hat{U}_{{\rm target}}-\hat{U}(T)\right)\right]\\ & ={\rm Tr}\left[\hat{U}_{{\rm target}}^{\dagger}\hat{U}_{{\rm target}}+\hat{U}^{\dagger}(T)\hat{U}(T)-\hat{U}_{{\rm target}}^{\dagger}\hat{U}(T)-\hat{U}^{\dagger}(T)\hat{U}_{{\rm target}}\right]\\ & =2\left(d-{\rm Re}{\rm Tr}\hat{U}_{{\rm target}}^{\dagger}\hat{U}(T)\right)\end{aligned}$$ where $d$ is the underlying Hilbert space dimension. Thus, we see that minimizing the error corresponds to maximizing the overlap ${\rm Re}{\rm Tr}\hat{U}_{{\rm target}}^{\dagger}\hat{U}(T)$ . Now the real part looks suspicious – if we have the gate right up to a global phase, $\hat{U}(T)=e^{i\phi}\hat{U}_{{\rm target}}$ this overlap indicates a non-perfect result. In fact, numerical experimentation shows that this would be a serious drawback. We can trace this error back to the original distance measure. The high-brow step to take now would be to elevate the description to full quantum channels. Pragmatically, we move from real part to absolute square and thus the most common performance index for gates is $$J=\left\|{\rm Tr}\left(\hat{U}_{{\rm target}}^{\dagger}\hat{U}(T)\right)\right\|^{2}.$$ This quantity can be interpreted in a somewhat operational fashion: First apply the gate you have, then undo the gate you want. If everything goes right you have but a global phase – the same one on all vectors of the standard basis. If not, you measure the deviation from unity for the complete standard basis. There are other possible choices (and good reasons to think about them), which we will discuss later. With this quantity, we can proceed in a way similar to state transfer, only that now we of course start at the unit matrix. We again use piecewise constant controls and define both the intermediate propagator and the intermediate back-propagated target $$\hat{X}_{j}=\hat{U}_{j}\cdots\hat{U}_{1}\quad\hat{P}_{j}=\hat{U}_{j+1}^{\dagger}\cdots\hat{U}_{N}^{\dagger}\hat{U}_{{\rm target}}$$ allowing us to rewrite $J=\left\|{\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right\|^{2}$ for all values of $j$ . We can now apply the same identities as before and find $$\begin{aligned} \frac{\partial J}{\partial u_{i}(j)} & =\frac{\partial}{\partial u_{i}(j)}\left({\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right)\left({\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right)^{\ast}\\ & =2{\rm Re}\left[\left(\frac{\partial}{\partial u_{i}(j)}{\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right)\left({\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right)\right]\\ & =-2i\delta t{\rm Re}\left[\left({\rm Tr}\hat{P}_{j}^{\dagger}\hat{H}_{i}\hat{X}_{j}\right)\left({\rm Tr}\hat{P}_{j}^{\dagger}\hat{X}_{j}\right)\right].\end{aligned}$$ With this analytical gradient, the GRAPE algorithm can be applied as above. The Krotov algorithms --------------------- The [Krotov algorithm]{} [@Koch-Krotov-Main; @Krotov1; @krotov1983iteration; @Krotov2] has been formulated before the GRAPE algorithm. Some of its presentations are historically based on applications in chemistry and emphasizes constraints more than its core. Looking back on how GRAPE is applied, we are blessed with an analytical gradient formula which in each iteration allows us to calculate the gradient of the cost function(al) with respect to all controls at all times and then by walking against it look for improved controls. Notably, the gradient is always computed at a point in parameter space given by the controls computed in the previous iteration. There are two different algorithms which carry the name “Krotov” - a fact which can be quite confusing, even for experts in the field. The first Krotov, prides itself with its monotonic convergence, which is achieved by propagating the forward state using the old control field, while the backward-propagating state makes use of the new field. A detailed description, with Python implementation, can be found in [@KrotovPython]. The second Krotov can be considered a greedy version of GRAPE, and is described in detail in [@Shai-PRA]: In this version of the Krotov algorithm, all previously computed knowledge is used, i.e., once an entry to the gradient is computed, it is applied right away and the next element of the gradient is computed with that correction already applied. This approach of not leaving any information behind in general lowers the number of iterations needed to reach convergence and it comes with proven monotonic convergence. On the other hand, each iteration step takes more time. The various update strategies are visualized in figure \[fig:update\]. Benchmarking of the various optimal control algorithms is a topic of ongoing research. ![Overview on the update schemes of gradient-based optimal control algorithms in terms of the set of time slices T (q) = {k(q), k(q), . . . k(q) } for which the control amplitudes are concurrently updated in each iteration. Subspaces are enumerated by q, gradient-based steps within each subspace by s, and r is the global step counter. In grape (a) all the M piecewise constant control amplitudes are updated at every step, so T (1) = {1,2,...M} for the single iteration q$\equiv$1. Sequential update schemes (b) update a single time slice once, in the degenerate inner-loop s$\equiv$1, be- fore moving to the subsequent time slice in the outer loop, q; therefore here T (q) = {q mod M}. Hybrid versions (c) follow the same lines: for instance, they are devised such as to update a (sparse or block) subset of p different time slices before moving to the next (disjoint) set of time slices. \[fig:update\]](updates){width="0.9\columnwidth"} Modern numerical issues ----------------------- ### Control landscapes A gradient search with an analytical gradient as outlined is the best way to find a local extremum of an [optimization landscape]{}. If the optimization landscape has multiple local minima, it can get stuck in a local minimum and needs to be enhanced. In a seminal series of papers, Rabitz has shown (see e.g. [@Rabitz2004]) that there is indeed only one extremum in the control landscape and that it is global. This theorem is a correct derivation of its assumptions – one of which is the absence of constraints in pulse amplitude and temporal resolution. In practice, these constraints exist and multiple local extrema exist – the more constrained the optimization, the more local extrema. Specifically in situations close to the quantum speed limit (see below), with low control resolution (Ref [@Liebermann16] looks at a single bit of amplitude resolution and required genetic algorithms to converge) or with complex many-body dynamics and only few controls, these call for more advanced methods. If one has a good intuition about the optimal pulse say, by solving a model that is very close to the desired model or by rescaling a solution that works at a longer gate duration, one can often stay close to the global extremum and otherwise requires a gradient search. If that is not the case, one needs to first start with a more global search method covering a large parameter space. Known systems for such gradient-free approaches are GROUP [@Sorensen18], genetic algorithms ([@Judson1992; @Liebermann16]), they are part of CRAB (see section \[ch:CRAB\]) and simulated annealing [@Yang17]. ### Fidelities We would like to come back to the choice of [fidelity]{} based on the 2-norm described above. It has been argued that the most appropriate way to characterize quantum processes is the use of the diamond norm [@quan-comp-accuracy1]. It can be expressed for a quantum operation $\mathcal{E}$ compared to an ideal operation $\mathcal{U}$ as $$\|\|\mathcal{U}_{\rm ideal}-\mathcal{E}\|\|_{\diamond}=\sup_q \max_{\psi} \left\|\rm{Tr}\left[\mathcal{U}_{\rm ideal}(\|\psi\rangle\langle\psi\|)-\mathcal{E}(\|\psi\rangle\langle\psi\|)\right]\right\|$$ This involves two generalizations of the 2-norm: On the one hand, rather than taking the 2-norm distance which is equivalent to averaging over all possible input states to the operation, we are taking the maximum over $\|\psi\rangle$, i.e., we choose the input state that maximizes the distance. On the other hand, rather than directly using the unitary operation, we enhance the Hilbert space by adding another space of dimension $q$ on which the identity operation is performed. The diamond norm is then the supremum over $q.$The latter may sound rather academic, but it is not if, e.g., the initial state is entangled between the original and the auxiliary system. For the purposes of quantum optimal control, the diamond norm is rather impractical – it is hard to compute (as it contains a supremum) and it can be non-differentiable (as it contains taking a maximum over states, the state at which it has reached can jump in state space). What does this mean for the applicability of quantum optimal control in the context of fault tolerance? There are two answers to this question. On the one hand, one can at least find performance indices that emphasize the worst case more strongly while being differentiable. A straightforward option is [@Hu08] $$\begin{aligned} J_{q}&={\rm \max_{\alpha\in[0,2\pi)}}\left\Vert \hat{U}_{{\rm target}}-e^{i\alpha}\hat{U}(T)\right\Vert _{2q}^{2q}\\ &=\max_{\alpha\in[0,2\pi)}{\rm Tr}\left[\left(\hat{U}_{{\rm target}}^{\dagger}-e^{-i\alpha}\hat{U}^{\dagger}(T)\right)\left(\hat{U}_{{\rm target}}-e^{i\alpha}\hat{U}(T)\right)\right]^{q}\end{aligned}$$ which can be implemented in a straightforward fashion yet does not have a known extension that avoids optimizing the global phase. On the other hand, it is pragmatically not very crucial to go through these steps as long as the algorithm converges properly: Our goal is to get the error as close to zero as possible and, as these norms can be continuously mapped onto each other, one pragmatically searches for controls that reduce the error in the 2-norm to an extremely low value which guarantees that even in the desired norm the error is low enough – using the paradigm to control and verify with two different measures. ### Increasing precision of GRAPE The GRAPE algorithm above defines a straightforward gradient algorithm for optimal control. There are a few known measures to speed up its convergence. One measure is the improvement of the use of the gradient by moving to a quasi-Newton method, the Broyden, Fletcher, Goldfard, and Shanno (BFGS) method [@L-BFGS]. Newton’s method, as the reader may have learned in an elementary introduction to numerical mathematics, rely on approximating the function whose zero we desire to find by its tangent – in our case, we desire to find the zero of the gradient, i.e., we need to approximate the functions up to its second derivative. As we are optimizing a scalar that depends on many parameters – all the controls taken at all the times of interest – the matrix of second derivatives is a high-dimensional object. In order to approximate the zero of the gradient, one would have to invert that matrix, which is numerically hard and would likely negate the potential computational advantage. The BFGS method instead relies on directly approximating the inverse Hessian ### \[subsec:The-gradient-of-a-matrix-exponential\]The gradient of a matrix exponential Expanding on the discussion surrounding eq. (\[eq:PWC-Uk\]), (\[eq:Deriv-of-time-slice\]), any gradient-driven optimal control optimization, such as GRAPE or Krotov, which treats the control fields as piecewise constant, will describe the coherent propagator of time slice $m$ as $$U_{m}=\exp\left(-\frac{i}{\hbar}\delta_{t}H\left(\bar{\alpha},t_{m}\right)\right)\label{eq:PWC-U-m}$$ where $\bar{\alpha}$ parameterizes the control functions $u\left(t\right)$. We are searching for the value of $\bar{\alpha}$ which will minimize the infidelity. At step $j$ of the optimization, to compute the gradient of the goal function with respect to $\bar{\alpha}$, we must compute $\partial_{\bar{\alpha}}U_{m}\left(\bar{\alpha}\right)\vert_{\bar{\alpha}=\bar{\alpha}_{j}}$ . At this point we can rewrite eq. (\[eq:PWC-U-m\]) as in eq. (\[eq:Deriv-of-time-slice\]), $$U_{m}=\exp\left(-\frac{i}{\hbar}\delta_{t}\left(H_{m,j}+\epsilon_{\bar{\alpha}}\tilde{H}_{m,j}\right)\right)$$ where $\epsilon_{\bar{\alpha}}$ is small and we seek $\partial_{\epsilon_{\bar{\alpha}}}U_{m}\left(\epsilon_{\bar{\alpha}}\right)\vert_{\epsilon_{\bar{\alpha}}=\bar{0}}$. Following [@grad-exp-1-aizu1963parameter; @exp-grad-2-levante1996pulse], and their summary in Appendix A of [@Shai-PRA], we denote the eigenvalues and eigenvectors of $H_{j}$ by $e_{k}$ and $\ket{e_{k}}$, respectively, then using the spectral theorem $$\Braket{e_{l}}{\partial_{\bar{\epsilon_{\bar{\alpha}}}}U_{m}}{e_{k}}=\left\{ \begin{array}{ccc} -\frac{i}{\hbar}\delta_{t}\Braket{e_{l}}{\tilde{H}_{m,j}}{e_{k}}\exp\left(-\frac{i}{\hbar}\delta_{t}e_{l}\right) & & \textrm{if\,\,}e_{l}=e_{k}\\ -\frac{i}{\hbar}\delta_{t}\Braket{e_{l}}{\tilde{H}_{m,j}}{e_{k}}\dfrac{\exp\left(-\frac{i}{\hbar}\delta_{t}e_{l}\right)-\exp\left(-\frac{i}{\hbar}\delta_{t}e_{k}\right)}{-\frac{i}{\hbar}\delta_{t}\left(e_{l}-e_{m}\right)} & & \textrm{if\,\,}e_{l}\neq e_{k} \end{array}\right.$$ one may invoke the spectral theorem in a standard way and calculate matrix functions via the eigendecomposition. To simplify notation, we shall look at $\partial_{x}\;e^{A+xB}$, with $A,B$ being an arbitrary pair of Hermitian (non-commuting) matrices and $x\in\mathbb{R}$. As previously $\{\ket{e_{l}}\}$ as the orthonormal eigenvectors to the eigenvalues $\{e_{l}\}$ of $A$ . We then obtain the following straightforward, if somewhat lengthy, derivation: $$\begin{aligned} D & = & \braket{e_{l}}{\partial_{x}\;e^{A+xB}\|e_{k}}\Big\|_{x=0}\\ & & \braket{e_{l}}{\partial_{x}\;\sum_{n=0}^{\infty}\frac{1}{n!}\big(A+xB\big)^{n}\|e_{k}}\Big\|_{x=0}\\ & & \braket{e_{l}}{\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{q=1}^{n}\big(A+xB\big)^{q-1}B\big(A+xB\big)^{n-q}\|e_{k}}\Big\|_{x=0}\\ & & \braket{e_{l}}{\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{q=1}^{n}A^{q-1}BA^{n-q}\|e_{k}}\\ & & {\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{q=1}^{n}e_{l}^{q-1}\braket{e_{l}}{B\|e_{k}}e_{k}^{n-q}}\\ & & \braket{e_{l}}{B\|e_{k}}\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{q=1}^{n}e_{l}^{q-1}e_{k}^{n-q}\end{aligned}$$ This provides the answer for in the case where $e_{l}=e_{k}$. For $e_{l}\neq e_{k}$ a bit more work is needed: $$\begin{aligned} D & = & \braket{e_{l}}{B\|e_{k}}\sum_{n=0}^{\infty}\frac{1}{n!}e_{k}^{n-1}\sum_{q=1}^{n}\left(\frac{e_{l}}{e_{k}}\right)^{q-1}\\ & & \braket{e_{l}}{B\|e_{k}}\sum_{n=0}^{\infty}\frac{1}{n!}e_{k}^{n-1}\frac{(e_{l}/e_{k})^{n}-1}{(e_{l}/e_{k})-1}\\ & & \braket{e_{l}}{B\|e_{k}}\sum_{n=0}^{\infty}\frac{1}{n!}\frac{e_{l}^{n}-e_{k}^{n}}{e_{l}-e_{k}}\\ & & \braket{e_{l}}{B\|e_{k}}\frac{e^{e_{l}}-e^{e_{k}}}{e_{l}-e_{k}}\end{aligned}$$ Note that we have explicitly made use of the orthogonality of eigenvectors to different eigenvalues in normal matrices. Applied optimal quantum control =============================== While quantum optimal control is a well-developed field and has been very successful in atomic and molecular systems, its track record in solid-state quantum technologies is somewhat less developed. The reason has to do with the accuracy of the models, i.e., the precision at which we know every ingredient of the Hamiltonian. First of all, a quantum-technological device (specifically, but not exclusively, in the solid state) has human-made components which contain some fabrication uncertainty. This affects the drift Hamiltonian – even if its eigenvalues can be accurately determined using spectroscopy, it is much more involved to find its eigenvectors. These naturally also affect the matrix elements of the control Hamiltonians. On top of that, some solid-state quantum devices need to be extremely well isolated from their environments including high-temperature black-body radiation. This means, that an applied control signal will get distorted on its way to the sample in a way that can be measured only to a limited degree, see fig. \[fig:chain\] for a summary. While one can improve hardware and characterization to meet these challenges, it is hard to get this to the precision required by, say, fault-tolerant quantum computing. Thus, other approaches are called for. ![\[fig:chain\] Typical sources of inaccuracy in quantum control for superconducting qubits including the transmission from the generator to the sample and inaccuracy of the Hamiltonian model. Right: Typical error sensitivity for a gate between superconducting qubits. ](Error_Sources_V2.eps) Closing the loop for pulse calibration -------------------------------------- One possible approach to handle uncertainties would be to use a robust control methodology inspired by magnetic resonance in ensembles. While this method can be useful, it slightly misses the point: It still requires a good estimate for the uncertainty and then it improves performance across the relevant parameter interval. Here, the situation is different, we do not have a parameter distribution but a single set of parameters – we just cannot find it or even the relevant model a priori. One way to still find good pulses are hybrid control methods such as Adaptive Hybrid Optimal Control (AdHOC, [@AdHOC]), Optimized Randomized Benchmarking for Immediate Tuneup (ORBIT, [@ORBIT]), and Adaptive Control via Randomized Optimization Nearly Yielding Maximization (ACRONYM, [@Ferrie15]). The idea of these methods is rather similar: After an initial design phase that may or may not contain traditional optimal control, a set of pulses is constructed based on models that are believed to approximate the actual system but its parameterization is left open to some corrections. These corrections are then determined in a closed loop – the fidelity is measured and the pulses are updated based on these fidelity measurements. ![\[fig:adhoc\] Typical two-stage control workflow with an open loop modeling stage](Protocol.eps) In the example of AdHOC, the pulse measurement is based on randomized benchmarking (described below) and the optimization that determines the corrections is based on the [Nelder-Mead]{} simplex algorithm, which is available in most numerical mathematics toolboxes. What is crucial is that this is a gradient-free algorithm in order to avoid issues with taking gradients of measurement data. Is that as a simplex algorithm, the search for a pulse described by $n$ parameters needs to be initialized using $n+1$ initial guesses. This raises the important question how the number $n$ can be kept as small as possible (but not smaller, see below) by finding an efficient parameterization. This is not an easy problem. So far, we have always assumed that the pulses are parameterized in piecewise constant fashion and have argued that this is naturally compatible with arbitrary wave form generators. However, this parameterization does not naturally lend itself to reduction of the number of parameters – simple, sparse controls in quantum physics are typically sine and cosine functions with smooth, Gaussian-derived envelopes. On the other hand, the piecewise constant parameterization was instrumental in deriving the gradient formula in an analytical way and cannot be easily removed. CRAB\[ch:CRAB\] --------------- Albeit originally developed from a different motivation, the optimization of many-body dynamics, the [Chopped RAndom Basis (CRAB)]{}[^5] algorithms serves that purpose, [@CRAB]. It introduced the concept of simple and sparse pulse parameterizations, i.e., finding a pulse parameterization that is not necessarily piecewise constant but rather can be written as $$H\left(\bar{\alpha},t\right)=H_{0}+\sum_{k=1}^{C}c_{k}\left(\bar{\alpha},t\right)H_{k}\,, \label{eq:Ham_tot}$$ where the functions $c_k$ can e.g. be harmonic functions characterized by amplitude, frequency and phase or a sequence of Gaussians $$c_{k}\left(\bar{\alpha},t\right)=\sum_{j=1}^{m}A_{k,j}\exp\left(-(t-\tau_{k,j})^2/\sigma_{k,j}^2\right). \label{eq:Fourier-controls}$$ In complex systems that were the initial motivation for CRAB, one has very little prior knowledge about a suitable basis and it is at best chosen random, hence the name. CRAB utilizes a gradient-free search, specifically Nelder-Mead (although other algorithms could be used), similarly to what we have already described for AdHOC[^6] . The fact CRAB is model-free, with the gradient-free search treating the quantity to be optimized as a black box, provides a distinct advantage in situations a precise model is unknown or when the model is know, but the gradient cannot be computed due to numerical complexity or other reasons. This makes CRAB appropriate for closed-loop experimental calibration of control fields in system ranging from nitrogen vacancy centers in nano-diamonds [@CRAB-Appl-1-binder2017qudi] and cancer treatment formulations [@CRAB-Appl-2-angaroni2019personalized], to DMRG-based simulations [@CRAB-Appl-3-silvi2014lattice]. Further, CRAB enjoys huge success in studying quantum phase transitions, preparing large Schrödinger cat states, sensing and many more. A variant of CRAB, known as dCRAB [@dCRAB], deals with a situation where the control parameterization has a higher dimensionality than can be optimized by Nelder-Mead, by iteratively optimizing different subsets (or low-dimension projections) of the high-dimension full parameter space. GOAT ---- [Gradient Optimization of Analytic conTrols (GOAT)]{} is a recently [@Machnes18] proposed optimal control algorithm which does not derive from the variational formulation of optimal control, defined earlier. Rather, GOAT finds the equations of motion for the gradient of the goal function with respect to the control parameters, integrating as you would the Schrödinger equation (as piece-wise-constant approximation, or using standard ODE tools such as Runge-Kutta optimizers). For our purpose, the goal function to minimize is defined as the projective $SU$ distance (infidelity) between the desired gate, $U_{\textrm{goal}}$, and the implemented gate, $U\left(T\right)$, [@PalaoKosloff2002] (also [@TESCH2001633]) $$g\left(\bar{\alpha}\right):=1-\tfrac{1}{\textrm{dim}\left(U\right)}\left\|\textrm{Tr}\left(U_{\textrm{goal}}^{\dagger}U\left(T\right)\right)\right\|\,,\label{eq:goal}$$ where $U\left(t\right)$ is the time ordered ($\mathbb{T}$) evolution operator $$U\left(\bar{\alpha},T\right)=\mathbb{T}\exp\left(\int_{0}^{T}-\frac{i}{\hbar}H\left(\bar{\alpha},t\right)dt\right).\label{eq:Upt}$$ GOAT’s ability to use any control ansatz makes it feasible to find drive shapes described by a small number of parameters, suitable for closed-loop calibration. A gradient-based optimal control algorithm requires two ingredients: an efficient computation of $\partial_{\bar{\alpha}}g\left(\bar{\alpha}\right)$ and a gradient-based search method over parameter space. GOAT presents a novel method for the former, while using any standard algorithm for the latter, such as BFGS. Consider the gradient of the goal function eq. (\[eq:goal\]) with respect to $\bar{\alpha}$, $$\partial_{\bar{\alpha}}g\left(\bar{\alpha}\right)=-\textrm{Re}\left(\frac{g^{}}{\left\|g\right\|}\frac{1}{\textrm{dim}\left(U\right)}\textrm{Tr}\left(U_{\textrm{goal}}^{\dagger}\partial_{\bar{\alpha}}U\left(\bar{\alpha},T\right)\right)\right)\,.\label{eq:goal-func-grad}$$ Neither $U\left(\bar{\alpha},T\right)$ nor $\partial_{\bar{\alpha}}U\left(\bar{\alpha},T\right)$ can be described by closed form expressions. $U$ evolves under the equation of motion $\partial_{t}U\left(\bar{\alpha},t\right)=-\frac{i}{\hbar}H\left(\bar{\alpha},t\right)U\left(\bar{\alpha},t\right)$. By taking the derivative of the $U$ equation of motion with respect to $\bar{\alpha}$ and swapping derivation order, we arrive at a coupled system of equations of motion for the propagator and its gradient, $$\partial_{t}\left(\begin{array}{c} U\\ \partial_{\bar{\alpha}}U \end{array}\right)=-\frac{i}{\hbar}\left(\begin{array}{cc} H & 0\\ \partial_{\bar{\alpha}}H & H \end{array}\right)\left(\begin{array}{c} U\\ \partial_{\bar{\alpha}}U \end{array}\right).\label{eq:joint-eom}$$ As $\bar{\alpha}$ is a vector, $\partial_{\bar{\alpha}}U$ represents multiple equations of motion, one for each component of $\bar{\alpha}$. $\partial_{\bar{\alpha}}H$ is computed using the chain rule. GOAT optimization proceeds as follows: Starting at some initial $\bar{\alpha}$ (random or educated guess), initiate a gradient driven search (e.g. L-BFGS [@L-BFGS]) to minimize eq. (\[eq:goal\]). The search algorithm iterates, requesting evaluation of eqs. (\[eq:goal\],\[eq:goal-func-grad\]) at various values of $\bar{\alpha}$, and will terminate when the requested infidelity is reached or it fails to improve $g$ further. Evaluation of $g\left(\bar{\alpha}\right)$, $\partial_{\bar{\alpha}}g\left(\bar{\alpha}\right)$ requires the values of $U\left(\bar{\alpha},T\right)$ and $\partial_{\bar{\alpha}}U\left(\bar{\alpha},T\right)$. These are computed by numerical forward integration of eq. (\[eq:joint-eom\]), by any mechanism for integration of ordinary differential equations that is accurate and efficient for time-dependent Hamiltonians, such as adaptive Runge-Kutta. Initial conditions are $U\left(t=0\right)=\mathcal{I}$ and $\partial_{\bar{\alpha}}U\left(t=0\right)=0$. Note that no back propagation is required. Experimental constraints can be easily accommodated in GOAT by mapping the optimization from an unconstrained space to a constrained subspace, and computing the gradient of the goal function using the chain rule. For example, $\bar{\alpha}$ components may be constrained by applying bounding functions, e.g. $\alpha_{k}\longrightarrow\frac{1}{2}\left(v_{\text{max}}-v_{\text{min}}\right)\sin\left(\bar{\alpha}_{k}\right)+\frac{1}{2}\left(v_{\text{max}}+v_{\text{min}}\right)$ which imposes $\alpha_{k}\in\left[v_{\text{min}}\ldots v_{\text{max}}\right]$. Amplitude constraints and a smooth start and finish of the control pulse can be enforced by passing the controls through a window function which constrains them to a time-dependent envelope. Gradients for $\partial_{\bar{\alpha}}H$ flow via the chain rule. Evaluating fidelity with randomized benchmarking. ------------------------------------------------- The closed-loop approaches mentioned above crucially rely on a measurement of success. While in state-transfer problems, e.g. creating an ordered state quickly or steering a chemical reaction, there may be generic tools to determine this success with a given experimental apparatus. In the case of a quantum gate, this is not so simple. While classic textbooks like first label quantum [process tomography]{}, this has a number of drawbacks, and is now replaced by more efficient methods. ### The trouble with tomography To understand this, let’s first take a look at quantum state tomography [@NielsenChuang]. This is, in a nutshell, the reconstruction of a quantum state (characterized by its density matrix) by performing a complete set of observable measurements. Next to some practical drawbacks having to do with guaranteeing a positive density matrix [@Paris04], this is also impractical: A typical quantum device can be read out with a single machine – an electric or optical measurement. Formally this corresponds to measuring in one basis (we will assume that we are dealing with qubits, so recording the expectation value completely characterizes the output distribution). In order to measure a complete set of operators, one has to first perform a basis change in the shape of performing a coherent operation. As this operation itself is prone to error, this will falsify the result. Together with the intrinsic imperfection of the readout device this constitutes measurement error. From state tomography, it is another step to process tomography, i.e., the reconstruction of a quantum channel – linear map from input to output density matrices – from measured. Formally, one can using the Choi-Jamiolkowski isomorphism [@NielsenChuang] map the process matrix of the channel onto the density matrix of a state and treat the problem of process tomography as one of state tomography. Practically, process tomography involves to now measure complete sets of both initial and final states that undergo the channel. Similar to measurement, also state preparation is usually possible only in one distinct basis – if state preparation is performed by measurement it is the measurement basis, if state preparation is performed via thermalization or optical pumping it is the drift Hamiltonian’s eigenbasis – and it is imperfect – both of these give rise to state preparation errors. Thus, in total, the quantum channel that one would like to characterize is masked by state preparation and measurement (SPAM) errors. On top of that, full process tomography is also forbiddingly labourious. The state of a $d$ dimensional quantum system is characterized by a $d^{2}$ entries in a densitry matrix that, accounting for hermiticity and norm boil down to $d^{2}-d+1$ real numbers. This has to be squared again to describe a quantum channel, leading to $O\left(d^{4}\right)$ numbers – which then are recombined to compute a single fidelity. In an $n$ - Qubit system, we have $d=2^{n}$ making full tomography forbiddingly data intensive. On top of that, we would like to ensure complete positivity of the measured channel, which gives rise to inequality constraints that are practically hard to meet specifically when the map is close to unitary. Now there are several methods such as compressed sensing and Monte Carlo sampling [@Gross10; @Chasseur17] that reduce that problem, but with SPAM still included, there is strong motivation to look for an independent method to evaluate fidelity in an experiment. Here, randomized benchmarking and its descendants (RB+) have appeared as a quasi-standard. A comprehensive review of RB+ has currently not been published. We are going to mention key papers on the way and otherwise refer to the work of J. Emerson. ### Randomization of quantum channels Let’s first lay the foundation of how we describe a [quantum channel]{} [@NielsenChuang]: A linear map hat takes any valid density matrix onto another valid density matrix, i.e., with $$\rho\mapsto\mathcal{E}\left[\rho\right]$$ we demand that if $\rho$ is hermitian, positively semidefinite, and has a normalized trace, so is $\mathcal{E}\left[\rho\right]$. This is satisfied by the iffindex[Kraus representation]{} $$\mathcal{E}\left[\rho\right]=\sum_{k}\hat{A}_{k}\hat{\rho}\hat{A}_{k}^{\dagger}\quad\sum_{k}\hat{A}_{k}^{\dagger}\hat{A}_{k}=\mathbb{I}.$$ The (non-unique) Kraus operators* $A_{k}$ characterize the channel. It can be easily shown that the Kraus representation leads to a valid channel and it takes a bit more attention to show that the validity of the channel also requires the Kraus representation. Now to estimate the average fidelity over a channel relative to a desired unitary $\hat{U}$ we apply the channel to a pure initial state, then undo the ideal channel, compute the overlap with the pure state and average over all pure inputs $$F=\int d\psi\ \left\langle \psi\left\|U^{\dagger}\mathcal{E}\left[\left\|\psi\rangle\langle\psi\right\|U\right]\right\|\psi\right\rangle$$ where the integral runs over a suitable uniform distribution of all states called the Haar measure. We now aim at replacing the average in this formula by another randomization procedure [@Emerson_2005]. We now decompose the real operation into an ideal operation followed by an error channel and Kraus-decompose the error channel $$\mathcal{E}=\Lambda\circ\mathcal{U}\quad\Lambda=\sum_{k}A_{k}\rho A_{k}^{\dagger}.$$ Plugging this into the expression for the average gets us $$F=\int d\psi\ \left\langle \psi\left\|U^{\dagger}\Lambda\left[U\left\|\psi\rangle\langle\psi\right\|U^{\dagger}\right]U\right\|\psi\right\rangle .$$ We can read this expression as implementing the motion-reversal transformation $U^{\dagger}\cdot U$ with an error $\Lambda$ occuring in the middle. Now instead of going for $F$ directly, let us average the fidelity over all unitaries that can enter the motion-reversal map – assuming tacitly that we have the same $\Lambda$ at all times. We now compute a at first glance very different average – we keep a single initial state $\rho=\|\psi\rangle\langle\psi\|$ and instead average over all unitaries $$E=\int dU\ {\rm Tr}\left[\rho U^{\dagger}\Lambda\left(U\rho U^{\dagger}\right)U\right].$$ Now we exchange the order of integration and change the order under the trace and write this as $$E={\rm Tr}\left(\rho\left[\int dU\ U\Lambda U^{\dagger}\right]\rho\right)$$ We can now read this exchanged expression at face value – in the center is noise averaged over all unitaries $$\Lambda_{{\rm ave}}=\int dU\ U\Lambda U^{\dagger}.$$ Building on the operations of unitary maps as generalized rotations, this is called a twirled channel. It can be mathematically shown what is physically rather obvious – this channel must be highly symmetric, it cannot prefer any basis over the other. The only channel compatible with this is the depolarizing channel $$\Lambda_{{\rm ave}}\left[\rho\right]=p\rho+\frac{1-p}{d}\mathbb{I}$$ which has a single error probability $p.$ With this the error averaged over all unitaries equals the fidelity of the twirled channel computed for a single input state $$E={\rm Tr}\left(\rho\Lambda_{{\rm ave}}\rho\right)=F.$$ where the last equality requires some more involved math to show that this is also the same as the average fidelity of a unitary averaged over all states. The fact that a single input state is enough – we have delegated the need for averaging from all states to twirling the channel – addresses the problem of SPAM errors. Now what is needed is an efficient way to implement $\Lambda_{{\rm ave}}$. We need to replace the integral over all unitaries by a sum over random elements that converges to this integral. This brings in the concept of a unitary 2-design: a set that correctly reproduces the full unitary set in polynomials of degree 2 . It can be shown [@Dankert09; @Knill09] (in a rather pedestrian way) that the Clifford group is sufficient. The Clifford group [@NielsenChuang] is formally defined as the normalizer of the Pauli group. For n qubits, this Pauli $P_{n}=\left\{ \text{\ensuremath{\sigma_{n}}}\right\} $ group consists of all direct products of Pauli matrices $\sigma_{n}=\otimes_{j=1}^{n}\sigma_{i_{j}}$, $i_{j}\in\left\{ 0,1,2,3\right\} $ so the corresponding Clifford group is the set of all unitaries that map all n-qubit Pauli matrices onto Pauli matrices $$C_{n}=\left\{ U\in SU\left(2^{n}\right):\forall\sigma_{n}\in P_{n}\ \exists\sigma_{m}\text{\ensuremath{\in}}P_{m}:\sigma_{m}=U\sigma_{n}U^{\dagger}\right\} .$$ For a single qubit, this group is generatd by all quarter-turns around the Bloch sphere. The Clifford group is a discrete group and quantum algorithms consisting of only Clifford gates can be efficiently classically simulated . These together lead to the remarkably simple protocol of randomized benchmarking. ### Randomized Benchmarking Let’s pull all of these ingredients together into a handy protocol: 1. Repeat for a few representative sequences 1. Draw a random set of Clifford gates 2. Compute the resulting operation and its inverse. Add the inverse to the end of the sequence 3. Repeat the following to establish an estimate for the final probability for survival of the initial state 1. initialize the system in a convenient state 2. run the sequence 3. measure if the outcome is the same state or not 4. Average to estimate the survival probability for the given sequence 2. Average to estimate the survival probability averaged of the Clifford groups. As a function of sequence length, the result will have the form $$p(n)=p_{0}+\lambda^{n}.$$ Here, $\lambda$ is the average Clifford gate fidelity and can be determined by fitting, whereas $p_0$ is the SPAM error. It turns out practically and can be reasoned analytically that the need for averaging is acceptable, artifacts of ensemble sizes vanish quickly [@Chasseur15]. In this basic version of RB, there are a lot of assumptions that can be questioned. The theory of randomized benchmarking has been extended to adapt most of the demands resulting from weakening these assumptions. We cannot do the vast literature full justice here but mention a few highlights. First of all, standard RB finds the fidelity averaged over the whole Clifford group. If one instead desires to characterize a single Clifford gate, the technique of interleaved randomized benchmarking (IRB) [@Magesan12] can be applied. There, one first performs regular RB. Then, one takes the sequences used for RB and interleaves the desired Clifford gate between any two of the gates from the sequence. The inverse to the resulting sequence needs to be re-computed. The comparison between the interleaved and the regular frequencies gives the average fidelity of that special Clifford gate. In a similar vein, issues like leakage out of the computational subspace, gate-dependent error and others can be taken into account [@Chasseur15], leading to the modern concept of cycle benchmarking. Including non-Clifford gates, however, can only be done at the cost of significant overhead, as the inverting operation is hard to compute as well as hard to invert - it is an arbitrarily quantum gate encompassing the whole system and not part of the Clifford group. A combination of RB with Monte Carlo sampling can be applied to still keep parts of the benefits of IRB [@Chasseur17]. That being said, in many practical architectures, the only non-Clifford gate is the T-gate, a $\pi/4$ z-axis rotation which can be done in software to high precision, so it is not crucial to calibrate it with optimal control. Also, as the two-qubit CNOT gate is a Clifford gate, one cannot claim that natively and without error correction Cliford gates are easier than non-Clifford. Approximating time evolutions with the Magnus expansion \[ch:Magnus\] --------------------------------------------------------------------- Control calculations involve solving the time-dependent Schrödinger equation. While this can be done analytically in, e.g., rotating wave situations or approximations, this can quickly become hard – even for a system as simple as a harmonically driven two-state-system this is a daunting task [@Gangopadhyay2010]. If we would like to proceed analytically with optimal control as far as possible, computing the final gate analytically is a key ingredient to which the Magnus expansion is an important ingredient. Numerically, techniques for coupled ordinary differential equations like Runge-Kutta can be used as well as split-operator techniques. For analytical calculations, one can use the Dyson series familiar from regular advanced quantum mechanics as systematic perturbation theory. In many cases, it is however more effective to use the [Magnus expansion]{}, an asymptotic expansion that used the number of nested commutators as a small parameter. It is exact but usually truncated at low order. Our treatment mostly follows [@WarrenWarren]. The problem at hand is to start from a Hamiltonian that has a (hopefully) large but solvable component and a perturbation $\hat{H}=\hat{H}_{0}(t)+\hat{V}(t)$. A clever choice of this division is key and there is no need for the former to be time-independent. We can transfer to the interaction picture with respect to $\hat{H}_{0}$ . The resulting transformed perturbation $\hat{V}^{I}(t)$ will then acquire additional time-dependence, often in the form of large oscillating terms. The objective is now to approximately calculate the time evolution $$\hat{U}^{I}(t)=\mathbb{T}\exp\left(-\frac{i}{\hbar}\int_{0}^{t}d\tau\ \hat{V}^{I}\left(\tau\right)\right)$$ where $\mathbb{T}$ is the usual time ordering operator. The Dyson expansion of this term starts as $$\hat{U}^{I}\left(t\right)=1-\frac{i}{\hbar}\int_{0}^{t}d\tau\ \hat{V}^{I}\left(\tau\right)-\frac{1}{\hbar^{^{2}}}\int_{0}^{t}d\tau\ \int_{0}^{\tau}d\tau^{\prime}\ \hat{V}^{I}(\tau)\hat{V}^{I}\left(\tau^{\prime}\right)+\dots$$ which we can expect to converge quickly if the perturbation combined with oscillations are so small that the integration over (potentially) long times does not hinder convergence. If this is not the case, one could resort to self-energy techniques as they are known in quantum field theory. For these time-dependent systems, the Magnus expansion is a related route. It provides an expansion $$\hat{U}^{I}\left(t\right)=e^{-i\sum_{n=0}^{\infty}\bar{H}_{n}(t)}\label{eq:Magnus_Expansion}$$ thus truncating this series happens in the exponent and maintains unitarity and is compatible with going to long times. Its lowest orders can be understood as follows: We start with the average Hamiltonian $$\bar{H}_{0}(t)=\int_{0}^{t}d\tau\ \hat{V}^{I}(\tau)$$ i.e. the expression that collects the classical part and ignores all commutators. The next order contains one commutator $$\bar{H}_{1}(t)=-\frac{i}{2}\int_{0}^{t}d\tau_{1}\,d\tau_{2}\ \left[\hat{V}^{I}(\tau_{2}),\hat{V}^{I}\left(\tau_{1}\right)\right]$$ but as it is in the exponent, it collects terms from all orders of the Dyson series (you can convince yourself by expanding the exponential in eq. (\[eq:Magnus\_Expansion\]). The next order of the expansion is $$\bar{H}_{2}\left(t\right)=-\frac{1}{6}\int_{0}^{t}d\tau_{1}\,d\tau_{2}\,d\tau_{3}\,\left\{ \left[\hat{V}^{I}\left(\tau_{3}\right),\left[\hat{V}^{I}\left(\tau_{2}\right),\hat{V}^{I}\left(\tau_{1}\right)\right]\right]+\left[\hat{V}^{I}\left(\tau_{1}\right),\left[\hat{V}^{I}\left(\tau_{2}\right),\hat{V}^{I}\left(\tau_{3}\right)\right]\right]\right\}$$ i.e. it contains two nested commutators . We will only be able to appreciate this expansion when we apply it, but we can already see that the different orders will inherit different operator structures from the different commutators and that stacking on more integrals will create ever more demanding resonance conditions, so higher orders likely oscillate out. That notwithstanding, the Magnus expansion is asymptotic in nature: Its formal radius of convergence is zero hence adding higher orders does not always improve the accuracy. Real-world limitations ---------------------- When applying (quantum) optimal control to real-world systems, we have to contend with the fact that all parameters under our control have practical limitations: power, frequency, timing, etc. are all constrained by the capabilities of the equipment through which we apply said control. Moreover, any feedback scheme (such as Ad-HOC), must account for experimental noise, uncertainties in the experimental system (both gaps in system characterization, and “random walk”-like drifts of experimental parameters) and imperfections in both control and readout. These issues above are complex and have to be dealt with simultaneously in real-word scenarios. There is no known textbook solution to these problems, and they are subject to ongoing research. We shall therefore limit ourselves to a very brief review of some of the approaches currently available: Constraints on applicable controls: Two approaches can be taken: Either the space of possible controls can be defined such all points in the search space are valid, appplicable, controls, or the optimization space is defined more liberally, and we penalize controls which fail to conform. For the first approach, limiting the control subspace, a partial solution is to choose and fix some parameters, such as control field frequency, ahead of time. This is the solution suggested by the CRAB optimal control algorithm [@CRAB]. A more general approach is to use bounded functions, such as cosine or inverse tangent, to transform an unconstrained physical parameter to a constrained one. For example, the search parameter $\alpha$ may be unconstrained and $\mathcal{O}\left(1\right)$, and we transform it to a constrained field amplitude via $A:=500\textrm{MHz}\times\cos\left(\alpha\right)$, which is subsequently used in the system Hamiltonian. Sometimes, the approaches above are insufficient as constraints are complex and include multiple parameters; or perhaps such substitutions are not a good fit to the optimal control problem. In such cases, we can impose a penalty term which will modify the functional for which we seek a minimum. For example, if we wish to impose a low-bandwidth solution on the control field $c\left(t\right)$, we may add a penalty term proportional to $\int_0^T {\left\vert \partial_t c\left(t\right)\right\vert}^2 dt$, which will be significant for highly oscillatory functions and zero for the DC component. Robust controls: Experiments are often noisy environments, which noise appearing both on control fields and on the underlying system Hamiltonians. To provide a control scheme which provides consistently good performance, once must add the robustness requirement of the optimization requirements. This can be done using “ensemble optimization”, where each optimization step averages over multiple manifestations of the dynamics, each with a different noise realization. The specific noise manifestations can be either fixed for the duration of the optimization of varied with each iteration step. The former approach is simpler to implement, but runs the risk of the optimization solving the problem only to the small subset of noises it encountered. The latter approach tends to result in more robust controls, but introduces a noisy goal function, which is harder to optimize reliably. In either case, ensemble optimization tends to be expensive in terms of computational resources. In some cases, it is possible to replace it with the a penalty term which is proportional to the absolute value of the gradient of the standard optimization goal with respect to the noisy variable (i.e. require that the control’s performance will be weakly dependent on the noisy parameter). In all cases, robust controls often exhibit the “no free lunch” rule of control theory – robust controls often require more time, more bandwidth, or provide a worst average-case performance than their non-robust counterparts [@Khani12]. Examples ======== Optimal control of a qubit -------------------------- Let’s start with a really elementary analytical example: A single qubit with Hamiltonian $\hat{H}(t)=u(t)\hat{\sigma}_{x}$ looking at the fastest state transfer possible from $\|0\rangle$ to $e^{i\phi}\|1\rangle$ . We can parameterize the state as $\|\psi\rangle(t)=\left(x_{0}+iy_{0}\right)\|0\rangle+\left(x_{1}+iy_{1}\right)\|1\rangle.$ The Schrödinger equation can be expressed in these real parameters as $$\dot{x}_{0}=uy_{1}\quad\dot{y}_{0}=-ux_{1}\quad\dot{x}_{1}=uy_{0}\quad\dot{y}_{1}=-ux_{0}$$ which are coupled in two sets of two that do not talk to the other components, already telling us that $\phi=\pm\pi/2$. Keep in mind, however, that $u$ can be time-dependent. Now we clearly see that the speed of evolutions scales with the control amplitude $u$ so our initial question was not even well-posed. We need to at least limit the amplitude of the control field. We make this dimensionless $\left\|u\right\|\le u_{{\rm max}}$. The optimal solution exhausts that amplitude and, indeed, plugging in $u=u_{{\rm max}}$ we find $$\ddot{x}_{0}+u_{{\rm max}}^{2}x_{0}=0$$ the harmonic oscillator equation of motion which leads to the desired solution $x_{0}=0$ after time $t_{{\rm min}}=\pi/2u_{max}$ . Solutions of this kind are called “bang” solutions. More generally, in strictly bilinear control problems like this one, the optimal solution jumps between its boundaries (which in the case of multiple controls can be quite intricate), then called “bang-bang”-control. It is interesting to study the physical significance of this result. A real system in its laboratory frame always has an attached drift $$\hat{H}_{1}(t)=\frac{E}{2}\hat{\sigma}_{z}+u(t)\hat{\sigma}_{x}$$ Now if $u_{{\rm max}}\gg\|E\|$ we can expect the previous solution to still hold approximately. If this condition is violated, the situation is different: The vectors $\left(\pm u_{{\rm max}},y,z\right)^{T}$ define two non-collinear axes on the Bloch sphere and a given initial state can reach all final states that are on the circle around that axis including that state. In general, we will need up to three “bangs” to reach out goal. The limitation of $u_{{\rm max}}$ may Exploring the speed limit with high parameter counts ---------------------------------------------------- The [quantum speed limit]{} (QSL) is defined as the minimal time that is needed to evolve a system from a given state $\rho_{0}$ to another state $\rho(t)$ with a specific fidelity $\Phi(\rho_{0,}\rho(t))$ [@QSL0]. This is relevant e.g. for qubit gate implementations, because it limits the minimal gate time (for unrestricted controls). When the control bandwidth is restricted, then the dimension of the set of reachable states $D_{\mathcal{{W}}}$ and the available bandwidth $\Delta\Omega$ give a lower bound for the evolution time [@LLoyd14]: $$T\geq\frac{D_{\mathcal{{W}}}}{\Delta\Omega}$$ This is a continuous version of the Solovay-Kitaev theorem. The set of reachable states consists of all states that can be written as $$\ket{\psi(t)}=U(t_{0},t)\ket{\psi_{0}}$$ where $U(t_{0},t)$ is the propagation operator of the system. A system is called completely controllable if one can choose the control parameters in such a way that the propagation operator is equal to any specific operator [@Controllability]. A method to explore the QSL for a gate is the following [@QSL2]: For different given gate times one optimizes the gate and plots the fidelity $\Phi_{\text{{goal}}}$ or the error $g(T)=1-\Phi_{\text{{goal}}}$ (see ) of the optimized gates against the gate times. If a QSL exists, there will be minimal time for which the error is small. For shorter gate times the error is significantly larger. This time is the QSL. The result depends on the chosen optimization method, concretely we show an example: In fig. \[fig:QSL\_PWC\] and fig. \[fig:QSLFourier\] the error $g$ is plotted against gate duration for two different parameterizations. The system is a CR-gate implementation of a CNOT gate [@QSL3]. ![Gate error as a function of gate time. The optimization was done using GRAPE with a PWC parameterization with 500 pieces. The QSL is around $10$ns.[]{data-label="fig:QSL_PWC"}](quantum_speed_lim_PWC_unbounded) ![Gate error as a function of gate time. The optimization was done using GOAT with a Fourier decomposition into 167 pieces. The QSL is around $40$ns.[]{data-label="fig:QSLFourier"}](quantum_speed_lim_167) In fig. \[fig:QSL\_PWC\] the QSL is shown for a piecewise constant (PWC) parameterization with 500 pieces and unconstrained controls. One can see that there is a jump around 10ns which indicates that this is the QSL in this case. Fig. \[fig:QSLFourier\] shows the same, but with a Fourier decomposition into 167 components. The QSL is here around 40ns and is reached more slowly. The difference is related to optimally have the controls interact with redirecting the drift. A key step to a theory of this phenomenon has been undertaken in [@LLoyd14]. Open systems ------------ In these notes we have mostly concentrated on the optimal control for closed quantum systems. One can ask related questions for open quantum systems as well. A treatment of this situation would go way beyond the scope of these lecture notes. Here, the space of potentially reachable states / of reachable time evolutions is much larger than in the unitary case. The theory of controllability and reachability is thus more involves, it is for example not at all clear, if the impact of decoherence can be reduced to zero, i.e., if the subset of unitary time evolutions is reachable. We would thus like to describe a pragmatic approach and refer the reader to the literature. For a Lindblad equation, it can be shown that the control fields cannot cancel dissipation effect and the system is not completely controllable. This is still an open question in the non-Markovian regime. As a first rule of thumb, there are situations when the decoherence experienced by the quantum subsystem has no or very little structure – e.g. in the case of uniform decoherence leading to a fully depolarizing channel and, at least for the synthesis of gates, for most Markovian decoherence models. These do not give an open system optimal control algorithm any space to actually exploit the structure of the decoherence to perform an optimization, rather, we can expect that the fastest solution of the closed system also is close to an optimal solution for the open system. Thus, running a closed-system version of optimal control and benchmarking it in a realistic open system is a good initial approach. If one suspects that the decoherence mechanism contains exploitable structure, or if one tries to accomplish a task that actively uses decoherence – such as tasks changing the entropy of the state, e.g., cooling, it is possible to generalize the aforementioned methods of optimal control. More specifically, e.g., in [OpenGRAPE]{}, one simply replaces the Schrödinger equation as the dynamical constraint by a suitable description of open systems dynamics, such as a master equation. One caveat lies in the need for backwards-in-time propagation: Open system dynamics is asymptotically irreversible, which can make back-propagation unstable. Practically, this can be handled by either focusing on decoherence rates that are not too large or by suitable initial guesses. As a well-defined example, let us consider a single qubits perturbed by a two-level fluctuator, i.e., a second two-state system that is coupled to a heat bath. This is a common situation in superconducting qubits [@Rebentrost07]. We specifically model a qubit coupled to a single TLF by ${H}={H}_{S}+{H}_{I}+{H}_{B}$. ${H}_{S}$ consists of the qubit and the coupled two-state system, i.e. $${H}_{S}=E_{1}(t){\sigma}_{z}+\Delta{\sigma}_{x}+E_{2}{\tau}_{z}+\Lambda{\sigma}_{z}{\tau}_{z}$$ where ${\sigma}_{i}$ and ${\tau}_{i}$ are the usual Pauli matrices operating in qubit and fluctuator Hilbert space respectively. $E_{1}(t)$ is time-dependent and serves as an external control. The source of decoherence is the coupling of the fluctuator to the heat bath, which leads to incoherent transitions between the fluctuator eigenstates, ${H}_{I}=\sum_{i}\lambda_{i}({\tau}^{+}{b}_{i}+{\tau}^{-}{b}_{i}^{\dagger}),\quad{H}_{B}=\sum_{i}\hbar\omega_{i}b_{i}^{\dagger}b_{i}.$ We introduce an Ohmic bath spectrum $J(\omega)=\sum_{i}\lambda_{i}^{2}\delta(\omega-\omega_{i})=\kappa\omega\Theta(\omega-\omega_{c})$ containing the couplings $\lambda_{i}$, the dimensionless damping $\kappa$, and a high-frequency cutoff $\omega_{c}$ (which we assume to be the largest frequency in the system). Now depending on the bath damping constant $\kappa$the fluctuator can flip fast or slow – and in the limit of slow flipping, the qubit sees noise with strong temporal correlation leading to highly non-Markovian qubit dynamics. To formally treat this system, we can on the other hand still set up a Markovian master equation for the augmented system of qubit and fluctuator and only after its solution trace over the fluctuator to get the effective density matrix of the qubit alone. We formulate the control approach by rewriting the master equation as $\dot{\rho}(t)=-\big(i\mathcal{H}(E_{1}(t))+\Gamma(E_{1}(t))\big)\rho(t)$ with the Hamiltonian commutator superoperator $\mathcal{H}(E_{1}(t))(\cdot)=[H(E_{1}(t)),\cdot]$ and the relaxation superoperator $\Gamma$, both time-dependent via the control $E_{1}(t)$. The formal solution to the master equation is a linear quantum map operating on a physical initial state according to $\rho(t)=F(t)\rho(0)$. Thus $F$ itself follows the operator equation of motion $$\dot{F}=-\left(i\mathcal{H}+\Gamma\right)F\label{eq:FTimeEvolution}$$ with initial condition $F(0)=\mathbb{I}$, as in ref. [@Tosh06]. Here, multiplication of quantum maps denotes their concatenation. The task is to find control amplitudes $E_{1}(t)$ with \$$t\in[0,t_{g}]$\$, \$$t_{g}$\$ being a fixed final time, such that the difference \$$\delta F=F_{U}-F(t_{g})$\$ between dissipative time evolution \$$F(t_{g})$\$ obeying eqn. (\[eq:FTimeEvolution\]) and a target unitary map \$$F_{U}$\$ is minimized with respect to the Euclidean distance $\|\|\delta F\|\|_{2}^{2}\equiv{\rm tr}\left\{ \delta F^{\dagger}\delta F\right\} $. Clearly, this is the case, when the trace fidelity $$\phi={\rm {Re\,tr}\left\{ F_{U}^{\dagger}\;F(t_{g})\right\} }\label{eq:FidelityOpenSystem}$$ is maximal. Note, that in an open system, one cannot expect to achieve zero distance to a unitary evolution $F_{U}$[@Tosh06]. The goal is to come as close as possible. On this setting, we find optimal pulses by gradient search. It is interesting to investigate the resulting pulses and performance limits. We see in figure ... that optimal control pulses allow to reach great gate performance after overcoming a quantum speed limit. Remarkably, the dependence on gate duration is non-monotonic at least in the regime of low $\kappa$ when the two settings of the TLS can be resolved. At some magic times, the frequency split from the TLS naturally refocuses, constraining the optimization much less than at other times. ![Top: Gate error versus pulse time $t_{g}$ for optimal Z-gate pulses in the presence of a non-Markovian environment with dissipation strength $\kappa$. A periodic sequence of minima at around $t_{n}=n\pi/\Delta$, where $n\ge1$, is obtained. Middle: The gate error of optimized pulses approaches a limit set by $T_{1}$ and $2T_{1}$, as shown with $\kappa=0.005$. Bottom: Optimized pulses reduce the error rate by approximately one order of magnitude compared to Rabi pulses for $\kappa=0.005$. Pulses starting from zero bias and with realistic rise times (penalty) require only a small additional gate time. In all figures the system parameters are $E_{2}=0.1\Delta$, $\Lambda=0.1\Delta$ and $T=0.2\Delta$.](0612165/Figure1){width="0.9\linewidth"} More remarkable, the maximally attainable fidelity also has a non-monotonic dependence on $\kappa$. At hindsight, this can be understood as follows: At low $\kappa$ there is no randomness of the system, it is fully reversible. The optimal control algorithm just has to deal with the fact that the setting of the TLS is unknown, which it perfectly accomplishes. On the other hand, at high $\kappa$, the phenomenon of motional narrowing occurs: Fast motion of the impurity broadens its spectrum thus reducing its spectral weight at low frequencies. ![Gate error versus TLF rate $\gamma$ for various temperatures for an optimized pulse with $t_{g}=5.0/\Delta$. The left inset is a magnification of the low-$\gamma$ part of the main plot and reveals the linear behaviour. The right inset shows the maximum of the curves of the main plot versus temperature. ($E_{2}=0.1\Delta$ and $\Lambda=0.1\Delta$](0612165/Figure3){width="0.9\columnwidth"} DRAG and its derivatives ------------------------ In general a quantum system will contain additional states outside of a specific subspace we want to operate in. If our control couples also to transitions out of the subspace we will leak population and degrade the performance of our operation. The [Derivative Removal with Adiabatic Gate (DRAG) ]{}method provides a framework to identify these leakages and to modify the control signals to counteract them. We will review the basic idea along the procedure shown in [@Motzoi2009]. Consider a 3-level-system that is controlled by a signal $u(t)=u_{x}(t)\cos(\omega_{d}t)+u_{y}(t)\sin(\omega_{d}t)$. The first two levels make up the computational subspace $\ket 0,\ket 1$ with transition frequency $\omega_{1}$ that we want to operate in and $\ket 2$ accounts for the leakage. It is modeled by the Hamiltonian $$H/\hbar=\omega_{1}\ketbra 11+(2\omega_{1}+\Delta)\ketbra 22+u(t)\hat{\sigma}_{0,1}^{x}+\lambda u(t)\hat{\sigma}_{1,2}^{x}\label{eq:ham_qutrits}$$ where the Pauli operators are $\hat{\sigma}_{j,k}^{x}=\ketbra jk+\ketbra kj$ and $\lambda$ describes the coupling of the drive to the 1-2 transition. We expressed the second transition frequency by the anharmonicity $\Delta=\omega_{2}-2\omega_{1}$. Let’s say we want to implement a simple Rabi pulse by choosing $u_{x}(t)=\Omega(t)$ and $u_{y}(t)=0$. This gives rise to unwanted leakage out of the computational subspace with the term $\lambda\Omega(t)\hat{\sigma}_{1,2}^{x}$. The DRAG idea shows how we can counteract this leakage by choosing $u_{y}(t)$ appropriately. We first express the Hamiltonian in the rotating frame with $R=\exp(i\omega_{d}\ketbra 11+2i\omega_{d}\ketbra 11)$ following the rule $H^{R}=RHR^{\dagger}+i\hbar\dot{R}R^{\dagger}$ which gives $$H^{R}/\hbar=\delta_{1}\ketbra 11+\delta_{2}\ketbra 22+\sum_{\alpha=x,y}\frac{u_{\alpha}}{2}(t)\hat{\sigma}_{0,1}^{\alpha}+\lambda\frac{u_{\alpha}}{2}(t)\hat{\sigma}_{1,2}^{\alpha}\:,$$ using the detunings $\delta_{1}=\omega_{1}-\omega_{d}$ and $\delta_{2}=\Delta+2\delta_{1}$ between the drive and transition frequencies. Applying an adiabatic transformation $V(t)$ by calculating $H^{V}=VHV^{\dagger}+i\hbar\dot{V}V^{\dagger}$ allows us to look at the system in a frame where the leakage and the $y$-component necessary to counteract it are visible. We take $$V(t)=\exp\left[-i\frac{u_{x}(t)}{2\Delta}(\hat{\sigma}_{0,1}^{y}+\lambda\hat{\sigma}_{1,2}^{y})\right],$$ a transformation that depends on our intended signal $u_{x}$, and apply it to first order in $u_{x}/\Delta$ to find $$\begin{aligned}H^{V}/\hbar & =\left(\delta_{1}-\frac{(\lambda^{2}-4)u_{x}^{2}}{4\Delta}\right)\ketbra 11+\left(\delta_{2}+\frac{(\lambda^{2}+2)u_{x}^{2}}{4\Delta}\right)\ketbra 22\\ & +\frac{u_{x}}{2}\hat{\sigma}_{0,1}^{x}+\lambda\frac{u_{x}^{2}}{8\Delta}\hat{\sigma}_{0,2}^{x}+\left[\frac{u_{y}}{2}+\frac{\dot{u}_{x}}{2\Delta}\right](\hat{\sigma}_{0,1}^{y}+\lambda\hat{\sigma}_{1,2}^{y}) \end{aligned}$$ From this expression we can see that our intended drive is unchanged $u_{x}/2\hat{\sigma}_{0,1}^{x}$ but if we also choose $u_{y}=-\dot{u}_{x}/\Delta$ we cancel the last term that is responsible for driving out of the computational subspace $\propto\lambda\hat{\sigma}_{1,2}^{y}$. The transformation also suggest detuning the drive by $\delta_{1}=(\lambda^{2}-4)u_{x}^{2}/4\Delta$ to avoid stark shifting of the 0-1 transition. This example illustrates the main working principle of DRAG which can be generalized to account for more than just leakage to a third level. By modifying $V(t)$, for example adding terms $\propto\hat{\sigma}_{0,2}^{y}$, or iteratively performing transformations $V_{j}(t)$ the intertial terms, the inertial terms $i\hbar\dot{V}_{j}V_{j}^{\dagger}$ generate more conditions on the control signals and its derivatives. ![\[fig:DRAG\_perform\](a): Performance of non-optimized DRAG variants as a function of gate time, derived from an iterative Schrieffer-Wolff expansion to higher orders. Target : $\hat\sigma_{x}$ rotation of a single qubit described by the lowest three levels of Hamiltonian (\[eq:ham\_qutrits\]). – (b): Performance of the DRAG pulses used in (a) for a fixed gate time $t_{g}=4\pi/\Delta_{2}$ as a function of coupling strength $\lambda$ to the leakage level.](drag/DRAG_performance_combined_iterative_AB.pdf){width="80.00000%"} The performance of solutions to different orders, obtained via iterative transformations, is depicted in Fig.\[fig:DRAG\_perform\]a as a function of pulse length, and in Fig.\[fig:DRAG\_perform\]b as a function of coupling strength $\lambda$ for a fixed gate time $t_{g}=4\pi/\Delta_{2}$. Higher order solutions are taken from [@Motzoi2013]. Note also that when the $\ket 0\leftrightarrow\ket 2$ transition is controlled via an additional corresponding frequency component, exact solutions to the three-level system exist (cf. chapter 8 in [@Motzoi2012]). Turning to the experimental implementation [@Lucero2010; @Chow2010] of DRAG pulses: In practice, actual system parameters differ somewhat from those assumed in theory due to characterization gaps, system drift, or unknown transfer functions affecting the input field shapes [@Motzoi2011]. As a simplification, we assume the low order terms in DRAG are easier to implement as their shape will be mostly maintained on entry into the dilution fridge. Even so, many different low-order variants of DRAG have been found in the literature for third-level leakage [@Motzoi2009; @Gambetta2011a; @Motzoi2013; @Lucero2010]. This reduced functional form can further be optimized theoretically [@Theis2016] and/or through a closed-loop process experimentally [@AdHOC; @ORBIT] to account for the effect of higher order terms and experimental uncertainties (preferably using more advanced gradient-free algorithms such as CMA-ES [@Hansen2003]). A systematic experimental study of the tune-up of the prefactors in front of the functional forms for the control operators was performed in [@Chen2016]. In writing up these optimizations and adapting them, the Magnus expansion, see chapter \[ch:Magnus\] is typically used. ![\[fig:calibration-landscape\]A slice of the 3D calibration landscape for DRAG solution up to the first order in the small parameter to the qubit $\sigma_{x}$-gate leakage problem. Point A and B denote [@Motzoi2009]’s and [@Gambetta2011a]’s first-order solutions, respectively. Point C is the optimum for this control function subspace (here $\alpha_{x}=-0.0069$), with infidelity of $10^{-6.63}$. A successful calibration process will typically start at a known DRAG solution, i.e. points A or B, and conclude in point C. The inset illustrates the associated pulse shapes: markers represent the unoptimized shapes ($u_{x}$: ${\color{blue}\bullet}$, $u_{y}$: ${\color{red}\blacksquare}$, $\delta$: ${\color{green}{\color{lime}{\color{green}\blacklozenge}}}$) whereas solid lines depict the corresponding optimal solution (C).](drag/DRAG_calibration_pulses_inset.pdf){width="0.8\linewidth"} For instance, let us denote the Gaussian pulse implementing a $\hat{\sigma}_{x}$ gate for the qubit by $G(t)$. Then the first order solutions described in [@Motzoi2009; @Gambetta2011a; @Motzoi2013] are parameterized by the limited functional basis $u_{x}\propto G$, $u_{y}\propto\partial_{t}G$ and $\delta\propto G^{2}$, which mimics the limited shaping control that can exist in experiment. None of the reported solutions are optimal within this functional basis: For typical example parameters, infidelities may be further reduced from $10^{-5.28}$ to $10^{-6.63}$ by slightly adjusting the prefactors of the control fields. For example, [@Motzoi2009]’s first order DRAG solution may be transformed according to $u_{x}\to(1+\alpha_{x})u_{x}$ and similarly for $u_{y}$ and $\delta$, and then the constants $\alpha_{x}$, $\alpha_{y}$ and $\alpha_{\delta}$ are optimized. A discussion for why optimization within a severely restricted functional subspace may often be sufficient is given in [@Caneva2011] and follow-up publications. A schematic of the optimization task involved in the calibration, as well as the shape of the associated controls, is shown in Fig.\[fig:calibration-landscape\]. Summary and outlook =================== Optimal control is a mature discipline of theoretical physics and related fields. In experimentation, it has remarkable success in situations in which physical systems are well characterized. Reaching out to engineered systems requires a close integration with characterization and benchmarking. Experimentalists and users of quantum control should have taken home an introduction of concepts, jargon, and results of the field. Theorists should feel motivated to embrace these challenges and to fashion their results into tools that can be used efficiently and scalably so quantum control and quantum technology applications can mutually benefit from their potential. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge collaboration with the optimal control group at Saarland University (and its previous locations), including Daniel Egger, Likun Hu, Kevin Pack, Federico Roy, Ioana Serban, and Lukas Theis as well as continuous collaboration with Tommaso Calarco, Steffen Glaser, Christiane Koch, Simone Montangero, and Thomas Schulte-Herbrüggen. Some of this work is sponsored by the Intelligence Advanced Research Projects Activity (IARPA) through the LogiQ Grant No. W911NF-16-1-0114, by the European Union under OpenSuperQ and the ITN Qusco. [^1]: where female attributions are the default, male is considered included [^2]: but not necessarily output, as the output is typically smoothed and filtered [^3]: although not always optimal, see below [^4]: we will later, under the Magnus expansion, study related steps more carefully [^5]: pronounced with a rolling ’r’ and a voiced ’b’ [^6]: note that CRAB was proposed before AdHOC
--- abstract: 'Parametric Interval Markov Chains ([[[pIMC]{.nodecor}]{}]{}s) are a specification formalism that extend Markov Chains ([[[MC]{.nodecor}]{}]{}s) and Interval Markov Chains ([[[IMC]{.nodecor}]{}]{}s) by taking into account imprecision in the transition probability values: transitions in [[[pIMC]{.nodecor}]{}]{}s are labeled with parametric intervals of probabilities. In this work, we study the difference between [[[pIMC]{.nodecor}]{}]{}s and other Markov Chain abstractions models and investigate the two usual semantics for [[[IMC]{.nodecor}]{}]{}s: once-and-for-all and at-every-step. In particular, we prove that both semantics agree on the maximal/minimal reachability probabilities of a given [[IMC]{.nodecor}]{}. We then investigate solutions to several parameter synthesis problems in the context of [[[pIMC]{.nodecor}]{}]{}s – consistency, qualitative reachability and quantitative reachability – that rely on constraint encodings. Finally, we propose a prototype implementation of our constraint encodings with promising results.' author: - Anicet Bart - Benoît Delahaye - Didier Lime - \| \ Éric Monfroy - Charlotte Truchet bibliography: - 'biblio.bib' title: Reachability in Parametric Interval Markov Chains using Constraints --- Introduction ============ Discrete time Markov chains ([[[MC]{.nodecor}]{}]{}s for short) are a standard probabilistic modeling formalism that has been extensively used in the litterature to reason about software [@whittaker1994markov] and real-life systems [@Husmeier2010]. However, when modeling real-life systems, the exact value of transition probabilities may not be known precisely. Several formalisms abstracting [[[MC]{.nodecor}]{}]{}s have therefore been developed. Parametric Markov chains [@Alur93] ([[[pMC]{.nodecor}]{}]{}s for short) extend [[[MC]{.nodecor}]{}]{}s by allowing parameters to appear in transition probabilities. In this formalism, parameters are variables and transition probabilities may be expressed as polynomials over these variables. A given [[[pMC]{.nodecor}]{}]{} therefore represents a potentially infinite set of [[[MC]{.nodecor}]{}]{}s, obtained by replacing each parameter by a given value. [[[pMC]{.nodecor}]{}]{}s are particularly useful to represent systems where dependencies between transition probabilities are required. Indeed, a given parameter may appear in several dinstinct transition probabilities, therefore requiring that the same value is given to all its occurences. Interval Markov chains [@JonssonL91] ([[[IMC]{.nodecor}]{}]{} for short) extend [[[MC]{.nodecor}]{}]{}s by allowing precise transition probabilities to be replaced by intervals, but cannot represent dependencies between distinct transitions. [[[IMC]{.nodecor}]{}]{}s have mainly been studied with two distinct semantics interpretation. Under the [once-and-for-all]{} semantics, a given [[[IMC]{.nodecor}]{}]{} represents a potentially infinite number of [[[MC]{.nodecor}]{}]{}s where transition probabilities are chosen inside the specified intervals while keeping the same underlying graph structure. The [at-every-step]{} semantics, which was the original semantics given to [[[IMC]{.nodecor}]{}]{}s in [@JonssonL91], does not require [[[MC]{.nodecor}]{}]{}s to preserve the underlying graph structure of the original [[[IMC]{.nodecor}]{}]{} but instead allows an “unfolding” of the original graph structure where different probability values may be chosen (inside the specified interval) at each occurence of the given transition. Model-checking algorithms and tools have been developed in the context of [[[pMC]{.nodecor}]{}]{}s [@Prophesy; @DBLP:conf/cav/HahnHWZ10; @DBLP:conf/cav/KwiatkowskaNP11] and [[[IMC]{.nodecor}]{}]{}s with the once-and-for-all semantics [@Chakraborty2015; @benedikt2013ltl]. State of the art tools [@Prophesy] for [[[pMC]{.nodecor}]{}]{} verification compute a rational function on the parameters that characterizes the probability of satisfying a given property, and then use external tools such as SMT solving [@Prophesy] for computing the satisfying parameter values. For these methods to be viable in practice, the number of parameters used is quite limited. On the other hand, the model-checking procedure for [[[IMC]{.nodecor}]{}]{}s presented in [@benedikt2013ltl] is adapted from machine learning and builds successive refinements of the original [[[IMC]{.nodecor}]{}]{}s that optimize the probability of satisfying the given property. This algorithm converges, but not necessarilly to a global optimum. It is worth noticing that existing model checking procedures for [[[pMC]{.nodecor}]{}]{}s and [[[IMC]{.nodecor}]{}]{}s strongly rely on their underlying graph structure. As a consequence, to the best of our knowledge, no solutions for model-checking [[[IMC]{.nodecor}]{}]{}s with the at-every-step semantics have been proposed yet. In this paper, we focus on Parametric interval Markov chains [@DelahayeLP16] ([[[pIMC]{.nodecor}]{}]{}s for short), that generalize both [[[IMC]{.nodecor}]{}]{}s and [[[pMC]{.nodecor}]{}]{}s by allowing parameters to appear in the endpoints of the intervals specifying transition probabilities, and we provide four main contributions. First, we formally compare abstraction formalisms for [[[MC]{.nodecor}]{}]{}s in terms of succinctness: we show in particular that [[[pIMC]{.nodecor}]{}]{}s are [strictly more succinct]{} than both [[[pMC]{.nodecor}]{}]{}s and [[[pIMC]{.nodecor}]{}]{}s when equipped with the right semantics. In other words, everything that can be expressed using [[[pMC]{.nodecor}]{}]{}s or [[[IMC]{.nodecor}]{}]{}s can also be expressed using [[[pIMC]{.nodecor}]{}]{}s while the reverse does not hold. Second, we prove that the once-and-for-all and the at-every-step semantics are equivalent w.r.t. rechability properties, both in the [[[IMC]{.nodecor}]{}]{} and in the [[[pIMC]{.nodecor}]{}]{} settings. Notably, this result gives theoretical backing to the generalization of existing works on the verification of [[[IMC]{.nodecor}]{}]{}s to the at-every-step semantics. Third, we study the parametric verification of fundamental properties at the [[[pIMC]{.nodecor}]{}]{} level: consistency, qualitative reachability, and quantitative reachability. Given the expressivity of the [[[pIMC]{.nodecor}]{}]{} formalism, the risk of producing a [[[pIMC]{.nodecor}]{}]{} specification that is incoherent and therefore does not model any concrete [[[MC]{.nodecor}]{}]{} is high. We therefore propose constraint encodings for deciding whether a given [[[pIMC]{.nodecor}]{}]{} is consistent and, if so, synthesizing parameter values ensuring consistency. We then extend these encodings to qualitative reachability, [[i.e.]{}, ]{}ensuring that given state labels are reachable in [all]{} (resp. [none]{}) of the [[[MC]{.nodecor}]{}]{}s modeled by a given [[[pIMC]{.nodecor}]{}]{}. Finally, we focus on the quantitative reachability problem, [[i.e.]{}, ]{}synthesizing parameter values such that the probability of reaching given state labels satisfies fixed bounds in [at least one]{} (resp. [all]{}) [[[MC]{.nodecor}]{}]{}s modeled by a given [[[pIMC]{.nodecor}]{}]{}. While consistency and qualitative reachability for [[[pIMC]{.nodecor}]{}]{}s have already been studied in [@DelahayeLP16], the constraint encodings we propose in this paper are significantly smaller (linear instead of exponential). To the best of our knowledge, our results provide the first solution to the quantitative reachability problem for [[[pIMC]{.nodecor}]{}]{}s. Our last contribution is the implementation of all our verification algorithms in a prototype tool that generates the required constraint encodings and can be plugged to any SMT solver for their resolution. Due to space limitation, the proofs of our results are given in Appendix. Background {#sec:background} ========== In this section we introduce notions and notations that will be used throughout the paper. Given a finite set of variables $X = \{x_1, \ldots, x_k\}$, we write $D_x$ for the domain of the variable $x \in X$ and $D_X$ for the set of domains associated to the variables in $X$. A valuation $v$ over $X$ is a set $v = \{(x,d) \| x \in X, d \in D_x\}$ of elementary valuations $(x,d)$ where for each $x \in X$ there exists a unique pair of the form $(x, d)$ in $v$. When clear from the context, we write $v(x) = d$ for the value given to variable $x$ according to valuation $v$. A rational function $f$ over $X$ is a division of two (multivariate) polynomials $g_1$ and $g_2$ over $X$ with rational coefficients, [[i.e.]{}, ]{}$f = g_1 / g_2$. We write ${{\ensuremath{\mathbb{Q}}}}$ the set of rational numbers and ${{\ensuremath{\mathbb{Q}}}}_X$ the set of rational functions over $X$. The evaluation $v(g)$ of a polynomial $g$ under the valuation $v$ replaces each variable $x \in X$ by its value $v(x)$. An [atomic constraint]{} over $X$ is a Boolean expression of the form $f(X) \bowtie g(X)$, with ${\bowtie} \in \{\le, \ge, <, >, =\}$ and $f$ and $g$ two functions over variables in $X$ and constants. A constraint is [linear]{} if the functions $f$ and $g$ are linear. A [constraint]{} over $X$ is a Boolean combination of atomic constraints over $X$. Given a finite set of states $S$, we write ${\ensuremath{\mathsf{Dist}}}(S)$ for the set of probability distributions over $S$, [[i.e.]{}, ]{}the set of functions $\mu : S \to [0,1]$ such that $\sum_{s\in S}\mu(s) = 1$. We write ${\ensuremath{{\mathbb{I}}}}$ for the set containing all the interval subsets of $[0,1]$. In the following, we consider a universal set of symbols $A$ that we use for labelling the states of our structures. We call these symbols [ atomic propositions]{}. We will use Latin alphabet in state context and Greek alphabet in atomic proposition context. [[Constraints.]{}]{} Constraints are first order logic predicates used to model and solve combinatorial problems [@Rossi2006HCP]. A problem is described with a list of variables, each in a given domain of possible values, together with a list of constraints over these variables. Such problems are then sent to solvers which decide whether the problem is satisfiable, [[i.e.]{}, ]{}if there exists a valuation of the variables satisfying all the constraints, and in this case computes a solution. Checking satisfiability of constraint problems is difficult in general, as the space of all possible valuations has a size exponential in the number of variables. Formally, a Constraint Satisfaction Problem ([[CSP]{.nodecor}]{}) is a tuple $\Omega = (X, D, C)$ where $X$ is a finite set of variables, $D = D_X$ is the set of all the domains associated to the variables from $X$, and $C$ is a set of constraints over $X$. We say that a valuation over $X$ satisfies $\Omega$ if and only if it satisfies all the constraints in $C$. We write $v(C)$ for the satisfaction result of the valuation of the constraints $C$ according to $v$ ([[i.e.]{}, ]{}true or false). In the following we call [[[[CSP]{.nodecor}]{}]{} encoding]{} a scheme for formulating a given problem into a [[[CSP]{.nodecor}]{}]{}. The size of a [[[CSP]{.nodecor}]{}]{} corresponds to the number of variables and atomic constraints appearing in the problem. Note that, in constraint programming, having less variables or less constraints during the encoding does not necessarily imply faster solving time of the problems. [[Discrete Time Markov Chains.]{}]{} A Discrete Time Markov Chain ([DTMC]{} or [[[MC]{.nodecor}]{}]{} for short) is a tuple $\mathcal{M}$ $=$ $(S,$ $s_0,$ $p,$ $V)$, where $S$ is a finite set of states containing the initial state $s_0$, $V : S \to 2^A$ is a labelling function, and $p : S \rightarrow {\ensuremath{\mathsf{Dist}}}(S)$ is a probabilistic transition function. We write [[$\tt{MC}$]{}]{} for the set containing all the discrete time Markov chains. A Markov Chain can be seen as a directed graph where the nodes correspond to the states of the [[[MC]{.nodecor}]{}]{} and the edges are labelled with the probabilities given by the transition function of the [[[MC]{.nodecor}]{}]{}. In this representation, a missing transition between two states represents a transition probability of zero. As usual, given a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$, we call a [path]{} of $\mathcal{M}$ a sequence of states obtained from executing $\mathcal{M}$, [[i.e.]{}, ]{}a sequence $\omega = s_1, s_2,\ldots $ s.t. the probability of taking the transition from $s_i$ to $s_{i+1}$ is strictly positive, $p(s_i)(s_i+1) >0$, for all $i$. A path $\omega$ is finite iff it belongs to $S^$, [[i.e.]{}, ]{}it represents a finite sequence of transitions from $\mathcal{M}$. \[ex:mc\] Figure \[fig:example\_mc\] illustrates the Markov chain $\mathcal{M}_1 = (S, s_0, p, V) \in {\ensuremath{\tt{MC}}}$ where the set of states $S$ is given by $\{s_0,s_1,s_2,s_3,s_4\}$, the atomic proposition are restricted to $\{\alpha, \beta\}$, the initial state is $s_0$, and the labelling function $V$ corresponds to $\{(s_0,\emptyset), (s_1,\alpha), (s_2,\beta), (s_3,\{\alpha, \beta\}), (s_4,\alpha)\}$. The sequences of states $(s_0,s_1,s_2)$, $(s_0,s_2)$, and $(s_0,s_2,s_2,s_2)$, are three (finite) paths from the initial state $s_0$ to the state $s_2$. [[Reachability.]{}]{} A Markov chain $\mathcal{M}$ defines a unique probability measure $\mathbb{P}^{\mathcal{M}}$ over the paths from $\mathcal{M}$. According to this measure, the probability of a finite path $\omega = s_0, s_1, \ldots, s_n$ in $\mathcal{M}$ is the product of the probabilities of the transitions executed along this path, [[i.e.]{}, ]{}$\mathbb{P}^{\mathcal{M}}(\omega) = p(s_0)(s_1) \cdot p(s_1)(s_2)\cdot \ldots \cdot p(s_{n-1})(s_n)$. This distribution naturally extends to infinite paths (see [@Baier2008PMC]) and to sequences of states over $S$ that are not paths of $\mathcal{M}$ by giving them a zero probability. Given a [[MC]{.nodecor}]{} $\mathcal{M}$, the overall probability of reaching a given state $s$ from the initial state $s_0$ is called the [ reachability probability]{} and written $\mathbb{P}^{\mathcal{M}}_{s_0}({\ensuremath{\Diamond}}s)$ or $\mathbb{P}^{\mathcal{M}}({\ensuremath{\Diamond}}s)$ when clear from the context. This probability is computed as the sum of the probabilities of all finite paths starting in the initial state and reaching this state for the first time. Formally, let $\mathsf{reach}_{s_0}(s) = \{\omega \in S^{} {\ensuremath{\ \| \ }}\omega = s_0, \ldots s_n \mbox{ with } s_n = s \mbox{ and } s_i \ne s \ \forall 0 \le i < n\}$ be the set of such paths. We then define $\mathbb{P}^{\mathcal{M}}({\ensuremath{\Diamond}}s) = \sum_{\omega \in \mathsf{reach}_{s_0}(s)} \mathbb{P}^{\mathcal{M}}(\omega)$ if $s \ne s_0$ and $1$ otherwise. This notation naturally extends to the reachability probability of a state $s$ from a state $t$ that is not $s_0$, written $\mathbb{P}^{\mathcal{M}}_{t}({\ensuremath{\Diamond}}s)$ and to the probability of reaching a label $\alpha \subseteq A$ written $\mathbb{P}^{\mathcal{M}}_{s_0}({\ensuremath{\Diamond}}\alpha)$. In the following, we say that a state $s$ (resp. a label $\alpha \subseteq A$) is reachable in $\mathcal{M}$ iff the reachability probability of this state (resp. label) from the initial state is strictly positive. In Figure \[fig:example\_mc\] the probability of the path $(s_0,$ $s_2,$ $s_1,$ $s_1,$ $s_3)$ is $0.3 \cdot 0.5 \cdot 0.5 \cdot 0.5 = 0.0375$ and the probability of reaching the state $s_1$ is ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_1}({\ensuremath{\Diamond}}s_1) = p(s_0)(s_1) + \Sigma_{i=0}^{+\infty}{p(s_0)(s_2){\cdot}p(s_2)(s_2)^i{\cdot}p(s_2)(s_1)} = p(s_0)(s_1) + p(s_0)(s_2){\cdot}p(s_2)(s_1){\cdot}(1/(1-p(s_2)(s_2))) = 1$. Furthermore, the probability of reaching $\beta$ corresponds to the probability of reaching the state $s_2$. Markov Chains Abstractions {#sec:abstraction-models} ========================== Modelling an application as a Markov Chain requires knowing the exact probability for each possible transition of the system. However, this can be difficult to compute or to measure in the case of a real-life application ([[e.g.]{}, ]{}precision errors, limited knowledge). In this section, we start with a generic definition of Markov chain abstraction models. Then we recall three abstraction models from the literature, respectively [[[pMC]{.nodecor}]{}]{}, [[[IMC]{.nodecor}]{}]{}, and [[[pIMC]{.nodecor}]{}]{}, and finally we present a comparison of these existing models in terms of succinctness. \[def:abstract\_model\] A Markov chain abstraction model (an abstraction model for short) is a pair $({\ensuremath{{\tt L}}}, {\ensuremath{\models}})$ where ${\ensuremath{{\tt L}}}$ is a nonempty set and ${\ensuremath{\models}}$ is a relation between ${{\ensuremath{\tt{MC}}}}$ and ${\ensuremath{{\tt L}}}$. Let $\mathcal{P}$ be in ${\ensuremath{{\tt L}}}$ and $\mathcal{M}$ be in [[$\tt{MC}$]{}]{} we say that $\mathcal{M}$ implements $\mathcal{P}$ iff $(\mathcal{M}, \mathcal{P})$ belongs to ${\ensuremath{\models}}$ ([[i.e.]{}, ]{}$\mathcal{M} {\ensuremath{\models}}\mathcal{P}$). When the context is clear, we do not mention the satisfaction relation ${\ensuremath{\models}}$ and only use ${\ensuremath{{\tt L}}}$ to refer to the abstraction model $({\ensuremath{{\tt L}}}, {\ensuremath{\models}})$. A [Markov chain Abstraction Model]{} is a specification theory for [[[MC]{.nodecor}]{}]{}s. It consists in a set of abstract objects, called [ specifications]{}, each of which representing a (potentially infinite) set of [[[MC]{.nodecor}]{}]{}s – [implementations]{} – together with a satisfaction relation defining the link between implementations and specifications. As an example, consider the powerset of [[$\tt{MC}$]{}]{} ([[i.e.]{}, ]{}the set containing all the possible sets of Markov chains). Clearly, $(2^{{\ensuremath{\tt{MC}}}}, \in)$ is a Markov chain abstraction model, which we call the [canonical abstraction model]{}. This abstraction model has the advantage of representing all the possible sets of Markov chains but it also has the disadvantage that some Markov chain abstractions are only representable by an infinite extension representation. Indeed, recall that there exists subsets of $[0,1] \subseteq {\ensuremath{\mathbb{R}}}$ which cannot be represented in a finite space (e.g., the Cantor set [@Cantor1883]). We now present existing [[[MC]{.nodecor}]{}]{} abstraction models from the literature. Existing MC Abstraction Models ------------------------------ [[Parametric Markov Chain]{}]{} is a [[[MC]{.nodecor}]{}]{} abstraction model from [@Alur93] where a transition can be annotated by a rational function over [parameters]{}. We write ${\ensuremath{\tt{pMC}}}$ for the set containing all the parametric Markov chains. \[def:pmc\] A Parametric Markov Chain ([[[pMC]{.nodecor}]{}]{} for short) is a tuple $\mathcal{I} = (S,s_0,P,V,Y)$ where $S$, $s_0$, and $V$ are defined as for [[MC]{.nodecor}]{}[s]{}, $Y$ is a set of variables (parameters), and $P: S \times S \to {\ensuremath{\mathbb{Q}}}_Y$ associates with each potential transition a parameterized probability. Let $\mathcal{M} = (S,s_0,p,V)$ be a ${\textnormal{MC}}$ and $\mathcal{I} = (S,s_0,P,V,Y)$ be a ${\textnormal{pMC}}$. The satisfaction relation ${\ensuremath{\models_{\tt{p}}}}$ between ${\ensuremath{\tt{MC}}}$ and ${\ensuremath{\tt{pMC}}}$ is defined by $\mathcal{M} {\ensuremath{\models_{\tt{p}}}}\mathcal{I}$ iff there exists a valuation $v$ of $Y$ s.t. $p(s)(s^\prime)$ equals $v(P(s,s^\prime))$ for all $s,s^\prime$ in $S$. Figure \[fig:example\_pmc\] shows a [[[pMC]{.nodecor}]{}]{} $\mathcal{I}^\prime = (S,s_0,P,V,Y)$ where $S$, $s_0$, and $V$ are similar to the same entities in the [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ from Figure \[fig:example\_mc\], the set of variable $Y$ contains only one variable $p$, and the parametric transitions in $P$ are given by the edge labelling ([[e.g.]{}, ]{}$P(s_0,s_1) = 0.7$, $P(s_1,s_3) = p$, and $P(s_2,s_2) = 1 - p$). Note that the [[[pMC]{.nodecor}]{}]{} $\mathcal{I}^\prime$ is a specification containing the [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ from Figure \[fig:example\_mc\]. [[Interval Markov Chains]{}]{} extend [[[MC]{.nodecor}]{}]{}s by allowing to label transitions with intervals of possible probabilities instead of precise probabilities. We write ${\ensuremath{\tt{IMC}}}$ for the set containing all the interval Markov chains. \[def:imc\] An Interval Markov Chain ([[[IMC]{.nodecor}]{}]{} for short) is a tuple $\mathcal{I} = (S,s_0,P,V)$, where $S$, $s_0$, and $V$ are defined as for [[MC]{.nodecor}]{}[s]{}, and $P : S \times S \to {\ensuremath{{\mathbb{I}}}}$ associates with each potential transition an interval of probabilities. \[ex:imc\] Figure \[fig:example\_imc\] illustrates [[IMC]{.nodecor}]{} $\mathcal{I} = (S, s_0, P, V)$ where $S$, $s_0$, and $V$ are similar to the [[MC]{.nodecor}]{} given in Figure \[fig:example\_mc\]. By observing the edge labelling we see that $P(s_0,s_1)=[0,1]$, $P(s_1,s_1)=[0.5,1]$, and $P(s_3,s_3)=[1, 1]$. On the other hand, the intervals of probability for missing transitions are reduced to $[0,0]$, e.g., $P(s_0,s_0)=[0,0]$, $P(s_0,s_3)=[0,0]$, $P(s_1,s_4)=[0,0]$. In the literature, [[[IMC]{.nodecor}]{}]{}s have been mainly used with two distinct semantics: [at-every-step]{} and [once-and-for-all]{}. Both semantics are associated with distinct satisfaction relations which we now introduce. The [once-and-for-all]{} [[[IMC]{.nodecor}]{}]{} semantics ([@Prophesy; @tulip; @puggelli13]) is alike to the semantics for [[pMC]{.nodecor}]{}, as introduced above. The associated satisfaction relation ${\ensuremath{\models^{o}_{\tt{I}}}}$ is defined as follows: A [[[MC]{.nodecor}]{}]{} $\mathcal{M} = (T, t_0, p, V^M)$ satisfies an [[[IMC]{.nodecor}]{}]{} $\mathcal{I} = (S,s_0,P,V^I)$ iff $(T,t_0,V^M) = (S,s_0,V^I)$ and for all reachable state $s$ and all state $s' \in S$, $p(s)(s') \in P(s,s')$. In this sense, we say that [[[MC]{.nodecor}]{}]{} implementations using the once-and-for-all semantics need to have the same structure as the [[[IMC]{.nodecor}]{}]{} specification. On the other hand, the [at-every-step]{} [[[IMC]{.nodecor}]{}]{} semantics, first introduced in [@JonssonL91], operates as a simulation relation based on the transition probabilities and state labels, and therefore allows [[[MC]{.nodecor}]{}]{} implementations to have a different structure than the [[[IMC]{.nodecor}]{}]{} specification. The associated satisfaction relation ${\ensuremath{\models^{a}_{\tt{I}}}}$ is defined as follows: A [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ $=$ $(T, t_0,$ $p,$ $V^M)$ satisfies an [[[IMC]{.nodecor}]{}]{} $\mathcal{I} = (S,s_0,P,V^I)$ iff there exists a relation $\mathcal{R} \subseteq T \times S$ such that $(t_0, s) \in \mathcal{R}$ and whenever $(t,s) \in \mathcal{R}$, we have the labels of $s$ and $t$ correspond: $V^M(t) = V^I(s)$, there exists a correspondence function $\delta: T \to (S \to [0, 1])$ s.t. $\forall t^\prime \in T$ if $p(t)(t^\prime) > 0$ then $\delta(t^\prime)$ is a distribution on $S$ $\forall s^\prime \in S: (\Sigma_{t^\prime \in T} p(t)(t^\prime) \cdot \delta(t^\prime)(s^\prime)) \in P(s,s^\prime)$, and $\forall (t^\prime,s^\prime) \in T \times S$, if $\delta(t^\prime)(s^\prime) > 0$, then $(t^\prime, s^\prime) \in \mathcal{R}$. By construction, it is clear that ${\ensuremath{\models^{a}_{\tt{I}}}}$ is more general than ${\ensuremath{\models^{o}_{\tt{I}}}}$, [[i.e.]{}, ]{}that whenever $\mathcal{M} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$, we also have $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$. The reverse is obviously not true in general, even when the underlying graphs of $\mathcal{M}$ and $\mathcal{I}$ are isomorphic (see Appendix \[ap:compare\_imcs\_satisfaction\_relations\] for details). \[ex:mcs\_satify\_imc\] Consider the [[[MC]{.nodecor}]{}]{} $\mathcal{M}_1$ with state space $S$ from Figure \[fig:example\_mc\] and the [[[MC]{.nodecor}]{}]{} $\mathcal{M}_2$ with state space $T$ from Figure \[fig:example\_mc\_not\_iso\]. They both satisfy the [[IMC]{.nodecor}]{} $\mathcal{I}$ with state space $S$ given in Figure \[fig:example\_imc\]. Furthermore, $\mathcal{M}_1$ satisfies $\mathcal{I}$ with the same structure. On the other hand, for the [[[MC]{.nodecor}]{}]{} $\mathcal{M}_2$ given in Figure \[fig:example\_mc\_not\_iso\], the state $s_3$ from $\mathcal{I}$ has been “split” into two states $t_3$ and $t_{3^\prime}$ in $\mathcal{M}_2$ and the state $t_1$ from $\mathcal{M}_2$ “aggregates” states $s_1$ and $s_4$ in $\mathcal{I}$. The relation $\mathcal{R} \subseteq T \times S$ containing the pairs $(t_0,s_0)$, $(t_1,s_1)$, $(t_1,s_4)$, $(t_2,s_2)$, $(t_3,s_3)$, and $(t_{3^\prime}, s_3)$ is a satisfaction relation between $\mathcal{M}_2$ and $\mathcal{I}$. [[Parametric Interval Markov Chains]{}]{}, as introduced in [@DelahayeLP16], abstract [[[IMC]{.nodecor}]{}]{}s by allowing (combinations of) parameters to be used as interval endpoints in [[[IMC]{.nodecor}]{}]{}s. Under a given parameter valuation the [[[pIMC]{.nodecor}]{}]{} yields an [[[IMC]{.nodecor}]{}]{} as introduced above. [[[pIMC]{.nodecor}]{}]{}s therefore allow the representation, in a compact way and with a finite structure, of a potentially infinite number of [[[IMC]{.nodecor}]{}]{}s. Note that one parameter can appear in several transitions at once, requiring the associated transition probabilities to depend on one another. Let $Y$ be a finite set of parameters and $v$ be a valuation over $Y$. By combining notations used for ${\textnormal{IMC}}$s and ${\textnormal{pMC}}$s the set ${\ensuremath{{\mathbb{I}}}}({\ensuremath{\mathbb{Q}}}_Y)$ contains all parametrized intervals over $[0,1]$, and for all $I = [f_1, f_2] \in {\ensuremath{{\mathbb{I}}}}({\ensuremath{\mathbb{Q}}}_Y)$, $v(I)$ denotes the interval $[v(f_1), v(f_2)]$ if $0 \le v(f_1) \leq v(f_2) \le 1$ and the empty set otherwise[^1]. We write ${\ensuremath{\tt{pIMC}}}$ for the set containing all the parametric interval Markov chains. \[def:pimc\] A Parametric Interval Markov Chain ([[[pIMC]{.nodecor}]{}]{} for short) is a tuple $\mathcal{P} = (S,s_0,P,V,Y)$, where $S$, $s_0$, $V$ and $Y$ are defined as for [[[pMC]{.nodecor}]{}]{}s, and $P : S \times S \to {\ensuremath{{\mathbb{I}}}}({\ensuremath{\mathbb{Q}}}_Y)$ associates with each potential transition a (parametric) interval. In [@DelahayeLP16] the authors introduced [[[pIMC]{.nodecor}]{}]{}s where parametric interval endpoints are limited to linear combination of parameters. In this paper we extend the [[[pIMC]{.nodecor}]{}]{} model by allowing rational functions over parameters as endpoints of parametric intervals. Given a [[pIMC]{.nodecor}]{} $\mathcal{P} =(S,s_0,P,V,Y)$ and a valuation $v$, we write $v(\mathcal{P})$ for the [[IMC]{.nodecor}]{} $(S,s_0,P_v,V)$ obtained by replacing the transition function $P$ from $\mathcal{P}$ with the function $P_v : S \times S \to {\ensuremath{{\mathbb{I}}}}$ defined by $P_v(s,s^\prime) = v(P(s,s^\prime))$ for all $s,s^\prime \in S$. The [[IMC]{.nodecor}]{} $v(\mathcal{P})$ is called an [instance]{} of [[pIMC]{.nodecor}]{} $\mathcal{P}$. Finally, depending on the semantics chosen for [[[IMC]{.nodecor}]{}]{}s, two satisfaction relations can be defined between [[[MC]{.nodecor}]{}]{}s and [[[pIMC]{.nodecor}]{}]{}s. They are written ${\ensuremath{\models^a_{\tt{pI}}}}$ and ${\ensuremath{\models^o_{\tt{pI}}}}$ and defined as follows: $\mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ (resp. ${\ensuremath{\models^o_{\tt{pI}}}}$) iff there exists an [[IMC]{.nodecor}]{} $\mathcal{I}$ instance of $\mathcal{P}$ s.t. $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$ (resp. ${\ensuremath{\models^{o}_{\tt{I}}}}$). Consider the [[[pIMC]{.nodecor}]{}]{} $\mathcal{P} = (S,$0$,P,V,Y)$ given in Figure \[fig:example\_pimc\]. The set of states $S$ and the labelling function are the same as in the ${{\textnormal{MC}}}$ and the ${{\textnormal{IMC}}}$ presented in Figures \[fig:example\_mc\] and \[fig:example\_imc\] respectively. The set of parameters $Y$ has two elements $p$ and $q$. Finally, the parametric intervals from the transition function $P$ are given by the edge labelling ([[e.g.]{}, ]{}$P(s_1,s_3)=[0.3, q]$, $P(s_2,s_4)=[0,0.5]$, and $P(s_3,s_3)=[1,1]$). Note that the [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$ from Figure \[fig:example\_imc\] is an instance of $\mathcal{P}$ (by assigning the value $0.6$ to the parameter $p$ and $0.5$ to $q$). Furthermore, as said in Example \[ex:mcs\_satify\_imc\], the Markov Chains $\mathcal{M}_1$ and $\mathcal{M}_2$ (from Figures \[fig:example\_mc\] and \[fig:example\_mc\_not\_iso\] respectively) satisfy $\mathcal{I}$, therefore $\mathcal{M}_1$ and $\mathcal{M}_2$ satisfy $\mathcal{P}$. In the following, we consider that the size of a [[[pMC]{.nodecor}]{}]{}, [[[IMC]{.nodecor}]{}]{}, or [[[pIMC]{.nodecor}]{}]{} corresponds to its number of states plus its number of transitions not reduced to $0$, $[0,0]$ or $\emptyset$. We will also often need to consider the predecessors ([${\tt{Pred}}$]{}), and the successors ([${\tt{Succ}}$]{}) of some given states. Given a [[pIMC]{.nodecor}]{} with a set of states $S$, a state $s$ in $S$, and a subset $S^\prime$ of $S$, we write: - [ ${\ensuremath{{\tt{Pred}}}}(s) = \{s^\prime \in S \mid P(s^\prime, s) \notin \{\emptyset, [0, 0]\}\}$]{} - [ ${\ensuremath{{\tt{Succ}}}}(s) = \{s^\prime \in S \mid P(s, s^\prime) \notin \{\emptyset, [0, 0] \}\}$]{} <!-- --> - [ ${\ensuremath{{\tt{Pred}}}}(S^\prime) = \bigcup_{s^\prime \in S^\prime} {\ensuremath{{\tt{Pred}}}}(s^\prime)$]{} - [ ${\ensuremath{{\tt{Succ}}}}(S^\prime) = \bigcup_{s^\prime \in S^\prime} {\ensuremath{{\tt{Succ}}}}(s^\prime)$]{} Abstraction Model Comparisons ----------------------------- ${{\textnormal{IMC}}}$, ${{\textnormal{pMC}}}$, and ${{\textnormal{pIMC}}}$ are three Markov chain Abstraction Models. In order to compare their expressiveness and compactness, we introduce the comparison operators ${\ensuremath{\sqsubseteq}}$ and $\equiv$. Let $({\ensuremath{{\tt L}}}_1, \models_1)$ and $({\ensuremath{{\tt L}}}_2,\models_2\nobreak)$ be two Markov chain abstraction models containing respectively $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $\mathcal{L}_1$ is entailed by $\mathcal{L}_2$, written $\mathcal{L}_1 {\ensuremath{\sqsubseteq}}\mathcal{L}_2$, iff all the [[[MC]{.nodecor}]{}]{}s satisfying $\mathcal{L}_1$ satisfy $\mathcal{L}_2$ modulo bisimilarity. ([[i.e.]{}, ]{} $\forall \mathcal{M} \models_1 \mathcal{L}_1, \exists \mathcal{M}^\prime \models_2 \mathcal{L}_2$ s.t. $\mathcal{M}$ is bisimilar to $\mathcal{M}^\prime$). We say that $\mathcal{L}_1$ is (semantically) equivalent to $\mathcal{L}_2$, written $\mathcal{L}_1 \equiv \mathcal{L}_2$, iff $\mathcal{L}_1 {\ensuremath{\sqsubseteq}}\mathcal{L}_2$ and $\mathcal{L}_2 {\ensuremath{\sqsubseteq}}\mathcal{L}_1$. Definition \[def:succinctness\] introduces succinctness based on the sizes of the abstractions. \[def:succinctness\] Let $({\ensuremath{{\tt L}}}_1, \models_1)$ and $({\ensuremath{{\tt L}}}_2, \models_2)$ be two Markov chain abstraction models. ${\ensuremath{{\tt L}}}_1$ is at least as succinct as ${\ensuremath{{\tt L}}}_2$, written ${\ensuremath{{\tt L}}}_1 \leq {\ensuremath{{\tt L}}}_2$, iff there exists a polynomial $p$ such that for every $\mathcal{L}_2 \in {\ensuremath{{\tt L}}}_2$, there exists $\mathcal{L}_1 \in {\ensuremath{{\tt L}}}_1$ s.t. $\mathcal{L}_1 \equiv \mathcal{L}_2$ and $\|\mathcal{L}_1\| \leq p(\|\mathcal{L}_2\|)$.[^2] Thus, ${\ensuremath{{\tt L}}}_1$ is strictly more succinct than ${\ensuremath{{\tt L}}}_2$, written ${\ensuremath{{\tt L}}}_1 < {\ensuremath{{\tt L}}}_2$, iff ${\ensuremath{{\tt L}}}_1 \leq {\ensuremath{{\tt L}}}_2$ and ${\ensuremath{{\tt L}}}_2 \not\leq {\ensuremath{{\tt L}}}_1$. We start with a comparison of the succinctness of the [[[pMC]{.nodecor}]{}]{} and [[[IMC]{.nodecor}]{}]{} abstractions. Since [[[pMC]{.nodecor}]{}]{}s allow the expression of dependencies between the probabilities assigned to distinct transitions while [[[IMC]{.nodecor}]{}]{}s allow all transitions to be independant, it is clear that there are [[[pMC]{.nodecor}]{}]{}s without any equivalent [[[IMC]{.nodecor}]{}]{}s (regardless of the [[[IMC]{.nodecor}]{}]{} semantics used), therefore $({\ensuremath{\tt{IMC}}},{\ensuremath{\models^{o}_{\tt{I}}}}) \not \le {\ensuremath{\tt{pMC}}}$ and $({\ensuremath{\tt{IMC}}},{\ensuremath{\models^{a}_{\tt{I}}}}) \not \le {\ensuremath{\tt{pMC}}}$. On the other hand, [[[IMC]{.nodecor}]{}]{}s imply that transition probabilities need to satisfy linear inequalities in order to fit given intervals. However, these types of constraints are not allowed in [[[pMC]{.nodecor}]{}]{}s. It is therefore easy to exhibit [[[IMC]{.nodecor}]{}]{}s that, regardless of the semantics considered, do not have any equivalent [[[pMC]{.nodecor}]{}]{} specification. As a consequence, $ {\ensuremath{\tt{pMC}}}\not \le ({\ensuremath{\tt{IMC}}},{\ensuremath{\models^{o}_{\tt{I}}}})$ and $ {\ensuremath{\tt{pMC}}}\not \le ({\ensuremath{\tt{IMC}}},{\ensuremath{\models^{a}_{\tt{I}}}})$. We now compare [[[pMC]{.nodecor}]{}]{}s and [[[IMC]{.nodecor}]{}]{}s to [[[pIMC]{.nodecor}]{}]{}s. Recall that the [[$\tt{pIMC}$]{}]{} model is a Markov chain abstraction model allowing to declare parametric interval transitions, while the [[$\tt{pMC}$]{}]{} model allows only parametric transitions (without intervals), and the [[$\tt{IMC}$]{}]{} model allows interval transitions without parameters. Clearly, any [[[pMC]{.nodecor}]{}]{} and any [[[IMC]{.nodecor}]{}]{} can be translated into a [[[pIMC]{.nodecor}]{}]{} with the right semantics (once-and-for-all for [[[pMC]{.nodecor}]{}]{}s and the chosen [[[IMC]{.nodecor}]{}]{} semantics for [[[IMC]{.nodecor}]{}]{}s). This means that $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}})$ is more succinct than [[$\tt{pMC}$]{}]{} and [[$\tt{pIMC}$]{}]{} is more succinct than [[$\tt{IMC}$]{}]{} for both semantics. Furthermore, since [[$\tt{pMC}$]{}]{} and [[$\tt{IMC}$]{}]{} are not comparable due to the above results, we have that the [[$\tt{pIMC}$]{}]{} abstraction model is strictly more succinct than the [[$\tt{pMC}$]{}]{} abstraction model and than the [[$\tt{IMC}$]{}]{} abstraction model with the right semantics. Our comparison results are presented in Proposition \[prop:sunccinctness\_pimc\_pmc\_imc\]. Further explanations and examples are given in Appendix \[ap:model\_comparison\]. \[prop:sunccinctness\_pimc\_pmc\_imc\] The Markov chain abstraction models can be ordered as follows w.r.t. succinctness: $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}}) < ({\ensuremath{\tt{pMC}}}, {\ensuremath{\models_{\tt{p}}}})$, $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}}) < ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}})$ and $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^a_{\tt{pI}}}}\nobreak) < ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}})$. Note that $({\ensuremath{\tt{pMC}}}, {\ensuremath{\models_{\tt{p}}}}) \le ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}})$ could be achieved by adding unary constraints on the parameters of a [[[pMC]{.nodecor}]{}]{}, which is not allowed here. However, this would not have any impact on our other results. Qualitative Properties {#sec:qualitative-reachability} ====================== As seen above, [[[pIMC]{.nodecor}]{}]{}s are a succinct abstraction formalism for [[[MC]{.nodecor}]{}]{}s. The aim of this section is to investigate qualitative properties for [[[pIMC]{.nodecor}]{}]{}s, [[i.e.]{}, ]{}properties that can be evaluated at the specification ([[[pIMC]{.nodecor}]{}]{}) level, but that entail properties on its [[[MC]{.nodecor}]{}]{} implementations. [[[pIMC]{.nodecor}]{}]{} specifications are very expressive as they allow the abstraction of transition probabilities using both intervals and parameters. Unfortunately, as it is the case for [[[IMC]{.nodecor}]{}]{}s, this allows the expression of incorrect specifications. In the [[[IMC]{.nodecor}]{}]{} setting, this is the case either when some intervals are ill-formed or when there is no probability distribution matching the interval constraints of the outgoing transitions of some reachable state. In this case, no [[[MC]{.nodecor}]{}]{} implementation exists that satisfies the [[[IMC]{.nodecor}]{}]{} specification. Deciding whether an implementation that satisfies a given specification exists is called the consistency problem. In the [[[pIMC]{.nodecor}]{}]{} setting, the consistency problem is made more complex because of the parameters which can also induce inconsistencies in some cases. One could also be interested in verifying whether there exists an implementation that reaches some target states/labels, and if so, propose a parameter valuation ensuring this property. Both the consistency and the consistent reachability problems have already been investigated in the [[[IMC]{.nodecor}]{}]{} and [[[pIMC]{.nodecor}]{}]{} setting [@Delahaye15; @DelahayeLP16]. In this section, we briefly recall these problems and propose new solutions based on CSP encodings. Our encodings are linear in the size of the original [[[pIMC]{.nodecor}]{}]{}s whereas the algorithms from [@Delahaye15; @DelahayeLP16] are exponential. Existential Consistency ----------------------- A [[[pIMC]{.nodecor}]{}]{} $\mathcal{P}$ is existential consistent iff there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{P}$ ([[i.e.]{}, ]{}there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying an [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$ instance of $\mathcal{P}$). As seen in Section \[sec:background\], [[[pIMC]{.nodecor}]{}]{}s are equipped with two semantics: once-and-for-all (${\ensuremath{\models^o_{\tt{pI}}}}$) and at-every-step (${\ensuremath{\models^a_{\tt{pI}}}}$). Recall that ${\ensuremath{\models^o_{\tt{pI}}}}$ imposes that the underlying graph structure of implementations needs to be isomorphic to the graph structure of the corresponding specification. In contrast, ${\ensuremath{\models^a_{\tt{pI}}}}$ allows implementations to have a different graph structure. It therefore seems that some [[[pIMC]{.nodecor}]{}]{}s could be inconsistent w.r.t ${\ensuremath{\models^o_{\tt{pI}}}}$ while being consistent w.r.t ${\ensuremath{\models^a_{\tt{pI}}}}$. On the other hand, checking the consistency w.r.t ${\ensuremath{\models^o_{\tt{pI}}}}$ seems easier because of the fixed graph structure. In [@Delahaye15], the author firstly proved that both semantics are equivalent w.r.t. existential consistency, and proposed a [[[CSP]{.nodecor}]{}]{} encoding for verifying this property which is exponential in the size of the [[[pIMC]{.nodecor}]{}]{}. Based on this result of semantics equivalence w.r.t. existential consistency from [@Delahaye15] we propose a new [[[CSP]{.nodecor}]{}]{} encoding, written [[${\bf C_{{\exists}c}}$]{}]{}, for verifying the existential consistency property for [[[pIMC]{.nodecor}]{}]{}s. Let $\mathcal{P}$ $=$ $(S,$$s_0,$$P,$$V,$$Y)$ be a [[[pIMC]{.nodecor}]{}]{}, we write [${\bf C_{{\exists}c}}$]{}($\mathcal{P}$) for the [[[CSP]{.nodecor}]{}]{} produced by [[${\bf C_{{\exists}c}}$]{}]{} according to $\mathcal{P}$. Any solution of [${\bf C_{{\exists}c}}$]{}($\mathcal{P}$) will correspond to a [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{P}$. In [${\bf C_{{\exists}c}}$]{}($\mathcal{P}$), we use one variable $\pi_p$ with domain $[0,1]$ per parameter $p$ in $Y$; one variable ${\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}}$ with domain $[0, 1]$ per transition $(s, s^\prime)$ in $\{ \{s\} \times {\ensuremath{{\tt{Succ}}}}(s) \mid s \in S\}$; and one Boolean variable $\rho_s$ per state $s$ in $S$. These Boolean variables will indicate for each state whether it appears in the [[[MC]{.nodecor}]{}]{} solution of the [[[CSP]{.nodecor}]{}]{} ([[i.e.]{}, ]{}in the [[[MC]{.nodecor}]{}]{} satisfying the [[[pIMC]{.nodecor}]{}]{} $\mathcal{P}$). For each state $s \in S$, Constraints are as follows: 1. [$\rho_{s}$, if $s = s_0$]{}\[encoding\_ec\_init\_state\] 2. [$\neg \rho_s \Leftrightarrow \Sigma_{s^\prime \in {\ensuremath{{\tt{Pred}}}}(s) \setminus \{s\}} {\ensuremath{{\ensuremath{\theta_{s^\prime}}}^{s}}} = 0$, if $s \ne s_0$]{}\[encoding\_ec\_cstr\_reach\_propag\] 3. [ $\rho_s \Rightarrow {\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} \in P(s,s^\prime)$, for all $s^\prime \in {\ensuremath{{\tt{Succ}}}}(s)$]{}\[encoding\_ec\_cstr\_intervals\] <!-- --> 1. [ $\rho_s \Leftrightarrow \Sigma_{s^\prime \in {\ensuremath{{\tt{Succ}}}}(s)} {\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} = 1$]{}\[encoding\_ec\_sum\_to\_one\] 2. [ $\neg \rho_s \Leftrightarrow \Sigma_{s^\prime \in {\ensuremath{{\tt{Succ}}}}(s)} {\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} = 0$]{}\[encoding\_ec\_sum\_to\_zero\] Recall that given a [[[pIMC]{.nodecor}]{}]{} $\mathcal{P}$ the objective of the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$ is to construct a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{P}$. Constraint \[encoding\_ec\_init\_state\] states that the initial state $s_0$ appears in $\mathcal{M}$. Constraint \[encoding\_ec\_cstr\_reach\_propag\] ensures that for each non-initial state $s$, variable $\rho_s$ is set to [[$\mathsf{false}$]{}]{} iff $s$ is not reachable from its predecessors. Constraint \[encoding\_ec\_sum\_to\_one\] ensures that if a state $s$ appears in $\mathcal{M}$, then its outgoing transitions form a probability distribution. On the contrary, Constraint \[encoding\_ec\_sum\_to\_zero\] propagates non-appearing states ([[i.e.]{}, ]{}if a state $s$ does not appear in $\mathcal{M}$ then all its outgoing transitions are set to zero). Finally, Constraint \[encoding\_ec\_cstr\_intervals\] states that, for all appearing states, the outgoing transition probabilities must be selected inside the specified intervals. \[ex:model\_consistency\] Consider the [[[pIMC]{.nodecor}]{}]{} $\mathcal{P}$ given in Figure \[fig:example\_pimc\]. Figure \[fig:variables\_consistency\] describes the variables in ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$: one variable per transition ([[e.g.]{}, ]{}${\ensuremath{{\ensuremath{\theta_{0}}}^{1}}}$, ${\ensuremath{{\ensuremath{\theta_{0}}}^{2}}}$, ${\ensuremath{{\ensuremath{\theta_{1}}}^{1}}}$), one Boolean variable per state ([[e.g.]{}, ]{}$\rho_0$, $\rho_1$), and one variable per parameter ($\pi_p$ and $\pi_q$). The following constraints correspond to the Constraints \[encoding\_ec\_sum\_to\_one\], \[encoding\_ec\_cstr\_reach\_propag\], \[encoding\_ec\_sum\_to\_zero\], and \[encoding\_ec\_cstr\_intervals\] generated by our encoding ${\ensuremath{{\bf C_{{\exists}c}}}}$ for the state $2$ of $\mathcal{P}$: 1. $\neg \rho_2 \Leftrightarrow {\ensuremath{{\ensuremath{\theta_{0}}}^{2}}} = 0$ 2. $\neg \rho_2 \Leftrightarrow {\ensuremath{{\ensuremath{\theta_{2}}}^{1}}} + {\ensuremath{{\ensuremath{\theta_{2}}}^{2}}} + {\ensuremath{{\ensuremath{\theta_{2}}}^{4}}} = 0$ <!-- --> 1. $\rho_2 \Leftrightarrow {\ensuremath{{\ensuremath{\theta_{2}}}^{1}}} + {\ensuremath{{\ensuremath{\theta_{2}}}^{2}}} + {\ensuremath{{\ensuremath{\theta_{2}}}^{4}}} = 1$ 2. $\rho_2 \Rightarrow 0 \leq {\ensuremath{{\ensuremath{\theta_{2}}}^{1}}} \leq \pi_p$ <!-- --> 1. $\rho_2 \Rightarrow 0.2 \leq {\ensuremath{{\ensuremath{\theta_{2}}}^{2}}} \leq \pi_p$ 2. $\rho_2 \Rightarrow 0 \leq {\ensuremath{{\ensuremath{\theta_{2}}}^{4}}} \leq 0.5$ Finally, Figure \[fig:solution\_consistency\] describes a solution for the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$. Note that given a solution of a [[[pIMC]{.nodecor}]{}]{} encoded by ${\ensuremath{{\bf C_{{\exists}c}}}}$, one can construct a [[[MC]{.nodecor}]{}]{} satisfying the given [[[pIMC]{.nodecor}]{}]{} by keeping all the states $s$ s.t. $\rho_s$ is equal to [[$\mathsf{true}$]{}]{} and considering the transition function given by the probabilities in the ${\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}}$ variables. We now show that our encoding works as expected. \[prop:csp\_existential\_consistency\] A [[[pIMC]{.nodecor}]{}]{} $\mathcal{P}$ is existential consistent iff ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$ is satisfiable. Our existential consistency encoding is linear in the size of the [[[pIMC]{.nodecor}]{}]{} instead of exponential for the encoding from [@DelahayeLP16] which enumerates the powerset of the states in the [[[pIMC]{.nodecor}]{}]{} resulting in deep nesting of conjunctions and disjunctions. Qualitative Reachability ------------------------ Let $\mathcal{P} = (S,s_0,P,V,Y)$ be a [[pIMC]{.nodecor}]{} and $\alpha \subseteq A$ be a state label. We say that $\alpha$ is [existential reachable]{} in $\mathcal{P}$ iff there exists an implementation $\mathcal{M}$ of $\mathcal{P}$ where $\alpha$ is reachable ([[i.e.]{}, ]{}${\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)>0$). In a dual way, we say that $\alpha$ is [universal reachable]{} in $\mathcal{P}$ iff $\alpha$ is reachable in any implementation $\mathcal{M}$ of $\mathcal{P}$. As for existential consistency, we use a result from [@Delahaye15] that states that both [[[pIMC]{.nodecor}]{}]{} semantics are equivalent w.r.t. existential (and universal) reachability. We therefore propose a new CSP encoding, written ${\ensuremath{{\bf C_{{\exists}r}}}}$, that extends ${\ensuremath{{\bf C_{{\exists}c}}}}$, for verifying these properties. Formally, [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P}) = (X \cup X^\prime,D \cup D^\prime,C \cup C^\prime)$ is such that $(X,D,C) = {\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$, $X^\prime$ contains one integer variable $\omega_s$ with domain $[0, \|S\|]$ per state $s$ in $S$, $D^\prime$ contains the domains of these variables, and $C^\prime$ is composed of the following constraints for each state $s \in S$: 1. [$\omega_{s} = 1$, if $s = s_0$]{}\[encoding\_er\_init\_state\] <!-- --> 1. [ $\omega_s \neq 1$, if $s \neq s_0$]{}\[encoding\_er\_non\_init\_state\] <!-- --> 1. [ $\rho_s \Leftrightarrow (\omega_s \neq 0)$]{}\[encoding\_er\_bool\_var\] <!-- --> 1. [ $\omega_s > 1 \Rightarrow \bigvee_{s^\prime \in {\ensuremath{{\tt{Pred}}}}(s) \setminus \{ s \} }(\omega_s = \omega_{s^\prime} + 1) \wedge ({\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} > 0)$, if $s \neq s_0$]{}\[encoding\_er\_propag\_reach\] 2. [ $\omega_s = 0 \Leftrightarrow \bigwedge_{s^\prime \in {\ensuremath{{\tt{Pred}}}}(s) \setminus \{ s \} }(\omega_{s^\prime} = 0) \vee ({\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} = 0)$, if $s \neq s_0$]{}\[encoding\_er\_propag\_non\_reach\] Recall first that [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}c}}}}(P)$ constructs a Markov chain $\mathcal{M}$ satisfying $\mathcal{P}$. Informally, for each state $s$ in $\mathcal{M}$ the Constraints \[encoding\_er\_init\_state\], \[encoding\_er\_non\_init\_state\], \[encoding\_er\_propag\_reach\] and \[encoding\_er\_propag\_non\_reach\] in ${\ensuremath{{\bf C_{{\exists}r}}}}$ ensure that $\omega_s = k$ iff there exists in $\mathcal{M}$ a path from the initial state to $s$ of length $k-1$ with non zero probability; and state $s$ is not reachable in $\mathcal{M}$ from the initial state $s_0$ iff $\omega_s$ equals to $0$. Finally, Constraint \[encoding\_er\_bool\_var\] enforces the Boolean reachability indicator variable $\rho_s$ to bet set to [[$\mathsf{true}$]{}]{} iff there exists a path with non zero probability in $\mathcal{M}$ from the initial state $s_0$ to $s$ ([[i.e.]{}, ]{}$\omega_s \neq 0$). Let $S_\alpha$ be the set of states from $\mathcal{P}$ labeled with $\alpha$. ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$ therefore produces a Markov chain satisfying $\mathcal{P}$ where reachable states $s$ are such that $\rho_s = {\ensuremath{\mathsf{true}}}$. As a consequence, $\alpha$ is existential reachable in $\mathcal{P}$ iff ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$ admits a solution such that $\bigvee_{s \in S_\alpha} \rho_s$; and $\alpha$ is universal reachable in $\mathcal{P}$ iff ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$ admits no solution such that $\bigwedge_{s \in S_\alpha} \neg\rho_s$. This is formalised in the following proposition. \[prop:model\_existential\_reachability\] Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[pIMC]{.nodecor}]{}, $\alpha \subseteq A$ be a state label, $S_\alpha = \{s \ \| \ V(s) = \alpha\}$, and $(X,D,C)$ be the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$. - [[CSP]{.nodecor}]{} $(X,D,C \cup \bigvee_{s \in S_\alpha} \rho_s)$ is satisfiable iff $\alpha$ is existential reachable in $\mathcal{P}$ - [[CSP]{.nodecor}]{} $(X,D,C \cup \bigwedge_{s \in S_\alpha} \neg\rho_s)$ is unsatisfiable iff $\alpha$ is universal reachable in $\mathcal{P}$ As for the existential consistency problem, we have an exponential gain in terms of size of the encoding compared to [@DelahayeLP16]: the number of constraints and variables in [[${\bf C_{{\exists}r}}$]{}]{} is linear in terms of the size of the encoded [[[pIMC]{.nodecor}]{}]{}. [[Remark.]{}]{} In ${\ensuremath{{\bf C_{{\exists}r}}}}$ Constraints \[encoding\_ec\_cstr\_reach\_propag\] inherited from ${\ensuremath{{\bf C_{{\exists}c}}}}$ are entailed by Constraints \[encoding\_er\_bool\_var\] and \[encoding\_er\_propag\_non\_reach\] added to ${\ensuremath{{\bf C_{{\exists}r}}}}$. Thus, in a practical approach one may ignore Constraints \[encoding\_ec\_cstr\_reach\_propag\] from ${\ensuremath{{\bf C_{{\exists}c}}}}$ if they do not improve the solver performances. Quantitative Properties ======================= \[sec:quantitative\] We now move to the verification of quantitative reachability properties in [[[pIMC]{.nodecor}]{}]{}s. Quantitative reachability has already been investigated in the context of [[[pMC]{.nodecor}]{}]{}s and [[[IMC]{.nodecor}]{}]{}s with the once-and-for-all semantics. Due to the complexity of allowing implementation structures to differ from the structure of the specifications, quantitative reachability in [[[IMC]{.nodecor}]{}]{}s with the at-every-step semantics has, to the best of our knowledge, never been studied. In this section, we propose our main theoretical contribution: a theorem showing that both [[[IMC]{.nodecor}]{}]{} semantics are equivalent with respect to quantitative reachability, which allows the extension of all results from [@tulip; @benedikt2013ltl] to the at-every-step semantics. Based on this result, we also extend the CSP encodings introduced in Section \[sec:qualitative-reachability\] in order to solve quantitative reachability properties on [[[pIMC]{.nodecor}]{}]{}s regardless of their semantics. Equivalence of ${\ensuremath{\models^{o}_{\tt{I}}}}$ and ${\ensuremath{\models^{a}_{\tt{I}}}}$ w.r.t quantitative reachability {#sec:equiv_imc_semantics} ------------------------------------------------------------------------------------------------------------------------------ Given an [[[IMC]{.nodecor}]{}]{} $\mathcal{I} = (S,s_0,P,V)$ and a state label $\alpha \subseteq A$, a quantitative reachability property on $\mathcal{I}$ is a property of the type $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$, where $0<p<1$ and ${\sim} \in \{\le, <, >, \ge\}$. Such a property is verified iff there exists an [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{I}$ (with the chosen semantics) such that $\mathbb{P}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha) {\sim} p$. As explained above, all existing techniques and tools for verifying quantitative reachability properties on [[[IMC]{.nodecor}]{}]{}s only focus on the once-and-for-all semantics. Indeed, in this setting, quantitative reachability properties are easier to compute because the underlying graph structure of all implementations is known. However, to the best of our knowledge, there are no works addressing the same problem with the at-every-step semantics or showing that addressing the problem in the once-and-for-all setting is sufficiently general. The following theorem fills this theoretical gap by proving that both semantics are equivalent w.r.t quantitative reachability. In other words, for all [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ such that $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$ and all state label $\alpha$, there exist [[[MC]{.nodecor}]{}]{}s $\mathcal{M}_\le$ and $\mathcal{M}_{\ge}$ such that $\mathcal{M}_{\le} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$, $\mathcal{M}_{\ge} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$ and $\mathbb{P}^{\mathcal{M}_{\le}}({\ensuremath{\Diamond}}\alpha) \le \mathbb{P}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha) \le \mathbb{P}^{\mathcal{M_{\ge}}}({\ensuremath{\Diamond}}\alpha)$. This is formalized in the following theorem. \[thm:reachability-semantics-equivalence-imcs\] Let $\mathcal{I} = (S,s_0,P,V)$ be an [[[IMC]{.nodecor}]{}]{}, $\alpha \subseteq A$ be a state label, ${\sim} \in \{\le,<,>,\ge\}$ and $0<p<1$. $\mathcal{I}$ satisfies $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ with the once-and-for-all semantics iff $\mathcal{I}$ satisfies $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ with the at-every-step semantics. The proof is constructive (see Appendix \[ap:equiv\_imc\_semantics\]): we use the structure of the relation $\mathcal{R}$ from the definition of ${\ensuremath{\models^{a}_{\tt{I}}}}$ in order to build the [[[MC]{.nodecor}]{}]{}s $\mathcal{M}_{\le}$ and $\mathcal{M}_{\ge}$. Constraint Encodings -------------------- Note that the result from Theorem \[thm:reachability-semantics-equivalence-imcs\] naturally extends to [[[pIMC]{.nodecor}]{}]{}s. We therefore exploit this result to construct a [[[CSP]{.nodecor}]{}]{} encoding for verifying quantitative reachability properties in [[[pIMC]{.nodecor}]{}]{}s. As in Section \[sec:qualitative-reachability\], we extend the CSP ${\ensuremath{{\bf C_{{\exists}c}}}}$, that produces a correct ${\textnormal{MC}}$ implementation for the given [[[pIMC]{.nodecor}]{}]{}, by imposing that this ${\textnormal{MC}}$ implementation satisfies the given quantitative reachability property. In order to compute the probability of reaching state label $\alpha$ at the [[[MC]{.nodecor}]{}]{} level, we use standard techniques from [@Baier2008PMC] that require the partitioning of the state space into three sets $S_{\top}$, $S_{\bot}$, and $S_?$ that correspond to states reaching $\alpha$ with probability $1$, states from which $\alpha$ cannot be reached, and the remaining states, respectively. Once this partition is chosen, the reachability probabilities of all states in $S_?$ are computed as the unique solution of a linear equation system (see [@Baier2008PMC], Theorem 10.19, p.766). We now explain how we identify states from $S_\bot, S_\top$ and $S_?$ and how we encode the linear equation system, which leads to the resolution of quantitative reachability. Let $\mathcal{P} = (S,s_0,P,V,Y)$ be a [[pIMC]{.nodecor}]{} and $\alpha \subseteq A$ be a state label. We start by setting $S_\top = \{s \ \|\ V(s) = \alpha\}$. We then extend ${\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$ in order to identify the set $S_\bot$. Let ${\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}(\mathcal{P}, \alpha) = (X \cup X^\prime,D \cup D^\prime,C \cup C^\prime)$ be such that $(X,D,C) = {\ensuremath{{\bf C_{{\exists}r}}}}(\mathcal{P})$, $X^\prime$ contains one Boolean variable $\lambda_s$ and one integer variable $\alpha_s$ with domain $[0, \|S\|]$ per state $s$ in $S$, $D^\prime$ contains the domains of these variables, and $C^\prime$ is composed of the following constraints for each state $s \in S$: 1. [$\alpha_s = 1$, if $\alpha = V(s)$]{}\[encoding\_erprime\_target\_state\] <!-- --> 1. [ $\alpha_s \neq 1$, if $\alpha \ne V(s)$]{}\[encoding\_erprime\_non\_target\_state\] <!-- --> 1. [ $\lambda_s \Leftrightarrow (\rho_s \wedge (\alpha_s \neq 0))$]{}\[encoding\_erprime\_bool\_var\] <!-- --> 1. [ $\alpha_s > 1 \Rightarrow \bigvee_{s^\prime \in {\ensuremath{{\tt{Succ}}}}(s) \setminus \{ s \} }(\alpha_s = \alpha_{s^\prime} + 1) \wedge ({\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} > 0)$, if $\alpha \ne V(s)$]{}\[encoding\_erprime\_propag\_target\] 2. [ $\alpha_s = 0 \Leftrightarrow \bigwedge_{s^\prime \in {\ensuremath{{\tt{Succ}}}}(s) \setminus \{ s \} }(\alpha_{s^\prime} = 0) \vee ({\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} = 0)$, if $\alpha \ne V(s)$]{}\[encoding\_erprime\_propag\_non\_target\] Note that variables $\alpha_s$ play a symmetric role to variables $\omega_s$ from ${\ensuremath{{\bf C_{{\exists}r}}}}$: instead of indicating the existence of a path from $s_0$ to $s$, they characterize the existence of a path from $s$ to a state labeled with $\alpha$. In addition, due to Constraint \[encoding\_erprime\_bool\_var\], variables $\lambda_s$ are set to [[$\mathsf{true}$]{}]{} iff there exists a path with non zero probability from the initial state $s_0$ to a state labeled with $\alpha$ passing by $s$. Thus, $\alpha$ cannot be reached from states s.t. $\lambda_s = {\ensuremath{\mathsf{false}}}$. Therefore, $S_\bot = \{s \ \|\ \lambda_s = {\ensuremath{\mathsf{false}}}\}$. Finally, we encode the equation system from [@Baier2008PMC] in a last [[[CSP]{.nodecor}]{}]{} encoding that extends ${\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}$. Let ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P}, \alpha) = (X \cup X^\prime,D \cup D^\prime,C \cup C^\prime)$ be such that $(X,D,C) = {\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}(\mathcal{P}, \alpha)$, $X^\prime$ contains one variable $\pi_s$ per state $s$ in $S$ with domain $[0, 1]$, $D^\prime$ contains the domains of these variables, and $C^\prime$ is composed of the following constraints for each state $s \in S$: 1. [$\neg\lambda_s \Rightarrow \pi_s = 0$]{} <!-- --> 1. [ $\lambda_s \Rightarrow \pi_s = 1$, if $\alpha = V(s)$]{} <!-- --> 1. [ $ \lambda_s \Rightarrow \pi_s = \Sigma_{s^{\prime} \in {\ensuremath{{\tt{Succ}}}}(s)} \pi_{s^\prime} {\ensuremath{{\ensuremath{\theta_{s^\prime}}}^{s}}}$, if $\alpha \ne V(s)$]{} As a consequence, variables $\pi_s$ encode the probability with which state $s$ eventually reaches $\alpha$ when $s$ is reachable from the initial state and $0$ otherwise. Let $p \in [0, 1] \subseteq {\ensuremath{\mathbb{R}}}$ be a probability bound. Adding the constraint $\pi_{s_0} \leq p$ (resp. $\pi_{s_0} \geq p$) to the previous [[${\bf C_{{\exists}\bar{r}}}$]{}]{} encoding allows to determine if there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ such that $\mathbb{P}^{\mathcal{M}} ({\ensuremath{\Diamond}}\alpha) \le p$ (resp $\ge p$). Formally, let ${\sim} \in \{\leq, <, \geq, >\}$ be a comparison operator, we write $\not\sim$ for its negation ([[e.g.]{}, ]{}$\not\leq$ is $>$). This leads to the following theorem. \[thm:pimc\_reachability\_in\_cp\] Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[pIMC]{.nodecor}]{}, $\alpha \subseteq A$ be a label, $p \in [0, 1]$, ${\sim} \in \{\leq,<, \geq,>\}$ be a comparison operator, and $(X,D,C)$ be ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P}, \alpha)$: - [[CSP]{.nodecor}]{} $(X,D,C \cup (\pi_{s_0} \sim p))$ is satisfiable iff $\exists \mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ s.t. ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$ - [[CSP]{.nodecor}]{} $(X,D,C \cup (\pi_{s_0} \not\sim p))$ is unsatisfiable iff $\forall \mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$: ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$ Prototype Implementation ======================== Our results have been implemented in a prototype tool[^3] which generates the above CSP encodings, and CSP encodings from [@DelahayeLP16] as well. Given a [[[pIMC]{.nodecor}]{}]{} in a text format inspired from [@tulip], our tool produces the desired [[[CSP]{.nodecor}]{}]{} as a SMT instance with the QF\_NRA logic (Quantifier Free Non linear Real-number Arithmetic). This instance can then be fed to any solver accepting the SMT-LIB format with QF\_NRA logic [@BarFT-SMTLIB]. For our benchmarks, we chose Z3 [@Z3] (latest version: 4.5.0). QF\_NRA does not deal with integer variables. In practice, logics mixing integers and reals are harder than those over reals only. Thus we obtained better results by encoding integer variables into real ones. In our implementations each integer variable $x$ is declared as a real variable whose real domain bounds are its original integer domain bounds; we also add the constraint $x < 1 \Rightarrow x = 0$. Since we only perform incrementation of $x$ this preserves the same set of solutions. In order to evaluate our prototype, we extend the [[<span style="font-variant:small-caps;">nand</span>]{.nodecor}]{} model from [@NPKS05][^4]. The original [[[MC]{.nodecor}]{}]{} [[<span style="font-variant:small-caps;">nand</span>]{.nodecor}]{} model has already been extended as a [[[pMC]{.nodecor}]{}]{} in [@Prophesy], where the authors consider a single parameter $p$ for the probability that each of the $N$ $nand$ gates fails during the multiplexing. We extend this model to [[[pIMC]{.nodecor}]{}]{} by considering intervals for the probability that the initial inputs are stimulated and we have one parameter per $nand$ gate to represent the probability that it fails. [[[pIMC]{.nodecor}]{}]{}s in text format are automatically generated from the PRISM model. Table \[tab:xp\] summarizes the size of the considered instances of the model (in terms of states, transitions, and parameters) and of the corresponding CSP problems (in terms of number of variables and constraints). In addition, we also present the resolution time of the given CSPs using the Z3 solver. Our experiments were performed on a $2.4$ GHz Intel Core i5 processor with time out set to $10$ minutes and memory out set to $2$Gb. Conclusion and future work ========================== In this paper, we have compared several Markov Chain abstractions in terms of succinctness and we have shown that Parametric Interval Markov Chain is a strictly more succinct abstraction formalism than other existing formalisms such as Parametric Markov Chains and Interval Markov Chains. In addition, we have proposed constraint encodings for checking several properties over [[[pIMC]{.nodecor}]{}]{}. In the context of qualitative properties such as existencial consistency or consistent reachability, the size of our encodings is significantly smaller than other existing solutions. In the quantitative setting, we have compared the two usual semantics for [[[IMC]{.nodecor}]{}]{}s and [[[pIMC]{.nodecor}]{}]{}s and showed that both semantics are equivalent with respect to quantitative reachability properties. As a side effect, this result ensures that all existing tools and algorithms solving reachability problems in [[[IMC]{.nodecor}]{}]{}s under the once-and-for-all semantics can safely be extended to the at-every-step semantics with no changes. Based on this result, we have then proposed [[[CSP]{.nodecor}]{}]{} encodings addressing quantitative reachability in the context of [[[pIMC]{.nodecor}]{}]{}s regardless of the chosen semantics. Finally, we have developed a prototype tool that automatically generates our [[[CSP]{.nodecor}]{}]{} encodings and that can be plugged to any constraint solver accepting the SMT-LIB format as input. We plan to develop our tool for [[[pIMC]{.nodecor}]{}]{} verification in order to manage other, more complex, properties ([[e.g.]{}, ]{}supporting the LTL-language in the spirit of what Tulip [@tulip] does). We also plan on investigating a practical way of computing and representing the set of [all solutions]{} to the parameter synthesis problem. Complements to Section \[sec:abstraction-models\] ================================================= ${\ensuremath{\models^{a}_{\tt{I}}}}$ is More General than ${\ensuremath{\models^{o}_{\tt{I}}}}$ {#ap:compare_imcs_satisfaction_relations} ------------------------------------------------------------------------------------------------ The [at-every-step]{} satisfaction relation is more general than the [once-and-for-all]{} satisfaction relation. Let $\mathcal{I} = (S, s_0, P, V)$ be an [[[IMC]{.nodecor}]{}]{} and $\mathcal{M} = (T, t_0, p, V^\prime)$ be a [[[MC]{.nodecor}]{}]{}. We show that 1. $\mathcal{M} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I} \Rightarrow \mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$; 2. in general $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I} \not\Rightarrow \mathcal{M}~{\ensuremath{\models^{o}_{\tt{I}}}}~\mathcal{I}$. The following examples also illustrates that even if a Markov chain satisfies an [[[IMC]{.nodecor}]{}]{} with the same graph representation w.r.t. the [[$\models^{a}_{\tt{I}}$]{}]{} relation it may not verify the [[$\models^{o}_{\tt{I}}$]{}]{} relation. 1. If $\mathcal{M} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$ then by definition of ${\ensuremath{\models^{o}_{\tt{I}}}}$ we have that $T = S$, $t_0 = s_0$, and $p(s)(s^\prime) \in P(s,s^\prime)$. The relation $\mathcal{R} = \{ (s,s) \mid s \in S\}$ is a satisfaction relation between $\mathcal{M}$ and $\mathcal{I}$ (consider for each state in $S$ the correspondence function $\delta : S \to (S \to [0,1])$ s.t. $\delta(s)(s^\prime) = 1$ if $s = s^\prime$ and $\delta(s)(s^\prime) = 0$ otherwise). Thus $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$. 2. Consider [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$, [[[MC]{.nodecor}]{}]{}s $\mathcal{M}_1$, $\mathcal{M}_2$, and $\mathcal{M}_3$ from Figure \[fig:ex\_imcs\_more\_general\_semantic\]. 1. By definition of ${\ensuremath{\models^{o}_{\tt{I}}}}$ we have that $\mathcal{M}_1 {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$. Thus by the previous remark we have that $\mathcal{M}_1 {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$ ($t_0 = s_0$, $t_1 = s_1$ and $t_2 = s_2$). 2. By definition of ${\ensuremath{\models^{o}_{\tt{I}}}}$ we have that $\mathcal{M}_2 \not{\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$. However the relation $\mathcal{R}$ containing $(t_0,s_0)$, $(t_1,s_1)$, $(t_1,s_2)$, $(t_2,s_1)$ and $(t_2,s_2)$ is a satisfaction relation between $\mathcal{I}$ and $\mathcal{M}_2$. Consider the functions: $\delta$ from $T$ to $(S \to [0,1])$ s.t. $\delta(t_1)(s_1) = 4/5$, $\delta(t_1)(s_2) = 1/5$, $\delta(t_2)(s_2) = 1$, and $\delta(t)(s) = 0$ otherwise; $\delta^\prime$ with the same signature than $\delta$ s.t. $\delta^\prime(t_1)(s_1) = 1$, $\delta^\prime(t_2)(s_2) = 1$, and $\delta^\prime(t)(s) = 0$ otherwise. We have that $\delta$ is a correspondence function for the pair $(t_0,s_0)$ and $\delta^\prime$ is a correspondence function for all the remaining pairs in $\mathcal{R}$. Thus there exists a [[[MC]{.nodecor}]{}]{}s $\mathcal{M}$ s.t. $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$ and $\mathcal{M} \not{\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$. Model Comparison {#ap:model_comparison} ---------------- According to the given succinctness definition, Lemma \[lem:compare\_imc\_and\_pmc\] states that [[$\tt{IMC}$]{}]{} and [[$\tt{pMC}$]{}]{} are not comparable w.r.t. both satisfaction relation ${\ensuremath{\models^{o}_{\tt{I}}}}$ and ${\ensuremath{\models^{a}_{\tt{I}}}}$ and that both satisfaction relations for [[[IMC]{.nodecor}]{}]{}s are not comparable. \[lem:compare\_imc\_and\_pmc\] ${\ensuremath{\tt{pMC}}}$ and ${\ensuremath{\tt{IMC}}}$ abstraction models are not comparable: ${\ensuremath{\tt{pMC}}}\not\leq$ $({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}})$; ${\ensuremath{\tt{pMC}}}\not\leq ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}})$; $({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}}) \not\leq$ ${\ensuremath{\tt{pMC}}}$; $({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}}) \not\leq ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}})$; $({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}}) \not\leq {\ensuremath{\tt{pMC}}}$; $({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}}) \not\leq ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}})$; We give a sketch of proof for each statement. Let $({\ensuremath{{\tt L}}}_1, \models_1)$ and $({\ensuremath{{\tt L}}}_2, \models_2)$ be two Markov chain abstraction models. Recall that according to the succinctness definition ([cf. ]{}Definition \[def:succinctness\]) ${\ensuremath{{\tt L}}}_1 \not\leq {\ensuremath{{\tt L}}}_2$ if there exists $\mathcal{L}_2 \in {\ensuremath{{\tt L}}}_2$ s.t. $\mathcal{L}_1 \not\equiv \mathcal{L}_2$ for all $\mathcal{L}_1 \in {\ensuremath{{\tt L}}}_1$. 1. Consider [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$ and [[[pMC]{.nodecor}]{}]{}s $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_n$ (with $n \in {\ensuremath{\mathbb{N}}}$) from Figure \[fig:imc\_no\_equiv\_pmc\]. presents an [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$ verifying case (1) $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_n$ (with $n \in {\ensuremath{\mathbb{N}}}$) are entailed by $\mathcal{I}$ w.r.t. ${\ensuremath{\models^{a}_{\tt{I}}}}$ but none of them is equivalent to $\mathcal{I}$. Indeed one needs an infinite countable number of states for creating a ${{\textnormal{pMC}}}$ equivalent to $\mathcal{I}$ w.r.t. ${\ensuremath{\models^{a}_{\tt{I}}}}$. However states space must be finite. 2. Consider [[[IMC]{.nodecor}]{}]{} $\mathcal{I^\prime}$ similar to $\mathcal{I}$ from Figure \[fig:imc\_no\_equiv\_pmc\] excepted that the transition from $s_1$ to $s_0$ is replaced by the interval $[0.5, 1]$. Since the [[[pIMC]{.nodecor}]{}]{} definition does not allow to bound values for parameters there is no equivalent $\mathcal{I^\prime}$ w.r.t. ${\ensuremath{\models^{a}_{\tt{I}}}}$. 3. Note that parameters in ${{\textnormal{pMC}}}$s enforce transitions in the concrete ${{\textnormal{MC}}}$s to receive the same value. Since parameters may range over continuous intervals there is no hope of modeling such set of Markov chains using a single ${\textnormal{IMC}}$. Figure \[fig:imc\_no\_equiv\_pmc\] illustrates this statement. 4. Recall that $({\textnormal{IMC}}, {\ensuremath{\models^{a}_{\tt{I}}}})$ is more general than $({\textnormal{IMC}}, {\ensuremath{\models^{o}_{\tt{I}}}})$. One cannot restrict the set of Markov chains satisfying an [[[IMC]{.nodecor}]{}]{} w.r.t. ${\ensuremath{\models^{a}_{\tt{I}}}}$ to be bisimilar to the set of Markov chain satisfying an [[[IMC]{.nodecor}]{}]{} w.r.t. ${\ensuremath{\models^{o}_{\tt{I}}}}$. 5. Same remark than item (3) 6. Counter-example similar to which in statement (2) ([[i.e.]{}, ]{}in order to simulate the at-every-step satisfaction relation with the once-and-for-all satisfaction relation one needs an infinite uncountable number of states). [[Proposition \[prop:sunccinctness\_pimc\_pmc\_imc\].]{.nodecor}]{} The Markov chain abstraction models can be ordered as follows w.r.t. succinctness: $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}}) < ({\ensuremath{\tt{pMC}}}, {\ensuremath{\models_{\tt{p}}}})$, $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}}) < ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{o}_{\tt{I}}}})$ and $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^a_{\tt{pI}}}}\nobreak) < ({\ensuremath{\tt{IMC}}}, {\ensuremath{\models^{a}_{\tt{I}}}})$. Recall that the [[$\tt{pIMC}$]{}]{} model is a Markov chain abstraction model allowing to declare parametric interval transitions, while the [[$\tt{pMC}$]{}]{} model allows only parametric transitions (without intervals), and the [[$\tt{IMC}$]{}]{} model allows interval transitions without parameters. Clearly, any [[[pMC]{.nodecor}]{}]{} and any [[[IMC]{.nodecor}]{}]{} can be translated into a [[[pIMC]{.nodecor}]{}]{} with the right semantics (once-and-for-all for [[[pMC]{.nodecor}]{}]{}s and the chosen [[[IMC]{.nodecor}]{}]{} semantics for [[[IMC]{.nodecor}]{}]{}s). This means that $({{\ensuremath{\tt{pIMC}}}},{\ensuremath{\models^o_{\tt{pI}}}})$ is more succinct than [[$\tt{pMC}$]{}]{} and [[$\tt{pIMC}$]{}]{} is more succinct than [[$\tt{IMC}$]{}]{} for both semantics. Furthermore since [[$\tt{pMC}$]{}]{} and [[$\tt{IMC}$]{}]{} are not comparable (cf Lemma \[lem:compare\_imc\_and\_pmc\]), we have that the [[$\tt{pIMC}$]{}]{} abstraction model is strictly more succinct than the [[$\tt{pMC}$]{}]{} abstraction model and than the [[$\tt{IMC}$]{}]{} abstraction model with the right semantics. Complements to Section \[sec:qualitative-reachability\] ======================================================= [[[CSP]{.nodecor}]{}]{} Encoding for Qualitative Reachability ------------------------------------------------------------- Let $\mathcal{P} = (S,s_0,P,V,Y)$ be a ${\textnormal{pIMC}}$. $[\Rightarrow]$ The [[CSP]{.nodecor}]{} ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P}) = (X,D,C)$ is satisfiable implies that there exists a valuation $v$ of the variables in $X$ satisfying all the constraints in $C$. Consider the [[MC]{.nodecor}]{} $\mathcal{M} = (S, s_0, p, V)$ such that $p(s, s^\prime) = v({\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}})$, for all ${\ensuremath{{\ensuremath{\theta_{s}}}^{s^\prime}}} \in {\ensuremath{\Theta}}$ and $p(s, s^\prime) = 0$ otherwise. Firstly, we show by induction that for any state $s$ in $S$: “if $s$ is reachable in $\mathcal{M}$ then $v(\rho_s)$ equals to [[$\mathsf{true}$]{}]{}”. This is correct for the initial state $s_0$ thanks to the Constraint (1). Let $s$ be a state in $S$ and assume that the property is correct for all its predecessors. By the Constraints (2), $v(\rho_s)$ equals to [[$\mathsf{true}$]{}]{} if there exists at least one predecessor $s^{\prime\prime} \neq s$ reaching $s$ with a non-zero probability ([[i.e.]{}, ]{}$v({\ensuremath{{\ensuremath{\theta_{s^{\prime\prime}}}}^{s}}}) \neq 0$). This is only possible by the Constraint (3) if $v(\rho_{s^{\prime\prime}})$ equals to [[$\mathsf{true}$]{}]{}. Thus $v(\rho_s)$ equals to [[$\mathsf{true}$]{}]{} if there exists one reachable state $s^{\prime\prime}$ s.t. $v({\ensuremath{{\ensuremath{\theta_{s^{\prime\prime}}}}^{s}}}) \neq 0$. Secondly, we show that $\mathcal{M}$ satisfies the [[pIMC]{.nodecor}]{} $\mathcal{P}$. We use Theorem 4 from [@DelahayeLP16] stating that ${\ensuremath{\models^a_{\tt{pI}}}}$ and ${\ensuremath{\models^o_{\tt{pI}}}}$ are equivalent w.r.t. qualitative reachability. We proved above that for all reachable states $s$ in $\mathcal{M}$, we have $v(\rho_s)$ equals to [[$\mathsf{true}$]{}]{}. By the Constraints (5) it implies that that for all reachable states $s$ in $\mathcal{M}$: $p(s,s^\prime) \in P(s,s^\prime)$ for all $s$ and $s^\prime$.[^5] $[\Leftarrow]$ The [[pIMC]{.nodecor}]{} $\mathcal{P}$ is consistent implies by the Theorem 4 from [@DelahayeLP16] stating that ${\ensuremath{\models^a_{\tt{pI}}}}$ and ${\ensuremath{\models^o_{\tt{pI}}}}$ are equivalent w.r.t. qualitative reachability, that there exists an implementation of the form $\mathcal{M} = (S, s_0, p, V )$ where, for all reachable states $s$ in $\mathcal{M}$, it holds that $p(s,s^\prime) \in P(s,s^\prime)$ for all $s^\prime$ in $S$. Consider $\mathcal{M}^\prime = (S, s_0, p^\prime, V )$ s.t. for each non reachable state $s$ in $S$: $p^\prime(s, s^\prime) = 0$, for all $s^\prime \in S$. The valuation $v$ is s.t. $v(\rho_s)$ equals [[$\mathsf{true}$]{}]{} iff $s$ is reachable in $\mathcal{M}$, $v({\ensuremath{{\ensuremath{\theta_{s}}}^{s^{\prime}}}}) = p^\prime(s,s^\prime)$, and for each parameter $y \in Y$ a valid value can be selected according to $p$ and $P$ when considering reachable states. Finally, by construction, $v$ satisfies the [[CSP]{.nodecor}]{} ${\ensuremath{{\bf C_{{\exists}c}}}}(\mathcal{P})$. Complements to Section \[sec:quantitative\] =========================================== Equivalence of ${\ensuremath{\models^{o}_{\tt{I}}}}$ and ${\ensuremath{\models^{a}_{\tt{I}}}}$ w.r.t quantitative reachability {#ap:equiv_imc_semantics} ------------------------------------------------------------------------------------------------------------------------------ We first introduce some notations. Let $\mathcal{I} = (S,s_0,P,V^I)$ be an [[[IMC]{.nodecor}]{}]{} and $\mathcal{M}=(T,t_0,p,V^M)$ be an [[[MC]{.nodecor}]{}]{} s.t. $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$. Let $\mathcal{R} \subseteq T \times S$ be a satisfaction relation between $\mathcal{M}$ and $\mathcal{I}$. For all $t \in T$ we write $\mathcal{R}(t)$ for the set $\{s \in S \mid t~\mathcal{R}~s \}$, and for all $s \in S$ we write $\mathcal{R}^{-1}(t)$ for the set $\{t \in T \mid s~\mathcal{R}~t \}$. Furthermore we say that $\mathcal{M}$ satisfies $\mathcal{I}$ with degree $n$ (written [$\mathcal{M} { \mathrel{\mathop{{\ensuremath{\models^{a}_{\tt{I}}}}}\limits^{ \vbox to-0.5ex{\kern-2\ex@ \hbox{$\scriptstyle\text{\hspace{0.02cm}\small{$n$}}$}\vss}}}} \mathcal{I}$]{}) if $\mathcal{M}$ satisfies $\mathcal{I}$ with a satisfaction relation $\mathcal{R}$ s.t. each state $t \in T$ is associated by $\mathcal{R}$ to at most $n$ states from $S$ ([[i.e.]{}, ]{}$\|\mathcal{R}(t)\| \leq n$); $\mathcal{M}$ satisfies $\mathcal{I}$ with the same structure than $\mathcal{I}$ if $\mathcal{M}$ satisfies $\mathcal{I}$ with a satisfaction relation $\mathcal{R}$ s.t. each state $t \in T$ is associated to at most one state from $S$ and each state $s \in S$ is associated to at most one state from $T$ ([[i.e.]{}, ]{}$\|\mathcal{R}(t)\| \leq 1$ for all $t \in T$ and $\|\mathcal{R}^{-1}(s)\| \leq 1$ for all $s \in S$). \[prop:nsat\_to\_bisimilar\_1sat\] Let $\mathcal{I}$ be an [[IMC]{.nodecor}]{}. If a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfies $\mathcal{I}$ with degree $n \in {\ensuremath{\mathbb{N}}}$ then there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M^\prime}$ satisfying $\mathcal{I}$ with degree $1$ such that $\mathcal{M}$ and $\mathcal{M}^\prime$ are bisimilar. The main idea for proving Proposition \[prop:nsat\_to\_bisimilar\_1sat\] is that if an ${{\textnormal{MC}}}$ $\mathcal{M}$ with states space $T$ satisfies an ${{\textnormal{IMC}}}$ $\mathcal{I}$ with a states space $S$ according to a satisfaction relation $\mathcal{R}$ then, each state $t$ related by $\mathcal{R}$ to many states $s_1, \ldots, s_n$ (with $n > 1$) can be split in $n$ states $t^1, \ldots, t^n$. The derived ${{\textnormal{MC}}}$ will satisfy $\mathcal{I}$ with a satisfaction relation $\mathcal{R}^\prime$ where each $t_i$ is only associated by $\mathcal{R}^\prime$ to the state $s_i$ ($i \leq n$). This $\mathcal{M^\prime}$ will be bisimilar to $\mathcal{M}$ and it will satisfy $\mathcal{I}$ with degree $1$. Note that by construction the size of the resulting ${{\textnormal{MC}}}$ is in $O(\|\mathcal{M}\|\times\|\mathcal{I}\|)$. Let $\mathcal{I} = (S,s_0,P,V^I)$ be an [[[IMC]{.nodecor}]{}]{} and $\mathcal{M}=(T,t_0,p,V)$ be a [[MC]{.nodecor}]{}. If $\mathcal{M}$ satisfies $\mathcal{I}$ (with degree $n$) then there exists a satisfaction relation $\mathcal{R}$ verifying the ${\ensuremath{\models^{a}_{\tt{I}}}}$ satisfaction relation. For each association $t~\mathcal{R}~s$, we write $\delta_t^s$ the correspondence function chosen for this pair of states. $\mathcal{M}$ satisfies $\mathcal{I}$ with degree $n$ means that each state in $\mathcal{M}$ is associated by $\mathcal{R}$ to at most $n$ states in $\mathcal{I}$. To construct a [[[MC]{.nodecor}]{}]{} $\mathcal{M^\prime}$ satisfying $\mathcal{I}$ with degree $1$ we create one state in $\mathcal{M^\prime}$ per association $(t,s)$ in $\mathcal{R}$. Formally, let $\mathcal{M^\prime}$ be equal to $(U, u_0, p^\prime, V^\prime)$ such that $U = \{u_t^s \mid t~\mathcal{R}~s\}$, $u_0 = u_{t_0}^{s_0}$, $V^\prime = \{(u_t^s, v) \mid v = V(t)\}$, and $p^\prime(u_t^s)(u_{t^\prime}^{s^\prime}) = p(t)(t^\prime) \times \delta_t^s(t^\prime)(s^\prime)$. Following computation shows that the outgoing probabilities given by $p^\prime$ form a probability distribution for each state in $\mathcal{M^\prime}$ and thus that $\mathcal{M^\prime}$ is a [[[MC]{.nodecor}]{}]{}. $$\begin{aligned} & {\sum\limits_{\mathclap{t^\prime \mathcal{R} s^\prime}^{}}} p^\prime(u_t^s)(u_{t^\prime}^{s^\prime}) = {\sum\limits_{\mathclap{t^\prime \mathcal{R} s^\prime}^{}}} p(t)(t^\prime) \times \delta_t^s(t^\prime)(s^\prime) \\ = & {\sum\limits_{\mathclap{t^\prime \in T}^{}}} p(t)(t^\prime) \times {\sum\limits_{\mathclap{s^\prime \in S}^{}}} \delta_t^s(t^\prime)(s^\prime) = {\sum\limits_{\mathclap{t^\prime \in T}^{}}} p(t)(t^\prime) \times 1 = 1 \end{aligned}$$ Finally, by construction of $\mathcal{M^\prime}$ based on $\mathcal{M}$ which satisfies $\mathcal{I}$, we get that $\mathcal{R^\prime} = \{(u_t^s,s) \mid t \in T, s \in S \}$ is a satisfaction relation between $\mathcal{M^\prime}$ and $\mathcal{I}$. Furthermore $\|\{s \mid u~\mathcal{R^\prime}~s\}\|$ equals at most one. Thus, we get that $\mathcal{M^\prime}$ satisfies $\mathcal{I}$ with degree $1$. Consider the relation $\mathcal{B}^\prime = \{ (u_t^s, t) \subseteq U \times T \mid t~\mathcal{R}~s \}$. We note $\mathcal{B}$ the closure of $\mathcal{B}^\prime$ by transitivity, reflexivity and symetry ([[i.e.]{}, ]{}$\mathcal{B}$ is the minimal equivalence relation based on $\mathcal{B}^\prime$). We prove that $\mathcal{B}$ is a bisimulation relation between $\mathcal{M}$ and $\mathcal{M^\prime}$. By construction each equivalence class from $\mathcal{B}$ contains exactly one state $t$ from $T$ and all the states $u_t^s$ such that $t~\mathcal{R}~s$. Let $(u_t^s, t)$ be in $\mathcal{B}$, $t^\prime$ be a state in $T$, and $B$ be the equivalence class from $\mathcal{B}$ containing $t^\prime$ ([[i.e.]{}, ]{}$B$ is the set $\{ t^\prime \} \cup \{ u_{t^\prime}^{s^\prime} \in U \mid s^{\prime} \in S \text{ and } t^{\prime}~\mathcal{R}~s^{\prime} \}$). Firstly note that by construction the labels agree on $u_t^s$ and $t$: $V^\prime(u_t^s) = V(t)$. Secondly the following computation shows that $p^\prime(u_t^s)(B \cap U)$ equals to $p(t)(B \cap T)$ and thus that $u_t^s$ and $t$ are bisimilar: $$\begin{aligned} p^\prime(u_t^s)(B \cap U) & = {\sum\limits_{\mathclap{\hspace{0.6cm}u_{t^\prime}^{s^\prime} \in B \cap U}^{}}} p^\prime(u_t^s)(u_{t^\prime}^{s^\prime}) = {\sum\limits_{\mathclap{\hspace{0.6cm}u_{t^\prime}^{s^\prime} \in B \cap U}^{}}} p(t)(t^\prime) \times \delta_t^s(t^\prime)(s^\prime) \\ & = {\sum\limits_{\mathclap{\hspace{1cm}\{s^\prime \in S \mid s^\prime \mathcal{R} t^\prime \}}^{}}} p(t)(t^\prime) \times \delta_t^s(t^\prime)(s^\prime) = p(t)(t^\prime) \times {\sum\limits_{\mathclap{\hspace{1cm}\{s^\prime \in S \mid s^\prime \mathcal{R} t^\prime \}}^{}}} \delta_t^s(t^\prime)(s^\prime) \\ & = p(t)(t^\prime) \times 1 = p(t)(\{t^\prime\}) = p(t)(B \cap T) \end{aligned}$$ \[lem:1sat\_same\_proba\] Let $\mathcal{I}$ be an [[IMC]{.nodecor}]{}, $\mathcal{M}$ be a [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$, and $\gamma$ be a [[PCTL$^$]{}]{} formulae. There exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M^\prime}$ satisfying $\mathcal{I}$ with degree $1$ such that the probability ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^\prime}(\gamma)$ equals the probability ${\ensuremath{\mathbb{P}}}^{\mathcal{M}}(\gamma)$. Corollary \[lem:1sat\_same\_proba\] is derived from Proposition \[prop:nsat\_to\_bisimilar\_1sat\] joined with the probability preservation of the PCTL\* formulae on bisimilar Markov chains (see [@Baier2008PMC], Theorem 10.67, p.813). Corollary \[lem:1sat\_same\_proba\] allows to reduce to one the number of states in the [[[pIMC]{.nodecor}]{}]{} $\mathcal{I}$ satisfied by each state in the [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ while preserving probabilities. \[lem:reach\_with\_less\_states\] Let $\mathcal{I} = (S,s_0,P,V)$ be an [[[IMC]{.nodecor}]{}]{}, $\mathcal{M} = (T,t_0,p,V)$ be a [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$ with degree $1$, and $\alpha \subseteq A$ be a proposition. If $\mathcal{M}$ does not have the same structure than $\mathcal{I}$ then there exists an [[[MC]{.nodecor}]{}]{} $\mathcal{M}_1$ (resp. $\mathcal{M}_2$) satisfying $\mathcal{I}$ with a set of states $S_1$ (resp. $S_2$) s.t. $S_1 \subset S$ and ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_1}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$ (resp. $S_2 \subset S$ and ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_2}({\ensuremath{\Diamond}}\alpha) \geq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$). Lemma \[lem:reach\_with\_less\_states\] reduces the number of states in $\mathcal{M}$ while preserving the maximal or minimal reachability probability. This lemma has a constructive proof. Here is the main idea of the proof. We first select one state $s$ from the [[[IMC]{.nodecor}]{}]{} $\mathcal{I}$ which is satisfied by many states $t_1, \ldots, t_n$ in $\mathcal{M}$. The ${{\textnormal{MC}}}$ $\mathcal{M}^\prime$ keeping the state $t_k$ minimizing the probability of reaching $\alpha$ in $\mathcal{M}$ and removing all the other states $t_i$ ([[i.e.]{}, ]{}remove the states $t_i$ s.t. $i \neq k$ and move the transitions arriving to a state $t_i$ s.t. $i \neq k$ to arrive to the state $t_k$) will have less states than $\mathcal{M}$ and verifies ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_1}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$. Let $\mathcal{I} = (S,s_0,P,V^I)$ be an [[[IMC]{.nodecor}]{}]{} and $\mathcal{M} = (T,t_0,p,V)$ be a [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$ with degree $1$. We write $\mathcal{R}$ the satisfaction relation between $\mathcal{M}$ and $\mathcal{I}$ with degree $1$. The following proves in 3 steps the ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_1}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$ case. 1. We would like to construct a [[[MC]{.nodecor}]{}]{} $\mathcal{M}^\prime$ satisfying $\mathcal{I}$ with less states than $\mathcal{M}^\prime$ such that ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^\prime}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$. Since the degree of $\mathcal{R}$ equals to $1$ each state $t$ in $T$ is associated to at most one state $s$ in $S$. Furthermore, since $\mathcal{M}$ does not have the same structure than $\mathcal{I}$ then there exists at most one state from $S$ which is associated by $\mathcal{R}$ to many states from $T$. Let $\bar{s}$ be a state from $S$ such that $\|\mathcal{R}^{-1}(s)\| \geq 2$, $\bar{T} = \{t_1, \ldots, t_n\}$ be the set $\mathcal{R}^{-1}(s)$ where the $t_i$ are ordered by decreasing probability of reaching $\alpha$ ([[i.e.]{}, ]{} ${\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_i}({\ensuremath{\Diamond}}\alpha) \geq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_{i+1}}({\ensuremath{\Diamond}}\alpha)$ for all $1 \leq i < n$). In the following we refer $\bar{t}$ as $t_n$. We produce $\mathcal{M}^\prime$ from $\mathcal{M}$ by replacing all the transitions going to a state $t_1, \ldots, t_{n-1}$ by a transition going to $t_{n}$, and by removing the corresponding states. Formally $\mathcal{M}^\prime = (T^\prime,t_0,p^\prime,V^\prime)$ s.t. $T^\prime = (T \setminus \bar{T}) \cup \{\bar{t}\}$, $V^\prime$ is the restriction of $V$ on $T^\prime$, and for all $t,t^\prime \in T^\prime$: $p^\prime(t)(t^\prime) = p(t)(t^\prime)$ if $t^\prime \neq \bar{t}$ and $p^\prime(t)(t^\prime) = {\sum\limits_{\mathclap{t^\prime \in \bar{T}}^{}}} p(t)(t^\prime)$ otherwise. $$\begin{aligned} {\sum\limits_{\mathclap{t^\prime \in T^\prime}^{}}} p^\prime(t)(t^\prime) = & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p^\prime(t)(t^\prime) \hspace{0.1cm} + \hspace{0.1cm} p^\prime(t)(\bar{t}) \\ \stackrel{(1)}{=} & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p(t)(t^\prime) \hspace{0.1cm} + \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in \bar{T}}^{}}} p(t)(t^\prime) \\ = & {\sum\limits_{\mathclap{\hspace{1cm}t^\prime \in T^\prime \setminus \{\bar{t}\} \cup \bar{T}}^{}}} p(t)(t^\prime) \hspace{0.1cm} \stackrel{(2)}{=} \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in T}^{}}} p(t)(t^\prime) \hspace{0.1cm} = \hspace{0.1cm} 1 \end{aligned}$$ Previous computation holds for each state $t$ in $\mathcal{M^\prime}$. It shows that the outgoing probabilities given by $p^\prime$ form a probability distribution for each state in $\mathcal{M^\prime}$ and thus that $\mathcal{M^\prime}$ is a [[[MC]{.nodecor}]{}]{}. Note that step ($1$) comes from the definition of $p^\prime$ with respect to $p$ and that step ($2$) comes from the definition of $T^\prime$ according to $\bar{T}$ and $\bar{t}$. 2. We now prove that $\mathcal{M}^\prime$ implements $\mathcal{I}$. $\mathcal{M}$ satisfies $\mathcal{I}$ implies that there a exists a satisfaction relation $\mathcal{R}$ between $\mathcal{M}$ and $\mathcal{I}$. Let $\mathcal{R}^\prime \subseteq T \times S$ be s.t. $t~\mathcal{R}^\prime~s$ if $t~\mathcal{R}~s$ and $\bar{t}~\mathcal{R}^\prime~\bar{s}$ if there exists a state $t^\prime \in \bar{T}$ s.t. $t^\prime~\mathcal{R}~\bar{s}$. We prove that $\mathcal{R}^\prime$ is a satisfaction relation between $\mathcal{M}^\prime$ and $\mathcal{I}$. For each pair $(t,s) \in \mathcal{R}$ we note $\delta_{(s,t)}$ the correspondance function given by the satisfaction relation $\mathcal{R}$. Let $(t,s)$ be in $\mathcal{R}^\prime$ and $\delta^\prime : T^\prime \to (S \to [0, 1])$ be s.t. $\delta^\prime(t^\prime)(s^\prime) = \delta_{(t,s)}(t^\prime)(s^\prime)$ if $t^\prime \neq \bar{t}$ and $\delta^\prime(t^\prime)(s^\prime) = max_{t^\prime \in \bar{T}}(\delta_{(t,s)}(t^\prime)(s^\prime))$ otherwise. $\delta^\prime$ is a correspondence function for the pair $(t,s)$ in $\mathcal{R}^\prime$ such as required by the [[$\models^{a}_{\tt{I}}$]{}]{} satisfaction relation: 1. Let $t^\prime$ be in $T$. If $t^\prime \neq \bar{t}$ then $\delta^\prime(t^\prime)$ is equivalent to $\delta_{(t,s)}(t^\prime)(s^\prime)$ which is by definition a distribution on $S$. Otherwise $t^\prime = \bar{t}$ and the following computation proves that $\delta^\prime(\bar{t})$ is a distribution on $S$. For the step (1) remind that $\mathcal{R}$ is a satisfaction relation with degree $1$ and that $\bar{t}~\mathcal{R}~\bar{s}$. This implies that $\delta_{(t,s)}(\bar{t})(s^\prime)$ equals to zero for all $s^\prime \neq \bar{s}$. For the step (2), $\mathcal{R}$ is a satisfaction relation with degree $1$ implies that $\delta_{(t,s)}(t^{\prime})(s^\prime)$ equals to $0$ or $1$ for all $t^{\prime} \in T$ and $s^\prime \in S$. Finally the recursive definition of the satisfaction relation $\mathcal{R}$ implies that there exists at least one state $t^{\prime\prime} \in \bar{T}$ s.t. $\delta_{(t,s)}(t^{\prime\prime})(\bar{s})$ does not equal to zero ([[i.e.]{}, ]{}equals to one). $$\begin{aligned} {\sum\limits_{\mathclap{s^\prime \in S}^{}}} \delta^\prime(\bar{t})(s^\prime) = & {\sum\limits_{\mathclap{\hspace{0.5cm}s^\prime \in S \setminus \{\bar{s}\}}^{}}} \delta^\prime(\bar{t})(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} \delta^\prime(\bar{t})(\bar{s}) \\ = & {\sum\limits_{\mathclap{\hspace{0.5cm}s^\prime \in S \setminus \{\bar{s}\}}^{}}} \delta_{(t,s)}(\bar{t})(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} max_{t^{\prime\prime} \in \bar{T}}(\delta_{(t,s)}(t^{\prime\prime})(\bar{s})) \\ \stackrel{(1)}{=} &~ max_{t^{\prime\prime} \in \bar{T}}(\delta_{(t,s)}(t^{\prime\prime})(s^\prime)) \\ \stackrel{(2)}{=} &~ 1 \end{aligned}$$ 2. Let $s^\prime$ be in $S$. Step (1) uses the definition of $p^\prime$ according to $p$. Step (2) uses the definition of $\delta^\prime$ according to $\delta_{(t,s)}$. Step (3) comes from the fact that for all $t, t^\prime \in T \times \bar{T}$, we have by the definition of the satisfaction relation $\mathcal{R}$ with degree $1$ and by construction of $\bar{T}$ that if $p(t,t^\prime) \neq 0$ then $\delta_{(t)(s)}(t^\prime, \bar{s}) = 1$ and $\delta_{(t,s)}(t^\prime)(s^\prime) = 0$ for all $s^\prime \neq \bar{s}$. Finally, step (4) comes from the definition of the correspondence function $\delta_{(t,s)}$ for the pair $(t,s)$ in $\mathcal{R}$. $$\begin{aligned} & {\sum\limits_{\mathclap{t^\prime \in T^\prime}^{}}} p^\prime(t)(t^\prime) \times \delta^\prime(t^\prime)(s^\prime) \\ = & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p^\prime(t)(t^\prime) \times \delta^\prime(t^\prime)(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} p^\prime(t, \bar{t}) \times \delta^\prime(\bar{t})(s^\prime) \\ \stackrel{(1)}{=} & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p(t)(t^\prime) \times \delta^\prime(t^\prime)(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in \bar{T}}^{}}} p(t)(t^\prime) \times \delta^\prime(\bar{t})(s^\prime) \\ \stackrel{(2)}{=} & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p(t)(t^\prime) \times \delta_{(t,s)}(t^\prime)(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in \bar{T}}^{}}} p(t)(t^\prime) \times max_{t^{\prime\prime} \in \bar{T}}(\delta_{(t,s)}(t^{\prime\prime})(s^\prime)) \\ \stackrel{(3)}{=} & {\sum\limits_{\mathclap{\hspace{0.7cm}t^\prime \in T^\prime \setminus \{\bar{t}\}}^{}}} p(t)(t^\prime) \times \delta_{(t,s)}(t^\prime)(s^\prime) \hspace{0.1cm} + \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in \bar{T}}^{}}} p(t)(t^\prime) \times \delta_{(t,s)}(t^{\prime})(s^\prime) \\ = & {\sum\limits_{\mathclap{\hspace{1cm}t^\prime \in T^\prime \setminus \{\bar{t}\} \cup \bar{T}}^{}}} p(t)(t^\prime) \times \delta_{(t,s)}(t^\prime)(s^\prime) \hspace{0.1cm} {=} \hspace{0.1cm} {\sum\limits_{\mathclap{\hspace{0cm}t^\prime \in T}^{}}} p(t)(t^\prime) \times \delta_{(t,s)}(t^\prime)(s^\prime) \\ \stackrel{(4)}\in & \hspace{0.1cm} P(s, s^\prime) \end{aligned}$$ 3. Let $t^\prime$ be in $T^\prime$ and $s^\prime$ be in $S$. We have by construction of $\mathcal{R}^\prime$ from $\mathcal{R}$ that if $\delta^\prime(t^\prime)(s^\prime) > 0$ then $(t^\prime, s^\prime) \in \mathcal{R}$. 3. Ne now prove that the probability of reaching $\alpha$ from $\bar{t}$ is lower in $\mathcal{M^\prime}$ than in $\mathcal{M}$. We consider the [[[MC]{.nodecor}]{}]{} $\mathcal{M}^{\prime\prime}$ from $\mathcal{M}$ where the states containing the label $\alpha$ are replaced by absorbing states. Formally $\mathcal{M}^{\prime\prime} = (T,t_0,p^{\prime\prime},V)$ such that for all $t,t^\prime \in T$: $p^{\prime\prime}(t, t^\prime) = p(t, t^\prime)$ if $\alpha \not\subseteq V(t)$ else $p^{\prime\prime}(t, t^\prime) = 1$ if $t=t^\prime$ and $p^{\prime\prime}(t, t^\prime) = 0$ otherwise. By definition of the reachability property we get that ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_t({\ensuremath{\Diamond}}\alpha)$ equals to ${\ensuremath{\mathbb{P}}}^{\mathcal{M}}_t({\ensuremath{\Diamond}}\alpha)$ for all state $t$ in $T^\prime$. Following computation concludes the proof. Step (1) comes from Lemma \[lem:proba\_with\_loops\]. Step (2) comes by construction of $\mathcal{M}^\prime$ from $\mathcal{M}$. Step (3) comes by construction of $\mathcal{M}^{\prime\prime}$ from $\mathcal{M}$ where states labeled with $\alpha$ are absorbing states. Step (4) comes from the fact that ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha)$ is equal to ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}(\neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}\alpha) + \Sigma_{1 \leq i \leq n} {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}( \neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}t_{i}) \times {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_i}({\ensuremath{\Diamond}}\alpha)$. Step (5) uses the fact that ${\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_i}({\ensuremath{\Diamond}}\alpha) \geq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_{n}}({\ensuremath{\Diamond}}\alpha)$ for all $1 \leq i \leq n$ and by construction this is also correct in $\mathcal{M}^{\prime\prime}$. Last steps are straightforward. $$\begin{aligned} & {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime}}_{\bar{t}}({\ensuremath{\Diamond}}\alpha) \\ \stackrel{(1)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime}}_{\bar{t}}(\neg \bar{t} {\textnormal{ U }}\alpha) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime}}_{\bar{t}}(\neg\alpha {\textnormal{ U }}\bar{t}) } \\ \stackrel{(2)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_n}(\neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}\alpha) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{t_n}(\neg\alpha {\textnormal{ U }}(t_1 \lor \ldots \lor t_n)) } \\ \stackrel{(3)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}(\neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}\alpha) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) } \\ \stackrel{(4)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) - {\sum\limits_{\mathclap{1 \leq i \leq n}^{}}} {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}( \neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}t_{i}) \times {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_i}({\ensuremath{\Diamond}}\alpha) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) } \\ \stackrel{(5)}{\leq} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) - {\sum\limits_{\mathclap{1 \leq i \leq n}^{}}} {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}( \neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}t_{i}) \times {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) } \\ \stackrel{(6)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) \times (1 - {\sum\limits_{\mathclap{1 \leq i \leq n}^{}}} {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}( \neg(t_1 \lor \ldots \lor t_n) {\textnormal{ U }}t_{i}) ) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) } \\ \stackrel{(7)}{=} & \frac{ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) \times (1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) ) }{ 1 - {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}(t_1 \lor \ldots \lor t_n)) } \\ = & ~ {\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime\prime}}_{t_n}({\ensuremath{\Diamond}}\alpha) \\ = & ~ {\ensuremath{\mathbb{P}}}^{\mathcal{M}}_{\bar{t}}({\ensuremath{\Diamond}}\alpha) \end{aligned}$$ The same method can be used for proving the ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_2}({\ensuremath{\Diamond}}\alpha) \geq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$ case by defining $\bar{T} = \{t_1, \ldots, t_n\}$ be the set $\mathcal{R}^{-1}(s)$ s.t. the $t_i$ are ordered by [increasing]{} probability of reaching $\alpha$. Thereby the $\leq$ symbol at step (5) for the computation of ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^{\prime}}_{\bar{t}}({\ensuremath{\Diamond}}\alpha)$ is replaced by the $\geq$ symbol. \[lem:proba\_with\_loops\] Let $\mathcal{M} = (S, s_0 , p, V)$ be a [[[MC]{.nodecor}]{}]{}, $\alpha \subseteq A$ be a proposition, and $s$ be a state from $S$. Then $${\ensuremath{\mathbb{P}}}^\mathcal{M}_s({\ensuremath{\Diamond}}\alpha) = \frac{ {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg s {\textnormal{ U }}\alpha)}{ 1 - {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg\alpha {\textnormal{ U }}s)}$$ Lemma \[lem:proba\_with\_loops\] is called in the proof of Lemma \[lem:reach\_with\_less\_states\]. Let $S^\prime$ be the subset of $S$ containing all the states labeled with $\alpha$ in $\mathcal{M}$. We write $\Omega_n$ with $n \in {\ensuremath{\mathbb{N}}}^$ the set containing all the paths $\omega$ starting from $s$ s.t. state $s$ appears exactly $n$ times in $\omega$ and no state in $\omega$ is labeled with $\alpha$. Formally $\Omega_n$ contains all the $\omega = s_1, \ldots, s_k \in S^k$ s.t. $k \in {\ensuremath{\mathbb{N}}}$, $s_1$ is equal to $s$, $\|\{ i \in [1,k] \mid s_i = s \}\| = n$, and $\alpha \not\subseteq V(s_i)$ for all $i \in [1,k]$. We get by (a) that $({\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_n \times S^\prime))_{n \geq 1}$ is a geometric series. In (b) we partition the paths reaching $\alpha$ according to the $\Omega_n$ sets and we use the geometric series of the probabilities to retrieve the required result. $$\begin{aligned} \text{(a)~} {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_n \times S^\prime) & = {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_1 \times \Omega_{n-1} \times S^\prime) \\ & = {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_1 \times \{ s \}) \times {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_{n-1} \times S^\prime) \\ & = {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg\alpha {\textnormal{ U }}s) \times {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_{n-1} \times S^\prime) \end{aligned}$$ $$\begin{aligned} \text{(b)~} {\ensuremath{\mathbb{P}}}^\mathcal{M}_s({\ensuremath{\Diamond}}\alpha) & = \Sigma_{n = 1}^{+\infty} {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_n \times S^\prime) \\ & = \frac{ {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\Omega_1 \times S^\prime)}{ 1 - {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg\alpha {\textnormal{ U }}s)} \\ & = \frac{ {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg s {\textnormal{ U }}\alpha)}{ 1 - {\ensuremath{\mathbb{P}}}^\mathcal{M}_s(\neg\alpha {\textnormal{ U }}s)} \end{aligned}$$ \[lem:min\_max\_reachability\] Let $\mathcal{I} = (S,s_0,P,V)$ be an [[[IMC]{.nodecor}]{}]{}, $\mathcal{M}$ be an [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$, and $\alpha \subseteq A$ be a proposition. There exist [[[MC]{.nodecor}]{}]{}s $\mathcal{M}_1$ and $\mathcal{M}_2$ satisfying $\mathcal{I}$ with the same structure than $\mathcal{I}$ such that ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_1}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}_2}({\ensuremath{\Diamond}}\alpha)$. Lemma \[lem:min\_max\_reachability\] is a consequence of Corollary \[lem:1sat\_same\_proba\] and Lemma \[lem:reach\_with\_less\_states\] and states that the maximal and the minimal probability of reaching a given proposition is realized by Markov chains with the same structure than the [[[IMC]{.nodecor}]{}]{}. Let $\mathcal{I}$ be an [[[IMC]{.nodecor}]{}]{} and $\mathcal{M}$ be a [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$. Consider the sequence of [[[MC]{.nodecor}]{}]{}s $(\mathcal{M}_n)_{n \in {\ensuremath{\mathbb{N}}}}$ s.t. $\mathcal{M}_0$ is the [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$ with degree $1$ obtained by Corollary \[lem:1sat\_same\_proba\] and for all $n \in {\ensuremath{\mathbb{N}}}$, $\mathcal{M}_{n+1}$ is the [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{I}$ with strictly less states than $\mathcal{M}_{n}$ and verifying ${\ensuremath{\mathbb{P}}}^{\mathcal{M}_{n+1}}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}_{n}}({\ensuremath{\Diamond}}\alpha)$ given by Lemma \[lem:reach\_with\_less\_states\] if $\mathcal{M}_{n}$ does not have the same structure than $\mathcal{I}$ and equal to $\mathcal{M}_{n}$ otherwise. By construction $(\mathcal{M}_n)_{n \in {\ensuremath{\mathbb{N}}}}$ is finite and its last element is a Markov chain $\mathcal{M}^\prime$ with the same structure than $\mathcal{I}$ s.t. ${\ensuremath{\mathbb{P}}}^{\mathcal{M}^\prime}({\ensuremath{\Diamond}}\alpha) \leq {\ensuremath{\mathbb{P}}}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha)$. [[Theorem \[thm:reachability-semantics-equivalence-imcs\]]{.nodecor}]{} Let $\mathcal{I} = (S,s_0,P,V)$ be an [[[IMC]{.nodecor}]{}]{}, $\alpha \subseteq A$ be a state label, ${\sim} \in \{\le,<,>,\ge\}$ and $0<p<1$. $\mathcal{I}$ satisfies $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ with the once-and-for-all semantics iff $\mathcal{I}$ satisfies $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ with the at-every-step semantics. Let $\mathcal{I} = (S,s_0,P,V)$ be an [[[IMC]{.nodecor}]{}]{}, $\alpha \subseteq A$ be a state label, ${\sim} \in \{\le,<,>,\ge\}$ and $0 < p < 1$. Recall that according to an [[[IMC]{.nodecor}]{}]{} satisfaction relation the property $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ holds iff there exists an [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{I}$ (with the chosen semantics) such that $\mathbb{P}^{\mathcal{M}}({\ensuremath{\Diamond}}\alpha) {\sim} p$. Recall also that ${\ensuremath{\models^{a}_{\tt{I}}}}$ is more general than ${\ensuremath{\models^{o}_{\tt{I}}}}$: for all [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ if $\mathcal{M} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$ then $\mathcal{M} {\ensuremath{\models^{o}_{\tt{I}}}}\mathcal{I}$ - Direct from the fact that ${\ensuremath{\models^{o}_{\tt{I}}}}$ is more general than ${\ensuremath{\models^{o}_{\tt{I}}}}$ ([cf. ]{}Appendix \[ap:compare\_imcs\_satisfaction\_relations\]) - $\mathbb{P}^{\mathcal{I}}({\ensuremath{\Diamond}}\alpha) {\sim} p$ with the at-every-step semantics implies that there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ s.t. $\mathcal{M} {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$ and $\mathcal{M} {\sim} p$. Thus by Lemma \[lem:min\_max\_reachability\] there exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M}^\prime$ s.t. $\mathcal{M}^\prime {\ensuremath{\models^{a}_{\tt{I}}}}\mathcal{I}$, $\mathcal{M}^\prime {\sim} p$, $\mathcal{M}^\prime$ has the same structure than $\mathcal{I}$. However if $\mathcal{M}^\prime$ has the same structure than $\mathcal{I}$ then $\mathcal{M}^\prime$ satisfies $\mathcal{I}$ with the one-and-for-all semantics. [[[CSP]{.nodecor}]{}]{} Encodings for Quantitative Reachability --------------------------------------------------------------- \[prop:model\_reach\_label\] Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[[pIMC]{.nodecor}]{}]{} and $\alpha \subseteq A$ be a state label. There exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ iff there exists a valuation $v$ solution of the [[CSP]{.nodecor}]{} ${\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}(\mathcal{P},\alpha)$ s.t. for each state $s \in S$: $v(\lambda_s)$ is equals to [[$\mathsf{true}$]{}]{} iff ${\ensuremath{\mathbb{P}}}^\mathcal{M}_s({\ensuremath{\Diamond}}\alpha) \neq 0$. Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[[pIMC]{.nodecor}]{}]{} and $\alpha \subseteq A$ be a state label. For each state $s \in S$, the variable $\alpha_s$ plays a symmetric role to the variable $\omega_s$ from ${\ensuremath{{\bf C_{{\exists}r}}}}$: instead of indicating the existence of a path from $s_0$ to $s$, it characterizes the existence of a path from $s$ to a state labeled with $\alpha$. As for the ${\ensuremath{{\bf C_{{\exists}r}}}}$ [[[CSP]{.nodecor}]{}]{} encoding, each solution of the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}r}}}}(P, \alpha)$ corresponds to a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{P}$ w.r.t. ${\ensuremath{\models^o_{\tt{pI}}}}$. Furthermore the constraints added in ${\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}$ ensures that $\alpha_s$ is equal to $k$ iff there exists a path of length $k-1$ with non zero probability from $s$ to a state label with $\alpha$ in $\mathcal{M}$. Thus by Constraint \[encoding\_erprime\_bool\_var\], variables $\lambda_s$ is equal to [[$\mathsf{true}$]{}]{} iff there exists a path with non zero probability from the initial state $s_0$ to a state labeled with $\alpha$ passing by $s$. \[prop:model\_quant\_reachability\] Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[[pIMC]{.nodecor}]{}]{} and $\alpha \subseteq A$ be a proposition. There exists a [[[MC]{.nodecor}]{}]{} $\mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ iff there exists a valuation $v$ solution of the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P},\alpha)$ s.t. $v(\pi_{s})$ is equal to ${\ensuremath{\mathbb{P}}}^\mathcal{M}_s({\ensuremath{\Diamond}}\alpha)$ if $s$ is reachable from the initial state $s_0$ in $\mathcal{M}$ and is equal to $0$ otherwise. Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[[pIMC]{.nodecor}]{}]{} and $\alpha \subseteq A$ be a state label. ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}$ extends the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C^{\prime}_{{\exists}r}}}}$ that produces a ${\textnormal{MC}}$ $\mathcal{M}$ satisfying $\mathcal{P}$ ([cf. ]{}Proposition \[prop:model\_reach\_label\]) by computing the probability of reaching $\alpha$ in $\mathcal{M}$. In order to compute this probability, we use standard techniques from [@Baier2008PMC] that require the partitioning of the state space into three sets $S_{\top}$, $S_{\bot}$, and $S_?$ that correspond to states reaching $\alpha$ with probability $1$, states from which $\alpha$ cannot be reached, and the remaining states, respectively. Once this partition is chosen, the reachability probabilities of all states in $S_?$ are computed as the unique solution of an equation system (see [@Baier2008PMC], Theorem 10.19, p.766). Recall that for each state $s \in S$ variable $\alpha_s$ is equal to [[$\mathsf{true}$]{}]{} iff $s$ is reachable in $\mathcal{M}$ and $s$ can reach $\alpha$ with a non zero probability. Thus we consider $S_\bot = \{s \ \|\ \alpha_s = {\ensuremath{\mathsf{false}}}\}$, $S_\top = \{s \ \|\ V(s) = \alpha\}$, and $S_? = S \setminus (S_\bot \cup S_\top)$. Finally constraints in ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}$ encodes the equation system from [@Baier2008PMC] according to chosen $S_\bot$, $S_\top$, and $S_?$. Thus $\pi_{s_0}$ is equal to the probability in $\mathcal{M}$ to reach $\alpha$. [[Theorem \[thm:pimc\_reachability\_in\_cp\].*]{.nodecor}]{} Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[pIMC]{.nodecor}]{}, $\alpha \subseteq A$ be a label, $p \in [0, 1]$, ${\sim} \in \{\leq,<, \geq,>\}$ be a comparison operator, and $(X,D,C)$ be ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P}, \alpha)$: - [[CSP]{.nodecor}]{} $(X,D,C \cup (\pi_{s_0} \sim p))$ is satisfiable iff $\exists \mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$ s.t. ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$ - [[CSP]{.nodecor}]{} $(X,D,C \cup (\pi_{s_0} \not\sim p))$ is unsatisfiable iff $\forall \mathcal{M} {\ensuremath{\models^a_{\tt{pI}}}}\mathcal{P}$: ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$ Let $\mathcal{P} = (S, s_0 , P, V, Y)$ be a [[[pIMC]{.nodecor}]{}]{}, $\alpha \subseteq A$ be a state label, $p \in [0, 1]$, and ${\sim} \in \{\leq,<, \geq,>\}$ be a comparison operator. Recall that ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P},\alpha)$ is a [[[CSP]{.nodecor}]{}]{} s.t. each solution corresponds to a ${\textnormal{MC}}$ $\mathcal{M}$ satisfying $\mathcal{P}$ where $\pi_{s_0}$ is equal to ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha)$. Thus adding the constraint $\pi_{s_0} \sim p$ allows to find a [[[MC]{.nodecor}]{}]{} $\mathcal{M}$ satisfying $\mathcal{P}$ s.t. ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$. This concludes the first item presented in the theorem. For the second item, we use Theorem \[thm:reachability-semantics-equivalence-imcs\] with the Proposition \[prop:model\_quant\_reachability\] which ensure that if the [[[CSP]{.nodecor}]{}]{} ${\ensuremath{{\bf C_{{\exists}\bar{r}}}}}(\mathcal{P},\alpha)$ plus the constraint $\pi_{s_0} \not\sim p$ is not satisfiable then there is no [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{P}$ w.r.t. ${\ensuremath{\models^a_{\tt{pI}}}}$ s.t. ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \not\sim p$; thus ${\ensuremath{\mathbb{P}}}^\mathcal{M}({\ensuremath{\Diamond}}\alpha) \sim p$ for all [[[MC]{.nodecor}]{}]{} satisfying $\mathcal{P}$ w.r.t. ${\ensuremath{\models^a_{\tt{pI}}}}$. [^1]: Indeed, when $0 \le v(f_1) \leq v(f_2) \le 1$ is not respected, the interval is inconsistent and therefore empty. [^2]: $\|\mathcal{L}_1\|$ and $\|\mathcal{L}_2\|$ are the sizes of $\mathcal{L}_1$ and $\mathcal{L}_2$, respectively. [^3]: All resources, benchmarks, and source code are available online as a Python library at <https://github.com/anicet-bart/pimc_pylib> [^4]: Available online at `http://www.prismmodelchecker.com` [^5]: As illustrated in Example \[ex:model\_consistency\], $\mathcal{M}$ is not a well formed [[[MC]{.nodecor}]{}]{} since some unreachable states do not respect the probability distribution property. However, one can correct it by simply setting one of its outgoing transition to $1$ for each unreachable state.
--- address: \| Department of Physics, University of Maryland,\ College Park, MD 20742, USA author: - 'M. VERZOCCHI [^1]' title: FIRST DATA FROM DØ IN RUN 2 --- Physics goals for Run 2 at the Tevatron {#sec:goals} ======================================= The Tevatron collider at Fermilab is the world’s highest energy accelerator, colliding protons and antiprotons at a centre of mass energy of $\sim\mathrm{2}$ TeV. Since the end of Run 1 the accelerator complex has been upgraded to raise the collision energy and to deliver larger integrated luminosities (2 $\mathrm{fb}^{\mathrm{-1}}$ before 2005, and 15 $\mathrm{fb}^{\mathrm{-1}}$ before the startup of the LHC), extending the physics reach of the experiments. The DØ detector has also been upgraded [@upgrade] to cope with the reduced time between bunch crossings, the increase in luminosity and backgrounds, and to extend the physics capabilities of the experiment. The physics goals of Run 2 include the investigation of the electroweak symmetry breaking mechanism and searches for physics beyond the Standard Model. The luminosity gain will also yield large statistics for precise measurements at lower mass scales, such as [b]{}–physics and QCD. The large increase in luminosity means that the reach for discoveries at the highest mass scales will be increased, while formerly rare processes, like the production of weak bosons and of the top quark, become the object of precision measurements. With 15 $\mathrm{fb}^{\mathrm{-1}}$ at the end of Run 2, the Tevatron experiments will be able to search for the Higgs boson in much of the phase space currently allowed by Standard Model fits [@run2hig; @lepewwg]. Meanwhile this allowed range can be reduced with improved precise measurements of the masses of the top quark and of the W boson, reducing the uncertainty in their determination respectively to 1–2 GeV and 20 MeV at the end of Run 2 [@ewrun2]. With 2 $\mathrm{fb}^{\mathrm{-1}}$ the mass reach in the search for supersymmetric particles will extend to 400 GeV for squark and gluinos, 200 GeV for stop and sbottom quarks and 180 GeV for charginos in the trilepton final state [@susy]. Other models (TeV–scale gravity and extra dimensions, technicolour, leptoquarks) will also be used to guide searches for new phenomena. In addition to these optimised searches, model independent searches for new physics will also be performed, following the approach pioneered by DØ in Run 1 [@sleuth]. Upgrade of the DØ detector {#sec:upgrade} ========================== The Run 2 upgrade builds on the strengths of the Run 1 DØ detector, its state of the art hermetic calorimeter system and its lepton identification capabilities over a large rapidity range. To achieve the Run 2 physics goals the detector has undergone a series of large changes, highlighted in the cross sectional view of the DØ detector shown in Fig. \[fig:d0dete\]. A 2 T superconducting solenoid has been installed in front of the calorimeter cryostat, surrounding a new tracking system comprising silicon microstrip and scintillating fibres detectors. Preshower detectors have been installed between the solenoid and the calorimeter in the central part of the detector, and in front of the forward calorimeter to compensate for the energy loss of electrons and photons in the solenoid and to improve the angular resolution for photons. The forward muon system has been completely rebuilt, separating the triggering function (using scintillator tiles), from the precision tracking (using mini-drift tubes). The forward muon detector benefits from the new shielding of the beam line, resulting in a large reduction of backgrounds compared to Run 1. In the central region of the detector additional layers of scintillators have been added to the muon system, allowing a more extensive reduction of out of time backgrounds. The readout electronics, the trigger and DAQ systems have all been rebuilt to cope with the reduced time between crossings and the increase in luminosity. Detector performance {#sec:perform} ==================== Calorimeter {#sub:calo} ----------- The DØ calorimeter is a 55k channels U/LAr calorimeter with fine longitudinal and transverse segmentation, uniform response and good energy resolution. New readout electronics with analog pipelines has been installed for Run 2. The calorimeter has been fully operational since the beginning of the run with less than 0.1% bad channels. The commissioning of the preshower detectors and of the new inter–cryostats detectors (scintillator tiles installed in the gaps between the barrel and endcap calorimeters to improve the resolution on the missing transverse momentum) was still ongoing at the time of the conference. A preliminary calibration of the calorimeter has been obtained investigating the invariant mass spectrum of dielectron events originating from the decay of $\mathrm{Z}^{\mathrm{0}}$ bosons, shown in Fig. \[fig:calo\]. The knowledge of the absolute calorimeter calibration is currently limited by the available statistics in the $\mathrm{Z}^{\mathrm{0}}$ sample. -- -- -- -- The energy scale for jets was then determined using $\gamma\mathrm{+jets}$ events. Also shown in Fig. \[fig:calo\] is a preliminary dijet invariant mass spectrum, which does not include correction for trigger threshold effects. Due to the increased centre of mass energy of the Tevatron collider, DØ has already observed events at invariant masses in excess of 500 GeV in a sample corresponding to an integrated luminosity of only about 1 $\mathrm{pb}^{\mathrm{-1}}$. While the resolution on the missing transverse momentum will improve with a better understanding of the calibration of the calorimeter, the Monte Carlo already provides a good description of the significance of the measurement of the missing transverse momentum, provided the error on this quantity is taken from data. This permits the use of the missing transverse momentum for analyses, one of the most important tools in the searches for physics beyond the Standard Model. Muon detector {#sub:muon} ------------- In DØ the trajectories of muons penetrating the iron toroid surrounding the calorimeters are measured by drift tubes. Trigger signals are obtained from scintillator counters, which provide a timing signal used to reject background from cosmic rays. In the central region the Run 1 drift tubes are used, with a faster gas mixture and new readout electronics, and new scintillator counters have been installed. The forward system has been completely rebuilt for Run 2: it includes 3 layers of mini drift tubes and scintillator pixel counters. The muon system has been fully operational since the beginning of Run 2 and thanks to the new shielding close to the beam line the detectors can easily discriminate muons from the low rate of out of time backgrounds. -- -- -- -- Two examples of the measurements performed so far to study the performance of the muon detector are shown in Fig. \[fig:muon\]. The left plot shows the invariant mass distribution for muons having $\mathit{p}_{\mathrm{T}}>\mathrm{3}$ GeV. The signal from the $\mathrm{J}/\psi$ resonance is clearly visible above the background. Its width is consistent with expectations for the muon system only, where the momentum resolution is dominated by the multiple scattering in the iron toroid. The muon momentum resolution improves with the use of the information obtained from the central tracking system (see Sec. \[sub:track\]). The right plot shows the spectrum of jets having a soft–muon [b]{}–tag. Muons having a transverse momentum relative to the jet axis in excess of 1 GeV most likely come from the decay of a [b]{} quark. Both the transverse momentum distribution of muons relative to the jet and the transverse momentum distribution of these jets agree with expectations, based on the Run 1 data. Tracker {#sub:track} ------- The DØ tracking system is entirely new: it is based on a silicon microstrip tracker (SMT) and a central fibre tracker (CFT) installed inside a 2 T solenoid. The SMT consists of 4 barrel layers of single and double–sided silicon microstrip detectors, interspread with disks arranged perpendicular to the beam direction. These, together with additional disks in the forward directions, allow efficient track reconstruction in the SMT up to $\|\eta\|=\mathrm{2.5}$ independently from the position of the primary vertex, which has a gaussian distribution along the beam axis with a RMS of 30 cm. Altogether the SMT comprises $\sim$800k readout channels. It is has been in continuous operation since the beginning of Run 2. Fig. \[fig:track\] shows the $\mathrm{K}^{\mathrm{0}}_{\mathrm{s}}$ peak obtained from the invariant mass of unlike sign tracks reconstructed in the SMT system alone. At larger radii (between 20 and 51 cm) tracking is performed in 8 double layers of 840 $\mu$m diameter scintillating fibres. Each layer has two axial and two $\mathrm{2}^o$ stereo fibres, read out through visible light photon counters operating at 9 K, with 85% quantum efficiency and good signal to noise ratio. The readout electronics for the fibre tracker (and the preshower detectors) has been completely installed in the spring of 2002. Fig. \[fig:track\] shows the hits and the reconstructed tracks in the DØ tracker for a typical two jet event. Track information is already being used in analyses, improving the resolution of the muon measurement, and providing a useful tool for the calibration of the electromagnetic calorimeter. Work is underway to improve the understanding of the vertexing algorithms and to develop impact parameter and vertex based [b]{} quark tags, which are crucial in the search for the Higgs boson and for reducing the background in top analyses. -- -- -- -- DAQ and trigger {#sub:trigdaq} --------------- The trigger and DAQ systems have been almost completely rebuilt for Run 2. High $\mathrm{p}_{\mathrm{T}}$ triggers have been running without prescales during the first year of Run 2 operations despite limitations in the DØ data taking capabilities due to delays in the delivery of L2 CPUs and of L3 components. Most of the L2 triggers and a new Ethernet based DAQ system are being commissioned: DØ will be capable of handling design trigger rates before the beginning of the summer. The L1 central tracking trigger (CTT), which uses the track measurements in the CFT, is being commissioned. Its use will result in sharper turn–on curves for muon triggers. The L2 silicon track trigger, an important addition for Higgs physics, will be installed and commissioned later in the fall. First physics results ===================== In addition to the physics signals already discussed in the previous section, a relatively clean sample of $\mathrm{W}\rightarrow\mathrm{e}\nu$ candidates was obtained by selecting events with a high $\mathrm{p}_{\mathrm{T}}$ electromagnetic cluster matched to a track and large missing transverse energy (see Fig. \[fig:w\]). The background has been estimated from data and consists mainly of QCD events with fake electrons. Also shown in Fig. \[fig:w\] is the E/p ratio for the candidate electrons. Those W candidates with additional jets will constitute the main background for top and Higgs analyses. Other preliminary results include the first candidates (most likely from background sources) in searches for trileptons and leptoquarks. More extensive results using higher quality data and larger luminosities are expected for the summer. -- -- -- -- Conclusions {#sec:concl} =========== The luminosity delivered by the Tevatron in the first year of Run 2 has been used by the DØ experiment mainly for detector commissioning purposes, allowing enormous progress in the understanding of the detector performance. Preliminary analyses have been performed using a subset of the delivered luminosity, indicating that the DØ collaboration will be able to fully exploit the physics opportunities presented by Run 2. Acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank all his DØ colleagues, who have helped in preparing the results, and the conference organisers for providing such an enjoyable and stimulating atmosphere. References {#references .unnumbered} ========== [99]{} T.LeCompte and H.T.Diehl, [Annu. Rev. Nucl. Part. Sci.]{} 50 (2000) 71. Run 2 Higgs and Supersymmetry Workshop (M.Carena [et al.]{}), hep–ph/0010228. LEP Electroweak Working Group, [`h`ttp://lepewwg.web.cern.ch/LEPEWWG]{}. M.Grünewald, U.Heintz, M.Narain and M.Schmitt, hep–ph/0111217. Run 2 Higgs and Supersymmetry Workshop (V.Barger [et al.]{}), hep–ph/0003154. DØ Collaboration (V.Abazov [et al.]{}), 64 (2001) 012004. [^1]: Representing the DØ Collaboration
--- abstract: 'In this talk I will briefly outline work in progress in two different contexts in astrophysical relativity, i.e. the study of rotating star spacetimes and the problem of reliably extracting gravitational wave templates in numerical relativity. In both cases the use of Weyl scalars and curvature invariants helps to clarify important issues.' author: - Marco Bruni - Andrea Nerozzi - Frances White title: 'Newman-Penrose quantities as valuable tools in astrophysical relativity' --- [ address=[Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK.]{} ]{} [ address=[Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK.]{} ]{} [ address=[Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK.]{} ]{} Introduction ============ The Weyl scalars - the components of the Weyl tensor over a null tetrad - were known and used in relativity before the introduction of the Newman-Penrose (NP) formalism, but within the latter they acquired a new relevance. Here I will summarise the use of Weyl scalars as tools in two different contexts in astrophysical relativity. First I will briefly summarize recent work [@bertietal] aimed at assessing the validity of the Hartle-Thorne (HT) slow-rotation approximation for describing stationary axisymmetric rotating Neutron Stars (NSs), introducing work in progress [@FWt] to extend the analysis. In this context the Weyl scalars are used to construct invariant measures of the deviation of the exterior spacetime from Petrov type D, in view of a possible development of a Teukolsky-like perturbative formalism for rotating NSs. In the second part I will outline how the Weyl scalars may be used in numerical relativity in order to construct a wave extraction formalism for simulations dealing with spacetimes that will settle to a perturbed black hole (BH) at late times. In this case the Teukolsky BH perturbation formalism [@teukolsky] is in principle applicable, but it is difficult to extract a BH background spacetime, i.e. the gravitational mass and angular momentum, from a given simulation. Introducing the notion of a quasi-Kinnersly frame (also used in [@bertietal]), in [@AN1; @AN2] a method was proposed that bypasses this difficulty, by not requiring a background, and allows direct wave extraction. I will present here work in progress [@ANt] where the method is directly applicable. \[Francesfig\] ![Contour plots of $(1-\|S\|)\rm{x}10^{4}$ for decreasing values (from bottom left) $5, 4, 3, 2, 1$, for a 1.4 $\rm{M}_\odot$ star, for a representative equation of state. Left panel: $\varepsilon=0.39258$; right panel: $\varepsilon=0.54440$.](HT_cont_14_04){height=".22\textheight"} ![Contour plots of $(1-\|S\|)\rm{x}10^{4}$ for decreasing values (from bottom left) $5, 4, 3, 2, 1$, for a 1.4 $\rm{M}_\odot$ star, for a representative equation of state. Left panel: $\varepsilon=0.39258$; right panel: $\varepsilon=0.54440$.](HT_cont_14_05){height=".22\textheight"} The spacetime of rotating stars =============================== Using a variety of models for NSs of different masses and equations of state, and comparing with full general relativistic numerical models, it was shown in [@bertietal] that the HT approximation to the metric of a rotating relativistic star is very good for most astrophysical applications, even at the rotation rates of the fastest known milli-second pulsar. For instance, the innermost stable circular orbit is predicted with an accuracy of $\sim 1\%$. It was also shown in [@bertietal] that although the spacetime of these stars is of Petrov type I (general; it would be type D for a spherical star), the deviation from type D is small, at least on the equatorial plane. The HT metric is obtained as a perturbative solution to second order in $\epsilon=\Omega/\Omega^$, where $\Omega$ is the star’s angular velocity and $\Omega^=(M/R^3)^{1/2}$ is a “Keplerian” rotational scale. The deviation from type D is measured by $1-\|S\|$, where $S=27J^2/I^3$ is an invariant curvature scalar, with $S=1$ for type D. Fig. \[Francesfig\] [@FWt] shows that $1-\|S\|$ is small also out of the equatorial plane. In fact, $1-\|S\|$ decreases more rapidly to zero out of the equatorial plane, as one would have expected. Thus in this sense $1-\|S\|$ in the equatorial plane is a good upper limit for the deviation from type D. Wave extraction =============== In [@AN1; @AN2] methods were introduced to identify, for a general numerical relativity implementation, what was dubbed the Quasi-Kinnersley frame, an equivalence class of tetrads that reduce, in the limit where the spacetime approaches type D, to the Kinnersly tetrad used in [@teukolsky] for the BH background. In this tetrad the Weyl scalar $\Psi_4$ carries information on outgoing gravitational radiation, and $\Psi_0$ on ingoing radiation. Work is in progress to identify one specific and physically significant tetrad from the equivalence class, appropriate for a generic numerical relativity code using an arbitrary ADM slicing. However, the method of [@AN1; @AN2] is already applicable when using a null slicing, in particular to the Bondi metric used in [@pap], where non-linear oscillations of a BH were analysed. In the case of the Bondi metric the gravitational wave signal can be extracted using the news function $\gamma_{,v}$ ($v$ is retarded time), directly related to the outgoing energy. In the linear regime, one expects $\Psi_4=-\gamma_{,vv}$. Thus this case is ideally suited to test our method, since we can compare [@ANt] the news function obtained directly from the Bondi metric with that obtained via $\Psi_4$ in the Quasi-Kinnersley frame. Fig. \[Andreafig\] shows the comparison of $\Psi_4$ with $\gamma_{,vv}$, where in our axisymmetric case $2\gamma=h^{TT}_{\theta\theta}$, i.e. the other polarization $h^{TT}_{\theta\phi}$ vanishes. Clearly the agreement is excellent at late times, as predicted, with an error $\Delta=\|\Psi_4+\gamma_{vv}\|\sim 10^{-6}$ at $v_4=80$. \[Andreafig\] ![Comparison (from [@ANt]) of $\Psi_4$ and $\gamma_{,vv}$ at retarded times $v_1=1$, $v_2=20$, $v_3=50$, $v_4=80$.](news "fig:"){height=".4\textheight"} Conclusions =========== Here we have shown how the method obtained in [@AN1; @AN2] works, reproducing very well results obtained in [@pap], when applied in the context of a code using the characteristic approach [@ANt]. Work is in progress to find a method to choose one particular Quasi-Kinnersly tetrad out of the general class in order to properly extract gravitational wave templates from any numerical relativity code using an arbitrary ADM slicing. For a rotating NS, we have shown that the deviation of the spacetime from Petrov type D is always very small, with $1-\|S\|$ rapidly decreasing out of the equatorial plan and with increasing distance from the star. [99]{} E. Berti, F. White, A. Maniopoulou and M. Bruni, Mon. Not. R. Astr. Soc. , submitted, (2004). F. White, PhD thesis, University of Portsmouth, [in preparation]{}. S. A. Teukolsky, Astrophys. J. 185, 635 (1973). A. Nerozzi, C. Beetle, M. Bruni, L. M. Burko and D. Pollney, Phys. Rev. D, submitted (2004). C. Beetle, M. Bruni, L. M. Burko and A. Nerozzi, Phys. Rev. D, submitted (2004). A. Nerozzi, PhD thesis, University of Portsmouth, (2004). P. Papadopoulos, Phys. Rev. D 65, 084016 (2002).
--- abstract: 'We consider an extension of the Weyl-Cartan-Weitzenböck (WCW) and teleparallel gravity, in which the Weitzenböck condition of the exact cancellation of curvature and torsion in a Weyl-Cartan geometry is inserted into the gravitational action via a Lagrange multiplier. In the standard metric formulation of the WCW model, the flatness of the space-time is removed by imposing the Weitzenböck condition in the Weyl-Cartan geometry, where the dynamical variables are the space-time metric, the Weyl vector and the torsion tensor, respectively. However, once the Weitzenböck condition is imposed on the Weyl-Cartan space-time, the metric is not dynamical, and the gravitational dynamics and evolution is completely determined by the torsion tensor. We show how to resolve this difficulty, and generalize the WCW model, by imposing the Weitzenböck condition on the action of the gravitational field through a Lagrange multiplier. The gravitational field equations are obtained from the variational principle, and they explicitly depend on the Lagrange multiplier. As a particular model we consider the case of the Riemann-Cartan space-times with zero non-metricity, which mimics the teleparallel theory of gravity. The Newtonian limit of the model is investigated, and a generalized Poisson equation is obtained, with the weak field gravitational potential explicitly depending on the Lagrange multiplier and on the Weyl vector. The cosmological implications of the theory are also studied, and three classes of exact cosmological models are considered.' author: - Zahra Haghani$^1$ - Tiberiu Harko$^2$ - Hamid Reza Sepangi$^1$ - Shahab Shahidi$^1$ title: 'Weyl-Cartan-Weitzenböck gravity through Lagrange multiplier' --- Introduction {#intro} ============ General relativity (GR) is considered to be the most successful theory of gravity ever proposed. Its classic predictions on the perihelion advance of Mercury, on the deflection of light by the Sun, gravitational redshift, or radar echo delay have been confirmed at an unprecedented level of observational accuracy. Moreover, predictions such as the orbital decay of the Hulse-Taylor binary pulsar, due to gravitational - wave damping, have also fully confirmed the observationally weak-field validity of the theory. The detection of the gravitational waves will allow the testing of the predictions of GR in the strong gravitational field limit, such as, for example, the final stage of binary black hole coalescence (for a recent review on the experimental tests of GR see [@Will]). Despite these important achievements, recent observations of supernovae [@Riess] and of the Cosmic Microwave Background radiation [@1] have suggested that on cosmological scales GR may not be the ultimate theory to describe the Universe. If GR is correct, in order to explain the accelerating expansion of the Universe, we require that the Universe is filled with some component of unknown nature, called dark energy, having some unusual physical properties. To find an alternative to dark energy to explain cosmological observations, in the past decade many modified theories of gravity, which deviate from the standard GR on cosmological scales have been proposed (see [@Clifton] for a recent review on modified gravity and cosmology). On the other hand, because of its prediction of space-time singularities in the Big Bang and inside black holes GR could be considered as an incomplete physical model. In order to solve the singularity problem it is generally believed that a consistent extension of GR into the quantum domain is needed. Since GR is essentially a geometric theory, formulated in the Riemann space, looking for more general geometric structures adapted for the description of the gravitational field may be one of the most promising ways for the explanation of the behavior at large cosmological scales of the matter in the Universe, whose structure and dynamics may be described by more general geometries than the Riemannian one, valid at the Solar System level. The first attempt to create a more general geometry is due to Weyl [@Weyl], who proposed a geometrized unification of gravitation and electromagnetism. Weyl abandoned the metric-compatible Levi-Civita connection as a fundamental concept, since it allowed the distant comparison of lengths. Substituting the metric field by the class of all conformally equivalent metrics, Weyl introduced a connection that would not carry any information about the length of a vector on parallel transport. Instead the latter task was assigned to an extra connection, a so-called length connection that would, in turn, not carry any information about the direction of a vector on parallel transport, but that would only fix, or gauge, the conformal factor. Weyl identified the length connection with the electromagnetic potential. A generalization of Weyl’s theory was introduced by Dirac [@Dirac], who proposed the existence of two metrics, one unmeasurable metric $ds_E$, affected by transformations in the standards of length, and a second measurable one, the conformally invariant atomic metric $ds_A$. In the development of the generalized geometric theories of gravity a very different evolution took place due to the work of Cartan [@Cartan], who proposed an extension of general relativity, which is known today as the Einstein-Cartan theory [@Hehl1]. The new geometric element of the theory, the torsion field, is usually associated from a physical point of view to a spin density [@Hehl1]. The Weyl geometry can be immediately generalized to include the torsion. This geomety is called the Weyl-Cartan space-time, and it was extensively studied from both mathematical and physical points of view [@WC]. To build up an action integral from which one can obtain a gauge covariant (in the Weyl sense) general relativistic massive electrodynamics, torsion was included in the geometric framework of the Weyl-Dirac theory in [@Isr] . For a recent review of the geometric properties and of the physical applications of the Riemann-Cartan and Weyl-Cartan space-times see [@Rev]. A third independent mathematical development took place in the work of Weitzenböck [@Weitz], who introduced the so-called Weitzenböck spaces. A Weitzenböck manifold has the properties $\nabla _{\mu }g_{\sigma \lambda }= 0$, $T^{\mu }_{\sigma \lambda }\neq 0$, and $R^{\mu }_{\nu \sigma \lambda }=0$, where $g_{\sigma \lambda }$, $T^{\mu }_{\sigma \lambda }$ and $R^{\mu }_{\nu \sigma \lambda }$ are the metric, the torsion, and the curvature tensors of the manifold, respectively. When $T^{\mu }_{\sigma \lambda }= 0$, the manifold is reduced to a Euclidean manifold. The torsion tensor possesses different values on different parts of the Weitzenböck manifold. Therefore, since their Riemann curvature tensor is zero, Weitzenböck spaces possess the property of distant parallelism, also known as absolute, or teleparallelism. Weitzenböck type geometries were first used in physics by Einstein, who proposed a unified teleparallel theory of gravity and electromagnetism [@Ein]. The basic idea of the teleparallel approach is to substitute, as a basic physical variable, the metric $g_{\mu \nu}$ of the space-time by a set of tetrad vectors $e^i_{\mu }$. In this approach the torsion, generated by the tetrad fields, can be used to describe general relativity entirely, with the curvature eliminated in favor of torsion. This is the so-called teleparallel equivalent of General Relativity (TEGR), which was introduced in [@11], and is also known as the $f(T)$ gravity model. Therefore, in teleparallel, or f(T) gravity, torsion exactly compensates curvature, and the space-time becomes flat. Unlike in $f(R)$ gravity, which in the metric approach is a fourth order theory, in the $f(T)$ gravity models the field equations are of second order. $f(T)$ gravity models have been extensively applied to cosmology, and in particular to explain the late-time accelerating expansion of the Universe, without the need of dark energy [@13]. An extension of the teleparallel gravity models, called WCW gravity, was introduced recently in [@WCW]. In this approach, the Weitzenböock condition of the vanishing of the sum of the curvature and torsion scalar is imposed in a background Weyl-Cartan type space-time. In contrast to the standard teleparallel theories, the model is formulated in a four-dimensional curved space-time, and not in a flat Euclidian geometry. The properties of the gravitational field are described by the torsion tensor and the Weyl vector fields, defined in a four-dimensional curved space-time manifold. In the gravitational action a kinetic term for the torsion is also included. The field equations of the model, obtained from a Hilbert-Einstein type variational principle, allow a complete description of the gravitational field in terms of two vector fields, the Weyl vector and torsion, respectively, defined in a curved background. The Newtonian limit of the model was also considered, and it was shown that in the weak gravitational field approximation the standard Poisson equation can be recovered. For a particular choice of the free parameters, in which the torsion vector is proportional to the Weyl vector, the cosmological applications of the model were investigated. A large variety of dynamical evolutions can be obtained in the WCW gravity model, ranging from inflationary/accelerated expansions to non-inflationary behaviors. The nature of the cosmological evolution is determined by the numerical values of the parameters of the cosmological model. In particular a de Sitter type late time evolution can be naturally obtained from the field equations of the model. Therefore the WCW gravity model leads to the possibility of a purely geometrical description of dark energy where the late time acceleration of the Universe is determined by the intrinsic nature of the space-time. Recently, the use of Lagrange multipliers in the formulation of dynamical gravity models has attracted considerable attention. The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality nonholonomic constraints, which are capable of reducing the dynamics [@20]. The extension of $f(R)$ gravity models via the addition of a Lagrange multiplier constraint has been proposed in [@fR]. This model can be considered as a new version of $f(R)$ modified gravity since dynamics, and the cosmological solutions, are different from the standard version of $f(R)$ gravity without such constraint. Cosmological models with Lagrange multipliers have been considered from different points of view in [@LM]. It is the purpose of the present paper to investigate a class of generalized WCW type gravity models, in which the Weitzenböck condition of the exact compensation of torsion and curvature is introduced into the gravitational action via a Lagrange multiplier approach. We start our analysis by considering the general action for a gravitational field in a Weyl-Cartan space-time, and we explicitly introduce the Weitzenböck condition into the action via a Lagrange multiplier. By taking the Weyl vector as being identically zero, we obtain the field equations of this gravity model in a Riemann-Cartan space time, with the Weitzenböck condition being described by a proportionality relation between the scalar curvature and torsion scalar, included in the gravitational action via a Lagrange multiplier method which mimics the teleparallel gravity. The weak field limit of the general theory is also investigated, and a generalized Poisson equation, explicitly depending on the Lagrange multiplier and the Weyl vector is obtained. The cosmological implications of the model are investigated for three classes of models. The solutions obtained describe both accelerating and decelerating expansionary phases of the Universe, and they may prove useful for modeling the early and late phases of cosmological evolution. The paper is organized as follows. The gravitational action of the WCW theory with Lagrange multiplier is introduced in Section \[model\]. The gravitational field equations are derived in Section \[feq\]. Some particular cases are also considered in detail. The field equations for the case of the zero Weyl vector are presented in Section \[w\]. The weak field limit of the theory is investigated in Section \[weak\], and the generalized Poisson equation is obtained. The cosmological implications of the theory are investigated in Section \[cos\], and several cosmological models are presented. We discuss and conclude our results in Section \[conclu\]. Some aspects of the Weyl invariance of the theory are considered in the Appendix. WCW gravity model with Lagrange multiplier {#model} ========================================== In this Section we formulate the action of the gravitational field in the WCW gravity with a Lagrange multiplier. A Weyl - Cartan space $CW_4$ is a four-dimensional connected, oriented, and differentiable manifold, having a metric with a Lorentzian signature chosen as $(-+++)$, curvature, torsion, and a connection which can be determined from the Weyl nonmetricity condition. Hence the Weyl-Cartan geometry has the properties that the connection is no longer symmetric, and the metric compatibility condition does not hold. The Weyl non-metricity condition is defined as $$\begin{aligned} \label{eq1} \nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu},\end{aligned}$$ where $w_\mu$ is the Weyl vector. Expanding the covariant derivative, we obtain the connection in the Weyl-Cartan geometry $$\begin{aligned} \label{eq2} \Gamma^\lambda_{~\mu\nu}=\left\{\begin{matrix}\lambda\\ \mu~\nu\end{matrix}\right\}+C^\lambda_{\mu\nu}+g_{\mu\nu}w^\lambda-\delta^\lambda_\mu w_\nu-\delta^\lambda_\nu w_\mu,\end{aligned}$$ where the first term in the LHS is the Christoffel symbol constructed out of the metric and the contorsion tensor $C^\lambda_{~\mu\nu}$ is defined as $$\begin{aligned} \label{eq3} C^\lambda_{~\mu\nu}=T^\lambda_{~\mu\nu}-g^{\lambda\beta}g_{\sigma\mu}T^\sigma_{~\beta\nu}-g^{\lambda\beta}g_{\sigma\nu}T^\sigma_{~\beta\mu},\end{aligned}$$ with torsion tensor $T^\lambda_{~\mu\nu}$ given by $$\begin{aligned} \label{eq4} T^\lambda_{~\mu\nu}={\frac{1}{2}}\left(\Gamma^\lambda_{~\mu\nu}-\Gamma^\lambda_{~\nu\mu}\right).\end{aligned}$$ One can then obtain the curvature tensor of the Weyl-Cartan space-time as $$\begin{aligned} \label{eq5} K^\lambda_{~\mu\nu\sigma}=\Gamma^\lambda_{~\mu\sigma,\nu}-\Gamma^\lambda_{~\mu\nu,\sigma}+\Gamma^\alpha_{~\mu\sigma}\Gamma^\lambda_{~\alpha\nu}-\Gamma^\alpha_{~\mu\nu}\Gamma^\lambda_{~\alpha\sigma}.\end{aligned}$$ Using equation (\[eq2\]) and contracting the curvature tensor with the metric, we obtain the curvature scalar $$\begin{aligned} \label{eq6} K=K^{\mu\nu}_{~~\mu\nu}&=R+6\nabla_\nu w^\nu-4\nabla_\nu T^\nu-6w_\nu w^\nu+8 w_\nu T^\nu\nonumber\\ &+T^{\mu\alpha\nu}T_{\mu\alpha\nu}+2T^{\mu\alpha\nu}T_{\nu\alpha\mu}-4T_\mu T^\mu,\end{aligned}$$ where $R$ is the curvature scalar constructed from the Christoffel symbols and we have defined $T_\beta=T^\alpha_{~\beta\alpha}$. Also all covariant derivatives are with respect to the Riemannian connection described by the Christoffel symbols constructed out of the metric $g_{\mu \nu}$. We also introduce two tensor fields $W_{\mu\nu}$ and $T_{\mu\nu}$, constructed from the Weyl vector and the torsion vector, respectively $$\begin{aligned} \label{eq8} W_{\mu\nu}&=\nabla_\nu w_\mu-\nabla_\mu w_\nu,\\ T_{\mu\nu}&=\nabla_\nu T_\mu-\nabla_\mu T_\nu,\end{aligned}$$ where $T=T_\mu T^\mu$. The most general action for a gravitational theory in the Weyl-Cartan space-time can then be formulated as $$\begin{aligned} \label{eq7} S=\int d^4x\sqrt{-g}\bigg({\frac{1}{\kappa^2}}K&-{\frac{1}{4}}W_{\mu\nu}W^{\mu\nu}\nonumber\\&+\hat\beta\nabla_\mu T\nabla^\mu T+\hat\alpha T_{\mu\nu}T^{\mu\nu}+L_m\bigg),\end{aligned}$$ where $L_m$ is the matter Lagrangian which depends only on the matter fields and the metric, and is independent on the torsion tensor and the Weyl vector. We have also added a kinetic term for the Weyl vector and two possible kinetic terms for the torsion tensor. In equation (\[eq7\]), $\hat{\alpha}$ and $\hat{\beta }$ are arbitrary numerical constants, and $\kappa ^2=16\pi G$. Substituting definition of the curvature scalar from equation , the action for the gravitational field becomes $$\begin{aligned} \label{eq9} S=&{\frac{1}{\kappa^2}}\int d^4x\sqrt{-g}\bigg(R+T^{\mu\alpha\nu}T_{\mu\alpha\nu}+2T^{\mu\alpha\nu}T_{\nu\alpha\mu}\nonumber\\ &-4T_\mu T^\mu-6w_\nu w^\nu+8 w_\nu T^\nu-{\frac{\kappa^2}{4}}W_{\mu\nu}W^{\mu\nu}\nonumber\\ &+\beta\nabla_\mu T\nabla^\mu T+\alpha T_{\mu\nu}T^{\mu\nu}+\kappa^2L_m\bigg),\end{aligned}$$ where we have defined $\alpha=\kappa^2\hat\alpha$ and $\beta=\kappa^2\beta$. The Weitzenböck condition $$\begin{aligned} \label{eq10} \mathcal{W}\equiv R+T^{\mu\alpha\nu}T_{\mu\alpha\nu}+2T^{\mu\alpha\nu}T_{\nu\alpha\mu}-4T_\mu T^\mu=0,\end{aligned}$$ requires that the sum of the scalar curvature and torsion be zero. In order to impose this condition on the gravitational field equations of the theory, we add it to the action by using a Lagrange multiplier $\lambda$. The gravitational action then becomes $$\begin{aligned} \label{eq11} S&={\frac{1}{\kappa^2}}\int d^4x\sqrt{-g}\bigg[-{\frac{\kappa^2}{4}}W_{\mu\nu}W^{\mu\nu}-6w_\nu w^\nu+8 w_\nu T^\nu\nonumber\\ &+(1+\lambda)\big(R+T^{\mu\alpha\nu}T_{\mu\alpha\nu}+2T^{\mu\alpha\nu}T_{\nu\alpha\mu}-4T_\mu T^\mu\big)\nonumber\\ &+\beta\nabla_\mu T\nabla^\mu T+\alpha T_{\mu\nu}T^{\mu\nu}+\kappa^2L_m\bigg].\end{aligned}$$ We note that for $\lambda=-1$ one gets the original WCW action [@WCW]. The gravitational field equations of the WCW gravity model with a Lagrange multiplier {#feq} ===================================================================================== Let us now derive the field equations of the WCW gravity with the Lagrange multiplier. By considering the irreducible decomposition of the torsion tensor and imposing a condition on the terms of the decomposition, we obtain an explicit representation of the Weitzenböck condition. The field equations of a simplified model in which the constant $\alpha =0$ are also obtained explicitly. The gravitational field equations and the effective energy-momentum tensor -------------------------------------------------------------------------- Variation of the action with respect to the Weyl vector and the torsion tensor results in the equations of motion $$\begin{aligned} \label{eq12} -{\frac{\kappa^2}{2}}\nabla_\nu W^{\nu\mu}-6w^\mu+4T^\mu=0,\end{aligned}$$ and $$\begin{aligned} \label{eq13} 4w^{[\rho}&\delta^{\beta]}_\mu+2\alpha\delta^{[\beta}_\mu\nabla_\alpha T^{\rho]\alpha}-2\beta T^{[\rho}\delta^{\beta]}_\mu\Box T\nonumber\\ &+(1+\lambda)\big(T_\mu^{~\rho\beta}+T^{\beta\rho}_{~~\mu}+T^{\rho~\beta}_{~\mu}-4T^{[\rho}\delta^{\beta]}_\mu\big)=0,\end{aligned}$$ respectively. Variation of the action with respect to the Lagrange multiplier $\lambda$ gives the Weitzenböck condition . Now, varying the action with respect to the metric and using the condition , we obtain the dynamical equation for the metric as $$\begin{aligned} \label{eq14} (1+\lambda)R_{\mu\nu}&={\frac{\kappa^2}{2}}T^m_{\mu\nu}+\nabla_\mu\nabla_\nu\lambda-g_{\mu\nu}\Box\lambda-(1+\lambda)\bigg(2T^{\alpha\beta}_{~~~\nu}T_{\alpha\beta\mu}-T_\mu^{~\alpha\beta}T_{\nu\alpha\beta}+2T_{\beta\alpha(\mu}T^{\alpha\beta}_{~~~\nu)}-4T_\mu T_\nu\bigg)\nonumber\\ &+{\frac{\kappa^2}{2}}\left(W_{\mu\alpha}W_\nu^{~\alpha}-{\frac{1}{4}}W_{\alpha\beta}W^{\alpha\beta}g_{\mu\nu}\right)-2\alpha\left(T_{\mu\alpha}T_\nu^{~\alpha}-{\frac{1}{4}}T_{\alpha\beta}T^{\alpha\beta}g_{\mu\nu}\right)+6\left(w_\mu w_\nu-{\frac{1}{2}}w^\alpha w_\alpha g_{\mu\nu}\right)\nonumber\\ &-\beta\left(\nabla_\mu T\nabla_\nu T-{\frac{1}{2}}g_{\mu\nu}\nabla_\alpha T\nabla^\alpha T-2 T_\mu T_\nu\Box T\right)-8\left(T_{(\mu}w_{\nu)}-{\frac{1}{2}}w^\alpha T_\alpha g_{\mu\nu}\right),\end{aligned}$$ where $T^m_{\mu\nu}$ is the energy-momentum of the ordinary-matter. The generalized Einstein field equation can be written as $$\begin{aligned} \label{eq14-1} G_{\mu\nu}=T^{eff}_{\mu\nu},\end{aligned}$$ where we have defined the effective energy-momentum tensor as $$\begin{aligned} \label{eq14-2} T^{eff}_{\mu\nu}&=(1+\lambda)^{-1}\bigg[ {\frac{\kappa^2}{2}}T^m_{\mu\nu}+\nabla_\mu\nabla_\nu\lambda-g_{\mu\nu}\Box\lambda-(1+\lambda)\bigg(2T^{\alpha\beta}_{~~~\nu}T_{\alpha\beta\mu}-T_\mu^{~\alpha\beta}T_{\nu\alpha\beta}+2T_{\beta\alpha(\mu}T^{\alpha\beta}_{~~~\nu)}-4T_\mu T_\nu\bigg)\nonumber\\ &+{\frac{\kappa^2}{2}}\left(W_{\mu\alpha}W_\nu^{~\alpha}-{\frac{1}{4}}W_{\alpha\beta}W^{\alpha\beta}g_{\mu\nu}\right)-2\alpha\left(T_{\mu\alpha}T_\nu^{~\alpha}-{\frac{1}{4}}T_{\alpha\beta}T^{\alpha\beta}g_{\mu\nu}\right)+6\left(w_\mu w_\nu-{\frac{1}{2}}w^\alpha w_\alpha g_{\mu\nu}\right)\nonumber\\ &-\beta\left(\nabla_\mu T\nabla_\nu T-{\frac{1}{2}}g_{\mu\nu}\nabla_\alpha T\nabla^\alpha T-2 T_\mu T_\nu\Box T\right)-8\left(T_{(\mu}w_{\nu)}-{\frac{1}{2}}w^\alpha T_\alpha g_{\mu\nu}\right)\nonumber\\&+{\frac{1}{2}}(1+\lambda)\left(T^{\gamma\beta\alpha}T_{\gamma\beta\alpha}+2T^{\gamma\beta\alpha}T_{\alpha\beta\gamma}-4T^\alpha T_\alpha\right)g_{\mu\nu}\bigg],\end{aligned}$$ and use has been made of equation . The decomposition of the torsion tensor --------------------------------------- The torsion tensor can be decomposed irreducibly into $$\begin{aligned} \label{eq15} T_{\mu\nu\rho}={\frac{2}{3}}(t_{\mu\nu\rho}-t_{\mu\rho\nu})+{\frac{1}{3}}(Q_\nu g_{\mu\rho}-Q_\rho g_{\mu\nu})+\epsilon_{\mu\nu\rho\sigma}S^\sigma,\end{aligned}$$ where $Q_\nu$ and $S^\rho$ are two vectors, and the tensor $t_{\mu\nu\rho}$ is symmetric under the change of the first two indices, and satisfies the following conditions $$\begin{aligned} \label{eq16} t_{\mu\nu\rho}+t_{\nu\rho\mu}+t_{\rho\mu\nu}=0,\\g^{\mu\nu}t_{\mu\nu\rho}=0=g^{\mu\rho}t_{\mu\nu\rho}.\end{aligned}$$ By contracting equation over $\mu$ and $\rho$ we obtain $Q_\mu=T_\mu$. Assuming that $t_{\mu\nu\rho}\equiv 0$ [@WCW], one may formulate the Weitzenböck condition as $$\begin{aligned} \label{eq17} R=-6S_\mu S^\mu+{\frac{8}{3}}T.\end{aligned}$$ Now in equations and the terms with coefficient $(1+\lambda)$ can be simplified to $$\begin{aligned} \label{eq18} T_\mu^{~\rho\beta}+T^{\beta\rho}_{~~\mu}+T^{\rho~\beta}_{~\mu}-4T^{[\rho}\delta^{\beta]}_\mu=-{\frac{8}{3}}T^{[\rho}\delta^{\beta]}_\mu-\epsilon_\mu^{~\rho\beta\sigma}S_\sigma,\end{aligned}$$ and $$\begin{aligned} \label{eq19} 2T^{\alpha\beta}_{~~~\nu}&T_{\alpha\beta\mu}-T_\mu^{~\alpha\beta}T_{\nu\alpha\beta}+2T_{\beta\alpha(\mu}T^{\alpha\beta}_{~~~\nu)}-4T_\mu T_\nu\nonumber\\ &=-{\frac{24}{9}}T_\mu T_\nu+2(S_\alpha S^\alpha g_{\mu\nu}-S_\mu S_\nu),\end{aligned}$$ respectively. Taking the trace of equation over indices $\beta$ and $\mu$, we have $$\begin{aligned} \label{eq20} \alpha\nabla_\alpha T^{\rho\alpha}-\beta T^\rho\Box T={\frac{4}{3}}(1+\lambda)T^\rho-2 w^\rho.\end{aligned}$$ Now, substituting the LHS of the above equation into we obtain $$\begin{aligned} \label{eq21} (1+\lambda)\epsilon_\mu^{~\rho\beta\sigma}S_\sigma=0.\end{aligned}$$ If one assumes $\lambda\neq -1$ then $S_\mu=0$. We note that from equation one has $R=8/3T$ which implies that the vector $T^\mu$ should be space-like for the accelerating Universe with $R=6(\dot{H}+2H^2)$, where $H$ is the Hubble parameter. The case $\alpha =0$ -------------------- In order to further simplify the gravitational field equations of the WCW model with a Lagrange multiplier, let us assume that $\alpha=0$, as in [@WCW]. In this case from equation we find $$\begin{aligned} \label{eq22} \Box T=-{\frac{4}{3\beta}}(1+\lambda)+{\frac{2}{\beta T}}w_\rho T^\rho,\end{aligned}$$ provided that $T\neq 0$. Substituting (\[eq20\]) into we obtain $$\begin{aligned} \label{eq23} T^\alpha T_\alpha(w^\rho\delta^\beta_\mu-w^\beta\delta^\rho_\mu)=w^\alpha T_\alpha(T^\rho\delta^\beta_\mu-T^\beta\delta^\rho_\mu),\end{aligned}$$ which implies that $T_\mu=Aw_\mu$, where $A$ is a constant. In order to obtain the value of the constant $A$, we take the covariant divergence of equation , with the result $$\begin{aligned} \label{eq24} \nabla_\mu(6w^\mu-4T^\mu)=0.\end{aligned}$$ The above equation implies that $A=3/2$, so we conclude that $$\begin{aligned} \label{eq25} T_\mu={\frac{3}{2}}w_\mu.\end{aligned}$$ Substituting the above equation into we obtain the dynamical field equation of the Weyl vector $$\begin{aligned} \label{eq26} \Box w_\mu-\nabla_\mu\nabla_\nu w^\nu-w^\nu R_{\nu\mu}=0.\end{aligned}$$ Now, using , we write equation as $$\begin{aligned} \label{eq27} \Box T=-{\frac{4}{3\beta}}\lambda,\end{aligned}$$ which implies $$\begin{aligned} \label{eq28} \lambda=-{\frac{27\beta}{16}}\Box w^2,\end{aligned}$$ where $w^2=w_\alpha w^\alpha$. Substituting $T^\mu$ and $\lambda$ from equations and into the metric field equation we obtain the effective energy-momentum tensor of the WCW model with the Lagrange multiplier, equation , as $$\begin{aligned} \label{eq29} T^{eff}_{\mu\nu}=\left(1-{\frac{27\beta}{16}}\Box w^2\right)^{-1}&\bigg[{\frac{\kappa^2}{2}}T^m_{\mu\nu}-{\frac{27\beta}{16}}(\nabla_\mu\nabla_\nu\Box w^2-\Box^2 w^2 g_{\mu\nu})+{\frac{\kappa^2}{2}}\left(W_{\mu\alpha}W_\nu^{~\alpha}-{\frac{1}{4}}W_{\alpha\beta}W^{\alpha\beta}g_{\mu\nu}\right)\nonumber\\ &+{\frac{81\beta}{32}}(2w^2\Box w^2 g_{\mu\nu}-2\nabla_\mu w^2\nabla_\nu w^2+g_{\mu\nu}\nabla_\alpha w^2\nabla^\alpha w^2)\bigg].\end{aligned}$$ In summary, one may obtain the Weyl vector from equation and then the Lagrange multiplier $\lambda$ from equation . The field equation , together with equation can then be used to obtain the evolution of the metric. Hence a complete solution of the gravitational field equations in the WCW model with a Lagrange multiplier can be constructed, once the thermodynamic parameters of the matter (energy density and pressure) are known. It is worth mentioning that because of the general covariance, the matter energy-momentum tensor should be conserved due to the Bianchi identity. One can easily prove this statement in the case $\alpha=0$. Using equation , one may write equation as $$\begin{aligned} \label{eq29-1} \left(1-{\frac{27\beta}{16}}\Box w^2\right)G_{\mu\nu}=&\bigg[{\frac{\kappa^2}{2}}T^m_{\mu\nu}-{\frac{27\beta}{16}}(\nabla_\mu\nabla_\nu\Box w^2-\Box^2 w^2 g_{\mu\nu})+{\frac{\kappa^2}{2}}\left(W_{\mu\alpha}W_\nu^{~\alpha}-{\frac{1}{4}}W_{\alpha\beta}W^{\alpha\beta}g_{\mu\nu}\right)\nonumber\\ &+{\frac{81\beta}{32}}(2w^2\Box w^2 g_{\mu\nu}-2\nabla_\mu w^2\nabla_\nu w^2+g_{\mu\nu}\nabla_\alpha w^2\nabla^\alpha w^2)\bigg].\end{aligned}$$ Taking the divergence of the above equation one obtains $$\begin{aligned} \label{eq29-2} -{\frac{27\beta}{16}}&\nabla^\mu\Box w^2 G_{\mu\nu}={\frac{\kappa^2}{2}}\nabla^\mu T^m_{\mu\nu}\nonumber\\&-{\frac{27\beta}{16}}\big(\Box\nabla_\nu\Box w^2-\nabla_\nu\Box^2 w^2\big)+{\frac{81\beta}{16}}w^2\nabla_\nu\Box w^2.\end{aligned}$$ Now, using the identity $$\begin{aligned} \label{eq29-3} \nabla^\mu\nabla_\nu A_\mu-\nabla_\nu\nabla^\mu A_\mu=R^\alpha_\nu A_\alpha,\end{aligned}$$ and considering the Weitzenböck condition which reads $R=6w^2$, where $R$ is the Ricci scalar, one easily finds $\nabla^\mu T^m_{\mu\nu}=0$. The limiting case $w^\mu=0$ and the teleparallel gravity {#w} ======================================================== In this Section we consider the limiting case in which the Weyl vector becomes zero. We also assume $\alpha=0$ for simplicity. The action of the theory becomes $$\begin{aligned} \label{w1} S&={\frac{1}{\kappa^2}}\int d^4x\sqrt{-g}\bigg[\beta\nabla_\mu T\nabla^\mu T+\kappa^2L_m\nonumber\\&+(1+\lambda)\big(R+T^{\mu\alpha\nu}T_{\mu\alpha\nu}+2T^{\mu\alpha\nu}T_{\nu\alpha\mu}-4T_\mu T^\mu\big)\bigg],\end{aligned}$$ One may then obtain the field equations for the torsion tensor and the metric as $$\begin{aligned} \label{w2} (1+\lambda)\big(T_\mu^{~\rho\beta}+T^{\beta\rho}_{~~\mu}+T^{\rho~\beta}_{~\mu}&-4T^{[\rho}\delta^{\beta]}_\mu\big)\nonumber\\&-2\beta T^{[\rho}\delta^{\beta]}_\mu\Box T=0,\end{aligned}$$ and $$\begin{aligned} \label{w3} G_{\mu\nu}=T^{eff}_{\mu\nu},\end{aligned}$$ with $$\begin{aligned} \label{w4} T^{eff}_{\mu\nu}&=(1+\lambda)^{-1}\bigg[ {\frac{\kappa^2}{2}}T^m_{\mu\nu}+\nabla_\mu\nabla_\nu\lambda-g_{\mu\nu}\Box\lambda-(1+\lambda)\bigg(2T^{\alpha\beta}_{~~~\nu}T_{\alpha\beta\mu}-T_\mu^{~\alpha\beta}T_{\nu\alpha\beta}+2T_{\beta\alpha(\mu}T^{\alpha\beta}_{~~~\nu)}-4T_\mu T_\nu\bigg)\nonumber\\ &-\beta\left(\nabla_\mu T\nabla_\nu T-{\frac{1}{2}}g_{\mu\nu}\nabla_\alpha T\nabla^\alpha T-2 T_\mu T_\nu\Box T\right)+{\frac{1}{2}}(1+\lambda)\left(T^{\gamma\beta\alpha}T_{\gamma\beta\alpha}+2T^{\gamma\beta\alpha}T_{\alpha\beta\gamma}-4T^\alpha T_\alpha\right)g_{\mu\nu}\bigg],\end{aligned}$$ The variation of the action with respect to the Lagrange multiplier gives the Weitzenböck condition . Now consider the decomposition of the torsion tensor, given by equation , with $t_{\mu\nu\rho}=0$. One can again obtain $S_\mu=0$ by the same trick as in Section III. We then obtain the Weitzenböck condition in the form $$\begin{aligned} \label{w5} R={\frac{8}{3}}T.\end{aligned}$$ From equation one can isolate the Lagrange multiplier $$\begin{aligned} \label{w6} \lambda=-{\frac{3\beta}{4}}\Box T-1.\end{aligned}$$ By substituting equation one can check that equation is automatically satisfied. The metric then equation becomes $$\begin{aligned} \label{w7} \Box T G_{\mu\nu}&=-{\frac{2\kappa^2}{3\beta}}T^m_{\mu\nu}+\nabla_\mu\nabla_\nu\Box T-g_{\mu\nu}\Box^2 T\nonumber\\&+{\frac{4}{3}}\nabla_\mu T\nabla_\nu T-{\frac{2}{3}}g_{\mu\nu}\nabla_\alpha T\nabla^\alpha T-{\frac{4}{3}}g_{\mu\nu}T\Box T.\end{aligned}$$ The case $\beta=0$ ------------------ For $\beta =0$ the torsion has no kinetic term. Putting $\beta=0$ in equation and using equation , we obtain $T^\rho=0$. The trace of equation then gives $S_\mu=0$. Therefore, from the field equations we obtain $T^\mu_{~\rho\nu}=0$ and the theory reduces to a Brans-Dicke type theory, with equations of motion $$\begin{aligned} \label{w8} G_{\mu\nu}=(1+\lambda)^{-1}\bigg[{\frac{\kappa^2}{2}}T^m_{\mu\nu}+\nabla_\mu\nabla_\nu\lambda-g_{\mu\nu}\Box\lambda\bigg],\end{aligned}$$ and $$\begin{aligned} \label{w9} \Box\lambda={\frac{\kappa^2}{6}}T^m,\end{aligned}$$ respectively, where $T^m$ is the trace of the energy-momentum tensor. We have used the Weitzenböck condition $R=0$ to obtain equation . The Newtonian Limit and the generalized Poisson equation {#weak} ======================================================== In this Section, we will obtain the generalized Poisson equation describing the weak field limit of the WCW theory with Lagrange multiplier. Taking the trace of equation , using the Weitzenböck condition , and noting that $S_\mu=0$ in our setup, we obtain $$\begin{aligned} {\frac{1}{2}}\kappa^2 T^m&-3\Box\lambda-6w^2+8T^\mu w_\mu\nonumber\\&+\beta(\nabla_\alpha T\nabla^\alpha T+2T\Box T)=0,\end{aligned}$$ Now, using equation to eliminate the $\Box T$ term, we find $$\begin{aligned} (1+\lambda)R&={\frac{1}{2}}\kappa^2 T^m-3\Box\lambda-6w^2+\beta\nabla_\mu T\nabla^\mu T\nonumber\\&+12w_\mu T^\mu+2\alpha T_\mu\nabla_\nu T^{\mu\nu}.\end{aligned}$$ In the limit of the weak gravitational fields the $(00)$ component of the metric tensor takes the form $g_{00}=-(1+2\phi)$, where $\phi$ is the Newtonian potential. In this limit we have $R=-\nabla^2\phi$, and obtain the generalized Poisson equation as $$\begin{aligned} \nabla^2\phi=(1+\lambda)^{-1}\bigg[{\frac{1}{4}}\kappa^2\rho&+{\frac{3}{2}}\Box\lambda+3w^2-6w_\mu T^\mu\nonumber\\&-\alpha T_\mu\nabla_\nu T^{\mu\nu}\bigg].\end{aligned}$$ In obtaining the above equation we have assumed that the matter content of the Universe is pressureless dust, and we have used the Weitzenböck equation to keep terms up to first order in $\phi$. In the particular case $\alpha=0$, from equation we find that the Lagrange multiplier is of the order of $\phi$. Using equation we obtain the generalized Poisson equation as $$\begin{aligned} \nabla^2\phi={\frac{1}{4}}\kappa^2\rho-{\frac{81}{32}}\beta\Box^2 w^2+6w^2.\end{aligned}$$ For $w=0$, we recover the standard Poisson equation of Newtonian gravity. Cosmological Solutions {#cos} ====================== In this Section we consider the cosmological solutions and implications of the WCW model with Lagrange multiplier. We assume that the metric of the space-time has the form of the flat Friedmann-Robertson-Walker (FRW) metric, $$\begin{aligned} \label{cos1} ds^2=-dt^2+a^2(t)\left(dx^2+dy^2+dz^2\right).\end{aligned}$$ Also in the following we suppose that the tensor $t_{\mu \nu \rho}$ vanishes, $t_{\mu\nu\rho}=0$. As we have mentioned in the previous Section, $S_\mu=0$ and $T_\alpha$ should be space-like in order to obtain an accelerating solution. We consider only models in which the Universe is filled with a perfect fluid, with the energy-momentum tensor given in a comoving frame by $$\begin{aligned} \label{cos6} T^\mu_\nu=\textmd{diag}(-\rho,p,p,p),\end{aligned}$$ where $\rho $ and $p$ are the thermodynamic energy density and pressure, respectively. The Hubble parameter is defined as $H=\dot{a}/a$. As an indicator of the accelerated expansion we will consider the deceleration parameter $q$, defined as q=-1. If $q<0$, the Universes experiences an accelerated expansion while $q>0$ corresponds to a decelerating dynamics. The case $\alpha=0$ ------------------- In this case the cosmological dynamics is described by equation which represents the dynamical equation for the Weyl vector together with equations and which determine the Lagrange multiplier and the scale factor, respectively. The Weitzenböck condition is $$\begin{aligned} \label{cos1-1} R=6w^2.\end{aligned}$$ Let us assume that the Weyl vector is of the form $$\begin{aligned} \label{cos1-2} w_\mu&=a(t)\psi(t)(0,1,1,1).\end{aligned}$$ The Weitzenböck equation reduces to $$\begin{aligned} \label{cos1-3} \dot H+2H^2-3\psi^2=0,\end{aligned}$$ and the Lagrange multiplier can be obtained as $$\begin{aligned} \label{cos1-4} \lambda={\frac{81\beta}{8}}\big(2\Psi^2+\dot\Psi+3\Psi H\big)\psi^2,\end{aligned}$$ where we have defined $\Psi=\dot\psi/\psi$. The dynamical equation for the Weyl vector is $$\begin{aligned} \label{cos1-5} \dot\Psi+\dot H+\Psi^2+2H^2+3\Psi H=0.\end{aligned}$$ The off diagonal elements of the metric field equation gives $$\begin{aligned} \label{cos1-6} \Psi+H=0.\end{aligned}$$ One can then check that the Weyl equation is automatically satisfied. By substituting $H$ from to the diagonal metric equations one obtains $$\begin{aligned} \label{cos1-7} {\frac{3}{8}}\beta\Psi\ddot\Psi+{\frac{3}{8}}\beta(3\psi^2&-\Psi^2)\dot\Psi-{\frac{3}{8}}\beta(3\psi^2-\Psi^2)\Psi^2\nonumber\\ &-{\frac{1}{27}}\psi^{-2}\Psi^2+{\frac{1}{162}}\kappa^2\psi^{-2}\rho=0,\end{aligned}$$ and $$\begin{aligned} \label{cos1-8} {\frac{1}{8}}\beta \dddot{\Psi}&-{\frac{1}{8}}\beta(2\dot\Psi+9\psi^2+\Psi^2)\dot\Psi-{\frac{2}{81}}\psi^{-2}\dot\Psi\nonumber\\&-{\frac{3}{8}}\beta\Psi^4+{\frac{1}{27}}\psi^{-2}\Psi^2+{\frac{1}{162}}\kappa^2\psi^{-2}p=0,\end{aligned}$$ We note that in this case we have four equations, , , and for four unknowns $a$, $\psi$, $\rho$ and $p$. The Lagrange multiplier can then be obtain from equation . Equation (\[cos1-6\]) can be immediately integrated to give a(t)(t)=[constant]{}=C\_00, where $C_0$ is an arbitrary constant of integration. With the use of $\psi (t)=C_0/a(t)$, the Weitzenböck condition, equation (\[cos1-3\]), becomes a+\^2-3C\_0\^2=0, or equivalently (a)=3C\_0\^2, which immediately leads to a\^2(t)=3C\_0\^2t\^2+C\_1t+C\_2, where $C_1$ and $C_2$ are arbitrary constants of integration. By assuming the initial conditions $a(0)=a_0$ and $H(0)=H_0$, respectively, we obtain $C_2=a_0^2$, and $C_1=2a_0^2H_0$. Thus for the Hubble parameter we obtain H(t)=. The energy density of the Universe can be obtained from equation (\[cos1-8\]) as $$\begin{aligned} \kappa ^2\rho (t)&=\frac{4374 C_0^{12} t^{10}}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\frac{2916 a_0^2 C_0^{10} t^8 (5 H_0 t+2)}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\frac{6 a_0^{12} H_0^2 (2 H_0 t+1)^4}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\nonumber\\ &\frac{36 a_0^{10} C_0^2 H_0 t (2 H_0 t+1)^3 (4 H_0 t+1)}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\frac{54 a_0^8 C_0^4 t^2 (2 H_0 t+1)^2 \left(26 H_0^2 t^2+12 H_0 t+1\right)}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\nonumber\\ &\frac{648 a_0^6 C_0^6 t^4 (2 H_0 t+1) \left(11 H_0^2 t^2+7 H_0 t+1\right)}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\frac{486 a_0^4 C_0^8 t^6 \left(41 H_0^2 t^2+32 H_0 t+6\right)}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}+\nonumber\\ &\frac{243\beta }{4 \left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^6}\times\Bigg\{2430 a_0^2 C_0^{10} H_0 t^4+324 a_0^2 C_0^8 t^3 \left(8 a_0^2 H_0^2-9 C_0^2\right)+\nonumber\\ &3 a_0^4 C_0^4 t^2 \Bigg[a_0^4 H_0^4+6 a_0^2 C_0^2 H_0^2 (84 H_0-1)+C_0^4 (9-972 H_0)\Bigg]+a_0^6 C_0^2 \Bigg[a_0^4 H_0^4 (48 H_0+1)-\nonumber\\ &6 a_0^2 C_0^2 H_0^2 (24 H_0+1)+9 C_0^4 (6 H_0+1)\Bigg]+2 a_0^4 C_0^2 t \Bigg[a_0^6 H_0^5+6 a_0^4 C_0^2 H_0^3 (36 H_0-1)+\nonumber\\ &9 a_0^2 C_0^4 (1-60 H_0) H_0+81 C_0^6\Bigg]+1458 C_0^{12} t^5\Bigg\}.\end{aligned}$$ The thermodynamic pressure is found in the form $$\begin{aligned} \kappa ^2p&=\frac{2 \left[a_0^4 H_0^2-6 a_0^2 C_0^2 H_0 t-6 a_0^2 C_0^2-9 C_0^4 t^2\right]}{\left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^2}+\frac{81 C_0^2\beta }{4 \left(2 a_0^2 H_0 t+a_0^2+3 C_0^2 t^2\right)^5}\times \nonumber\\ &\Bigg[-35 a_0^8 H_0^4+135 a_0^6 C_0^2 H_0^2-63 a_0^4 C_0^4+324 a_0^2 C_0^6 H_0 t^3+t \left(558 a_0^4 C_0^4 H_0-150 a_0^6 C_0^2 H_0^3\right)+\nonumber\\ &t^2 \left(837 a_0^2 C_0^6-117 a_0^4 C_0^4 H_0^2\right)+243 C_0^8 t^4\Bigg].\end{aligned}$$ For $t=0$ we obtain the initial values of the density and pressure as \[r0\] (0)=\_0=6 H\_0\^2+, and \[p0\] p(0)=p\_0= - , respectively. Once the initial conditions $\left(a_0,H_0,\rho_0,p_0\right)$ are known, from Eqs. (\[r0\]) and (\[p0\]) the values of the integration constants can be determined. The deceleration parameter can be obtained as q(t)=a\_0\^2. If the initial values of the scale factor and Hubble parameter satisfy the condition $a_0H_0<\sqrt{3}C_0$, $q<0$ for all times then the Universe is in an accelerated expansionary phase. If $a_0H_0=\sqrt{3}C_0$, $q(t)\equiv 0$ then the Universe is in a marginally inflating state. Finally, the Lagrange multiplier for this model can be obtained as (t)=. The case $\alpha\neq0$ ---------------------- We assume that the Weyl vector is space-like, mimicking the proportionality of the torsion and the Weyl vector as in the case $\alpha=0$. Let us assume that $$\begin{aligned} \label{cos2} T_\mu&=a(t)\phi(t)(0,1,1,1),\nonumber\\ w^\mu&={\frac{\psi(t)}{a(t)}}(0,1,1,1).\end{aligned}$$ By substituting these forms of the torsion and Weyl vector into equation we obtain, after some algebra, the Lagrange multiplier $$\begin{aligned} \label{eqlam} \lambda=&{\frac{3}{4}}(6\beta\phi^2-\alpha)\dot{\Phi}-{\frac{3}{4}}\alpha\dot{H}+{\frac{9}{2}}\beta \phi^2\left(2\Phi+3H\right)\Phi\nonumber\\ &-{\frac{3}{4}}\alpha(\Phi^2+2H^2+3 H\Phi)+{\frac{3}{2}}{\frac{\psi}{\phi}}-1,\end{aligned}$$ where we have defined $$\begin{aligned} \label{cos4} \Phi={\frac{\dot{\phi}}{\phi}}.\end{aligned}$$ By using equation , the field equation becomes $$\begin{aligned} \label{cos5} \ddot{\psi}+3H\dot{\psi}+(\dot{H}+2H^2+12\kappa^{-2})\psi-8\kappa^{-2}\phi=0.\end{aligned}$$ The Weitzenböck equation takes the form $$\begin{aligned} \label{eqdh} \dot{H}=-2H^2+{\frac{4}{3}}\phi^2.\end{aligned}$$ Substituting equations and into , one obtains $$\begin{aligned} \label{cos7} 9H(6\beta\phi^2-\alpha)\ddot{\Phi}&+6\bigg[(\alpha-6\beta\phi^2)(2\phi^2-6H^2-3H\Phi)+36\beta\phi^2 H\Phi\bigg]\dot{\Phi}-3\dot{\psi}\left(\kappa^2\dot{\psi}+2\kappa^2 H\psi-6{\frac{H}{\phi}} \right)\nonumber\\&+8\phi^4(2\alpha-9\beta\Phi^2) +216\beta \phi^2 H \Phi^2(\Phi+2H)+6\phi^2\Phi(4\alpha\Phi-27\beta H^3)+24\psi\phi+9\alpha H^2\Phi (3H-\Phi)\nonumber\\&-3\psi^2(\kappa^2 H^2+12)+18H{\frac{\psi}{\phi}}(H-\Phi)=2\kappa^2\rho,\end{aligned}$$ $$\begin{aligned} \label{cos8} -&9(\alpha-6\beta\phi^2)\phi\dddot\Phi-9\phi\big[\alpha(5H+2\Phi)-6\beta\phi^2(5H+8\Phi)\big]\ddot\Phi+18\ddot\psi+18(18\beta\phi^2-\alpha)\phi\dot\Phi^2\nonumber\\ &+9\big[40\beta\phi^5+6(24\beta\Phi^2+30\beta H\Phi-7\beta H^2-2\alpha)\phi^3+\alpha H(7H-4\Phi)\phi-2\psi\big]\dot\Phi+3 \kappa^2 \phi\dot\psi^2\nonumber\\&+8(117\beta\Phi^2-81\beta H\Phi-2\alpha)\phi^5+18\big[12\beta(2\Phi+5H)\Phi^3-2(4\alpha+21\beta H^2)\Phi^2+(27\beta H^2-2\alpha)H\Phi\big]\phi^3 \nonumber\\&+72\psi\phi^2+\big[6 \kappa^2 p+9\alpha H^2\Phi^2+3\kappa^2 H^2\psi^2-81\alpha H^3\Phi-36\psi^2\big]\phi\nonumber\\&+6(6H-6\Phi+\kappa^2\phi H\psi)\dot\psi+18\psi\Phi^2-36\psi H\Phi-18\psi H^2=0,\end{aligned}$$ and $$\begin{aligned} \label{cos9} \kappa^2\dot{\psi}(\dot{\psi}+2\psi H)+\psi^2 (\kappa^2H^2-12)+4\alpha \phi^2\left(\dot{\Phi}-H^2+H\Phi+{\frac{4}{3}}\phi^2\right)+8\psi\phi=0,\end{aligned}$$ Eqs. , , , and form a closed system of differential equations for five unknowns $\psi$, $\phi$, $H$, $p$ and $\rho$. Equation can then be used to determine the Lagrange multiplier. In the following we will look only for a de Sitter type solution of the field equations - with $H=H_0={\rm constant}$ and $a=\exp\left(H_0t\right)$, respectively. Then the Weitzenböck condition (\[eqdh\]) immediately gives \^2=H\_0\^2=[constant]{}, and $\Phi \equiv 0$, respectively. Equation (\[cos5\]) takes the form +3H\_0+(2H\_0\^2-12\^[-2]{})=-4\^[-2]{}H\_0\^2, with the general solution given by (t)&=&+c\_1 e\^[ t]{}+\ &&c\_2 e\^[t]{}, where $c_1$ and $c_2$ are arbitrary constants of integration. The simplest case corresponds to the choice $c_1=0$, $c_2=0$, giving ==[constant]{}. By substituting this form of $\psi $ into equation (\[cos9\]) we obtain the value of $\alpha $ as =. For the energy density of the Universe we obtain \^2=, $$\begin{aligned} \kappa ^2p =\frac{72H_0^2(2\kappa^2H_0^2-3)}{\left(6-H_0^2 \kappa ^2\right)^2}.\end{aligned}$$ One can see that the energy density and the pressure is positive if $H_0\geq 1/\kappa^2\sqrt{3/2}$. Cosmological models with $w^\mu=0$ ---------------------------------- Finally, we consider the cosmological implications of the WCW model with Lagrange multiplier with $w^\mu=0$. Assuming the following form for the torsion, $$\begin{aligned} T_\mu=a(t)\phi(t)\big[0,1,1,1],\end{aligned}$$ the Weitzenböck condition is formulated as $$\begin{aligned} R-8\phi^2=0.\end{aligned}$$ The Lagrange multiplier can be obtained in the form $$\begin{aligned} \lambda+1={\frac{9}{2}}\beta\phi^2\big(\dot\Phi+2\Phi^2+3H\Phi\big),\end{aligned}$$ where we have defined $\Phi=\dot\phi/\phi$. The metric field equations take the form $$\begin{aligned} \ddot\Phi&+2\dot\Phi(2H+3\Phi)-{\frac{4}{3}}{\frac{\phi^2}{H}}(\dot\Phi+\Phi^2+3H\Phi)\nonumber\\&+\Phi(4\Phi^2+3H^2+8H\Phi+3\dot H)-{\frac{\kappa^2}{27\beta}}{\frac{\rho}{H\Phi^2}}=0,\end{aligned}$$ and $$\begin{aligned} &\dddot\Phi+\ddot\Phi(5H+8\Phi)+3\Phi\ddot H-4\phi^2(\dot\Phi+3\Phi^2+3H\Phi)+8\Phi^4\nonumber\\&+4\dot H(2\dot\Phi+4\Phi^2+3H\Phi)+9H^2\Phi(H+2\Phi)+20H\Phi^3\nonumber\\&+3\dot\Phi(10H\Phi+8\Phi^2+3H^2+2\dot\Phi)+{\frac{\kappa^2}{9\beta}}\phi^{-2}p=0, \end{aligned}$$ respectively. Let us consider the case $a(t)=t^s$. In this case one obtains $$\begin{aligned} \label{} \phi(t)={\frac{\sqrt{3s(2s-1)}}{t}},\end{aligned}$$ and the energy density and pressure take the form $$\begin{aligned} \label{} \rho(t)&={\frac{81s^2(3s^2+8s-10)(2s-1)}{4\kappa^2}}{\frac{\beta}{t^6}},\\ p(t)&=-{\frac{81(3s^3+2s^2-26s+20)(2s-1)s}{4\kappa^2}}{\frac{\beta}{t^6}}.\end{aligned}$$ In order to have a consistent solution, $\phi$ should be real and $\rho$ and $p$ must be positive. This restricts the range of $s$ to $$\begin{aligned} \label{} {\frac{1}{3}}(\sqrt{46}-4)<s<2.\end{aligned}$$ For the deceleration parameter we obtain q=-1,-<q<0.078. In the case $a(t)=e^{H_0t}$ we have $$\begin{aligned} \label{} \phi(t)^2={\frac{3}{2}}H_0^2,\end{aligned}$$ with the matter energy density and the pressure becoming exactly zero =p=0. Conclusion {#conclu} ========== In this paper we have considered an extension of the Weitzenböck type gravity models formulated in a Weyl-Cartan space time. The basic difference between the present and the previous investigations is the way in which the Weitzenböck condition which in a Riemann-Cartan space time requires the exact cancellation of the Ricci scalar and the torsion scalar, is implemented. By starting with a general geometric framework, corresponding to a $CW_4$ space - time described by a metric tensor, torsion tensor and Weyl vector, we formulated the action of the gravitational field by including the Weitzenböck condition via a scalar Lagrange multiplier. With the use of this action the gravitational field equations have been explicitly obtained. They show the explicit presence in the field equations of a new degree of freedom, represented by the Lagrange multiplier $\lambda $. The field equations must be consistently solved together with the Weitzenböck condition which allows the unique determination of the Lagrange multiplier $\lambda $. The weak field limit of the model was also investigated and it was shown that the Newtonian approximation leads to a generalization of the Poisson equation where besides the matter energy-density, the weak field gravitational potential also explicitly depends on the Lagrange multiplier and the square of the Weyl vector. An interesting particular case is represented by the zero Weyl vector case. For this choice of the geometry the covariant divergence of the metric tensor is zero and the Weitzenböck condition takes the form of a proportionality relation between the Ricci scalar and the torsion scalar, respectively. When one neglects the kinetic term associated to the torsion, the model reduces to a Brans-Dicke type theory where the role of the scalar field is played by the Lagrange multiplier. The cosmological implications of the theory have also been investigated by considering a flat FRW background type cosmological metric. We have considered three particular models, corresponding to the zero and non-zero values of the coupling constant $\alpha $, and to the zero Weyl vector respectively. For $\alpha =0$ the field equations can be solved exactly, leading to a scale factor of the form $a(t)=\sqrt{3c_0^2t^2+2H_0a_0^2+a_0^2}$. The energy density and the pressure are monotonically decreasing functions of time and are both non-singular at the beginning of the cosmological evolution. The nature of the cosmological expansion - acceleration or deceleration - is determined by the values of the constants $\left(C_0, a_0,H_0\right)$ and three regimes are possible: accelerating, decelerating, or marginally inflating. In the case $\alpha \neq 0$, we have considered only a de Sitter type solution of the field equations. Such a solution does exist if the matter energy density and pressure are constants, or, more exactly, the decrease in the matter energy density and pressure due to the expansion of the Universe is exactly compensated by the variation in the energy and pressure due to the geometric terms in the energy-momentum tensor. In the case of the cosmological models with vanishing Weyl vector we have investigated two particular models corresponding to a power law and exponential expansion, respectively. In the case of the power law expansion, the energy density and pressure satisfy a barotropic equation of state, so that $p\sim \rho$ where both the energy and pressure decay as $t^{-6}$. Depending on the value of the parameter $s$, both decelerating and accelerating models can be obtained. On the other hand, for a vanishing Weyl vector, the de Sitter type solutions require a vanishing matter energy density and pressure and hence the accelerated expansion of the Universe is determined by the geometric terms associated with torsion which play the role of an effective cosmological constant. In the present paper we have introduced a theoretical model for gravity, defined in a Weyl-Cartan space-time, in which the Weitzenböck geometric condition has been included in the action via a Lagrange multiplier method. The field equations of the model have been derived by using variational methods, and some cosmological implications of the model have been explored. Further astrophysical and cosmological implications of this theory will be considered elsewhere. Note on Weyl gauge invariance ============================= Suppose that length of a vector at point $x$ is $l$. In the Weyl geometry, the length of the vector under parallel transportation to the nearby point $x^\prime$ is $l^\prime=\xi l$. On the other hand, the change in the length of the vector can be written as $$\begin{aligned} \label{ap1} \delta l=lw_\mu\delta x_\mu.\end{aligned}$$ So, the change in the Weyl vector is $$\begin{aligned} \label{ap2} w_\mu\rightarrow w^\prime_\mu=w_\mu+\partial_\mu\log\xi,\end{aligned}$$ From the above relations, one obtains the change in the metric tensor $$\begin{aligned} \label{ap3} g_{\mu\nu}&\rightarrow g^\prime_{\mu\nu}=\xi^2 g_{\mu\nu},\\ g^{\mu\nu}&\rightarrow g^{\prime\mu\nu}=\xi^{-2} g^{\mu\nu}.\end{aligned}$$ The torsion tensor is invariant under the above gauge transformation, i.e., $$\begin{aligned} \label{ap4} T^\mu_{~\rho\sigma}\rightarrow T^{\prime\mu}_{~\rho\sigma}=T^\mu_{~\rho\sigma}\end{aligned}$$ We note that the curvature tensor is covariant with the power $-2$, which means $$\begin{aligned} \label{ap5} K^\prime=\xi^{-2}K.\end{aligned}$$ and the metric determinant has power $4$. Naturally, one demands to make the Lagrangian gauge-invariant. 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--- abstract: \| It has been suggested that the ejection to interplanetary space of terrestrial crustal material accelerated in a large impact, may result in the interchange of biological material between Earth and other Solar System bodies. In this paper, we analyze the fate of debris ejected from Earth by means of numerical simulations of the dynamics of a large collection of test particles. This allows us to determine the probability and conditions for the collision of ejecta with other planets of the Solar System. We also estimate the amount of particles falling-back to Earth as a function of time after being ejected. The Mercury 6 code is used to compute the dynamics of test particles under the gravitational effect of the inner planets in the Solar System and Jupiter. A series of simulations are conducted with different ejection velocity, considering more than 10$^4$ particles in each case. We find that in general, the collision rates of Earth ejecta with Venus and the Moon, as well as the fall-back rates, are consistent with results reported in the literature. By considering a larger number of particles than in all previous calculations we have also determined directly the collision probability with Mars and, for the first time, computed collision probabilities with Jupiter. We find that the collision probability with Mars is greater than values determined from collision cross section estimations previously reported. address: - 'Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo Postal 877, Ensenada 22800, B.C., México' - 'Facultad de Ciencias, Universidad Autonoma de Baja California, Ensenada 22860, B.C., México' - 'Instituto de Estudios Avanzados de Baja California, Ensenada 22800, B.C., México' author: - 'M. Reyes-Ruiz' - 'C.E. Chavez' - 'M.S. Hernandez' - 'R. Vazquez' - 'H. Aceves' - 'P.G. Nuñez' title: Dynamics of escaping Earth ejecta and their collision probability with different Solar System bodies --- Astrobiology ,Impact processes ,Celestial mechanics Introduction ============ Presently, the collision of kilometer-scale bodies with Earth, such as comets or asteroids, is believed to occur on a timescale of the order of millions of years (Chapman, 1994). Impacts by even greater bodies, with diameter of tens of kilometers, such as the Chicxulub event (Kent et al. 1981), are thought to take place approximately every 10$^8$ years. During the Late Heavy Bombardment (LHB) epoch of the Solar System’s history, both the frequency and diameter of Earth impactors is believed to be much greater than these estimates (Strom et al. 2005). In addition to their catastrophic effect on the diversity of life-species on Earth, giant impacts may also lead to ejecta accelerated with velocities greater than the planetary escape velocity, $V_{\rm esc}$. Depending on the impactor energy, ejected debris may reach velocities significantly higher than $V_{\rm esc}$ reaching Jupiter crossing orbits (Gladman et al. 2005). In the impact spallation model of Melosh (1984, 1985), material from a thin surface layer of the Earth’s crust, can be lifted and accelerated to more than escape velocity by the interference of impact induced shock waves. Very low peak-shock pressures are predicted for such ejecta, offering a plausible explanation of the observed shock levels of meteors of Lunar and Martian origin. Once material is ejected to interplanetary space, it will travel in orbits that may, depending on the ejection velocity, cross the orbits of other planets in the Solar System. Gladman et al. (2005, and references therein) have analysed the dynamics of such ejecta modelling these as a large collection of test particles. They found that, for low ejection velocities, a fraction of particles return to Earth after approximately 5000 years, between 0.6–0.2$\%$ for $V_{\infty}$ between 1 and 2 km/s, where $V_{\infty}$ is the velocity reached by the particle at very large distances from the Earth. An even smaller percentage, of the order of 0.015% collides with Venus on a similar timescale. In principle, particles ejected with a velocity greater than the planet’s escape velocity may reach Mars or even Jupiter crossing orbits. However, no collisions with either body is reported in the calculations of Gladman et al. (2005). It has been suggested that such ejected crustal debris may carry along biologic material which may, if it collides with a suitable target, serve as seed for the development of life elsewhere in the Solar System (Mileikowsky et al. 2000 and Nicholson et al. 2000). Additionally, ejected material may return to Earth and ”reseed” terrestrial life after the sterilizing effect of a giant impact has passed (Gladman et al. 2005). For this to happen, additional constraints are imposed on the time the ejecta may remain in space, as well as on the size of the crustal fragments, as these factors impact the amount of high energy radiation to which the biologic material is exposed. Wells et al. (2003) have argued that, with the current characteristics of the space environment (cosmic rays, x-rays, EUV) exposure for times greater than a few thousand years would make biological material nonviable. In this paper, we analyse the dynamics of particles ejected from Earth in a manner similar to the analysis of Gladman et al. (2005) but improving the statistics by increasing the number of particles by more than a factor of three and using a different scheme and code to integrate the equations of motion. In section 2 we describe the numerical method used, the initial conditions and other details of our simulations. Results of the various simulations conducted are presented in section 3. In section 4 we discuss our results and we present our concluding remarks in section 5. Model description {#sec:problem} ================= We consider ejecta as test particles moving under the action of the gravitational field of the Sun, the Moon, and all planets of the Solar System. No other forces are considered in the present study. Particles are assumed not to collide with each other, but may impact any of the massive bodies. The dynamics of the planets and test particles system is calculated using the Mercury 6.2 code developed by Chambers (1999). The code offers several integrator choices, including a hybrid integrator option which combines a symplectic integrator with a Bulirsch-Stoer scheme appropriate for when particles approach any of the massive bodies in the simulation. We have used the hybrid option of the code to follow the movement of each particle for 30,000 years (using independent integrations for each test particle). Our choice of 30,000 years for the time that biological material can remain viable in space is adopted following Gladman (2005), who argue that unless the sensitivity of ancient microorganisms to radioactivity (the main killing factor in ejecta greater than a few meters) is 2-3 orders of magnitude greater than that of modern day bacteria, then survival times for viable biological material can range from 3,000 to 30,000 years. The code stops the integration if the test particle collides with a planet or the Moon, if it reaches distances smaller than 1$R_{\odot}$ from the Sun, or if it is ejected from the simulation domain (presently set at 40 AU). The base time step for the symplectic integrator is 24 hours, and it is decreased when the code switches to a Bulirsch-Stoer integrator during close encounters between test particles and massive objects to achieve a given accuracy. In the present simulations the change from one integrator to another is set to occur when test particles are within 3 Hill radii of a massive body. Initial conditions ------------------ We have analyzed several cases, each consisting of 10,242 particles, set off with a given speed, $V_{\rm ej}$. Particles are distributed uniformly over the surface of a sphere. To do so we use a Fuller spherical distribution (Prenis 1988 and Saff and Kuijlaars 1997), we begin with an icosahedron (by construction each of the 12 vertices of this figure is on the circumscribed sphere). An icosahedron has 30 possible lines between each point and its nearest neighbours. We take the middle point of each one of these lines and we project them into the sphere. Therefore obtaining a new figure with 30+12=42 vertices uniformly distributed on the sphere. The general formula to obtain the number of vertices $n_{p}$ is giving by the following recursive formula: $$n_{p}(j)=n_{lines}(j-1)+n_{p}(j-1) \label{npart}$$ Where $n_{p}(j)$ is the number of vertices in the step $j$, $n_{lines}(j-1)$ and $n_{p}(j-1)$ are the number of lines and number of vertices in the previous step $(j-1)$, respectively. It is important to notice that $n_{p}(1)=12$. It can be proven that $n_{lines}(j-1)=3 n_{p}(j-1) - 6$. Therefore we can simplify Eq. \[npart\] and obtain the following: $$n_{p}(j)=[3 n_{p}(j-1) - 6] + n_{p}(j-1)=4 n_{p}(j-1) - 6 \label{npartsimpl}$$ Therefore, using Eq. \[npartsimpl\] we obtain 10,242 vertices after 5 iterations, we place our particles on each one of these vertices. The points are uniformly distributed on the sphere. ![Initial spatial distribution of ejected particles at a height of 100 km over the surface of the planet. The bottom panel shows the initial location of particles as a function of latitude ($\phi$) and “longitude” ($\theta$ (measured from the midnight meridian).[]{data-label="fig1"}](fig1.eps){width="\columnwidth"} The initial ephemeris of the planets-moon system is taken from the Horizons website ([http://ssd.jpl.nasa.gov/horizons]{}) and correspond to the configuration of the Solar System on the 6th of July 1998 at 00:00:00.0 UT. This date is the default starting date defined in the Mercury 6.2 code examples, it does not represent any special configuration of the planetary bodies and is adopted as an arbitrary initial condition. In section 4 we discuss the effect of changing the initial ephemerides on some of our results. Test particles are set off from a height of 100 km over the surface of the Earth with an initial velocity that is purely radial. Of course, this velocity distribution does not correspond to any particular impact, as these would most likely result in the preferential ejection from one side of the planet (over a single quadrant) and ejecta with a distribution of velocities. Rather, by choosing the ejection velocity in this manner, and analysing several cases with a single ejection speed separately, we intend to sample all likely ejection conditions and identify trends indicating the effect of the ejection parameters, speed, launch position and direction. In the discussion section, we analyse the dependence of the collision probability with different bodies on the location from which a test particle is launched, although this can only be done with reasonable statistics for collisions with Earth and, to a lesser extent, with Venus and Jupiter. According to Gladman et al. (2005), in the impact spallation theory of Melosh (1985) the cumulative distribution of ejected mass from Earth, from the escape velocity up to a given ejection velocity, is given by: $$F(V_{\rm esc} < V < V_{\rm ej}) = \frac{1-(\frac{V_{\rm ej}}{V_{\rm esc}})^{-5/3}}{1-(\frac{U}{2 V_{\rm esc}})^{-5/3}}$$ where $F$ is the fraction of the total mass ejected that leaves Earth with velocity in the range $V_{\rm esc} < V < V_{\rm ej}$ and $U$ is the impactor speed. Note that the distribution is unity when $V_{\rm ej} = U/2$, meaning that the maximum velocity with which material is ejected is one half of the speed with which the impactor collides with Earth. The distribution function for Earth-crossing asteroids or comets drops steeply as the impactor velocity increases, so that ejecta with velocity much greater than the escape velocity are even less likely to occur. The range of ejection velocities we consider correspond to impacts with speed less than 33 km/s, covering most of the asteroid and comet impacts with the Earth and Moon (Chyba et al. 1994). In the present paper we do not consider ejecta moving with even higher velocity which may result from Earth impacts by objects moving on higher velocity, comet-like trajectories. Case A B C D E --------------------- --------- --------- --------- --------- --------- $V_{\rm ej}$ (km/s) 11.22 11.71 12.7 14.7 16.4 Earth 496 106 48 22 10 (4.84%) (1.03%) (0.47%) (0.21%) (0.1%) Moon 2 2 2 1 0 (0.02%) (0.02%) (0.02%) (0.01%) (0%) Venus 6 17 7 7 3 (0.06%) (0.17%) (0.07%) (0.07%) (0.03%) Mars 0 1 1 0 0 (0%) (0.01%) (0.01%) (0%) (0%) Jupiter 0 0 0 6 5 (0%) (0%) (0%) (0.06%) (0.05%) Sun 0 0 0 0 19 (0%) (0%) (0%) (0%) (0.19%) Ejected$^$ 0 0 0 254 691 (0%) (0%) (0%) (2.48%) (6.75%) $^$ Results ======= A series of numerical simulations with different ejection velocity were conducted, cases with low, intermediate and high ejection velocity are considered, corresponding to specific cases also reported in the study by Gladman et al. (2005). This allows a direct comparison to the results reported by these authors. We also report results for two additional cases taken at intermediate values of the ejection velocity, which do not exactly coincide with the cases studied by Gladman et al. (2005). Table 1 shows a summary of our results for each of these cases. ![Projection in the $x-y$ plane of the trajectory of 2 particles, one ejected from center of the leading face of the planet, along its direction of motion (left column of panels), and the other ejected from the center of the trailing face. Dots in each panel denote the position of the particle at each of the output times of our simulations. Top, middle and bottom rows correspond to different ejection velocities, Cases A, C and E (Table 1), respectively.[]{data-label="fig2"}](fig2.eps){width="\columnwidth" height="1.7\columnwidth"} Neglecting the effect of the other bodies in the simulation, the ejection velocity from Earth determines the maximum apoapsis and minimum periapsis in the orbit of the ejected particles around the Sun. The maximum apoapsis is reached by particles ejected from the leading face of the planet, where the net velocity with respect to the Sun is maximum. Oppositely, particles moving in the direction opposing the motion of the planet at the moment of ejection, will have the smallest heliocentric velocity and fall to the minimum periapsis. Since the particles we are studying are launched very close to the Earth we must take Earth’s gravitational potential into account. The velocity “at infinity” ($V_{\rm \infty}$) is defined as the speed that ejecta have after escaping the planet’s gravitational well and is related to the ejection speed ($V_{\rm ej}$) by the following: $$V_{\rm \infty}^2=V_{\rm ej}^2-V_{\rm esc}^2 , \label{vinfty}$$ where $V_{\rm esc}=\sqrt{2 G m_{E} / r_{E}}$, is the escape velocity from the planet, $G$ is the gravitational constant, $m_{E}$ is the mass of the Earth and $r_{E}$ is the distance from the ejected particle initial position to the centre of the Earth. In the present study we adopt $r_E$ = 6471 km, so that $V_{\rm esc}$ = 11.098 km/s. In order to estimate the periapsis and apoapsis we assume that the particles have already escaped the gravitational potential of the Earth and have the velocity given by Eq. (\[vinfty\]) but since they have been launched from Earth’s surface we need to add the velocity of our planet, taken as the average along its orbit, $V_{E}=29.29$ km/s, so the particle velocity is given by: $$V_{part}=V_{E}\pm V_{\rm \infty} \label{vpart}$$ In Eq. (\[vpart\]) we use the positive sign if the particle is ejected in the leading face and negative otherwise. Let us assume that the Earth is in a circular orbit, then the angular momentum of the particle is given by $h=r_{p} V_{part}$, where $r_p$ is the position of the particle with respect to the Sun. At the time of ejection, $r_{p} \approx 1$ AU, but with a velocity that is too big for staying in a circular orbit (or too small for the negative sign in Eq. (\[vpart\])), the new semi-major axis and eccentricity of the ejected particle can be calculated using Eqs. (2.134) and (2.135) from Murray and Dermott (1999): $$a=\Bigg( {2\over r_{p}} - \frac{V_{part}^2}{G M_{\odot}} \Bigg) \label{apart}$$ $$e=\sqrt{ 1 - \frac{h^2}{G M_{\odot} a} } \label{epart}$$ where $a$, and $e$ are the semi-major axis and eccentricity of the new orbit of the particle, and $M_{\odot}$ is the mass of the Sun. Then for a given velocity of the particle $V_{part}$ we can have the following maximum apoapsis (taking positive sign in Eq. (\[vpart\])) and minimum periapsis (taking negative sign in Eq. (\[vpart\])): $$Q_{max}=a_{max} (1 + e_{max}) \label{apomax}$$ $$q_{min}=a_{min} (1 - e_{min}) \label{perimin}$$ Where $a_{max}$ and $e_{max}$ are calculated from Eqs. (\[apart\]) and (\[epart\]) using the positive sign for $V_{part}$ and similarly $a_{min}$ and $e_{min}$ are calculated using the negative sign. According to these formulas, An ejection velocity of 11.62 km/s is needed to reach Mars and 14.28 km/s are required to reach the orbit of Jupiter. Figure 2 illustrates typical trajectories followed by particles in the low, intermediate and high ejection velocity cases (A, C and E in Table 1). Shown are the orbits of 2 particles during the 30,000 years integration for each case, one ejected from the central regions of the leading face of the planet (along its direction of motion) and the other from the center of the trailing face. These cases represent extrema in the orbital energy of ejected particles and are illustrate the range of possible trajectories. Dots in each panel of Figure 2 denote the position of the particle in the $x-y$ plane at each of the output times of our simulations, separated by $\Delta t = 5000$ days. Particles ejected with the lowest velocity considered, $V_{\rm ej} = 11.22$ km/s, generally remain in orbits close to that of the Earth, as shown in the top panels of Figure 2. Those ejected with the highest velocity, $V_{\rm ej} = 16.4$ km/s (shown in the bottom panels of Figure 2), have access to a wide range of orbits. In this case, many particles, as the one depicted in the bottom left panel of Figure 2, are launched to the periphery of the Solar System and spend a very short time in the inner Solar System, hence the absence of dots in the figure. Collision probability --------------------- As indicated in Table 1, the probability that particles collide with Earth or any other body of the Solar System, depends strongly on the velocity with which it is ejected from Earth. Particles ejected with a low velocity never cross the orbit of Mars or Jupiter, and they can only collide with Venus, the Moon or fall back to Earth. In the context of our calculations, particles ejected with a velocity just 1% greater than the escape velocity, $V_{\rm ej} = 11.22$ km/s (Case A in Table 1) have a maximum probability of falling back to Earth. Within 30,000 years, almost 5% of all particles fall back to Earth. Particles do not have enough energy to reach the orbits of Mars, as exemplified in the top panel of Figure 2 and hence, there are no collisions with Mars and Jupiter. There are two particles that impact the Moon, and six that impact Venus, representing 0.02 and 0.06% of the ejected population of test particles, respectively. In Case B of Table 1, characterized by ejection velocity $V_{\rm ej} = 11.71$ km/s, there are 106 particles that have fallen back to Earth (1.03% of the 10242 ejected particles) by the end of the 30,000 yr simulations, 2 that hit the Moon (0.02%) and 17 collide with Venus (0.17%). In contrast to case A, in case B ejected particles now have enough energy to reach Mars, and one particle collides with the planet, representing 0.01% of the ejecta. Ejection energy is still not enough to reach any of the outer planets. As the ejection velocity increases to $V_{\rm ej} = 12.7$ km/s, Case C, the number of particles falling back to Earth continues decreasing, in comparison to cases A and B, with only 0.47% of the 10242 particles returning to Earth. As in Case B, 2 particles impact the Moon, 7 particles collide with the planet Venus and 1 particle reaches Mars. Of the particles ejected with conditions of case D, $V_{\rm ej} = 14.7$ km/s, only 0.21 % return to Earth, continuing the trend observed in the previous cases. One particle impacts the Moon and 7 collide with Venus. Ejecta now can have enough energy to reach the orbit of Jupiter and many particles do so. As a result, 6 particles collide with this planet representing 0.06 % of the total number of ejected particles. The high velocity and low gravitational pull of Mars result in the absence of collisions with Mars for this case. Also, a significant amount of particles are launched into orbits reaching the periphery of the Solar System, 254 particles travel beyond 40 AU. Purely for numerical efficiency reasons, we consider these particles as ejected from the Solar System. Finally, only 0.1 % of the particles ejected with the maximum velocity we considered, $V_{\rm ej} = 16.4$ km/s, ever return to Earth, no particle impacts the Moon or Mars, and only 3 particles impact Venus. A great number of particles reach the outer planets region, 5 of them colliding with Jupiter and 691 “escaping” from the Solar System (as defined above). The low orbital velocity of particles ejected from the trailing face of the planet results in about 0.2% of particles colliding with the Sun. Similarly to the previous case, no particles colliding with the planet Mars are found. ![Collision time for particles ejected with $V_{\rm ej} = 11.22$ km/s, binned into 5000 yr intervals. Shades of gray correspond to particles falling back to Earth, colliding with the Moon, Venus Mars of Jupiter, as indicated in the Figure legend. Note that to make the plot clear, the values corresponding to collisions with Venus and the Moon in the top panel are 10 times their real value.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"} Collision time -------------- By collision time we mean the period a particle spends in space until it collides with the Earth or any other Solar System body. In Figure 3, we illustrate the collision time for particles falling back to Earth, colliding with the Moon, with Venus, with Mars or with Jupiter, for different values of the ejection velocity (Cases A, C and E). Results are binned into 5000 year intervals with each line indicating collisions with a different body as indicated in the figure legend. It must be pointed out that in general, the number of collisions with bodies other than Earth is very small, and the trends in the time evolution of collision rates with such bodies, are of questionable statistical significance. Further studies with a greater number of ejected particles, necessary to address this issue in greater depth, are beyond the scope of the present paper. The top panel of Figure 3 corresponds to Case A, with $V_{\rm ej} = 11.22$ km/s. In this case, more than 2/3 of the particles returning to Earth do so in less than 20,000 years. The number of fallback particles does not decrease strictly monotonically, but less than 10% of particles return in the last time bin, between 25 and 30 kyr after ejection. Also shown in the top panel, are the collision times for particles colliding with the Moon and Venus. The numbers shown in the figure are 10 times the actual values, which are too small to be plotted. A few of the particles colliding with Venus do so after less than 15 kyr but the majority take more than 20 kyr to reach the planet. The middle panel of Figure 3 shows the collision times for particles ejected with $V_{\rm ej} = 12.7$ km/s, Case B discussed above. The number of particles falling back to Earth peaks at early times, approximately 15% of particles return in the first 5000 years. The fall-back rate more or less remains constant after this and for the next 25 kyr. Only two particles collide with the Moon and they do so at widely different times, the first one after less than 10 kyr and the second one at the very last time bin, after 25 kyr. The single particle that hits Mars does so near the end of our simulation, after 20 kyr. All particles travelling to Venus do so in less than 20 kyr, with the peak in the collision time to the planet between 10 and 15 kyr. Finally, collision times for particles ejected from Earth with $V_{\rm ej} = 16.4$ km/s are shown in the bottom panel of Figure 3. Only a few particles fall-back to Earth, the peak in the number of these is between 10 and 15 kyr and by 20 kyr, 90% of the particles that do so, have returned to Earth. Only 3 particles collide with Venus, 1 of these impacts after less than 15 kyr and the rest do so before 25 kyr. Of the 5 particles reaching Jupiter, a group representing 40% of the colliders do so within just 10 kyr after being ejected, the remaining 60% collide with Jupiter towards the end of the simulation. Discussion ========== Gladman et al. (2005) has performed a similar calculation to the one conducted in this paper using a different numerical code, initial conditions and a smaller number of test particles in each simulation. In general, our results agree well with those they report. Two notable exceptions, most likely attributable to the greater number of test particles we follow in our simulations, are that we find collisions with Mars, one particle in Cases B and C, and also, we find collisions with Jupiter, 0.06% of all ejecta in Case D and 0.05% in Case E. Using an Öpik collision probability calculation, Gladman et al. (2005) estimated the collision rate with Mars to be about 2 orders of magnitude lower that found on the basis of our simulations. However, as also noted in their paper, our results for Mars are within the known typical errors of such probability estimations. No collisions with Jupiter are reported in Gladman et al. (2005). ![Initial location of particles that collide with different bodies in our simulation. Angle $\phi$ is the latitude of the launch position for each particle, $\theta$ is a longitude like angle but measured from the midnight meridian. The top panel corresponds to colliding particles ejected at low velocity (Case A), the middle panel corresponds to intermediate ejection velocity (Case C) and the bottom panel shows the initial location of colliding particles in the highest ejection velocity considered (Case E). The figure legend at the bottom right corner indicates the body with which a particle collides.[]{data-label="fig4"}](fig4.eps){width="\columnwidth" height="1.33\columnwidth"} Both results, definite collisions with Mars and Jupiter, are of astrobiological significance, owing to the possible presence of life sustaining environments in early Mars and in Jupiter’s moons Europa and Ganymedes. Also worth noting is the fact that the single particle colliding with Mars, does so towards the end of our simulation, between 25 and 30 thousand years after being ejected from Earth. Collisions with Jupiter are characterized by a wider range of collision times, one half reaching the giant planet in less than 10,000 years. In future studies we will extend our analysis of both cases to determine the statistical significance of these results. Effect of the Moon ------------------ In order to estimate the effect of the presence of the Moon on the dynamics of ejecta, we have performed a simulation with exactly the same initial conditions and run parameters as Case D (Table 1), but, as is done in Gladman et al. (2005), assuming that the Earth and the Moon are integrated into a single body with the combined mass and located at the center of mass of the system. Only minor differences in the results are found, in comparison to Case D with the Earth and the Moon as separate bodies. In the single body Earth-Moon simulation, 23 particles fallback to the Earth-Moon (22 in Case D), 8 particles impact Venus (7 in Case D), 4 collide with Jupiter (6 in Case D) and 2 reach the Sun (0 in Case D). A small difference is also found in the number of particles “ejected” from the system, 220 in this case (254 in Case D). Effect of ejection location --------------------------- In Figure 4, we plot the distribution of launch positions of fall-back and colliding particles. Each panel depicts the initial latitude (angle $\phi$) and a longitude-like angle ($\theta$ is measured from west to east but starting from the midnight meridian at the time of ejection). For example, the point $\phi = 0$ and $\theta = 90^{\rm o}$ corresponds approximately to the center of the leading face along the direction of motion of the planet. The top, middle and bottom rows of Figure 4, show the initial location of particles ejected with $V_{\rm ej} = 11.22$, $V_{\rm ej} = 12.7$ and $V_{\rm ej} = 16.4$ km/s, Cases A, C and E, respectively. No clear asymmetry is found in the distribution of ejection locations for particles falling back to Earth in the case with $V_{\rm ej} = 11.22$ km/s. This can be understood from the fact that ejecta with such a low velocity remain in orbits which are very close to Earth, as illustrated in Figure 2, thus increasing the chance of collision. As expected, there is also no asymmetry with respect to the equator. The slight asymmetries in the ejection location for particles eventually colliding with Venus and the Moon, are probably not statistically significant, and must be tested in future simulations with an even greater number of particles. In the case with intermediate ejection velocity, $V_{\rm ej} = 12.7$ km/s, shown in the middle panel of Figure 4, the number of particles falling back to Earth is slightly higher for those ejected from the trailing face, since orbits of these particles are less dispersed, i.e. more concentrated in the vicinity of the Earth, than for particles ejected from the leading face (see Figure 2). For the same reason, we also find that particles ejected from the trailing face are more likely to impact Venus. The opposite is true for particles traveling to Mars, as ejecta from the trailing face in this velocity range do not have enough energy to reach the planet. The highest ejection velocity we consider, $V_{\rm ej} = 16.4$ km/s, leads to ejecta capable of traveling outside the planetary region of the Solar System. We label these particles as ejected since they spend a very short amount of time in the inner Solar System, so that their collision probability with other planets is negligible. These are shown in the bottom panel of Figure 4 and they arise exclusively from the leading face of the planet. A similar asymmetry in the ejection location is found for particles colliding with Jupiter, since only particles ejected with a high total velocity are capable of reaching the planet. A few particles are found to fall-back to Earth and to collide with Venus, mostly ejected from the trailing face. These results suggest that the probability of collision with different Solar System bodies of Earth ejecta resulting from a giant impact, is clearly dependent on the particular place on Earth where the collision occurred. Impacts on the leading face of the planet along its direction of motion, which are statistically more likely, lead to ejecta that have a higher probability of colliding with Mars and Jupiter. Effect of initial ephemeris --------------------------- We have also performed a simulation with the same ejection velocity as Case A, but using a different initial ephemerides for the planets, corresponding to an initial time 6 months after the start of the rest of the simulations reported in this paper. This allows us to determine whether the initial planet configuration has a significant effect on the dynamics of ejecta and the resulting collision probabilities. Only small differences in our results are found in comparison to Case A. In the case of an initial ephemerides taken 6 months later than that considered in Case A, 531 particles fallback to Earth (496 in Case A), 8 particles impact Venus (6 in Case A) and 1 reaches the Moon (2 in Case A). A difference of 7% in the number of particles returning to Earth is found as a result of the different initial planetary configuration. The statistical significance of the differences in the number of collisions with the Moon and Venus, must be verified in future simulations with a greater number of ejected particles. Conclusions =========== We have computed the trajectory of an ensemble of particles representing ejecta from Earth, resulting from the giant impact of a comet or asteroid, in order to determine the collision probability with different Solar System bodies. Several ejection velocities, representing the different ejecta components in a given impact, as well as different initial planetary configurations, have been explored with simulations over a period of 30,000 years. In agreement with previous work (Gladman et al. 2005) we find that ejecta can collide with the Moon, Venus or fallback to Earth after a period of several thousand years in space. A novel result in our simulations is finding particles that collide with the planet Mars, for intermediate ejection velocities, and also with Jupiter, for high ejection velocity. Of course, a given impact will give rise to ejecta with a wide spectrum of velocities, the maximum determined by the speed of the impactor as it hits the Earth. This, together with other characteristics of the impact, also defines the amount of material ejected to space. In general, most of the escaping material does so with a velocity slightly greater than the Earth’s escape velocity, the number of particles ejected drops rapidly as ejection velocity increases (Eq. (1)). Hence, the calculation of the net collision probability with a given Solar System body, must take into account the sharply decreasing velocity distribution of ejecta. On the basis of our results, and considering the rule of thumb that the maximum ejection velocity is one half of the impactor speed, we must conclude that: 1) In collisions with impactor speed greater than 2 $V_{\rm esc}$, a significant amount of material, of the order of a few percent, will fall back to Earth after remaining in interplanetary space for less than 30 kyr, 2) Ejecta transfer to Venus and the Moon can occur as long as $U \gtrsim 2 V_{\rm esc}$, 3) Ejecta transfer to Mars requires an ejection speed only slightly greater, less than 5% more, than the escape velocity and 4) The transport of terrestrial crustal material to the vicinity of Jupiter requires an impactor speed of almost 3 times the escape speed, and can occur only if the impact is on the leading face of the planet as it orbits around the Sun. A more detailed calculation of the collision probability, taking into account the velocity distribution of ejecta, as well as computations with a greater number of particles to estimate statistically significant collision rates with the Moon, Mars and Jupiter, will be the subject of future contributions. [[Acknowledgements]{}]{} The authors acknowledge support from research grants IN109409 of DGAPA-UNAM and CONACYT-México grant No.128563. [99]{} Chambers, J.E. 1999, A hybrid symplectic integrator that permits close encounters between massive bodies, MNRAS, [[304]{}]{}, 793-799. Chapman, C.R., 1994, Impacts on the Earth by asteroids and comets: assessing the hazard, Nature, [[367]{}]{}, 33. Chyba, C.F., Owen,T.C., Ip,W.-H., 1994, Impact Delivery of Volatiles and Organic Molecules to Earth [[in]{}]{} Hazards due to comets and asteroids, eds. T.Gehrels, T., M.S.Matthews. and A.Schumann, University of Arizona Press, Tucson, p.9. 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--- abstract: 'In this paper, we introduce a definition of phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems, the frequency responses of which are cramped at all frequencies. This phase concept generalizes the notions of positive realness and negative imaginariness. We also define the half-cramped systems and provide a time-domain interpretation. As a starting point in an endeavour to develop a comprehensive phase theory for MIMO systems, we establish a small phase theorem for feedback stability, which complements the well-known small gain theorem. In addition, we derive a sectored real lemma for phase-bounded systems as a natural counterpart of the bounded real lemma.' author: - 'Wei Chen, Dan Wang, Sei Zhen Khong, and Li Qiu [^1] [^2]' title: 'Phase Analysis of MIMO LTI Systems [^3]' --- MIMO systems, phase response, small phase theorem, sectored real lemma, half-cramped systems Introduction ============ In the classical frequency domain analysis of single-input-single-output (SISO) systems, the magnitude (gain) response and phase response go hand in hand. In particular, the Bode magnitude plot and phase plot are always drawn shoulder to shoulder. The combined Bode plot of a loop transfer function provides a significant amount of useful information about the closed-loop stability and performance. The gain and phase crossover frequencies of a loop transfer function give salient information on the gain and phase margins of the feedback system. The famous Bode gain-phase integral relation binds the gain and phase together. In frequency domain controller synthesis, phase also plays an important role. Loop-shaping design techniques, such as lead and lag compensation, are rooted in the phase stabilization ideas. The inception of MIMO systems theory sees extension and thriving of the magnitude concept, but not equal flourishing in the phase concept. While the small gain theorem is widely known in the field of robust control, much less attention has been paid to the development of a small phase theorem. Moreover, the magnitude plot of a MIMO frequency response has been inbuilt to the computing environment MATLAB, a useful phase plot has not been available in practice. Several notable preliminary works on MIMO systems phases include [@Chen; @Freudenberg; @Anderson1988] and [@Macfarlane1981]. The references [@Chen; @Freudenberg; @Anderson1988] extended the Bode gain-phase integral relation for SISO systems to MIMO systems. The reference [@Macfarlane1981] proposed a definition of phases for MIMO systems, based on which a small phase theorem was formulated. However, the condition therein depends on both phase and gain information, which somewhat deviates from the initial purpose of finding a phase counterpart to the small gain theorem. An important line of research with a phasic point of view is on positive real (passive) and negative imaginary systems. Roughly speaking, one can think of positive real systems as those whose phases lie within $[-\frac{\pi}{2},\frac{\pi}{2}]$ and negative imaginary systems as those whose phases over positive frequencies lie within $[-\pi,0]$. Research on positive real systems can be traced back to more than half a century ago and has led to a rich theory through efforts of generations of researchers. See books [@Anderson1973; @BaoLee; @Brogliato; @DV1975] and the survey paper [@Kottenstette] for a review. Over the past two decades, negative imaginary systems [@lanzon2008stability; @petersen2010csm] and counter-clockwise dynamics [@Angeli2006] have attracted much attention. The abundant studies on these systems, concerning feedback stability, performance and beyond, provide valuable insights in developing a general phase theory for MIMO LTI systems. One main reason accounting for the underdevelopment of MIMO phases is the following. While the gains of a complex matrix are well described by its singular values, a universally accepted definition of matrix phases has been lacking over a long period. Very recently, we initiated to adopt the canonical angles introduced in [@FurtadoJohnson2001] as the phases of a cramped complex matrix whose numerical range does not contain the origin [@WCKQ2018]. We studied various properties of matrix phases, some of which are briefly reviewed later. This paves the ground for conducting a systematic study of phase analysis and design for MIMO LTI systems. In this paper, we first define the phase responses of MIMO LTI systems whose frequency responses are cramped at all frequencies. Such phase concept agrees with and generalizes the notions of positive realness and negative imaginariness. We then develop a small phase theorem for negative feedback interconnections of phase bounded systems, complementing the well known small gain theorem. We derive a sectored real lemma, which gives state space conditions for phase-bounded systems in terms of linear matrix inequalities (LMIs). This serves as a counterpart of bounded real lemma. In addition, we pay special attention to the class of half-cramped systems which exhibit a nice time-domain interpretation. We absorb much nutrition from the existing studies on positive real systems, negative imaginary systems, KYP lemma, generealized KYP lemma, integral quadratic constraints (IQCs), etc. along the way. The rest of the paper is organized as follows. A review of matrix phases is presented in Section II. The phase responses of MIMO LTI systems are defined in Section III, followed by the discussions on half-cramped systems in Section IV. A small phase theorem is presented in Section V. State-space conditions are derived for phase bounded systems in Section VI. The paper is concluded in Section VII. The notation used in this paper is more or less standard and will be made clear as we proceed. Phases of a Complex Matrix ========================== A nonzero complex scalar $c$ can be represented in the polar form as $c=\sigma e^{i\phi}$ with $\sigma >0$ and $\phi$ taking values in a half open $2\pi$-interval, typically $[0,2\pi)$ or $(-\pi,\pi]$. Here $\sigma=\|c\|$ is called the modulus or the magnitude and $\phi=\angle c$ is called the argument or the phase. The polar form is particularly useful when multiplying two complex numbers. We simply have $\|ab\| =\|a\| \|b\|$ and $\angle (ab) = \angle a + \angle b \mbox{ mod $2\pi$}$. It is well understood that an $n \times n$ complex matrix $C$ has $n$ magnitudes, served by the $n$ singular values $$\sigma (C) = \begin{bmatrix} \sigma_1 (C) & \sigma_2 (C) & \cdots & \sigma_n (C) \end{bmatrix}$$ with $\overline{\sigma}(C)=\sigma_1 (C) \geq \sigma_2 (C) \geq \cdots \geq \sigma_n(C) = \underline{\sigma}(C)$ [@HornJohnson]. The magnitudes of a matrix possess plentiful nice properties, among which the following majorization inequality regarding the magnitudes of matrix products are of particular interest to the control community. Given $x,y\in \mathbb{R}^n$, we denote by $x^\downarrow$ and $y^\downarrow$ the rearranged versions of $x$ and $y$ so that their elements are sorted in a non-increasing order. Then, $x$ is said to be majorized by $y$ [@Marshall], denoted by $x\prec y$, if $$\begin{aligned} \sum_{i=1}^k x^\downarrow_i\leq\sum_{i=1}^k y^\downarrow_i,\ k=1,\ldots, n-1,\;\;\; \text{and} \;\; \sum_{i=1}^n x^\downarrow_i=\sum_{i=1}^n y^\downarrow_i.\end{aligned}$$ When $x$ and $y$ are nonnegative, $x$ is said to be log-majorized by $y$, denoted by $x \prec_{\log} y$, if $$\begin{aligned} \prod_{i=1}^k x^\downarrow_i\leq\prod_{i=1}^k y^\downarrow_i,\ k=1,\ldots, n-1, \;\;\; \text{and} \;\; \prod_{i=1}^n x^\downarrow_i=\prod_{i=1}^n y^\downarrow_i.\end{aligned}$$ The magnitudes of matrix product satisfy [@Marshall] $$\begin{aligned} \sigma(AB) \prec_{\log} \sigma (A) \odot \sigma(B), \label{gainmajorization}\end{aligned}$$ where $\odot$ denotes the Hadamard product, i.e., the elementwise product. In contrast to the magnitudes of a complex matrix $C$, how to define the phases of $C$ appears to be an unsettled issue. An early attempt [@Macfarlane1981] defined the phases of $C$ as the phases of the eigenvalues of the unitary part of its polar decomposition. This definition was motivated by the seeming generalization of the polar form of a scalar to the polar decomposition of a matrix. However, phases defined this way do not have certain desired properties. Very recently, we discovered a more suitable definition of matrix phases based on numerical range [@WCKQ2018]. The numerical range, also called field of values, of a matrix $C \in \mathbb{C}^{n\times n}$ is defined as $W(C) = \{ x^Cx: x \in \mathbb{C}^n \mbox{ with } \\|x\\|=1\}$, which, as a subset of $\mathbb{C}$, is compact and convex, and contains the spectrum of $C$ [@horntopics]. If $0\notin W(C)$, then $W(C)$ is contained in an open half complex plane due to its convexity. In this case, $C$ is said to be a cramped matrix. It is known that a cramped $C$ is congruent to a diagonal unitary matrix that is unique up to a permutation [@Horn; @ZhangFuzhen2015], i.e., there exists a nonsingular matrix $T$ and a diagonal unitary matrix $D$ such that $C=T^DT$. This factorization is called sectoral decomposition in [@ZhangFuzhen2015]. Let $\delta(C)$ be the field angle of $C$, i.e., the angle subtended by the two supporting rays of $W(C)$ at the origin. We define the phases of $C$, denoted by $\phi_1(C),\phi_2(C),\dots,\phi_n(C)$, to be the phases of the eigenvalues of $D$, taking values in an interval $(\theta, \theta+\pi)$, where $\theta\in[-\pi,\delta(C))$. The phases defined in this fashion coincide with the canonical angles of $C$ introduced in [@FurtadoJohnson2001]. Assume without loss of generality that $$\overline{\phi}(C)=\phi_1(C)\geq \phi_2(C)\geq \dots\geq \phi_n(C)=\underline{\phi}(C).$$ Moreover, define $\phi (C) = [ \phi_1 (C) \ \ \phi_2 (C) \ \ \cdots \ \ \phi_n(C) ]$. The phases defined above admit the maximin and minimax expressions [@Horn]: $$\begin{split} \phi_i(C)&=\max_{\mathcal{M}: \mathrm{dim}\mathcal{M}=i}\min_{x\in \mathcal{M}, \\|x\\|=1} \angle x^Cx\\&=\min_{\mathcal{N}: \mathrm{dim}\mathcal{N}=n-i+1}\max_{x\in \mathcal{N}, \\|x\\|=1}\angle x^Cx. \end{split}$$ In particular, $$\begin{aligned} \overline{\phi}(C)&=\max_{x\in\mathbb{C}^n,\\|x\\|=1}\angle x^Cx,\\ \underline{\phi}(C)&=\min_{x\in\mathbb{C}^n,\\|x\\|=1}\angle x^Cx.\end{aligned}$$ A graphic interpretation of the phases is illustrated in . The two angles from the positive real axis to each of the two supporting rays of $W(C)$ are $\overline{\phi}(C)$ and $\underline{\phi}(C)$ respectively. The other phases of $C$ lie in between. ![Geometric interpretation of $\overline{\phi}(C)$ and $\underline{\phi}(C)$.[]{data-label="fig1"}](numerical_range.pdf) It is noteworthy that the notion of matrix phases subsumes the well-studied strictly accretive matrices [@Kato], i.e., matrices with positive definite Hermitian part. In particular, the phases of $C$ lie in $(- \pi/2, \pi/2)$ if and only if $C$ is strictly accretive. Given matrix $C$, we can check whether it is cramped or not by plotting its numerical range. From the plot of numerical range, we can also determine a $\pi$-interval $(\theta,\theta+\pi)$ in which the phases take values. How to efficiently compute $\phi(C)$ is an important issue. The following observation provides some insights along this direction. Suppose $C$ is cramped. Then it admits a sectoral decomposition $C=T^DT$ and thus $$\begin{aligned} C^{-1}C^=T^{-1}D^{-1}T^{-}T^D^T=T^{-1}D^{-2}T,\end{aligned}$$ indicating that $C^{-1}C^$ is similar to a diagonal unitary matrix. Hence, we can first compute $\angle\lambda(C^{-1}C^)$, taking values in $(-2\theta\!-\!2\pi,-2\theta)$, and then let $\phi(C)=-\frac{1}{2}\angle\lambda(C^{-1}C^)$. This gives one possible way to compute $\phi(C)$. We are currently exploring other methods, hopefully of lower complexity, for the computation of matrix phases. The matrix phases defined above have plentiful properties, of which a comprehensive study has been conducted in [@WCKQ2018]. First, note that the set of phase bounded matrices defined as $$\begin{gathered} \mathcal{C}[\alpha, \beta]\\=\left\{C\!\in\! \mathbb{C}^{n\times n}: C \text{ is cramped and }\overline{\phi}(C)\!\leq\! \beta,\ \underline{\phi}(C)\!\geq\! \alpha\right\},\nonumber\end{gathered}$$ where $0\leq \beta-\alpha<2\pi$, is a cone. In addition, the following lemma can be shown by exploiting the maximin and minimax expressions of phases. \[convexcone\] If $\beta-\alpha<\pi$, then $\mathcal{C}[\alpha, \beta]$ is a convex cone. Another important property pertinent to later developments in this paper is concerned with product of cramped matrices. In view of the magnitude counterpart in , one may expect $\phi (AB) \prec \phi(A)+\phi(B)$ to hold for cramped matrices $A$ and $B$. This, unfortunately, fails even for positive definite $A$ and $B$. Notwithstanding, if we consider instead $\lambda (AB)=\begin{bmatrix}\lambda_1(AB)&\dots&\lambda_n(AB)\end{bmatrix}$, i.e., the vector of eigenvalues of $AB$, the following weaker but useful result has been derived. \[thm: product\_majorization\] Let $A,B\in\mathbb{C}^{n\times n}$ be cramped matrices with phases in $(\theta_1, \theta_1+\pi)$ and $(\theta_2,\theta_2+\pi)$, respectively, where $\theta_1\in[-\pi,\delta(A))$ and $\theta_2 \in [-\pi,\delta(B))$. Let $\angle \lambda (AB)$ take values in $(\theta_1+\theta_2, \theta_1+\theta_2+2\pi)$. Then $$\begin{aligned} \angle\lambda(AB) \prec \phi(A) + \phi(B).\end{aligned}$$ The above majorization relation underlies the development of a small phase theorem, much in the spirit of being the foundation of the celebrated small gain theorem. To be more specific, recall that the singularity of matrix $I + AB$ plays an important role in the stability analysis of feedback systems. It is straightforward to see that if $\sigma(A)$ and $\sigma(B)$ are both sufficiently small, then $I+AB$ is nonsingular. By contrast, one can observe that if $\phi (A)$ and $\phi (B)$ are both sufficiently small in magnitudes, then $I+AB$ is nonsingular. Phase Response of MIMO LTI Systems ================================== Let $G$ be an $m\times m$ real rational proper stable transfer matrix, i.e., $G\in\mathcal{RH}^{m\times m}_\infty$. Then $\sigma(G(j\omega))$, the vector of singular values of $G(j \omega)$, is an $\mathbb{R}^m$-valued function of the frequency, which we call the [magnitude response]{} of $G$. The $\mathcal{H}_\infty$ norm of $G$, denoted by $\\|G\\|_\infty=\sup_{\omega \in \mathbb{R}} \overline{\sigma}(G(j \omega))$, is of particular importance. Suppose $G(j \omega)$ is cramped for all $\omega \in \mathbb{R}$. Such a system is called a frequency-wise cramped system. Also, assume for simplicity that $W(G(j\omega))$ does not intersect the negative real axis for all $\omega \in \mathbb{R}$. Then $\phi(G(j\omega))$, the vector of phases of $G(j\omega)$ with each element taking values in $(-\pi,\pi)$, is well defined as an $\mathbb{R}^m$-valued function of the frequency, which we call the [phase response]{} of $G$. We define the $\mathcal{H}_\infty$ phase of $G$, as the counterpart to its $\mathcal{H}_\infty$ norm, to be $$\begin{aligned} \Phi_\infty(G)= \displaystyle \sup_{\omega \in \mathbb{R}, \\|x\\| = 1} \angle x^* G(j\omega)x.\end{aligned}$$ Clearly, $\Phi_\infty(G)\!\leq\! \pi$. It is noteworthy that the set of phase bounded systems $$\begin{aligned} \mathfrak{C}[\alpha]=\{G\in\mathcal{RH}_\infty^{m\times m}:\Phi_\infty(G)\leq \alpha\},\label{pbs}\end{aligned}$$ where $\alpha\in[0,\pi)$, is a cone. Having defined the phase response of $G$, we can now plot $\sigma(G(j\omega))$ and $\phi(G(j\omega))$ together to complete the MIMO Bode plot of $G$, laying the foundation of a complete MIMO frequency-domain analysis. \[example1\] The Bode plot of system $$\begin{aligned} G(s)=\begin{bmatrix}\!\frac{1}{s^2+2s+200}\!&\frac{2}{s^2+2s+200}\!\vspace{3pt} \\\!\frac{2}{s^2+2s+200}\!&\frac{0.2s^3+0.5s^2+44.2s+24}{s^3+3s^2+202s+200}\!\end{bmatrix}\end{aligned}$$ is shown in Fig. \[mimo-response\]. ![MIMO Bode plot of a frequency-wise cramped system.[]{data-label="mimo-response"}](FC_gain.pdf "fig:") ![MIMO Bode plot of a frequency-wise cramped system.[]{data-label="mimo-response"}](FC_phase.pdf "fig:") Note that the well-known notions of positive real systems [@Anderson1973; @Brogliato; @Kottenstette] and negative imaginary systems [@lanzon2008stability; @petersen2010csm] can be characterized using their phase responses. For simplicity, here we briefly mention the strong and strict versions of these notions. A transfer function matrix $G \in \mathcal{RH}^{m\times m}_\infty$ is said to be strongly positive real if $G(j\omega)+G^(j\omega) > 0$ for all $\omega\in[-\infty,+\infty]$ [@LiuYao2016]. In the language of phase, $G \in \mathcal{RH}^{m\times m}_\infty$ is strongly positive real if and only if $$\Phi_\infty (G) < \frac{\pi}{2}.$$ On the other hand, a transfer function matrix $G$ is said to be strictly negative imaginary if $(G(j\omega)-G^(j \omega))/j< 0$ for all $\omega \in (0, \infty)$ [@lanzon2008stability]. This is equivalent to $$[\underline{\phi}(G(j\omega)), \overline{\phi} (G(j \omega))] \subset (-\pi, 0)$$ for all $\omega \in (0, \infty)$. The phase concept of MIMO LTI systems gives a way to unify these concepts, together with of course the trivial SISO systems phase, and more. The system shown in Fig. \[mimo-response\] is neither positive real nor negative imaginary but it has well-defined phase response. Half-cramped Systems ==================== Let $G\in\mathcal{RH}^{m\times m}_\infty$. Then, $G(j\omega)$ is conjugate symmetric, i.e., $$G(-j\omega)=\overline{G(j\omega)},$$ and hence $W(G(j\omega))$ and $W(G(-j\omega))$ are symmetric about the real axis. This property hints that in dealing with many problems such as feedback stability, one only has to examine the frequency response for nonnegative frequency, while the other half frequency range will be automatically taken care of due to the symmetry. Following this hint, we define half-cramped systems, and provide a time-domain interpretation for such systems. A system $G$ is said to be half-cramped if $$\mathrm{cl.\;Co}\{W(G(j\omega)),\omega\geq 0\}$$ is contained in an open half plane and does not intersect the negative real axis, where cl. denotes closure and Co denotes convex hull. Whether a system is half-cramped or not can be read out from its phase plot. For instance, the system in Example \[example1\] is not half-cramped as its positive frequency phase response has a spread larger than $\pi$. Below we give an example of a half-cramped system. Consider the system $$\begin{aligned} G(s)=\begin{bmatrix}\frac{s^3+6.5s^2+10s+6}{s^3+1.5s^2+1.5s+1}&\frac{s+2}{s+1}\vspace{3pt}\\\frac{s+2}{s+1}&\frac{s+2}{s+1}\end{bmatrix}.\end{aligned}$$ Its Bode plot is shown in Fig. 3, from which one can easily see that the system is half-cramped, but is neither positive real nor negative imaginary. ![MIMO Bode plot of a half-cramped system.[]{data-label="HC"}](HC_gain.pdf "fig:") ![MIMO Bode plot of a half-cramped system.[]{data-label="HC"}](HC_phase.pdf "fig:") Interestingly, there is a nice time-domain interpretation for half-cramped systems. For preparation, we briefly introduce some background knowledge on signal spaces and Hilbert transform. The Hilbert transform has been used extensively in signal processing, especially in the time-frequency domain analysis. It has also been applied in the control field, mostly in gain-phase relationship and system identification, etc. We refer interested readers to [@Hahn] for more details. Let $\mathcal{F}$ be the usual Fourier transform on $\mathcal{L}^T_2(-\infty, \infty)$, the Hilbert space of complex-valued bilateral time functions $$[\mathcal{F} x ] (j\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt .$$ Note that $\mathcal{F}$ is an isometry onto $\mathcal{L}^\Omega_2(-\infty, \infty)$, the Hilbert space of complex-valued bilateral frequency functions. If we decompose $\mathcal{L}_2^\Omega (-\infty, \infty)$ into a positive frequency signal space and a negative frequency signal space as $$\mathcal{L}_2^\Omega(-\infty, \infty)= \mathcal{L}_2^\Omega(0, \infty) \oplus \mathcal{L}_2^\Omega(-\infty, 0),$$ then clearly this is an orthogonal decomposition. Let $P$ be the orthogonal projection onto $\mathcal{L}^\Omega_2 (0,\infty)$. Then we naturally have the orthogonal decomposition $$\mathcal{L}_2^T(-\infty, \infty)= \mathcal{F}^{-1} \mathcal{L}_2^\Omega(0, \infty) \oplus \mathcal{F}^{-1} \mathcal{L}_2^\Omega(-\infty, 0).$$ Let us call the first space above $\mathcal{A}$ and hence the second space $\mathcal{A}^\perp$. Let $Q$ be the orthogonal projection onto $\mathcal{A}$. Then the commutative diagram in Fig. \[commutativediagram\] gives a complete picture of the relationships among these spaces. Recall the Hilbert transform $\mathcal{H}: \mathcal{L}^T_2(-\infty, \infty) \rightarrow \mathcal{L}^T_2(-\infty, \infty)$ defined as $$[\mathcal{H} x](t)= \frac{1}{\pi}\int_{-\infty}^\infty \frac{x(\tau)}{t - \tau} \, d\tau.$$ It then turns out that $Qx=\frac{1}{2} (x+j \mathcal{H} x)$ and $(I-Q)x=\frac{1}{2} (x-j \mathcal{H} x)$, the analytic part and the skew-analytic part of $x$ respectively. (40,70) (25,64)[(-1,0)[10]{}]{} (15,66)[(1,0)[10]{}]{} (35,45)[(10,10)[$P$]{}]{} (-5,45)[(10,10)[$Q$]{}]{} (35,40)[(0,1)[20]{}]{} (30,60)[(10,10)[$\mathcal{L}^\Omega_2(0, \infty)$]{}]{} (15,65)[(10,10)[$\mathcal{F}$]{}]{} (15,55)[(10,10)[$\mathcal{F}^{-1}$]{}]{} (0,60)[(10,10)[$\mathcal{A}$]{}]{} (5,40)[(0,1)[20]{}]{} (0,30)[(10,10)[$\mathcal{L}^T_2(-\infty, \infty)$]{}]{} (5,30)[(0,-1)[20]{}]{} (-6,15)[(10,10)[$I-Q$]{}]{} (0,0)[(10,10)[$\mathcal{A}^\perp$]{}]{} (25,34)[(-1,0)[10]{}]{} (15,36)[(1,0)[10]{}]{} (15,35)[(10,10)[$\mathcal{F}$]{}]{} (15,25)[(10,10)[$\mathcal{F}^{-1}$]{}]{} (25,4)[(-1,0)[10]{}]{} (15,6)[(1,0)[10]{}]{} (15,5)[(10,10)[$\mathcal{F}$]{}]{} (15,-5)[(10,10)[$\mathcal{F}^{-1}$]{}]{} (30,30)[(10,10)[$\mathcal{L}^\Omega_2(-\infty,\infty)$]{}]{} (35,30)[(0,-1)[20]{}]{} (36,15)[(10,10)[$I-P$]{}]{} (30,0)[(10,10)[$\mathcal{L}^\Omega_2(-\infty, 0)$]{}]{} Now let $\mathbf{G}: \mathcal{L}^T_2(-\infty, \infty)\rightarrow\mathcal{L}^T_2(-\infty, \infty)$ be the linear operator corresponding to $G(s)\in\mathcal{RH}_\infty$. Clearly, both $\mathcal{A}$ and $\mathcal{A}^\perp$ are invariant subspaces of $\mathbf{G}$. We define the positive frequency numerical range and negative frequency numerical range as $$\begin{aligned} W_+(\mathbf{G})&:= \!\{\langle Qu, \mathbf{G}u \rangle:\ u \!\in \mathcal{L}_2^T (-\infty, \infty),\\|u\\|_2=1\}\\ W_-(\mathbf{G})&:= \!\{\langle (I-Q)u, \mathbf{G}u \rangle:\ u \!\in \mathcal{L}_2^T (-\infty, \infty),\\|u\\|_2=1\}\end{aligned}$$ respectively. It can be easily seen that $W_+(\mathbf{G})$ and $W_-(\mathbf{G})$ are symmetric with respect to the real axis. Also, note that $$\begin{aligned} \langle Qu, \mathbf{G}u \rangle&=\int_{-\infty}^{+\infty}[Qu]^(t)[\mathbf{G}u](t) dt\\ &=\int_{0}^{+\infty} [\mathcal{F} u]^(j\omega) G(j\omega)[\mathcal{F} u](j\omega)d\omega,\end{aligned}$$ which suggests that $$\begin{aligned} \mathrm{cl.}\ W_+(\mathbf{G})\subset\mathrm{cl.\;Co}\{W(G(j\omega)),\omega\geq 0\}.\end{aligned}$$ In fact, one can further show $$\begin{aligned} \mathrm{cl.}\ W_+(\mathbf{G})= \mathrm{cl.\;Co}\{W(G(j\omega)),\omega\geq 0\},\end{aligned}$$ and thus $\Phi_\infty(G)=\max \{\sup_{z \in W_+(G)} \angle z , \sup_{z \in W_-(G)} \angle z \}$. The detailed proof is omitted for brevity and will be available in a longer version of this paper. Small Phase Theorem =================== Suppose $G$ and $H$ are $m\times m$ real rational proper transfer function matrices. The feedback interconnection of $G$ and $H$, as depicted in Fig. \[fdbk\], is said to be stable if the Gang of Four matrix $$\begin{aligned} G\#H=\begin{bmatrix} (I + HG)^{-1} & (I + HG)^{-1}H\\ G(I + HG)^{-1} & G(I + HG)^{-1}H \end{bmatrix}\end{aligned}$$ is stable, i.e., $G\#H \in \mathcal{RH}^{2m \times 2m}_\infty$. (50,25) (0,20)[(1,0)[8]{}]{} (10,20) (12,20)[(1,0)[8]{}]{} (20,15)[(10,10)[$G$]{}]{} (30,20)[(1,0)[10]{}]{} (40,20)[(0,-1)[13]{}]{} (38,5)[(-1,0)[8]{}]{} (40,5) (50,5)[(-1,0)[8]{}]{} (20,0)[(10,10)[$H$]{}]{} (20,5)[(-1,0)[10]{}]{} (10,5)[(0,1)[13]{}]{} (5,10)[(5,5)[$y_1$]{}]{} (40,10)[(5,5)[$y_2$]{}]{} (0,20)[(5,5)[$w_1$]{}]{} (45,0)[(5,5)[$w_2$]{}]{} (13,20)[(5,5)[$u_1$]{}]{} (32,0)[(5,5)[$u_2$]{}]{} (10,10)[(6,10)[$-$]{}]{} The celebrated small gain theorem [@Zhou; @LiuYao2016] is one of the most used results in robust control theory over the past half a century. A version of it states that for $G, H \in \mathcal{RH}^{m\times m}_\infty$, the feedback system $G\#H$ is stable if $$\begin{aligned} \overline{\sigma}(G(j\omega)) \overline{\sigma}(H(j\omega))< 1\end{aligned}$$ for all $\omega \in \mathbb{R}$. There was an attempt to formulate a small phase theorem by using phases defined from the matrix polar decomposition [@Macfarlane1981]. However, the condition therein involves both phase and gain information and thus deviates from the initial purpose of having a phase counterpart of the small gain theorem. Armed with the new definition of matrix phases $\phi(C)$, we work out a version of the small phase theorem. \[smallphase\] For frequency-wise cramped $G, H \in \mathcal{RH}^{m\times m}_\infty$, the feedback system $G\#H$ is stable if $$\begin{aligned} \overline{\phi} (G(j\omega))+ \overline{\phi}(H(j\omega)) < \pi \label{spi}\end{aligned}$$ for all $\omega \in \mathbb{R}$. Since $G, H \in \mathcal{RH}_\infty$, it follows that $G\#H$ is stable if and only if $(I + HG)^{-1}\in\mathcal{RH}_\infty$. Hence, it suffices to show that $\det[I+G(s)H(s)]\neq 0$ for all $s\in\mathbb{C}^+ \cup \{\infty\}$, where $\mathbb{C}^+$ denotes the closed right half plane. To this end, observe that when (\[spi\]) is satisfied, by symmetry, the inequality $\underline{\phi} (G(j\omega))+ \underline{\phi}(H(j\omega)) > -\pi$ also holds for all $\omega\in\mathbb{R}$. Applying Lemma \[thm: product\_majorization\], we have $$\begin{gathered} \underline{\phi} (G(j\omega))+ \underline{\phi}(H(j\omega))\leq \angle\lambda_i(G(j\omega)H(j\omega))\\\leq \overline{\phi} (G(j\omega))+ \overline{\phi}(H(j\omega))\nonumber\end{gathered}$$ and thus $-\pi<\angle\lambda_i(G(j\omega)H(j\omega))<\pi$ for all $\omega\in\mathbb{R},i=1,2,\dots,m$. Now, let $\tau$ be an arbitrary number in $[0,1]$. From Lemma \[convexcone\], it follows that $$\begin{aligned} &\overline{\phi}(\tau G(j\omega)+(1-\tau)I)\leq \overline{\phi}(G(j\omega)),\\ &\overline{\phi}(\tau H(j\omega)+(1-\tau)I)\leq \overline{\phi}(H(j\omega)),\\ &\underline{\phi}(\tau G(j\omega)+(1-\tau)I)\geq \underline{\phi}(G(j\omega)),\\ &\underline{\phi}(\tau H(j\omega)+(1-\tau)I)\geq \underline{\phi}(H(j\omega)),\end{aligned}$$ for all $\omega\in\mathbb{R}$. Then, following the same arguments as above, we can show $$\begin{aligned} -\pi< \angle\lambda_i[(\tau G(j\omega)\!+\!(1-\tau)I)(\tau H(j\omega)\!+\!(1-\tau)I)]<\pi,\end{aligned}$$ which in turn yields that $$\begin{aligned} \det[I+(\tau G(j\omega)\!+\!(1-\tau)I)(\tau H(j\omega)\!+\!(1-\tau)I)]\neq 0\end{aligned}$$ for all $\omega\in\mathbb{R}$. Since when $\tau=0$, $$\begin{aligned} \det[I+(\tau G(s)\!+\!(1-\tau)I)(\tau H(s)\!+\!(1-\tau)I)]\neq 0\end{aligned}$$ for all $s\in\mathbb{C}^+$, it follows by continuity that the same holds for all $\tau\!\in\![0,1]$. Particularly, when $\tau\!=\!1$, there holds $\det[I\!+\!G(s)H(s)]\neq 0$ for all $s\!\in\!\mathbb{C}^+$. Finally, note that $\det[I\!+\!G(\infty)H(\infty)]\neq 0$ due to the well-posedness of the feedback system. This completes the proof. We wish to mention that the small phase theorem can also be established via IQCs. Specifically, when the condition (\[spi\]) is satisfied, one can find a dynamic multiplier of the form $$\begin{aligned} \Pi(s)=\begin{bmatrix}0&e^{j\theta(s)}\\e^{-j\theta(s)}&0\end{bmatrix}\end{aligned}$$ so that $\Pi(s)\in \mathcal{L}_\infty$ is continuous on the imaginary axis and the following quadratic constraints $$\begin{aligned} &\begin{bmatrix}I\\G(j\omega)\end{bmatrix}^\Pi(j\omega)\begin{bmatrix}I\\G(j\omega)\end{bmatrix}\geq 0,\\ &\begin{bmatrix}H(j\omega)\\I\end{bmatrix}^\Pi(j\omega)\begin{bmatrix}H(j\omega)\\I\end{bmatrix}<0\end{aligned}$$ are satisfied for all $\omega\!\in\!\mathbb{R}$. The feedback stability then follows from the result in [@MR]. From this perspective, the small phase theorem provides a nice phasic interpretation of the condition obtained from IQCs. The small phase theorem generalizes a stronger version of the passivity theorem [@DV1975; @LiuYao2016], which states that for $G, H \!\in\! \mathcal{RH}^{m\times m}_\infty$, the feedback system $G\#H$ is stable if $G$ and $H$ are strongly positive real. Note that the small gain theorem provides a quantifiable tradeoff between the gains of $G$ and $H$, while the above small phase theorem does the same with respect to the phases of $G$ and $H$. In the literature, the notions of input feedforward passivity index and output feedback passivity index [@Vidyasagar; @Wen; @BaoLee; @Kottenstette] have been used to characterize the tradeoff between the surplus and deficit of passivity in open-loop systems. It is our belief that the concept of MIMO system phases is more suited to this task. Specifically, $\frac{\pi}{2}-\Phi_\infty(G)$ gives a natural measure of passivity of system $G$, which we call the angular passivity index. The small phase theorem above implies that if the sum of the angular passivity indexes of $G$ and $H$ are positive, then $G\#H$ is stable. In addition, one can see that $\pi-\Phi_\infty(GH)$ yields a natural phase stability margin of $G\#H$. It is well known that the condition given in the small gain theorem is necessary in the following sense [@Zhou]. Suppose $G\!\in\!\mathcal{RH}^{m\times m}_\infty$ and let $\mathfrak{B}[\gamma]\!=\!\{H\in\mathcal{RH}^{m\times m}_\infty: \!\\|H\\|_\infty \!\leq\! \gamma\}$, where $\gamma>0$. Then, the feedback system $G\#H$ is stable for all $H\in\mathfrak{B}[\gamma]$ if and only if $\\|G\\|_\infty < \frac{1}{\gamma}$. Regarding the necessity of small phase theorem, we observe evidences supporting the following conjecture. Recall the set of phase bounded systems $\mathfrak{C}[\alpha]$ defined in (\[pbs\]), where $\alpha \in[0,\pi)$. Suppose $G\!\in\!\mathcal{RH}^{m\times m}_\infty$. Then, the feedback system $G\#H$ is stable for all $H\in\mathfrak{C}[\alpha]$ if and only if $\Phi_\infty(G)<\pi-\alpha$. Evidently, this conjecture holds in the SISO case in light of the Nyquist stability criterion. A rigorous proof in the MIMO case appears technically challenging and is under our current investigation. State-space Conditions for Phase Bounded Systems ================================================ The $\mathcal{H}_\infty$ norm of an LTI system can be determined by the well-known bounded real lemma. The efficient computation of $\mathcal{H}_\infty$ norm is specifically useful as evidenced in small gain theorem and facilitates robust control design. The bounded real lemma [@Zhou] states that for $G\!\in\!\mathcal{RH}_\infty^{m\times m}$ with a minimal realization $\left[\begin{array}{c\|c}A & B \\ \hline C & D\end{array} \right]$, $\\|G\\|_{\infty}\!<\!\gamma$ if and only if there exists $X>0$ satisfying the LMI $$\begin{aligned} \begin{bmatrix}A'X+XA&XB&C'\\B'X&-\gamma I &D'\\C&D&-\gamma I\end{bmatrix}<0.\end{aligned}$$ One would naturally wish to see an analogous state-space condition for phase bounded systems. It is equally important to have an LMI characterization for a system $G$ satisfying $\Phi_\infty(G)\!<\!\alpha$, where $\alpha\in(0,\pi]$. Along this direction, we obtain a sectored real lemma, a natural counterpart of the bounded real lemma. Before proceeding, we introduce some preliminary knowledge on KYP lemma and generalized KYP lemma. KYP lemma and generalized KYP lemma ----------------------------------- The well known KYP lemma builds the equivalence between infinite many frequency domain inequalities over the entire frequency range and a finite dimensional LMI. Let $A\!\in\!\mathbb{C}^{n\times n}$, $B\!\in\!\mathbb{C}^{n\times m}$, $M\!=\!M^\in\mathbb{C}^{(n+m)\times (n+m)}$. Assume that $A$ has no eigenvalues on the imaginary axis. Then the inequality $$\begin{aligned} \begin{bmatrix}(j\omega I-A)^{-1}B\\I\end{bmatrix}^M\begin{bmatrix}(j\omega I-A)^{-1}B\\I\end{bmatrix}<0\end{aligned}$$ holds for all $\omega\in\mathbb{R}\cup\{\infty\}$ if and only if there exists a Hermitian matrix $X$ satisfying the LMI $$\begin{aligned} M+\begin{bmatrix}A^X+XA&XB\\B^X&0\end{bmatrix}<0.\end{aligned}$$ In contrast to the KYP lemma which copes with frequency domain inequalities over the entire frequency, the generalized KYP lemma [@IH] has the capability to address the frequency domain inequalities over partial frequency ranges. Specifically, the generalized KYP lemma builds the equivalence between inequalities on curves in the complex plane and LMIs. Consider the curves characterized by the set $$\begin{aligned} \mathrm{\bold{\Lambda}}(\Sigma,\Psi)=\left\{\lambda\in\mathbb{C}\left\|\begin{bmatrix}\lambda\\1\end{bmatrix}^\!\Sigma\begin{bmatrix}\lambda\\1\end{bmatrix}=0,\begin{bmatrix}\lambda\\1\end{bmatrix}^\!\Psi\begin{bmatrix}\lambda\\1\end{bmatrix}\geq0\right.\right\},\end{aligned}$$ where $\Sigma,\Psi$ are given $2\times 2$ Hermitian matrices. By appropriately choosing $\Sigma$ and $\Psi$, $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ can represent the partial or whole segment(s) of a straight line or a circle in the complex plane. When $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ is unbounded, it is extended with $\infty$. Denote by $\otimes$ the Kronecker product of matrices. A version of the generalized KYP lemma is as follows. , $B\!\in\!\mathbb{C}^{n\times m}$, $M\!=\!M^\!\in\!\mathbb{C}^{(n+m)\times(n+m)}$, and $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ be curves in the complex plane. Let $\mathrm{\bold{\Omega}}$ be the set of eigenvalues of $A$ in $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$. Then the inequality $$\begin{aligned} \begin{bmatrix}(\lambda I-A)^{-1}B\\I\end{bmatrix}^M\begin{bmatrix}(\lambda I-A)^{-1}B\\I\end{bmatrix}<0\end{aligned}$$ holds for all $\lambda\in \mathrm{\bold{\Lambda}}(\Sigma,\Psi)\backslash \mathrm{\bold{\Omega}}$ if and only if there exist two Hermitian matrices $X$ and $Y$ such that $$\begin{aligned} Y>0,\quad \begin{bmatrix}A&B\\I&0\end{bmatrix}^(\Sigma\otimes X+\Psi\otimes Y)\begin{bmatrix}A&B\\I&0\end{bmatrix}+M<0.\end{aligned}$$ By choosing $\Sigma$ and $\Psi$ appropriately, one can use $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ to define a variety of frequency ranges. For instance, when $\Sigma=\begin{bmatrix}0&1\\1&0\end{bmatrix},\Psi=0$, $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ is simply the imaginary axis, and the generalized KYP lemma reduces to the classical KYP lemma. Sectored real lemma ------------------- The following theorem gives a state space characterization for frequency-wise cramped systems satisfying $\Phi_\infty(G)<\alpha$, where $\alpha\in(0,\frac{\pi}{2}]$. Let $G\!\in\!\mathcal{RH}_\infty^{m\times m}$ with a minimal realization $\left[\begin{array}{c\|c}A & B \\ \hline C & D\end{array} \right]$ and $\alpha\in(0,\frac{\pi}{2}]$. Then $\Phi_\infty(G)\!<\!\alpha$ if and only if there exists $X>0$ satisfying the LMI $$\begin{aligned} \begin{bmatrix} A'X + XA & XB \!-\! e^{-j(\frac{\pi}{2} - \alpha)}C' \\ B'X \!-\! e^{j(\frac{\pi}{2} - \alpha)}C & - e^{j(\frac{\pi}{2} - \alpha)}D\!-\!e^{-j(\frac{\pi}{2} - \alpha)}D' \end{bmatrix} \!<\! 0.\label{srllmi}\end{aligned}$$\[SRL\] Note that $\Phi_\infty(G)<\alpha$, $\alpha\in(0,\frac{\pi}{2}]$ is equivalent to requiring $e^{j(\frac{\pi}{2}-\alpha)}G$ to be strongly positive real, i.e., $$\begin{aligned} e^{j(\frac{\pi}{2}-\alpha)}G(j\omega)+e^{-j(\frac{\pi}{2}-\alpha)}G^(j\omega)>0 \label{srli}\end{aligned}$$ for all $\omega\in\mathbb{R}\cup\{\infty\}$. The inequality (\[srli\]) can be rewritten as $$\begin{aligned} \begin{bmatrix}(j\omega I\!-\!A)^{-1}B\\I\end{bmatrix}^\!M\! \begin{bmatrix}(j\omega I\!-\!A)^{-1}B\\I\end{bmatrix}\!<\!0,\end{aligned}$$ where $M=\begin{bmatrix}0&-e^{-j(\frac{\pi}{2}-\alpha)}C'\\-e^{j(\frac{\pi}{2}-\alpha)}C&-e^{j(\frac{\pi}{2} - \alpha)}D-e^{-j(\frac{\pi}{2} - \alpha)}D'\end{bmatrix}$. Then, it follows from KYP lemma that $\Phi_\infty(G)<\alpha$ if and only if the LMI (\[srllmi\]) has a Hermitian solution $X$. Finally, the positive definiteness of $X$ follows from the stability of $A$ and $A'X+XA<0$. When $\alpha=\frac{\pi}{2}$, the above sectored real lemma reduces to the strongly positive real lemma [@SKS1994]. The case when $\alpha\!\in\!(\frac{\pi}{2},\pi]$ appears much more complicated. Nevertheless, for half-cramped systems, we are able to derive an LMI condition by employing the generalized KYP lemma. In dealing with half-cramped real systems, one only needs to concern the frequency domain characterization for positive frequency, i.e., $\{j\omega\|\omega\in[0,\infty]\}$. This frequency range can be captured by $\mathrm{\bold{\Lambda}}(\Sigma,\Psi)$ with $\Sigma=\begin{bmatrix}0&1\\1&0\end{bmatrix},\Psi=\begin{bmatrix}0&j\\-j&0\end{bmatrix}$. Now we present the state-space condition for half-cramped systems. Let $G\in\mathcal{RH}_\infty^{m\times m}$ with a minimal realization $\left[\begin{array}{c\|c}A & B \\ \hline C & D\end{array} \right]$ and $\alpha\!\in\!(\frac{\pi}{2},\pi]$. Then $G$ is half-cramped and $\Phi_\infty(G)\!<\!\alpha$ if and only if there exist Hermitian matrices $X$ and $Y$ satisfying either $$\begin{aligned} Y>0, \begin{bmatrix}A&B\\I&0\end{bmatrix}'\!\begin{bmatrix}0&X+jY\\X-jY&0\end{bmatrix}\!\begin{bmatrix}A&B\\I&0\end{bmatrix}+M<0,\label{LMI1}\end{aligned}$$ where $M\!=\!\begin{bmatrix}0&-e^{-j(\alpha-\frac{\pi}{2})}C'\\-e^{j(\alpha-\frac{\pi}{2})}C&-e^{j(\alpha-\frac{\pi}{2})}D-e^{-j(\alpha-\frac{\pi}{2})}D'\end{bmatrix}$, or $$\begin{aligned} Y>0, \begin{bmatrix}A&B\\I&0\end{bmatrix}'\!\begin{bmatrix}0&X+jY\\X-jY&0\end{bmatrix}\!\begin{bmatrix}A&B\\I&0\end{bmatrix}+N<0,\end{aligned}$$ where $N\!=\!\begin{bmatrix}0&-e^{j(\alpha-\frac{\pi}{2})}C'\\-e^{-j(\alpha-\frac{\pi}{2})}C&-e^{-j(\alpha-\frac{\pi}{2})}D-e^{j(\alpha-\frac{\pi}{2})}D'\end{bmatrix}$. By definition, we know $G$ is half-cramped and $\Phi_\infty(G)\!<\!\alpha$ for $\alpha\in(\frac{\pi}{2},\pi]$ if and only if either $$\begin{aligned} e^{j(\alpha-\frac{\pi}{2})}G(j\omega)+e^{-j(\alpha-\frac{\pi}{2})}G^(j\omega)>0 \label{halfcrampedfdi}\end{aligned}$$ or $$\begin{aligned} e^{-j(\alpha-\frac{\pi}{2})}G(j\omega)+e^{j(\alpha-\frac{\pi}{2})}G^(j\omega)>0\end{aligned}$$ holds for all $\omega\in[0,\infty]$. For brevity, we consider the case when (\[halfcrampedfdi\]) holds for all $\omega\in[0,\infty]$. The other case can be shown similarly. The inequality (\[halfcrampedfdi\]) can be rewritten into $$\begin{aligned} \begin{bmatrix}(j\omega I-A)^{-1}B\\I\end{bmatrix}^M\begin{bmatrix}(j\omega I-A)^{-1}B\\I\end{bmatrix}<0,\;\omega\in[0,\infty].\end{aligned}$$ Then, applying the generalized KYP lemma with $\Sigma\!=\!\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $\Psi=\begin{bmatrix}0&j\\-j&0\end{bmatrix}$ yields that the above frequency domain inequalities hold if and only if there exist Hermitian matrices $X$ and $Y$ satisfying LMIs (\[LMI1\]). This completes the proof. Conclusion {#ending} ========== In this paper, we define the phase responses of frequency-wise cramped MIMO LTI systems. The combined magnitude and phase plots constitute a complete MIMO Bode plot. We obtain a small phase theorem for closed-loop stability, a counterpart of the well-known small gain theorem. We also derive a sectored real lemma for phase-bounded systems, a counterpart of the bounded real lemma. This paper focuses on the analysis of MIMO systems. 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Khong, and L. Qiu, “Phases of a complex matrix and their properties," working paper, 2019. J. T. Wen, “Robustness analysis based on passivity,” in [Proceedings of 1988 American Control Conference]{}, pp. 1207-1213, 1988. F. Zhang, “A matrix decomposition and its applications," [Linear and Multilinear Algebra]{}, vol. 63, pp. 2033-2042, 2015. K. Zhou, J. C. Doyle, and K. Glover, [Robust and Optimal Control]{}, Prentice Hall, New Jersey, 1996. [^1]: W. Chen, D. Wang, and L. Qiu are with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. Email: wchenust@gmail.com, dwangah@connect.ust.hk, eeqiu@ust.hk [^2]: S. Z. Khong is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China. Email: szkhong@hku.hk [^3]: This work was supported in parts by the Research Grants Council of Hong Kong Special Administrative Region, China, under the Theme-Based Research Scheme T23-701/14-N.
--- abstract: 'A large deviations approach is introduced, which calculates the probability density and outage probability of the MIMO mutual information, and is valid for large antenna numbers $N$. In contrast to previous asymptotic methods that only focused on the distribution close to its [most probable]{} value, this methodology obtains the [full]{} distribution, including its non-Gaussian tails. The resulting distribution interpolates between the Gaussian approximation for rates $R$ close its mean and the asymptotic distribution for large signal to noise ratios $\rho$ [@Zheng2003_DiversityMultiplexing]. For large enough $N$, this method provides the outage probability over the whole $(R, \rho)$ parameter space. The presented analytic results agree very well with numerical simulations over a wide range of outage probabilities, even for small $N$. In addition, the outage probability thus obtained is more robust over a wide range of $\rho$ and $R$ than either the Gaussian or the large-$\rho$ approximations, providing an attractive alternative in calculating the probability density of the MIMO mutual information. Interestingly, this method also yields the eigenvalue density constrained in the subset where the mutual information is fixed to $R$ for given $\rho$. Quite remarkably, this eigenvalue density has the form of the Marčenko-Pastur distribution with square-root singularities.' author: - 'Pavlos Kazakopoulos, Panayotis Mertikopoulos, Aris L. Moustakas and Giuseppe Caire [^1]' title: 'Living at the Edge: A Large Deviations Approach to the Outage MIMO Capacity' --- Diversity–multiplexing tradeoff (DMT), Gaussian approximation, information capacity, large-system limit, multiple-input multiple-output (MIMO) channels. Introduction {#Introduction} ============ Considerable interest has arisen from the initial prediction[@Foschini1998_BLAST1; @Telatar1995_BLAST1] that the use of multiple antennas in transmitting and receiving signals can lead to substantial gains in information throughput. To analyze the theoretical limits of such a MIMO (Multiple Input Multiple Output) system, it has been convenient to focus on the case of i.i.d. Gaussian noise and input. For the MIMO channel model $$\label{eq:MIMO_ch_model} \mathbf{y= Hx + z}$$ with coherent detection and no channel state information at the transmitter [@Foschini1998_BLAST1; @Telatar1995_BLAST1], the mutual information $I_N$ for a given value of the channel matrix $\bH$ takes the familiar form: $$\label{eq:logdet_def} I_N=\log\det\left(\bI + \rho\bH^\dagger\bH\right).$$ where “$\log$” signifies the natural logarithm, $\rho$ is the signal to noise ratio and $\bH$ is the $M\times N$ channel matrix whose elements are independent ${\cal CN}(0,1/N)$ random variables. This corresponds to the case of $N$ transmitting and $M$ receiving antennas, which is captured by the ratio $\beta=M/N$. Without loss of generality we assume that $\beta\geq 1$; otherwise, if $\beta<1$, we may simply replace $\rho$ with $\rho_{\text{new}}=\rho\beta$ in (\[eq:logdet\_def\]) and interchange the roles of $M$ and $N$. If the channel matrix $\bH$ varies in time according to a stationary ergodic process, and coding spans an arbitrarily large number of fading states, then the “ergodic” channel capacity is given by the mutual information expected value $\operatorname{\mathbb{E}}\left[I_N\right]$ [@Telatar1995_BLAST1]. Initially, this quantity was calculated asymptotically for large $N$, with $\beta$ remaining fixed and finite. In particular, in this case, $\bH$ can be viewed as a large random matrix. Then, by applying ideas and methods from the theory of random matrices, it was shown in [@Rapajic2000_InfoCapacityOfARandomSignatureMIMOChannel] that the value of the mutual information per antenna $I_N(\rho,\bH)/N$ “freezes” to a deterministic value in the large $N$ limit, the so-called [ergodic average]{} $r_{\text{erg}}(\rho)$. Underlying this result is the fact that the very eigenvalue distribution of $\bH^\dagger\bH$ freezes to the celebrated Marčenko-Pastur distribution: $$\label{eq:MP_def} p(x) = \frac{\sqrt{(b-x)(x-a)}}{2\pi x}$$ where $a, b= (\sqrt{\beta}\pm 1)^2$ are the end-points of its support. Even though later the closed form solution of $\operatorname{\mathbb{E}}[I_N]$ for general $M$, $N$ was found[@Wang2002_OutageMutualInfoOfSTMIMOChannels], the asymptotic form of $r_{erg}(\rho)$ was particularly popular due to its simplicity and accuracy, even for small number of antennas. Another more relevant regime is when the channel matrix is random, but varies in time much more slowly than the typical coding delay. In this case (usually referred to as the “quasi-static” fading channel) $\bH$ can be considered as a random constant and the mutual information $I_N(\bH)$ is a random variable. In this regime, the relevant performance metric is the “rate versus outage probability” tradeoff [@Biglieri1998_FadingChannels], captured by the cumulative distribution function of $I_N(\bH)$. Various approaches [@Moustakas2003_MIMO1; @Hochwald2002_MultiAntennaChannelHardening; @Hachem2006_GaussianCapacityKroneckerProduct; @Taricco2006_MIMOCorrelatedCapacity; @Taricco2008_MIMOCorrelatedCapacity] have shown that the mutual information $I_N(\bH)$ becomes asymptotically Gaussian for large $N$, with mean equal to the ergodic capacity $R_{\text{erg}} = N r_{\text{erg}}(\rho)$ and a variance of order ${\mathcal{O}}(1)$ in $N$. This Gaussian variability of the mutual information is due to the fluctuations of the eigenvalues of the matrix around the most probable distribution described by the Marčenko-Pastur law. Since this Gaussian approximation is essentially a variation of the central limit theorem, it only applies within a small number of standard deviations away from the mean $R_{\text{erg}}$. As a result, this approximation fails to capture the tails of the distribution, e.g. the probability of the mutual information $I_N$ falling below half its ergodic value $R_{\text{erg}}/2$, because this event only occurs ${\mathcal{O}}(N)$ standard deviations away from the mean. Nevertheless, the tails of the distributions of the mutual information are important, because they correspond to regions with low outage probability, where one would want to operate a MIMO system. This is particularly important when, for large $\rho$, the slopes of the outage curves are large. The interplay between low outage and multiplexing gain was exemplified in the seminal paper [@Zheng2003_DiversityMultiplexing] where the authors analyzed the asymptotics of the distribution of the mutual information in the limit of large $\rho$ (keeping $R/\log\rho$ fixed). They found that the asymptotic form of the logarithm of the outage probability of the mutual information ${P_{\text{out}}}(R) \equiv \operatorname{\mathbb{P}}(I_{N}(\bH)\leq R)$ is a piecewise linear function of $R/\log\rho$, interpolating between the discrete set of values: $$\label{eq:DMT_def} \log {P_{\text{out}}}(R_n) \sim - \log\rho\left(\frac{R_n}{\log\rho} - M\right)\left(\frac{R_n}{\log\rho} - N\right)$$ where $R_n = n \log\rho$ for integer $n\leq N\leq M$. When, in addition to $\rho$, $N$ is also large, $\log {P_{\text{out}}}(R)$ in (\[eq:DMT\_def\]) becomes (to leading order) a continuous function of $R/N$. It should be pointed out that this approach generalizes the large $N$ asymptotics discussed above, since it provides insight in the distribution of the mutual information quite far from its peak, which for large $\rho$ (and large $N$) is situated at $I_N\approx N\log\rho$. More recently, in [@Azarian2007_finite_rate_DMT] the authors recast the DMT problem providing a formula to calculate $\log P_{out}$ as a function of $R$ when $R$ lies in each linear subsegment of (\[eq:DMT\_def\]). Nevertheless both approaches [@Zheng2003_DiversityMultiplexing; @Azarian2007_finite_rate_DMT] do not provide the offset to the leading, $O(\log\rho)$ behavior of (\[eq:DMT\_def\]). As a result, these approaches, while quite intuitive fail, often by a large margin, to provide an acceptable quantitative estimate of $P_{out}$ unless $\log\rho$ is extremely large. In the meantime, all variants [@Moustakas2003_MIMO1; @Hochwald2002_MultiAntennaChannelHardening; @Hachem2006_GaussianCapacityKroneckerProduct] of the large $N$ Gaussian approximation of the mutual information fail for large $\rho$. Specifically, they all predict that the outage probability is given asymptotically by: $$\label{eq:largeN_large_rho} \log {P_{\text{out}}}(R) \sim \frac{(\log\rho)^2}{2\log\left(1-\beta^{-1}\right)}\left(\frac{R}{\log\rho} - N\right)^2$$ where $\beta=M/N>1$, an expression which is in striking disagreement with (\[eq:DMT\_def\]). Even though for $\beta=1$ the asymptotic form of (\[eq:DMT\_def\]) is recovered within the Gaussian approximation[@Moustakas2003_MIMO1; @Hachem2006_GaussianCapacityKroneckerProduct], the discrepancy for $\beta\neq 1$ indicates that the limits $N\rightarrow\infty$ and $\rho\rightarrow\infty$ cannot be naïvely interchanged. In the Gaussian approximation, one focuses on the most probable eigenvalue distribution, which converges vaguely to the Marčenko-Pastur distribution (\[eq:MP\_def\]). However, as can be seen in (\[eq:MP\_def\]), this distribution (almost surely) produces no eigenvalues of $\bH^\dagger\bH$ close to zero when $\beta>1$. Nevertheless, the analysis for large $\rho$ focuses at the regime where the eigenvalues are of order $O(\rho^{-1})$. As a result, it is not surprising that the large-$N$ Gaussian approximation of the mutual information distribution misses the correct behavior. In summary, we have two methods, the large-$N$, fixed-$\rho$ Gaussian approximation on the one hand and the large-$\rho$, fixed-$N$ limit on the other, both having their own regions of validity, and both failing to produce quantitative results for the outage probability outside their respective regions. Thus, one still needs an approach that correctly describes the outage behavior of the mutual information distribution for arbitrary $\rho$ and $R$. In this paper, we introduce a large deviations approach to calculate the full asymptotic distribution of $R$. It is formally valid for large $N$, but works over the whole range of values of $R$ and $\rho$. This method bridges the two regions of small/intermediate and large signal to noise ratios within a single framework and, in effect, it amounts to calculating the rate function of the logarithm of the average moment generating function of the mutual information. Our approach was first introduced in the context of random matrix theory by Dyson [@Dyson1962_DysonGas] and has been more recently applied in a variety of problems [@Majumdar2006_LesHouches; @Vivo2007_LargeDeviationsWishart; @Vivo2008_DistributionsConductanceShotNoise; @Nadal2009_NonIntersectingBrowianInterfaces]. It is quite intuitive because it interprets the eigenvalues of $\bH^\dagger\bH$ as point charges on a line repelling each other logarithmically. This is the first time this approach has been applied in information theory and communications. As a byproduct of this approach, we obtain the most probable eigenvalue distribution constrained on the subset of channel matrices $\bH^\dagger\bH$ that have fixed total rate $R$ and signal to noise ratio $\rho$. This is a generalized Marčenko-Pastur distribution that gives the constrained eigenvalue distribution for values of $R$ even far from its ergodic value. It is worth pointing out that many of the results presented here could be set on a more formal mathematical footing using tools developed in [@Johansson1998_2ndOrderRMTFluctuations]. However, we will follow the less formal but more intuitive approach developed by Dyson. This generalized Marčenko-Pastur distribution can also be seen as the inverse of the so-called [Shannon transform]{} [@Tulino2004_RMTInfoTheoryReview] in the following sense: while the Shannon transform produces the value of normalized mutual information $I_N/N$ as a functional of the asymptotic eigenvalue distribution of $\bH^\dagger\bH$ (the Marčenko-Pastur distribution), the generalized Marčenko-Pastur distribution introduced here boils down to the asymptotic eigenvalue distribution of $\bH^\dagger\bH$ for a given value of the mutual information $R=Nr$, i.e., when $\bH^\dagger\bH$ is constrained on the subset defined by $r= I_N(\bH)/N$. Outline {#sec:Outline} ------- In the next section we will introduce the necessary mathematical methodology. In particular, Section \[sec:map\_coulomb\_gas\] describes the mapping of the joint probability distribution of eigenvalues of the Wishart matrix to a Coulomb gas of charges with a continuous density (discussed in more detail in Appendix \[app:Coulomb\_gas\]) and the large-deviations analysis of the problem. Next, section \[sec:integral\_equation\] deals with the solution of the resulting integral equation that produces the most-likely eigenvalue distribution at the tails of the full distribution. If one is not particularly interested in the details of our derivation, Section \[sec:methodology\] may be skipped in favor of section \[sec:results\] where we present our main results. Specifically, in Section \[sec:MP\_Distribution\] we rederive the Marčenko-Pastur distribution (that is, the most likely distribution without the mutual information constraint) to highlight the efficacy of our method. Subsequently, Sections \[sec:beta>1\] and \[sec:beta=1\] contain our results for the cases $\beta>1$ and $\beta=1$ respectively, while in Section \[sec:OutageProbability\] we show how to calculate the outage probability directly by means of the results of the previous sections. In Section \[sec:limiting\_cases\] we analytically obtain previous results as limiting cases of this method, and also examine a number of different limiting cases. In Section \[sec:numerical simulations\] we provide numerical comparisons of our method to other approximations previously outlined and to Monte Carlo simulations. The proofs of the properties of [tame]{} distributions (introduced in section \[sec:map\_coulomb\_gas\]) are given in appendix \[app:tameness\] and we discuss Dyson’s original construction of the Coulomb gas model in appendix \[app:Coulomb\_gas\]. Appendices \[app:solution\] and \[app:uniqueness\_ab\] have been reserved for the exposition of some technical issues that cropped up during our calculations. Finally, Appendix \[app:1/Ncorrections\] discusses higher order $O(1/N)$ corrections to our model and comparisons with Monte Carlo simulations. Methodology {#sec:methodology} =========== Our approach can roughly be divided in two main parts. First, in section \[sec:map\_coulomb\_gas\] we reduce the original problem of finding the probability distribution of the mutual information to harvesting the minimum energy of a gas of charged particles (among other things we show here that the minimum energy configuration is unique). Then, in section \[sec:integral\_equation\], we will solve the integral equation that comes up and actually obtain the minimum energy configuration of the charges. Mapping the Problem to a Coulomb Gas {#sec:map_coulomb_gas} ------------------------------------ We begin by establishing the mathematical methodology, treading on the elegant footsteps of [@Vivo2007_LargeDeviationsWishart; @Dean2008_ExtremeValueStatisticsEigsGaussianRMT]. Our overall aim will be to calculate the probability distribution of the mutual information (\[eq:logdet\_def\]), which can be written in terms of the eigenvalues $\lambda_{k}$ of the Wishart matrix $\bH^\dagger\bH$ as: $$\begin{aligned} \label{eq:MI_def} I_N({\hbox{\boldmath$\lambda$}}) &=& \sum_{k=1}^N \log\left(1+\rho\lambda_k\right)\end{aligned}$$ Note that the aforementioned probability distribution of the mutual information thus depends on the joint probability distribution function of the eigenvalues $\lambda_1\ldots \lambda_{N}$ of $\bH^\dag\bH$. In its turn, this distribution takes the well-known form: $$\begin{aligned} \label{eq:P_lambda_k} P_{{{\hbox{\boldmath$\lambda$}}}}(\lambda_{1}\ldots\lambda_{N}) &=& A_N \Delta({\hbox{\boldmath$\lambda$}})^2 \prod_{k=1}^N \lambda_k^{M-N} e^{- N \lambda_k} \\ &=& A_N e^{-N^2 E\left({\hbox{\boldmath$\lambda$}}\right)} \label{eq:P_lambda_k_exp}\end{aligned}$$ where $A_N$ is a normalization constant and $\Delta({\hbox{\boldmath$\lambda$}}) = \prod_{i>j} (\lambda_i-\lambda_j)$ is the Vandermonde determinant of the eigenvalues $\lambda_k$. The exponent $E({\hbox{\boldmath$\lambda$}})$ is an energy function of the eigenvalues $\{\lambda_i\}$ that will become very useful later: $$\begin{aligned} \label{eq:E_lambda} E({\hbox{\boldmath$\lambda$}})&=& \frac{1}{N} \sum_k \left(\lambda_k - (\beta-1) \log\lambda_k\right) \\ \nonumber &+& \frac{2}{N^2} \sum_{j>k} \log\left\|\lambda_j-\lambda_k\right\|\end{aligned}$$ Note that the normalization we have chosen is such that $E({\hbox{\boldmath$\lambda$}})$ corresponds roughly to the energy per eigenvalue. The cumulative probability distribution (CDF) of the normalized mutual information $I_N/N$ can then be written as a ratio of two volumes in ${\hbox{\boldmath$\lambda$}}$-space: $$\begin{aligned} \label{eq:CDF_vol_ratio} F_{N}(r)&=& \operatorname{\mathbb{P}}(I_N/N\leq r) = \frac{V_r}{V_{\text{tot}}} \\ \nonumber &=&\int P_{{\hbox{\boldmath$\lambda$}}}({\hbox{\boldmath$\lambda$}})\, \Theta(r-I_N/N){\,d}{\hbox{\boldmath$\lambda$}}$$ where $I_N$ is given by (\[eq:MI\_def\]), [$\Theta(x)$ is the Heaviside step function ($\Theta(x)=1$ if $x>0$ and $\Theta(x)=0$ if $x<0$) and the integrals are taken with respect to the ordinary $N$-dimensional Lebesgue measure $d{\hbox{\boldmath$\lambda$}}=\prod_i d\lambda_i$]{}. The above CDF is by definition the outage probability, i.e. the probability that the normalized mutual information falls below $r$. Its corresponding probability density (PDF) can be obtained from (\[eq:CDF\_vol\_ratio\]) by taking the derivative with respect to $r$ [@Papoulis_book]: $$\label{eq:P_R_vol_ratio} P_{N}(r) = F_{N}'(r) =\int P_{{\hbox{\boldmath$\lambda$}}}({\hbox{\boldmath$\lambda$}}) \, \delta(r-I_N/N) {\,d}{\hbox{\boldmath$\lambda$}}$$ where we have used the fact that the (distributional) derivative of the step function is the Dirac $\delta$-function: $\Theta'(x) = \delta(x)$. Our primary goal will be to use (\[eq:P\_R\_vol\_ratio\]) in order to obtain an analytic expression for the probability distribution function of the mutual information $I_{N}$. However, in general there is no standard way to evaluate integrals like $V_r$ (except for some special cases [@Simon2002_TIMO1]). Nevertheless, in the large-$N$ limit it is possible to analyze such integrals in a systematic way. This so-called Coulomb-gas approach [@Mehta_book] is based on the intuitive idea to interpret the eigenvalues $\lambda$ as the positions of $N$ positive unit charges located on a line, a picture first proposed by Dyson [@Dyson1962_DysonGas]. Within this interpretation, the last term in the exponent $E({\hbox{\boldmath$\lambda$}})$ in (\[eq:E\_lambda\]) corresponds to the logarithmic repulsion energy, while the first term is the potential due to a constant field and the second term is the repulsion of a point charge located at the origin.[^2] Now, it is instructive to look at the form of $E({\hbox{\boldmath$\lambda$}})$ to get an intuitive understanding of the minimum energy configuration of ${\hbox{\boldmath$\lambda$}}$ in the absence of the constraint $I_N/N=r$. As discussed above, the first two terms in $E({\hbox{\boldmath$\lambda$}})$ correspond to the external forces acting on the charges, while the last term represents the repulsion between charges. In the absence of the charge repulsion the minimum energy configuration will correspond to all charges settling at the minimum of the external potential, i.e. $\lambda_k=\beta-1$ for all $k=1,\ldots,N$. However, the repulsion between charges will make them move away from that point but still, from simple electrostatics considerations, the external forces will not allow this repulsion to carry charges too far away from the minimum. As a result, we expect that at the minimum of $E({\hbox{\boldmath$\lambda$}})$ all charges will be concentrated in the neighborhood of $\beta-1$. As the number of charges increases, it will make sense, at least for configurations with energy $E({\hbox{\boldmath$\lambda$}})$ close to the minimum, to expect that the charge distribution will be approximately a [continuous]{} distribution. As a result, all sums over ${\hbox{\boldmath$\lambda$}}$ in $E({\hbox{\boldmath$\lambda$}})$ may be replaced by integrals, and we expect that this will also be true in the presence of constraints as in (\[eq:P\_R\_vol\_ratio\]). To make this continuum limit more precise, one begins by conditioning the probability law $\operatorname{\mathbb{P}}$ of the eigenvalues of the Wishart matrix $\bH^{\dagger}\bH$ on the set $J_{r}=\{\lambda:I_{N}(\lambda)/N=r\}$, i.e. by considering the conditional probability law $\operatorname{\mathbb{P}}(\cdot\|I_{N}/N=r)$ and the corresponding PDF. As $N\to\infty$, large deviations theory suggests that this density function will be sharply concentrated around its most probable value, i.e. the minimum of the energy functional (\[eq:E\_lambda\]). Then, according to Dyson, this minimum can be asymptotically recovered by looking at the minimum of the [continuous]{} version of (\[eq:E\_lambda\]): As $N\to\infty$, the empirical distribution of charges/eigenvalues under the rate constraint $I_{N}/N = r$ converges vaguely to an absolutely continuous density $p(x)$ which minimizes the continuous energy functional: $$\begin{aligned} \label{eq:energy} {\mathcal{E}}[p] &=& \int \!x p(x) {\,d}x- (\beta-1)\int\! p(x) \log x {\,d}x \\ \nonumber &-&\iint p(x) p(y) \!\log\|x-y\| {\,d}x dy\end{aligned}$$ over the space of densities which satisfy the constraint $\int_{0}^{\infty}p(x)\log(1+\rho x) {\,d}x=r$. In other words, as $N\to\infty$, the total charge in any interval $I\subseteq{\mathbb{R}}$ will be given by: $$\sigma(I) = \int_{I} p(x) dx,$$ with $p$ as above. This assumption is essentially identical to the one in Mehta’s book [@Mehta_book] and has been extensively employed in the literature [@Vivo2007_LargeDeviationsWishart; @Dyson1962_DysonGas; @Majumdar2006_LesHouches]. Unfortunately, despite its simple and intuitive nature, this assumption has resisted most attempts at a rigorous proof, thereby giving birth to different approaches, such as the one in [@Johansson1998_2ndOrderRMTFluctuations]. Nevertheless, the results obtained there are in agreement with the ones obtained with the help of the Coulomb Gas assumption and, hence, we feel that our posit here is rather mild (see also appendix \[app:Coulomb\_gas\] for a more detailed discussion). At any rate, to make proper use of the energy functional ${\mathcal{E}}$ (\[eq:energy\]) we must first make sure that it remains finite over a reasonably large class of densities $p(x)$. This leads us to the concept of “tameness”: \[dfn:tame\] An [integrable]{} function $p:{\mathbb{R}}_{+}\to{\mathbb{R}}$ will be called ${\varepsilon}$-[tame]{} when: - the “absolute mean” of $p$ is finite: $$\label{eq:center} \int_{0}^{\infty} x \|p(x)\| {\,d}x<\infty;$$ - there exists some ${\varepsilon}>0$ such that $p$ is $L^{1+{\varepsilon}}$-integrable, i.e. $$\label{eq:1+e_integrable_def} \int_{0}^{\infty}\|p(x)\|^{1+{\varepsilon}} {\,d}x < \infty.$$ The phrasing of condition (i) simply reflects our interest in tame functions $p\equiv p_{X}$ that are probability densities of random variables $X$ with values in ${\mathbb{R}}_{+}$. In that case, condition (i) simply states that $X$ has finite mean: $$\tag{\ref{eq:center}'} \operatorname{\mathbb{E}}\left[X\right]= \int_0^\infty x p(x){\,d}x <\infty.$$ Condition (ii) will be crucial to our analysis. At first, it might appear as a mere technical necessity (see e.g. section \[sec:integral\_equation\] and appendix \[app:solution\]) but, in fact, it has a very deep physical interpretation: a probability density with finite mean might still fail to have finite energy, making it inadmissible on physical grounds. Condition (ii) ensures that ${\mathcal{E}}[p]$ will be finite (see lemma \[lem:tame\] below). When it is not necessary to make explicit mention of the exponent ${\varepsilon}$, we will simply say that $p$ is [tame]{}. Similarly, an absolutely continuous (signed) measure $\sigma$ on ${\mathbb{R}}_{+}$ will be called [tame]{} when its Lebesgue derivative $p(x) = \frac{d\sigma(x)}{dx}$ is tame. Given this equivalence between continuous measures and Lebesgue derivatives, we will use the two terms interchangeably. Going back to the energy functional ${\mathcal{E}}$ of (\[eq:energy\]), we can see that condition (i) guarantees that the first term in (\[eq:energy\]) is finite, while (ii) bounds the second and third terms. This is captured in the following: \[lem:tame\] Let ${\Omega}$ be the space of tame functions on ${\mathbb{R}}_{+}$ and let ${\mathcal{E}}$ be defined as in (\[eq:energy\]). Then, ${\mathcal{E}}[p]<\infty$ for all $p\in{\Omega}$ and the restriction of ${\mathcal{E}}$ to any subspace of $L^{1+{\varepsilon}}$-integrable functions with finite mean is continuous (in the $L^{1+{\varepsilon}}$ norm). In other words, tame densities have finite energy and tame variations in density induce small variations in energy. We prove this lemma in Appendix \[app:tameness\] where we also give some background information on the $L^{r}$ norms. For now, it will be more useful to express the probability density $P_{N}(r)$ as the ratio: $$P_{N}(r) = \frac{{\mathcal{Z}}_{r}}{{\mathcal{Z}}}$$ where, in accordance with (\[eq:P\_lambda\_k\_exp\]), (\[eq:P\_R\_vol\_ratio\]) and (\[eq:energy\]), ${\mathcal{Z}}_{r}$ and ${\mathcal{Z}}$ are the (un-normalized) [partition functions]{}:[^3] $$\begin{aligned} \label{partition_functions_def} {\mathcal{Z}}_{r} = \int_{{\mathcal{X}}_{r}}\!\!\!{\,\mathcal{D}}\!p \,\, e^{-N^{2}{\mathcal{E}}[p]}\\ {\mathcal{Z}}= \int_{{\mathcal{X}}}\!\!\!{\,\mathcal{D}}\!p \, \, e^{-N^{2}{\mathcal{E}}[p]}\end{aligned}$$ and ${\,\mathcal{D}}p$ denotes the path-integral measure over the domains of tame densities ${\mathcal{X}},{\mathcal{X}}_{r}\subseteq{\Omega}$: $$\begin{aligned} {\mathcal{X}}= \left\{p\in{\Omega}: p\geq 0 \text{ and } \int\! p(x){\,d}x =1\right\}\\ {\mathcal{X}}_{r} = \left\{p\in{\mathcal{X}}: \int\! p(x) \log(1+\rho x) {\,d}x = r\right\}.\end{aligned}$$ Of course, from a mathematical point of view, constructing a measure ${\,\mathcal{D}}p$ over the infinite-dimensional space of functions is an intricate process which is far from trivial. Path integrals were first introduced by R. Feynman [@Feynman1965_QM_PathIntegrals] in physics and have been used there extensively over the last 70 years. We prefer not to introduce them formally but, rather, to follow a more intuitive approach instead, in Appendix \[app:Coulomb\_gas\]. With all these considerations taken into account, we may take the large $N$ limit and write: $$\lim_{N\to\infty} \frac{1}{N^{2}} \log P_{N}(r) = \lim_{N\to\infty} \frac{1}{N^{2}} \left(\log{\mathcal{Z}}_{r} - \log{\mathcal{Z}}\right)$$ and, by invoking Varadhan’s lemma[@Dembo_book_LargeDeviationsTechniques], we obtain: $$\lim_{N\to\infty} \frac{1}{N^{2}} \log P_{N}(r) = {\mathcal{E}}_0 - {\mathcal{E}}_1(r)$$ or, equivalently: $$P_{N}(r) \sim e^{-N^{2} \left({\mathcal{E}}_{1}(r) - {\mathcal{E}}_{0}\right)} \label{eq:P_N(r)}$$ where $$\begin{aligned} \label{eq:E0=infE} {\mathcal{E}}_{0} &=& \inf_{p\in{\mathcal{X}}} {\mathcal{E}}[p] \\ \label{eq:E1=infE} {\mathcal{E}}_{1}(r) &=& \inf_{p\in{\mathcal{X}}_{r}} {\mathcal{E}}[p]\end{aligned}$$ In other words, we have reduced the problem of determining the asymptotic behavior of $P_{N}(r)$ to finding the minimum of the convex functional ${\mathcal{E}}$ over the two convex domains ${\mathcal{X}}$ and ${\mathcal{X}}_{r}$. To that end, we have: \[lem:convex\] Let ${\mathcal{X}}\subseteq\Omega$ be the set of tame probability measures: ${\mathcal{X}}= \left\{p\in\Omega: p\geq 0 \text{ and } \int\!p(x) {\,d}x = 1\right\}$. Then, ${\mathcal{X}}$ is a convex subset of the topological vector space ${\Omega}$ and ${\mathcal{E}}$ is (strictly) convex on ${\mathcal{X}}$. Again, we will postpone the proof of this lemma until appendix \[app:tameness\]. However, an immediate corollary is that there exists a unique charge density $p$ which minimizes (\[eq:E0=infE\]) and (\[eq:E1=infE\]). To find this unique solution - and the corresponding (global) minima ${\mathcal{E}}_{0}, {\mathcal{E}}_{1}(r)$ - it turns out to be more convenient to work over the whole space of tame measures ${\Omega}$ and introduce Lagrange multipliers for the two domains ${\mathcal{X}}$ and ${\mathcal{X}}_{r}$. This leads to the Lagrangian functions: $$\begin{aligned} \label{eq:L0} {\mathcal{L}}_{0}[p,\nu,c] &=& {\mathcal{E}}[p] - c\left(\int_0^\infty\!\!\! p(x) {\,d}x -1\right) \nonumber\\ &-& \int_0^\infty \!\!\!\nu(x)p(x) {\,d}x\end{aligned}$$ $$\begin{aligned} \label{eq:L1} {\mathcal{L}}_{1}[p,\nu,c,k] &=& {\mathcal{L}}_0[p,\nu,c]\nonumber\\ &-& k\left(\int_0^\infty\!\!\! p(x)\log(1+\rho x) {\,d}x-r\right)\end{aligned}$$ from which we obtain ${\mathcal{E}}_{0}$ and ${\mathcal{E}}_{1}(r)$ by maximizing over the dual parameters $\nu$ (non-negativity constraint), $c$ (normalization constraint) and $k$ (mutual information constraint): $$\begin{aligned} \label{eq:minimum_0} {\mathcal{E}}_0 &=& \sup_{\nu\geq 0; \, c} \inf_p {\mathcal{L}}_0[p,\nu,c]\\ {\mathcal{E}}_1(r) &=& \sup_{\nu\geq 0; \, c,k} \inf_p {\mathcal{L}}_1[p,\nu,c,k] \label{eq:minimum_1}\end{aligned}$$ The convexity of ${\mathcal{L}}_0$, ${\mathcal{L}}_1$ over $p$ ensures that it suffices to find a local minimum $p(x)$ for the corresponding Lagrangian ${\mathcal{L}}$, for fixed $\nu$, $c$, $k$. Then, any value of $k$, $c$ that satisfies the constraints of $p$ will be unique [@Boyd_book]. It is also worth pointing out that the only difference between ${\mathcal{E}}_0$ and ${\mathcal{E}}_1$ above is that the former can be seen as the maximum over ${\mathcal{L}}_1[p,\nu,c,k]$ keeping $k=0$; this relation will come in handy later, because it allows us to work with ${\mathcal{L}}_1$ and at the very last step set $k=0$ to obtain ${{\cal E}}_0$. We are now left to find a local minimum of ${\mathcal{L}}_1$ and the easiest way to do this is by looking at its functional derivative w.r.t. $p$. Indeed, recall that the functional derivative of ${\mathcal{L}}_{1}$ at $p\in{\mathcal{X}}_{r}$ is the distribution $\delta{\mathcal{L}}_{1}[p, \nu, c, k]$ whose action on test functions $\phi\in{\Omega}$ is given by:[^4] $$\label{eq:funcderiv} \left\langle\delta{\mathcal{L}}_{1}[p],\phi\right\rangle = \frac{d}{dt}\bigg\|_{t=0} \!\!{\mathcal{L}}_{1}[p + t\phi].$$ Note now that the expression ${\mathcal{L}}_{1}[p+t\phi]$ is well-defined for all $p\in{\mathcal{X}}_{r}$, $\phi\in{\Omega}$, thanks to lemma \[lem:tame\] so that, at least, it makes sense to study its behavior as $t\to0$. In addition to that, our convexity result (lemma \[lem:convex\]) simplifies things even more because, if $\delta{\mathcal{L}}_{1}[p]=0$ for some $p\in{\mathcal{X}}_{r}$, it immediately follows that ${\mathcal{L}}_{1}$ will be attaining its global minimum at $p$.[^5] Then, maximizing the result with respect to $k$ and $c$ simply corresponds to enforcing the normalization and mutual information constraints that appear in (\[eq:L0\]) and (\[eq:L1\]): $$\begin{aligned} \label{eq:norm_condition} \int_0^\infty p(x){\,d}x &=& 1 \\ \int_0^\infty p(x) \log(1+\rho x){\,d}x &=& r \label{eq:mutual_information_condition}\end{aligned}$$ Furthermore, we must also maximize with respect to $\nu$, in order to ensure that $p(x)$ be non-negative in ${\mathbb{R}}_{+}$. This optimization constraint can be enforced by observing that $\nu(x)=0$ when $p(x)>0$ and vice-versa, as we shall see below. As a result, once we manage to find a solution to the above optimization problem, we will have: \[prop:uniquenes\] Assume that the tame probability measure $p$ satisfies the stationarity condition: $$\label{eq:delta_L_cond} \delta{\mathcal{L}}[p]=0 \quad\text{(resp. }\delta{\mathcal{L}}_{1}[p]=0)$$ along with the constraint (\[eq:norm\_condition\]) (resp. (\[eq:norm\_condition\]), (\[eq:mutual\_information\_condition\])). Then, $p$ is the unique global minimum point of (\[eq:E0=infE\]) (resp. (\[eq:E1=infE\])). This proposition stems directly from the convexity of ${\mathcal{E}}$ and will be of considerable help to us in what follows because it ensures that any stationary point of ${\mathcal{L}},{\mathcal{L}}_{1}$ which satisfies the relevant constraints will be the (unique) solution to our original minimization problem. Solving the Integral Equation {#sec:integral_equation} ----------------------------- Our task now will be to actually [find]{} the solution of (\[eq:funcderiv\]), subject to the constraints (\[eq:norm\_condition\]), (\[eq:mutual\_information\_condition\]). The solution for ${\mathcal{E}}_0$ in (\[eq:minimum\_0\]) can then be obtained by relaxing the constraint (\[eq:mutual\_information\_condition\]) and setting $k=0$ in the final result. To that end, a brief calculation (see appendix \[app:solution\]) for the functional derivative for the functional derivative $\delta{\mathcal{L}}_{1}[p]$ of (\[eq:funcderiv\]) yields the integral equation: $$\begin{aligned} \label{eq:func_deriv_E1_result} 2\int_0^\infty p(x')\log\|x-x'\| {\,d}x'&=& x - (\beta-1)\log x \\ \nonumber &-& c - k\log(1+\rho x)-\nu(x).\end{aligned}$$ The role of $\nu(x)$ in the above equation is to enforce the inequality constraint $p(x)\geq 0$ for all $x\geq 0$. It is well known [@Boyd_book] that $\nu(x)>0$ only when the probability density $p(x)$ vanishes, while when the probability density is positive, $\nu(x)$ has to be zero. The solution of the integral equation involves the inversion of the integral operator in the left-hand-side of (\[eq:func\_deriv\_E1\_result\]), which is no simple task, because the inversion process depends on the support $\operatorname{supp}(p)$ of the density $p(x)$[@Tricomi_book_IntegralEquations]. As discussed in the previous subsection (and with a fair amount of hindsight gained from the Coulomb gas analogy), we will be looking for compactly supported solutions that are continuous in $(0,\infty)$; in other words, we will be assuming that $\operatorname{supp}(p) = [a,b]$ where $0\leq a <b<\infty$. There is one important issue that must be mentioned here: when the dimensions of the channel matrix attain the critical value $\beta=1$, we will see that $p$ exhibits two different behaviors depending on the values of $r$ and $\rho$ in constraint (\[eq:mutual\_information\_condition\]). On one hand, we could have $a>0$ which, by continuity, introduces the constraint $p(a)=0$; on the other hand, we could also have solutions with $a=0$ (which impose no extra constraints because $p$ is assumed continuous only on $(0,\infty)$). If the rate $r$ is less than some critical value $r_{c}(\rho)$, it turns out that solutions with $a>0$ must be rejected because they attain negative values. In that case, we are led to solutions with $a=0$ which have no such problems; the converse happens when $r>r_{c}$, while when $r=r_{c}$ the two solutions coincide. Having said that, we may return to (\[eq:func\_deriv\_E1\_result\]), where we have $\nu(x)>0$ if and only if $p(x)=0$. By restricting $x$ to lie in the interval $[a,b]$, we may henceforth ignore $\nu(x)$ altogether. Furthermore, to eliminate $c$ for the moment, a differentiation of (\[eq:func\_deriv\_E1\_result\]) with respect to $x$ yields: $$\label{eq:func_deriv_E1_result_3} 2{\cal P} \int_a^b \frac{p(x')}{x-x'} dx' = 1-\frac{\beta-1}{x}-\frac{k\rho}{1+\rho x} \equiv f(x)$$ where ${\cal P}$ denotes the Cauchy principal value of the integral.[^6] The above equation has a straightforward physical meaning: it represents a balance of forces at every location $a\leq x <b$, because the repulsion from all other charges of the distribution located at $x'$ (the LHS expression) is equal to the external forces (RHS). For $\beta>1$, we intuitively expect that $p(x)$ must vanish at $x=0$ because in this case the force from the finite charge density located at $x=0$ (the second term of (\[eq:func\_deriv\_E1\_result\_3\])) would be infinite. As a result, we intuitively expect that $a>0$ for all $\beta>1$; this expectation will be vindicated shortly. Indeed, the solution of this integral equation for general $f(x)$ can be obtained using standard methods from the theory of integral equations [@Mikhlin_book_IntegralEquations; @Tricomi_book_IntegralEquations]. So as not to interrupt the presentation, we will postpone the details until appendix \[app:solution\] and will only give the final result here: $$\begin{aligned} \label{eq:gen_solution_int_eq0} p(x) &=& \frac{{\cal P} \int_a^b \frac{\sqrt{(y-a)(b-y)}f(y)}{y-x} dy + C'}{2\pi^2\sqrt{(x-a)(b-x)}} \\ \nonumber &=& \frac{-x - \frac{k\sqrt{(1+a\rho)(1+b\rho)}}{1+\rho x} -\frac{(\beta-1)\sqrt{ab}}{x} + C}{2\pi\sqrt{(x-a)(b-x)}}\end{aligned}$$ where $C,C'$ are unknown constants to be determined by the condition $p(b)=0$. As we explain in Appendix \[app:solution\], this formula is valid only when the function $f$ is itself $L^{\eta}$-integrable for some $\eta>1$. This is always true if $\beta=1$, because the singular term proportional to $(\beta-1)$ is not present in the LHS of (\[eq:func\_deriv\_E1\_result\_3\]). However, as we have already mentioned, the case $\beta=1$ has its own set of subtleties, analyzed at length in section \[sec:beta=1\]. In particular, we obtain two different solutions depending on whether the support of $p$ extends to $0$ or not (imposing the constraints $a=0$ or $p(a)=0$ respectively), but only one of them is physically admissible (i.e. is a tame probability measure lying in the rate-constrained domain ${\mathcal{X}}_{r}$). On the other hand, this dichotomy ceases to exist when $\beta>1$. Indeed, if $\beta>1$ and $a=0$, the LHS of (\[eq:func\_deriv\_E1\_result\_3\]) is no longer integrable. However, the RHS of (\[eq:func\_deriv\_E1\_result\_3\]) [is]{} $L^{1+{\varepsilon}}$-integrable whenever $p$ is itself ${\varepsilon}$-tame, on account of the properties of the finite Hilbert transform [@Tricomi_book_IntegralEquations] (see also appendix \[app:solution\]). We thus conclude that any solution to (\[eq:func\_deriv\_E1\_result\_3\]) whose support extends to $0$ cannot be tame and will thus have to be rejected. As a result, the support of $p$ for $\beta>1$ has to be bounded away from $0$, thus leading to the constraint $p(a)=0$ and proving our intuitive expectation above. So, starting with the general case $a,b>0$, we find that the constraint of continuity requires that the distribution $p(x)$ vanish at the endpoints $a,b$ of its support. The condition $p(b)=0$ determines the value of $C$ in (\[eq:gen\_solution\_int\_eq0\]) resulting in the following form for $p(x)$: $$\label{eq:gen_solution_int_eq1} p(x) = \frac{\sqrt{b-x}}{2\pi\sqrt{x-a}} \left(1 - \frac{k\rho}{(1+\rho x)}\sqrt{\frac{1+a\rho}{1+b\rho}} -\frac{\beta-1}{x}\sqrt{\frac{a}{b}}\right)$$ The additional condition $p(a)=0$ (when $a>0$) results to $$\begin{aligned} \label{eq:p_x_beta>1} p(x) &=& \frac{1}{2\pi} \frac{\sqrt{(b-x)(x-a)}}{x(1+\rho x)} \left(\rho x+\frac{\beta-1}{\sqrt{ab}}\right)\end{aligned}$$ with the value of $a$ determined (as a function of $b$ and $k$) by the equation: $$\label{eq:p_a_=0} \frac{k\rho}{\sqrt{(1+\rho a)(1+ \rho b)}} + \frac{\beta-1}{\sqrt{ab}} = 1.$$ Demanding that $p$ be properly normalized as in (\[eq:norm\_condition\]), imposes the constraint: $$\begin{aligned} \label{eq:norm_integral} \int_a^b p(x) dx &=& \frac{a+b-2k-2(\beta-1)}{4} \\ \nonumber &+&\frac{k}{2\sqrt{(1+a\rho)(1+b\rho)}} =1.\end{aligned}$$ In Appendix \[app:uniqueness\_ab\] we show that (\[eq:p\_a\_=0\]) and (\[eq:norm\_integral\]) admit a unique solution $a,b$ for any given $k$ and, as a result, Proposition \[prop:uniquenes\] guarantees the existence of a (necessarily unique) density $p(x)$ that minimizes (\[eq:minimum\_1\]). Now, given the resulting solution $p(x)$ we can readily calculate the minimum energy ${\mathcal{E}}[p]$ itself: $$\begin{aligned} {\mathcal{E}}[p] &=& \int_a^b x p(x) {\,d}x - (\beta-1) \int_a^b p(x) \log x {\,d}x \nonumber \\ &-& \int_a^b \int_a^b p(x) p(y) \log\|x-y\| {\,d}y {\,d}x \nonumber \\ &=& \frac{1}{2}\int_a^b x p(x) dx - \frac{\beta-1}{2} \int_a^b p(x) \log x {\,d}x \nonumber \\ &+& \frac{k}{2}\int_a^b p(x) \log(1+\rho x) {\,d}x +\frac{c}{2} \label{eq:energy_functional_calc}\end{aligned}$$ where in the second line we eliminated the double integral by substituting it from (\[eq:func\_deriv\_E1\_result\]) [@Vivo2007_LargeDeviationsWishart]. As for the value of $c$ itself, it can be determined by evaluating (\[eq:func\_deriv\_E1\_result\]) at a fixed value of $x$, say $x=a$: $$\begin{aligned} \label{eq:c_calc} c &=& a - (\beta-1)\log a - k \log(1+\rho a) \\ \nonumber &-& 2 \int_a^b \log(x-a) p(x) dx\end{aligned}$$ Inserting this in (\[eq:energy\_functional\_calc\]) then yields: $$\begin{aligned} \label{eq:energy_functional_calc1} {\mathcal{E}}[p] &=& \frac{1}{2}\int_a^b x p(x) dx - \frac{\beta-1}{2} \int_a^b p(x) \log x dx \\ \nonumber &-& \int_a^b p(x)\log(x-a) dx \\ \nonumber &+& \frac{1}{2}\left(k\left(r-\log(1+\rho a)\right)+a-(\beta-1)\log a\right)\end{aligned}$$ Probability Distributions $P_N(r)$, $P_{\text{out}}(r)$ {#sec:results} ======================================================= The central aim of the paper is to evaluate the probability density of the rate $r$ for large $N$, namely $P_N(r)$ given by (\[eq:P\_N(r)\]) $$\label{eq:P_N(r)_norm_def} P_N(r) \approx B_N e^{-N^2({\mathcal{E}}_1(r)-{\mathcal{E}}_0)}$$ where $B_N$ is a normalization constant, while ${\mathcal{E}}_1(r)$ (\[eq:E1=infE\]) and ${\mathcal{E}}_0$ (\[eq:E0=infE\]) are the most probable values of the energy evaluated with and without the mutual information constraint (\[eq:mutual\_information\_condition\]), respectively. In this section we will calculate these values and derive the corresponding eigenvalue probability densities $p(x)$ that minimize the energy functional ${\mathcal{E}}[p]$. In Section \[sec:MP\_Distribution\], we will derive ${\mathcal{E}}_0$ and we will show how the corresponding density $p(x)$ is the Marčenko-Pastur Distribution. In Sections \[sec:beta>1\] and \[sec:beta=1\] we will calculate ${\mathcal{E}}_1(r)$ for the cases $\beta>1$ and $\beta=1$ respectively. Finally, in Section \[sec:OutageProbability\] we will show how one can calculate the outage probability $P_{out}(r)$. Evaluation of ${\mathcal{E}}_0$ {#sec:MP_Distribution} ------------------------------- As mentioned above, it is instructive to first calculate the most probable distribution of eigenvalues without the mutual information constraint (\[eq:mutual\_information\_condition\]), which will end up being the well-known Marčenko-Pastur distribution. This can be immediately extracted from the analysis in Section \[sec:integral\_equation\] by setting $k=0$. Solving for $a,b$ in (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) gives $$\begin{aligned} \label{eq:a_b_MP} a &=& \left(\sqrt{\beta}-1\right)^2 \\ \nonumber b &=& \left(\sqrt{\beta}+1\right)^2\end{aligned}$$ and (\[eq:p\_x\_beta>1\]) then takes the well-known form (\[eq:MP\_def\]). [^7] We may also evaluate the energy ${\mathcal{E}}_0$ by setting $k=0$ in (\[eq:energy\_functional\_calc1\]). Thus we get: $$\begin{aligned} \label{eq:energy_functional_MP} {\mathcal{E}}_0 &=& \frac{1}{2}\int_a^b x p(x) {\,d}x +\frac{1}{2}\left(a-(\beta-1)\log a\right) \\ \nonumber &-&\frac{\beta-1}{2} \int_a^b p(x) \log x {\,d}x - \int_a^b p(x) \log(x-a) {\,d}x\end{aligned}$$ and, after some algebra, we can rewrite the above expression in the closed form: $$\begin{aligned} \label{eq:energy_functional_MP_closed_form} {\mathcal{E}}_0 &=& \frac{\Delta^2}{32} + \frac{a}{2} -\log\Delta -\frac{\beta-1}{2}\log(a\Delta) \\ \nonumber &-& \frac{\Delta}{2}\left[G\left(0,\frac{a}{\Delta}\right) + \frac{\beta-1}{2} G\left(\frac{a}{\Delta},\frac{a}{\Delta}\right)\right]\end{aligned}$$ where $\Delta\equiv b-a$ and the function $G(x,y)$ is given by [@Chen1996_EigDistributionsLaguerre]: $$\begin{aligned} \label{eq:Q_fun} G(x,y) &=& \frac{1}{\pi}\int_0^1 \sqrt{t(1-t)} \frac{\log(t+x)}{t+y}{\,d}t \\ \nonumber &=& -2\sqrt{y(1+y)}\log\left[\frac{\sqrt{x(1+y)}+\sqrt{y(1+x)}}{\sqrt{1+y}+\sqrt{y}}\right] \\ \nonumber &+& \left(1+2y\right)\log\left[\frac{\sqrt{1+x}+\sqrt{x}}{2}\right] \\ \nonumber &-& \frac{1}{2}\left(\sqrt{1+x}-\sqrt{x}\right)^2\end{aligned}$$ When $\beta=1$, $a$, $b$ in (\[eq:a\_b\_MP\]) take the values $b=4$ and $a=0$, and hence (\[eq:energy\_functional\_MP\]) becomes ${\mathcal{E}}_0=3/2$. Evaluation of ${\mathcal{E}}_1(r)$: $\beta>1$ {#sec:beta>1} --------------------------------------------- We will now calculate ${\mathcal{E}}_1$ for the case $\beta>1$. To do so, we need to evaluate the constants $a$, $b$, $k$ as a function of $r$ and $\rho$ using (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) and (\[eq:mutual\_information\_condition\]). The values of these constants will determine the density of eigenvalues constrained on the subset with fixed total rate $R=Nr$ in the large $N$ limit. After inserting (\[eq:p\_x\_beta>1\]) into the last equation and integrating, (\[eq:mutual\_information\_condition\]) can be expressed explicitly as $$\begin{aligned} \label{eq:mut_info_beta>1} r& =& \int_a^b p(x) \log(1+\rho x) {\,d}x\\ \nonumber &=& \log\Delta\rho + \frac{\Delta k\rho}{2\sqrt{(1+\rho a)(1+\rho b)}} G\left(\frac{1+\rho a}{\Delta\rho},\frac{1+\rho a}{\Delta\rho}\right) \\ \nonumber &+& \frac{\Delta}{2}\left(1-\frac{k\rho}{\sqrt{(1+\rho a)(1+\rho b)}}\right) G\left(\frac{1+\rho a}{\Delta\rho},\frac{a}{\Delta}\right)\end{aligned}$$ where $G(x,y)$ is given in (\[eq:Q\_fun\]). Based on the arguments discussed in the previous section, it suffices to show that there exists a distribution $p(x)$ in the form of (\[eq:p\_x\_beta>1\]) satisfying the constraints (\[eq:norm\_condition\]), (\[eq:mutual\_information\_condition\]). This corresponds to finding values of $a$, $b$, and $k$ that satisfy (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) and (\[eq:mut\_info\_beta>1\]), while at the same time maintaining $p(x)\geq 0$ for all $x\in [a,b]$. If such a solution exists, then according to Theorem \[prop:uniquenes\] it will be unique. In Appendix \[app:uniqueness\_ab\] we show that equations (\[eq:p\_a\_=0\]) and (\[eq:norm\_integral\]) admit a unique solution for any $k$. We therefore only need to show that (\[eq:mut\_info\_beta>1\]) has a solution in $k$ for any $r>0$. It suffices to show that the function defined solely as a function of $k$ by the right-hand-side of (\[eq:mut\_info\_beta>1\]) (with $a$ and $b$ expressed in terms of $k$) takes all values in $(0,\infty)$. Hence by continuity it will attain the value $r$ for all positive rates $r>0$. We first see that as $k\rightarrow -\infty$ the solution of (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) is $a\approx (\sqrt{\beta}-1)^2/(\rho\|k\|)$ and $b\approx (\sqrt{\beta}+1)^2/(\rho \|k\|)$; then, inserting these solutions into (\[eq:mut\_info\_beta>1\]), we see that it may be written in leading order as $r\approx \beta/\|k\|$. On the other hand, for $k\rightarrow \infty$ (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) give $a\approx \sqrt{k+\beta-\rho^{-1}/2}-1$ and $b\approx \sqrt{k+\beta-\rho^{-1}/2}+1$, resulting to $r\approx \log k\rho$. This shows that the corresponding solution $p(x;r)$ is the unique minimizing distribution of ${\mathcal{E}}$ in ${\mathcal{X}}_r$. In Fig. \[fig:gen\_mp\_beta\_2\_comp\] we compare this distribution with the corresponding empirical probability distribution function obtained by numerical simulations. We see that the agreement is quite remarkable, indicating a quick convergence to the asymptotic distribution function of the eigenvalues constrained at the tails of the distribution of the mutual information. Furthermore, to get a feeling for the dependence of the eigenvalue distributions in terms of their parameters, in Fig. \[fig:gen\_mp\_beta\_4\] we plot a few representative examples. We may now calculate the value of ${\mathcal{E}}_1$. Inserting $p(x)$ from (\[eq:p\_x\_beta>1\]) into (\[eq:energy\_functional\_calc1\]) and integrating finally gives us: $$\begin{aligned} \label{eq:E1_beta>1} {\mathcal{E}}_1 &=& \frac{\Delta^2}{32} + \frac{a}{2} -\log\Delta -\frac{\beta-1}{2}\log(a\Delta) \\ \nonumber &+&\frac{k}{2}\left(r-\log(1+\rho a)-\frac{\left(\sqrt{1+\rho b}-\sqrt{1+\rho a}\right)^2}{4\rho\sqrt{(1+\rho a)(1+\rho b)}}\right) \\ \nonumber &-& \frac{\Delta k\rho}{2\sqrt{(1+\rho a)(1+\rho b)}} \\ \nonumber &\cdot& \left[ G\left(0,\frac{1+\rho a}{\Delta\rho}\right)+ \frac{\beta-1}{2}G\left(\frac{a}{\Delta},\frac{1+\rho a}{\Delta\rho}\right)\right] \\ \nonumber &-& \frac{\Delta}{2}\left(1-\frac{k\rho}{\sqrt{(1+\rho a)(1+\rho b)}}\right) \\ \nonumber &\cdot& \left[ G\left(0,\frac{a}{\Delta}\right) +\frac{\beta-1}{2} G\left(\frac{a}{\Delta},\frac{a}{\Delta}\right) \right]\end{aligned}$$ where $G(x,y)$ is given by (\[eq:Q\_fun\]). Plugging this together with ${\mathcal{E}}_0$ into (\[eq:P\_N(r)\_norm\_def\]) we obtain $P_N(r)$, up to the normalization constant. Evaluation of ${\mathcal{E}}_1(r)$: $\beta=1$ {#sec:beta=1} --------------------------------------------- The case $\beta=1$ deserves special attention. In this case the logarithmic repulsion from the $\delta$-function density of eigenvalues at the origin in (\[eq:energy\]) and (\[eq:func\_deriv\_E1\_result\]) is no longer present. As discussed in Section \[sec:methodology\].B, depending on the parameters $r$ and $\rho$ there are two distinct types of solutions, which we treat here separately. =1.0 ### Case $\beta=1$ and $r>r_c(\rho)$ {#sec:case r>r_c} We start by attempting to solve the problem as in the $\beta>1$ case, namely by looking for solutions of $0<a<b$ for the distribution’s support. It is straightforward to show that the conditions (\[eq:p\_a\_=0\]) and (\[eq:norm\_condition\]) yield the following values for $a$, $b$ when $\beta=1$: $$\begin{aligned} \label{eq:a_b_beta=1_r>r_c} a &=& \left(\sqrt{k+1}-1\right)^2 - \rho^{-1} \nonumber \\ b &=& \left(\sqrt{k+1}+1\right)^2 - \rho^{-1}.\end{aligned}$$ As a result, the probality density function $p$ becomes: $$\label{eq:p_x_beta=1_r>rc} p(x) = \frac{\rho}{2\pi}\frac{\sqrt{(b-x)(x-a)}}{1+\rho x}$$ The value of the parameter $k$ can be obtained in a unique way from the mutual information condition, which now reads: $$\label{eq:mut_info_beta=1_r>rc} r-\log\rho=(k+1)\log(k+1)-k\log k-1.$$ The monotonicity of the right-hand-side of this equation with respect to $k$ implies a unique $k(r)$ satisfying (\[eq:mut\_info\_beta=1\_r>rc\]) and hence a unique set of $a$,$b$ in (\[eq:a\_b\_beta=1\_r>r\_c\]), guaranteeing uniqueness of (\[eq:p\_x\_beta=1\_r>rc\]). In its turn, this can be used to evaluate the value of the outage exponent: $$\label{eq:E1_beta=1_r>rc} {{\cal E}}_1-{{\cal E}}_0= \frac{k-1}{2}(r-\log\rho) +k-\frac{1}{2}-\rho^{-1}-\frac{k\log k}{2}.$$ From (\[eq:a\_b\_beta=1\_r>r\_c\]) we can see that this solution can only be valid for $k\geq k_c(z)\equiv \rho^{-1}+2/\sqrt{\rho}$, or equivalently for $r>r_c(\rho)$ where $$\begin{aligned} \label{eq:r_c} r_c(\rho) &\equiv& \frac{1+2\sqrt{\rho}}{\rho}\log\left(1+\frac{\rho}{1+2\rho}\right) \\ \nonumber &+&2\log\left(1+\sqrt{\rho}\right)-1 >r_{\text{erg}}\end{aligned}$$ The reason is that for $k<k_c(\rho)$ (or $r<r_c(\rho)$) the value of $a$ becomes negative, which is unacceptable. ### Case $\beta=1$ and $r\leq r_c(\rho)$ {#sec:case r<r_c} In this case we can no longer treat $a$ as a free variable. Instead, because $p(x)=0$ for $x<0$, the charge density becomes confined at the boundary $x=0$. Thus, we need to look for solutions of (\[eq:func\_deriv\_E1\_result\]) with $a=0$, in which case the charge density has a square-root singularity at $x=0$ (instead of vanishing continuously). This is actually quite natural since we expect that, for $k=0$ (or, equivalently, for $r=r_{\text{erg}}$), the charge distribution should take the form of the $\beta=1$ Marčenko-Pastur density: $$\label{eq:p_x_beta=1_MP} p(x) = \frac{\sqrt{4-x}}{2\pi\sqrt{x}}.$$ Indeed, for general $b$, $k$, the distribution becomes: $$\label{eq:p_x_beta=1_r<r_c} p(x) = \frac{\sqrt{b-x}}{2\pi(1+\rho x)\sqrt{x}}\left(\rho x+1-\frac{k\rho }{\sqrt{1+\rho b}}\right),$$ and the normalization condition (\[eq:norm\_condition\]) implies $$\label{eq:norm_beta=1_r<r_c} k= \frac{\frac{b}{2}-2}{1-\frac{1}{\sqrt{1+\rho b}}}$$ It can easily be shown that the right-hand-side of (\[eq:norm\_beta=1\_r<r\_c\]) is increasing in $b$ and, hence, (\[eq:norm\_beta=1\_r<r\_c\]) has a unique solution in $b$ for all $k$. In the last case ($a=0$), the mutual information condition (\[eq:mutual\_information\_condition\]) can be integrated using (\[eq:p\_x\_beta=1\_r<r\_c\]) to give: $$\begin{aligned} \label{eq:mut_info_beta=1_r<r_c} r&=& 2(k+1)\log\frac{1+\sqrt{1+\rho b}}{2} \nonumber \\ &-&\frac{1}{4\rho}\left(\sqrt{1+\rho b}-1\right)^2-\frac{k}{2}\log\left(1+\rho b\right).\end{aligned}$$ We may use the same argument as in the previous subsection to show that this equation has at least one solution for any $0<r<r_c(\rho)$. Indeed when $k=k_c$, the right-hand-side above takes the value of $r_c$. In contrast, when $k\rightarrow -\infty$, (\[eq:norm\_beta=1\_r<r\_c\]) gives $b\approx 4/(\rho \|k\|)$, in which case the right-hand-side of (\[eq:mut\_info\_beta=1\_r<r\_c\]) becomes $\approx 1/\|k\|$. Thus all values between $(0, r_c(\rho))$ are taken when $k\in (-\infty,k_c(\rho))$. Hence by continuity it will attain the value $r\in (0,r_c)$. After solving for $b$ and $k$ as a function of $r$ and $\rho$, ${\mathcal{E}}_1$ can be calculated easily. Therefore, the exponent of the probability distribution $P_N(r)$ becomes: $$\begin{aligned} \label{eq:E1_beta=1_r<rc} {{\cal E}}_1-{{\cal E}}_0 &=& \frac{k}{2}\left(r-\frac{b}{4}\right)- \log\frac{b}{4} - k \log\frac{1+\sqrt{1+\rho b}}{2} \nonumber \\ &+& \frac{1}{32}\left(b - 4\right)\left(4\rho^{-1}+3b+12\right)\end{aligned}$$ We should point out that just as the solution (\[eq:p\_x\_beta=1\_r<r\_c\]) is not valid for $r>r_c(\rho)$, the solution (\[eq:p\_x\_beta=1\_r<r\_c\]), which we found to be valid for $r>r_c(\rho)$ is not valid for $r<r_c(\rho)$. To see this, it is straightforward to show that in this case the constant term in the last parenthesis in (\[eq:p\_x\_beta=1\_r<r\_c\]) (namely $1-k\rho/\sqrt{1+\rho b}$) is negative. As a result, (\[eq:p\_x\_beta=1\_r<r\_c\]) cannot be valid for $k<k_c(\rho)$ because the charge density becomes negative at some point $x>0$. As a result the solutions we found above are unique in their domains of validity. Interestingly there is a weak, third order discontinuity at the transition $r=r_c(\rho)$, in the sense that the first two derivatives of ${\mathcal{E}}_1(r)$ with respect to $r$ evaluated at $r=r_c$ are continuous, while the third is discontinuous. This is analogous to the phase transition observed in [@Vivo2008_DistributionsConductanceShotNoise]. Evaluation of the Outage Probability $P_{out}(r)$ {#sec:OutageProbability} ------------------------------------------------- In this section we will calculate the outage probability $P_{out}(r)=\operatorname{\mathbb{P}}(I_N<Nr)$ from ${\mathcal{E}}_1(r)$. To do this we need to integrate $\exp\left[-N^2({{\cal E}}_1(r)-{{\cal E}}_0)\right]$ over $r$. Generally it is impossible to evaluate this integral in closed form. Nevertheless, due to the presence of the factor $N$ in the exponent, $P_N(r)$ falls rapidly away from its peak and thus we may use Watson’s lemma[@Bender_Orszag_book] (a special case of Varadhan’s lemma), to evaluate the asymptotic value of the integral. First, we will calculate the normalization factor of the distribution. As we shall see in Section \[sec:limiting\_cases\] for $r$ close to $r_{\text{erg}}$, ${{\cal E}}_1(r)-{{\cal E}}_0\sim (r-r_{\text{erg}})^2/v_{\text{erg}}$, where $v_{\text{erg}}$ is the ergodic variance (\[eq:var\_erg\_all\_alpha\]) of the mutual information distribution. Therefore, we have $$\label{eq:Watson_lemma_int_P(R)} \int_0^\infty e^{-N^2({{\cal E}}_1(r)-{{\cal E}}_0)} {\,d}r \approx \int_0^\infty e^{-\frac{N^2(r-r_{\text{erg}})^2}{2v_{\text{erg}}}} {\,d}r \approx \frac{\sqrt{2\pi v_{\text{erg}}}}{N}$$ which then gives $$\label{eq:P_R_result} P_N(r) \approx \frac{N}{\sqrt{2\pi v_{\text{erg}}}} e^{-N^2\left({{\cal E}}_1(r)-{{\cal E}}_0\right)}$$ and fixes the normalization constant in (\[eq:P\_N(r)\_norm\_def\]). To calculate the outage probability ${P_{\text{out}}}(r)=\operatorname{\mathbb{P}}(I_N<Nr)$ to leading order in $N$, we first note that for $r<r_{\text{erg}}$ ($r>r_{\text{erg}}$), ${{\cal E}}_1(r)$ is a decreasing (increasing) function of $r$. Therefore, to leading order, the behavior will be dominated by the value of the exponent at $r$. Using Watson’s lemma once again we obtain the following expression for the outage probability: $$\label{eq:outage_prob} P_{out}(r) \approx \frac{e^{-N^2\left[{{\cal E}}_1(r)-{{\cal E}}_0-\frac{{{\cal E}}_1'(r)^2}{2{{\cal E}}_1''(r)}\right]}Q\left(\frac{N\left\|{{\cal E}}_1'(r)\right\|}{\sqrt{{{\cal E}}_1''(r)}} \right) }{\sqrt{{{\cal E}}_1''(r) v_{erg}}}$$ when $r<r_{\text{erg}}$ and $$\label{eq:outage_prob2} P_{out}(r) \approx 1 - \frac{e^{-N^2\left[{{\cal E}}_1(r)-{{\cal E}}_0-\frac{{{\cal E}}_1'(r)^2}{2{{\cal E}}_1''(r)}\right]}Q\left[\frac{N\left\|{{\cal E}}_1'(r)\right\|}{\sqrt{{{\cal E}}_1''(r)}} \right] }{\sqrt{{{\cal E}}_1''(r) v_{erg}}}$$ when $r>r_{\text{erg}}$. In the above, ${{\cal E}}_1'(r)$ and ${{\cal E}}_1''(r)$ are the first and second derivatives of ${{\cal E}}_1(r)$ with respect to $r$ and $Q(x)$ is given by $$\label{eq:Qx_def} Q(x) = \int_x^\infty \frac{dx}{\sqrt{2\pi}} e^{-\frac{t^2}{2}}$$ =1.0 Analysis of Limiting Cases {#sec:limiting_cases} ========================== We will now analyze the results of the previous section in specific limiting cases of the parameter space $(\rho, r, \beta)$. We will thereby be able to connect with already existing results in specific regions, and also to describe the behavior of the probability density of $P_N(r)$ in other regions, which hitherto have defied asymptotic analysis. Gaussian Region $r\approx r_{\text{erg}}(\rho)$ {#sec:GaussianApprox} ----------------------------------------------- The most relevant limiting case is the Gaussian regime: after all, the Gaussian approximation, as well as the present approach assume that the number of antennas $N$ is large. The difference is that our approach does not focus only in the region of $N\|r-r_{\text{erg}}\|={\mathcal{O}}(1)$, where the Gaussian approximation should be valid. To reach that limit, we need to analyze the small $k$ region of (\[eq:mut\_info\_beta>1\]), (\[eq:mut\_info\_beta=1\_r<r\_c\]) since, in the limit $k=0$, both equations reduce to $r=r_{\text{erg}}(\rho)$, where the normalized ergodic mutual information $r_{\text{erg}}$ is well known to be [@Moustakas2003_MIMO1; @Verdu1999_MIMO1; @Rapajic2000_InfoCapacityOfARandomSignatureMIMOChannel]: $$\begin{aligned} \label{eq:r_erg_all_alpha} r_{\text{erg}} = \log u + \beta \log\left[1+\frac{\rho}{u}\right] - \left(1-u^{-1}\right)\end{aligned}$$ with: $$\label{eq:u_def} u = \frac{1}{2}\left(1+\rho(\beta-1)+\sqrt{(1+\rho(\beta-1))^2+4\rho}\right)$$ By implicitly differentiating $a$, $b$, $k$ with respect to $r$ through the equations that define them, and expressing their values and the values of their derivatives at $r=r_{erg}$ we can obtain the following expansion $$\label{eq:E1-E0_small_k2} {\mathcal{E}}_1-{\mathcal{E}}_0 = \frac{\left(r-r_{\text{erg}}\right)^2}{2v_{\text{erg}}} + \frac{s_{erg}}{6}\left(r-r_{\text{erg}}\right)^3 + {\mathcal{O}}\left(\left(r-r_{\text{erg}}\right)^4\right)$$ where $$\begin{aligned} \label{eq:var_erg_all_alpha} v_{\text{erg}} = -\log\left[1-\frac{(1-u)^2}{\beta u^2}\right]\end{aligned}$$ coincides with the variance of the mutual information distribution as analyzed in [@Moustakas2003_MIMO1; @Hachem2006_GaussianCapacityKroneckerProduct], and $s_{erg}$ is the third [total]{} derivative of ${\mathcal{E}}_1$ with respect to $r$ and evaluated at $r=r_{erg}(\rho)$. Without the cubic term, (\[eq:E1-E0\_small\_k2\]) is exactly the Gaussian limit of the mutual information distribution discussed in various papers in the past. This Gaussian limit is valid as long as the cubic (as well as all higher order) terms in the exponent of the probability are smaller than unity. Since this condition depends on $s_{erg}$, it is worth looking its behavior with $\rho$. In Fig. \[fig:E3\_vs\_rho\] we plot $s_{erg}$ as a function of $\rho$. We see that it has a well-defined limit for large $\rho$. Specifically, it has the following asymptotic form $$\label{eq:E3_asymptotics} s_{erg}(\rho)\approx \left\{ \begin{array}{cc} -\frac{2}{\log(\rho)^3} & \beta=1 \\ -\frac{1}{\beta(\beta-1)\log\left(1-\beta^{-1}\right)^3} & \beta>1 \\ \end{array} \right.$$ Also, for small $\rho\ll 1$ we can show that $s_{erg}\approx -c_\beta/\rho^3$, where $c_{\beta}>0$ is a constant that depends on $\beta$. Thus the condition for validity of the Gaussian approximation is $$\label{eq:Gaussian_validity} \left\|r-r_{erg}(\rho)\right\| \ll \sqrt[3]{\frac{6}{\left\|s_{erg}\right\|}} N^{-2/3}$$ We therefore see that the Gaussian approximation should not be valid for significant deviations from $r_{erg}$, e.g. $r=r_{erg}/2$. In contrast our large deviations (LD) approximation continues to be valid in that rate region as well. Large $\rho$ Approximation: $r<r_{\text{erg}}$ {#sec:DMT_region_r<rerg} ---------------------------------------------- Next we analyze the behavior of the probability distribution of $r$ in the large $\rho$ limit, while keeping the ratio $r/\log\rho$ finite and less than $1$.[^8] Since in the large $\rho$ limit $r_{\text{erg}}\sim \log\rho$, the region $q\leq 1$ with $\rho\gg 1$ corresponds to $k<0$, equations (\[eq:norm\_integral\]), (\[eq:p\_a\_=0\]) will admit the following solutions for $a,b$: $$\begin{aligned} \label{eq:a_dmt_regime} a &\sim& \frac{(\beta-1)^2}{4\rho(1-q)(\beta-q)} \\ \label{eq:b_dmt_regime} b & \sim& 4q\end{aligned}$$ where $q=r/\log\rho$ and we are assuming that $0<q<1$. Now, note that the lower end of the spectrum has become of order $O(1/\rho)$, while the upper limit is still finite, just as expected. It is also interesting to calculate the proportion of eigenvalues that are in the neighborhood of $x=1/\rho$ when $\rho\rightarrow \infty$. Indeed, by integrating the probability distribution $p(x)$ (\[eq:p\_x\_beta>1\]) from $a={\mathcal{O}}(\rho^{-1})$ (\[eq:a\_dmt\_regime\]) to $L\rho^{-1}$ for some (arbitrarily) large $L$ we get $$\begin{aligned} \label{eq:probability_x=O(z)} \lim_{L\rightarrow \infty} \lim_{\rho\rightarrow \infty} \operatorname{\mathbb{P}}(\rho x<L) = 1-q\end{aligned}$$ Thus, the proportion of “small” eigenvalues is simply $1-q$, in agreement with [@Zheng2003_DiversityMultiplexing]. Plugging (\[eq:a\_dmt\_regime\]), (\[eq:b\_dmt\_regime\]) into the equation for ${\mathcal{E}}_1$ then gives the expected result for the exponent: $$\label{eq:E1-E0_dmt} {\mathcal{E}}_1-{\mathcal{E}}_0 \sim \log\rho\left[(1-q)(\beta-q)\right]$$ which is exactly the diversity exponent (divided by $N^2$) of [@Zheng2003_DiversityMultiplexing]. From the above, we see the difference between the two asymptotic analyses discussed above. In the previous section, the eigenvalue distribution did not deviate significantly from the most probable Marčenko-Pastur distribution, since $k$ was assumed to be small. In contrast, here, $k$ is finite, and in particular equal to $k=2q-1-\beta$, In addition, a significant portion of the eigenvalues in this subset of fixed $r=q\log \rho$ is now to become very small, of order $1/\rho$. In the above discussion, we see that generally the exponent ${\mathcal{E}}_1(r)$ is not only continuous, but also differentiable in $r$. This is in disagreement to the prediction by [@Zheng2003_DiversityMultiplexing; @Azarian2007_finite_rate_DMT] that when $\rho\rightarrow \infty$, the outage has a piecewise linear behavior. The length of these segments is $\Delta R\approx\log\rho$, or $\Delta r\approx\log\rho/N$. Thus for these segments to be pronounced we need $$\label{eq:LD_validity} N \ll \log\rho$$ for large $\rho$. This provides a limit on the formal limitations of our large deviations (LD) approach. In particular, clearly the antenna number $N$ has to be large, as in the Gaussian case. But, in contrast to the Gaussian approximation, there is no constraint here that the deviation of the rate from the ergodic rate has to be small, as in (\[eq:Gaussian\_validity\]). Thus the [scale ]{} of $N$ at which the method should break down is given by $\log\rho$ for large $\rho$, rather than $\rho$ itself. This is corroborated in the numerical results in the next section. Surprisingly, however, the analysis in this section shows that the [form]{} of the DMT exponent (\[eq:DMT\_def\]) is correctly predicted within the LD approach in (\[eq:E1-E0\_dmt\]). Large $\rho$ Approximation: $r>r_{\text{erg}}$ {#sec:DMT_region_r>rerg} ---------------------------------------------- The regime of large $\rho$ and fixed $q=r/\log\rho\leq 1$ is relevant in the analysis of the link-level outage probability. However, the opposite regime of $q>1$ is also of interest in a cellular setting with many multi-antenna users receiving data in a TDMA fashion from a single multi-antenna base-station.[^9] In this context, to analyze the system level throughput, it is the higher end of the probability distribution of the link-level mutual information that is important [@Hochwald2002_MultiAntennaChannelHardening; @Bender2000_HDR_ComMagReview]. Therefore, it is worthwhile to calculate the probability distribution of $r$ for large $\rho$ with $q>1$. Interestingly enough, the behavior here is quite different from the $q<1$ case. Here $k\sim \rho^{q-1}$ and $$\begin{aligned} \label{eq:a_q>1_regime} a &\sim& \left(\sqrt{k+\beta}-1\right)^2 \\ \label{eq:b_q>1_regime} b &\sim& \left(\sqrt{k+\beta}+1\right)^2\end{aligned}$$ resulting to $$\label{eq:E1-E0_q>1} {\mathcal{E}}_1-{\mathcal{E}}_0 \sim \rho^{q-1} = \frac{e^r}{\rho}$$ independent of $\beta$. The resulting probability distribution of $r$ is $$\label{eq:P(r)_r>>r_erg} P(r) \sim e^{-N^2 e^r/\rho}$$ We see that when $N$ is not too small, the probability of finding $I_N$ significantly larger than its ergodic value is extremely small (in fact, doubly exponentially small in $r$). This is the manifestation of the fact that scheduling the best user in a MAC-layer in a multi-antenna setting does not seem to provide any clear advantage. Interestingly, in [@Hochwald2002_MultiAntennaChannelHardening] the authors have the same conclusion, even though they assume a Gaussian distribution for $I_N$ even for its tails. Here we see that the distribution of $I_N$ goes to zero for $r>r_{erg}$ in a rate even faster than Gaussian, thereby making the above conclusion, to which they also reached even stronger. This result has the following intuitive explanation. For large $\rho$ and $r>r_{erg}$ all eigenvalues of the matrix $\bH^\dagger\bH$ will be large and the only constraint imposed upon them is (\[eq:mutual\_information\_condition\]). Thus, we may say that all of them are constrained by the condition $r\sim \log(1+\rho \lambda_i)\sim\log\rho\lambda_i$ i.e. $\lambda_i\sim e^r/\rho$. In this limit, the exponent is roughly $N$ times the sum of the eigenvalues. Limit $r\rightarrow 0$ {#sec:r->0} ---------------------- The final regime that is interesting to analyze is when $r\rightarrow 0$, independently of $\rho$. In this regime the solution of (\[eq:mut\_info\_beta>1\]) (\[eq:mut\_info\_beta=1\_r<r\_c\]) for small $r$ is $r\sim \beta/\|k\|$ for $k\rightarrow-\infty$ and the corresponding values of $a,b$ are: $$\begin{aligned} \label{eq:a_r->0_regime} a &\sim& \frac{r}{\rho \beta}\left(\sqrt{\beta}-1\right)^2 \\ \label{eq:b_r->0_regime} b &\sim& \frac{r}{\rho \beta}\left(\sqrt{\beta}+1\right)^2\end{aligned}$$ resulting in: $$\label{eq:E1-E0_r->0} {\mathcal{E}}_1-{\mathcal{E}}_0 \sim -\beta \log \left[\frac{e r}{\beta \rho}\right].$$ where $e$ is the Euler number. This means that the probability distribution $P_N(r)$ has a tail of the form $$\label{eq:P(r)_r->0} P(r) \sim \left(\frac{r e}{\rho \beta}\right)^{MN}$$ The above behavior of $P_N(r)$ for small $r$ is easy to understand: for $r$ to be small, we need all matrix elements of the matrix $\bH$ to be small. In fact, since $\bH$ appears in a quadratic way in the mutual information equation (\[eq:logdet\_def\]) we need all $MN$ elements of $\bH$ to be less than $O(\sqrt{r/\rho})$. However, there are $2MN$ real degrees of freedom in the $M\times N$ complex matrix $\bH$. Hence the allowed volume of space scales as $(r/\rho)^{MN}$ as above. It should also be noted that the behavior $P(r)\sim \rho^{-MN}$ of the mutual information cumulative distribution function is precisely what is known as the “full diversity” of error probability, i.e., the SNR exponent of error probability for fixed but very small rate $R$ while SNR $\rho$ increases is $\rho^{-MN}$, which corresponds to the left extreme point of the Zheng-Tse exponent[@Zheng2003_DiversityMultiplexing]. Numerical Simulations {#sec:numerical simulations} ===================== To test the applicability of this approach, we have performed a series of numerical simulations and have compared our large deviations (LD) approach to other popular approximations. We start with the case of small rates $r$. In this limit the Gaussian approximation is guaranteed to give misleading results. For example, the Gaussian approximation predicts a finite outage probability at zero rate, while this is clearly wrong. The LD approximation, on the other hand, correctly predicts that the outage probability goes to zero at small $r$, as seen in (\[eq:P(r)\_r->0\]). In Figs. \[fig:CDF\_N2M2\] and \[fig:CDF\_N3M3\] we plot the outage probability of the LD approach with the Gaussian and Monte Carlo simulations for low rates, small $\rho$ and small square ($2\times 2$ and $3\times 3$) antenna arrays. The comparison shows that while the Gaussian curves miss the correct outage, the LD curves remain close to the simulated ones, even for the $2\times 2$ MIMO system. It is worthwhile to mention that the Gaussian outage probability is consistently greater than the correct (simulated) one. The reason for this can be traced to the fact that for all $\beta=1$ and all values of $\rho$, the third derivative of the exponent ${\cal E}_1(r)-{\cal E}_0$ with respect to $r$ evaluated at $r_{erg}$, i.e. $s_{erg}(\rho)$ in (\[eq:E1-E0\_small\_k2\]) is negative. Disturbing away from the peaks of the distribution we have $$\label{eq:P_out_gaussian_approx} \log P_{out,Gaussian}(r)\approx -\frac{N^2(r-r_{erg})^2}{2v_{erg}}$$ while $$\label{eq:P_out_gaussian+3rd_approx} \log P_{out}(r)\approx -\frac{N^2(r-r_{erg})^2}{2v_{erg}} - \frac{s_{erg}N^2(r-r_{erg})^3}{6}$$ We may thus conclude that when $r<r_{erg}$ and $s_{erg}<0$ we should have $P_{out,Gaussian}>P_{out}$. From Fig. \[fig:E3\_vs\_rho\] we see that for increasing $\rho$, $s_{erg}$ decreases in absolute size, which correctly predicts that the discrepancy between the Gaussian and the Monte-Carlo curves (and LD) decreases for larger $\rho$. We have also analyzed the probability distribution for rates greater than the ergodic rate $r>r_{erg}$. Even though this region is not relevant for the outage probability evaluation, it is important in the analysis of the multiuser capacity for MIMO links in a multi-user setting with a greedy scheduler, such as a maximum rate scheduler.[@Hochwald2002_MultiAntennaChannelHardening] In such a case, the multiuser diversity gain comes from the opportunity the scheduler has to schedule transmission to users when their fading rates are greater than their mean. Thus it is important to understand the tails of the distribution in this region. In Fig. \[fig:CCDF\_N3M3\] we obtained the complementary CDF (CCDF) of the mutual information, i.e. $1-P_{out}(r)$, for a $3\times 3$ setting. Here the probability of finding users with high rates falls faster than the Gaussian, especially in Fig. \[fig:CCDF\_N3M3\]b for large $\rho$. We also find that the LD approximation follows the Monte Carlo simulations more accurately than the Gaussian curve, especially for lower outages. In this situation it is worth pointing out that the argument mentioned above regarding the sign of $s_{erg}$ would make $1-P_{out}(r)$ smaller in the Gaussian approximation compared to the correct result. We see that this only occurs for rates relatively close to the peak. In contrast, for rates greater than the critical rate $r_c(\rho)$ the behavior of the numerical and the LD outage probability changes markedly and they both become substantially smaller than the Gaussian curve. This is not surprising in view of the phase transition occurring at $r=r_c(\rho)$ as discussed in Section \[sec:case r<r\_c\]. We next analyzed the outage probability as a function of the SNR. The outage has been analyzed in the large SNR limit for finite rates in [@Azarian2007_finite_rate_DMT], where they have dubbed this analysis as throughput reliability tradeoff (TRT). This model provides a piecewise linear function of the outage probability, which for completeness is provided below: $$\begin{aligned} \label{eq:TRT_model} \log_2 P_{out} &\approx& c(k) R - g(k) \log_2\rho \\ \nonumber c(k) &=& M+N-2k-1 \\ \nonumber g(k) &=& MN -k(k+1)\end{aligned}$$ when $\rho$ is large and $k\log_2\rho <R<(k+1)\log_2\rho$. This piecewise linear behavior however is observable only at extremely high rates and SNRs, which may not necessarily be relevant for realistic MIMO systems. We analyzed the case of $3\times 6$, $3\times 3$ and $6\times 6$ arrays in Figs. \[fig:Pout\_N3M6\] and \[fig:Pout\_N3M3\]. In all three we have found that the LD approximation agrees with simulations over a wide region of rates $r$ and SNR $\rho$. Characteristic is Fig. \[fig:Pout\_N3M6\]b, where the TRT curve is accurate in large SNR, the Gaussian is accurate in low SNR, but the LD curve is consistently closer to the correct outage. For the $N=M=3$ case and extremely high SNRs and rates the piecewise linear behavior predicted by TRT starts becoming visible. Nevertheless, even in those high rates the TRT curve also fails to give quantitatively correct outage estimates and the LD curve is still closer to the correct outage. It is sensible to point out that here the Gaussian outage probability is consistently less than the simulated and the LD values. In this case the argument made above for $s_{erg}$ is reversed. As can be seen in Fig. \[fig:E3\_vs\_rho\] for $\beta=2$ and large $\rho$ the sign of $s_{erg}$ is opposite, i.e. we have $s_{erg}>0$ and hence indeed we should have $P_{out,Gaussian}<P_{out}$. =1.0 In Fig. \[fig:E1\_N5\_beta2\_SNR\_100\], we plot the logarithm of the appropriately normalized probability density function (PDF) $P_N(r)$ as a function of the throughput $r$ and we compare the result with the two other asymptotic forms, namely the Gaussian approximation of the mutual information [@Moustakas2003_MIMO1] and the large-$\rho$ asymptotic result given by (\[eq:DMT\_def\]) [@Zheng2003_DiversityMultiplexing]. We see that our result performs much better at low outage, even at moderately large $\rho =20dB$. As discussed in the Introduction, the LD method is the correct generalization of the Gaussian approximation to capture the tails of the distribution of the mutual information. As a result, it is expected to give increasingly accurate results as the antenna number $N$ increases. In the above comparisons we have compared the LD method with numerical simulations focusing on its tails (low outage $P_{out}$ or low values of $1-P_{out}$) for small antenna numbers. We have found that the LD approximation behaves well even at these values of $N$. The discrepancy between the LD approximation and Monte Carlo simulations becomes smaller for larger $N$ as seen in Fig. \[fig:E1\_N5\_beta2\_SNR\_100\]. In Appendix \[app:1/Ncorrections\] we provide an improved estimate on the probability distribution close its center. This estimate is a result of the inclusion of the $O(1/N)$ corrections to the distribution derived in [@Moustakas2003_MIMO1]. Fig. \[fig:PDF\_N2M4\] shows the normalized probability distribution function of the Gaussian approximation as well as the LD approximation with and without the $O(1/N)$ corrections. We see that the improved estimate behaves extremely well when the antenna numbers are quite small, in which cases the leading approximation (without the $O(1/N)$ correction), has some small discrepancies. (This should be contrasted with Fig. \[fig:E1\_N5\_beta2\_SNR\_100\], where $N=5$ and the $O(1/N)$ correction is no longer necessary to provide close agreement.) Conclusion {#sec:conclusion} ========== In this paper we have used a large deviation approach, first introduced in the context of statistical mechanics [@Dyson1962_DysonGas; @Vivo2007_LargeDeviationsWishart], to calculate the probability distribution of the mutual information of MIMO channels in the limit of large antenna numbers. In contrast to previous approaches that focused only close to the mean of the distribution,[@Moustakas2003_MIMO1; @Hochwald2002_MultiAntennaChannelHardening; @Hachem2006_GaussianCapacityKroneckerProduct], we also calculate the probability for [rare events]{} in the tails of the distribution, corresponding to instances where the observed mutual information differs by ${\mathcal{O}}(N)$ from the most probable value of the asymptotic distribution (where the Gaussian approximation for the mutual information is invalid). We find that the distribution in those tails is markedly different from what happens near the mean and our resulting probability distribution interpolates seamlessly between the Gaussian approximation for rates close to the ergodic mutual information and the results of [@Zheng2003_DiversityMultiplexing] for large signal to noise ratios (where the outage probability is given asymptotically by (\[eq:DMT\_def\])). Our method thus provides an analytic tool to calculate outage probabilities at any point in the $(R, \rho, N)$ parameter space, as long as $N$ is large enough. We performed numerical simulations that showed the robustness of our approximation over a wide range of parameters. Additionally, this approach also provides the probability distribution of eigenvalues constrained in the subset where the mutual information is fixed to $R$ for a given signal to noise ratio $\rho$. Interestingly, this eigenvalue density is of the form of the Marčenko-Pastur distribution with square-root singularities. Since the outage probability is an increasing function of the rate $r$ for fixed $\rho$, we may use our approach to evaluate the transmission rate $R$ for a required outage $P_{out}$ and $\rho$. Thus, if the channel is known at the transmitter, we can optimize the transmitted rate by waterfilling on the known eigenvalue density that corresponds to the required outage probability [@Ordonez2009_OrderedEigsMIMO]. This generalization is left for a future work. Finally, it is worth pointing out that, to our knowledge, this is the first time this methodology has been applied to information theory and communications, and it is our belief that it may find other applications in this field. We can corroborate this belief by pointing out that this Coulomb gas methodology can be generalized to other channel distributions, as long as the resulting distribution can be written as a product of functions of the eigenvalues of $\bH^\dagger\bH$. Another related generalization is, for example, to include the correlations of the channel, a problem which is considerably more difficult compared to the present one. Some preliminary mathematical tools have already been developed in [@Matytsin1994_LargeNLimitIZIntegral], and we will expand on this in the future. Properties of tame probability measures {#app:tameness} ======================================= This appendix is largely devoted to the study of the energy functional ${\mathcal{E}}$: $$\begin{aligned} \tag{\ref{eq:energy}} {\mathcal{E}}[p] &= \int \!x p(x) {\,d}x- (\beta-1)\int\! p(x) \log x {\,d}x \\ \nonumber &-\iint p(x) p(y) \!\log\|x-y\| {\,d}x {\,d}y\end{aligned}$$ where $p\in{\Omega}$ is a tame density. As evidenced by definition \[dfn:tame\] where the concept of tameness was introduced, an extremely important part in our analysis will be played by the so-called $L^{r}$ norm $\\|\cdot\\|_{r}$ defined by: $$\\|f\\|_{r} \equiv \left(\int \|f(x)\|^{r} {\,d}x\right)^{1/r}.$$ If a function $f$ has finite $L^{r}$ norm it is called $L^{r}$-integrable and the space of such functions constitutes a complete vector space (also denoted by $L^{r}$). The completeness of this space follows from [Hölder’s inequality]{} which we state without proof and which will be of great use to us [@Fo99]: $$\\|fg\\|_{1} \leq\\|f\\|_{r}\\|g\\|_{s}$$ whenever the exponents $r,s>1$ are [conjugate]{}, that is: $r^{-1} + s^{-1} = 1$. We will also make heavy use of the convolution $fg$ between two functions $f$ and $g$: $$(fg)(x) =\int f(x-y) g(y) {\,d}y.$$ If $f\in L^{1}$ and $g\in L^{r}$, [Young’s inequality]{} (pp. 240–241 in [@Fo99]) states that their convolution will be finite for almost every $x$ and also that: $$\\|fg\\|_{r} \leq\\|f\\|_{1}\\|g\\|_{r}.$$ We may now proceed with the proof of lemma \[lem:tame\] regarding the domain of ${\mathcal{E}}$ and its continuity properties: To show that ${\mathcal{E}}$ is finite for all tame functions $p\in{\Omega}$, we will study ${\mathcal{E}}[p]$ term by term. To that end, let $p:{\mathbb{R}}_{+}\to{\mathbb{R}}$ be tame for some exponent ${\varepsilon}>0$; that is, assume that $\int \|p\|^{1+{\varepsilon}} <\infty$ and that $\int x p(x) {\,d}x<\infty$. We then have: - The first term of ${\mathcal{E}}[p]$ is finite by definition. - The second term in (\[eq:energy\]) can be written as: $$\begin{gathered} \left\|\int p(x)\log x {\,d}x\right\| \leq \int \|p(x) \log x\|{\,d}x\notag\\ = \int_0^1 \|p(x) \log x\| {\,d}x + \int_1^\infty \|p(x) \log x\| {\,d}x.\end{gathered}$$ Since $\log x < x$ for $x>1$, the second integral will be bounded from above by $\int x \|p(x)\| {\,d}x<\infty$. As for the first integral, set $r=1+{\varepsilon}$ and $s=1+\frac{1}{{\varepsilon}}$ so that $r^{-1}+s^{-1}=1$. Now, if $\chi_{[0,1]}$ is the indicator function of $[0,1]$, note that $\int \|\chi_{[0,1]} \log x\|^{s} {\,d}x = \int_{0}^{1}\|\log x\|^{s} {\,d}x<\infty$ for all $s>-1$. As a result, Hölder’s inequality yields: $$\begin{aligned} \int_{0}^{1} \|p(x) \log x\| {\,d}x &= \\|p \cdot \chi_{[0,1]}\log\\|_{1}\notag\\ &\leq \\|p\\|_{1+{\varepsilon}} \cdot \\|\chi_{[0,1]}\log\\|_{1+1/{\varepsilon}}<\infty\end{aligned}$$ on account of $p$ being $L^{1+{\varepsilon}}$-integrable. - For the last term of ${\mathcal{E}}$, let $D_{+} = \{(x,y)\in{\mathbb{R}}^{2}: y>x\}$ and note that: $$\begin{gathered} \left\|\iint p(x) p(y) \log\|x-y\| {\,d}y{\,d}x\right\|\\ \shoveleft{\leq 2\iint_{D_{+}} \|p(x) p(y) \log\|x-y\|\| {\,d}y {\,d}x}\\ = 2\int_{0}^{\infty} \|p(x)\|\int_{x}^{\infty} \|p(y) \cdot \log(y-x)\| {\,d}y {\,d}x.\end{gathered}$$ Now, the innermost integral can be written in the form: $$\begin{gathered} \label{eq:estimate} \int_{x}^{\infty} \|p(y)\|\cdot \|\log(y-x)\| {\,d}y\\ \shoveleft =\int_{0}^{\infty} \|p(x+w)\cdot\log w\| {\,d}w\\ \shoveleft =\int_{0}^{\infty} \|p(x+w) K(w)\| {\,d}w + \int_{1}^{\infty} \|p(x+w)\log w\| {\,d}w\\ \leq \int_{0}^{\infty} \|p(y)\| K(y-x) {\,d}y \\ + \int_{0}^{\infty} \|p(1+x+w)\log (1+w)\| {\,d}w.\end{gathered}$$ where $K(w)$ is the kernel: $$K(w) = \begin{cases}\log\|w\|, &0<w\leq 1\\ 0, &\text{otherwise.}\end{cases}$$ As above, $K$ will be $L^{s}$-integrable for all $s>-1$ and, in particular, for $s=1+\frac{1}{{\varepsilon}}$. Therefore, we will have: $$\begin{gathered} \int_{0}^{\infty} \|p(x)\| \int_{0}^{\infty} \|p(y)\| K(x-y) {\,d}y {\,d}x \\ = \big\\|\|p\|\cdot \big(\|p\|\|K\|\big)\big\\|_{1}\\ \leq\\|p\\|_{1+{\varepsilon}}\cdot \\|pK\\|_{1+1/{\varepsilon}}\\ \leq \\|p\\|_{1+{\varepsilon}} \cdot \\|p\\|_{1} \cdot \\|K\\|_{1+1/{\varepsilon}}<\infty\end{gathered}$$ where the penultimate step is an application of Hölder’s estimate and the last one follows from Young’s inequality. Finally, the second integral of (\[eq:estimate\]) can be estimated by: $$\begin{gathered} \label{eq:estimate2} \int_{0}^{\infty} \|p(1+x+w) \log(1+w)\| {\,d}w\\ \leq \int_{0}^{\infty} \|p(1+x+w)\| w {\,d}w\\ \leq C x \int_{0}^{\infty} w \|p(w)\| {\,d}w\end{gathered}$$ for some sufficiently large $C>0$. Then, since $p$ is tame (i.e. $\int w \|p(w)\| {\,d}w <\infty$), we may integrate (\[eq:estimate2\]) over $x$ to finally obtain that ${\mathcal{E}}[p]<\infty$. This completes the proof that ${\mathcal{E}}[p]$ is finite for all tame functions $p\in{\Omega}$. To show that ${\mathcal{E}}$ is continuous on all subspaces of $L^{1+{\varepsilon}}$-integrable functions with finite absolute mean, it simply suffices to note that all our estimates of ${\mathcal{E}}[p]$ are bounded by the $L^{1+{\varepsilon}}$ norm of $p$. If a function is in $L^{r}$ for some $r>1$ and has finite mean, it will necessarily be in $L^{1}$ as well; in this way, tame measures form a (dense) subspace ${\Omega}$ of $L^{1}({\mathbb{R}}_{+})$ that is similar to the union $\bigcup_{{\varepsilon}>0}L^{1+{\varepsilon}}$. We will now prove Lemma \[lem:convex\] showing that ${\mathcal{E}}$ is not only continuous but also convex over the (convex) domain ${\mathcal{X}}$ of tame [probability]{} measures. Let $p,q\in{\mathcal{X}}$ be two tame probability measures and introduce the bilinear pairing: $$\langle p,q\rangle = -\int\int p(x) q(y) \log\|x-y\| {\,d}x {\,d}y$$ which is actually well-defined on the whole space ${\Omega}$ (as can be seen by the proof of lemma \[lem:tame\]). Since the first two terms of ${\mathcal{E}}$ are linear (and hence convex), it will suffice to show that: $$\label{eq:convexpair} \big\langle (1-t)p + tq, (1-t)p +tq\big\rangle < (1-t) \langle p, p\rangle + t \langle q, q \rangle$$ for all $t\in(0,1)$. Indeed, if we let $\phi = p-q\in{\Omega}$, equation (\[eq:convexpair\]) reduces to showing that the pairing $\langle\cdot,\cdot\rangle$ is an inner product on the subspace of densities with zero total charge, i.e. that: $$\langle \phi, \phi \rangle >0$$ for any nonzero tame $\phi\in{\Omega}$ with $\int \phi(x){\,d}x=\int \big(p(x)-q(x)\big) {\,d}x=0$. From the point of view of electrostatics, this is plain to see: after all $\langle \phi,\phi\rangle$ is just the self-energy of the charge density $\phi$. More specifically, let us define $D_{+}=\{(x,y): x<y\}$ as in the proof of lemma \[lem:tame\]. Then we will have: $$\begin{aligned} \langle \phi,\phi\rangle &=& -2\int_{D_{+}}\!\! \phi(x)\phi(y) \log\|x-y\| {\,d}x {\,d}y\nonumber\\ &=& -2\int_{0}^{\infty}\phi(x) \int_{0}^{x} \phi(y) \log(x-y) {\,d}y {\,d}x \nonumber\\ &>& 2\int_{0}^{\infty} \phi(x) \int_{0}^{x} \phi(y) (y-x) {\,d}y {\,d}x\end{aligned}$$ So, if we set $\Phi (x) = \int_{0}^{x} \phi(y){\,d}y$ and integrate by parts, we get: $$\begin{aligned} \langle\phi,\phi\rangle &>& \int_{0}^{\infty}\phi(x)\int_{0}^{x} y\phi(y) {\,d}y {\,d}x - \int_{0}^{\infty} x\phi(x) \Phi(x) {\,d}x \nonumber\\ &=& - \int_{0}^{\infty}\phi(x) \left(\int_{0}^{x}\Phi(y) {\,d}y\right){\,d}x\nonumber\\ &=& \int_{0}^{\infty}\Phi^{2}(x) {\,d}x-\Phi(\infty)\int_{0}^{\infty}\Phi(y) {\,d}y\nonumber\\ &>& 0\end{aligned}$$ since $\Phi(\infty)\equiv\int_{0}^{\infty}\phi(x){\,d}x = 0 = \Phi(0)$ on account of $\phi$ having zero total charge. Construction of the Coulomb gas model {#app:Coulomb_gas} ===================================== In this appendix we will briefly show how the transition from discrete to continuous eigenvalue measures discussed in Section \[sec:map\_coulomb\_gas\] occurs. As in the main text, we will not present any formal proof here either. However, we will argue that treating the formally discrete distribution of eigenvalues appearing in (\[eq:E\_lambda\]), as continuous in the large $N$ limit is quite reasonable. A more formal method showing the same result appears in [@Vivo2007_LargeDeviationsWishart]. The main reasoning, also discussed in the main text, is that the external confining potentials defined by the first two terms in (\[eq:E\_lambda\]) or (\[eq:energy\]) are strong enough to overcome the logarithmic repulsion between eigenvalues (third term in (\[eq:E\_lambda\])), and therefore guarantee that (with high probability) most of the eigenvalues will be confined in a finite width region near the minimum of the external potential. At the same time, this will mean that the eigenvalue density per unit length will be scaling with $N$ if $N$ is large enough. As a result, this can be seen as a high-density limit and therefore the continuous approximation for the measure will be valid, at least close to configurations whose energy is low enough. In the remainder of this section we will motivate the transition from the discrete to continuous eigenvalue densities and show what kind of terms we expect to see. We start by focusing in a finite region of eigenvalues of length $D$. We then divide the integration over $\lambda_k$ in (\[eq:CDF\_vol\_ratio\]) in $L$ segments of length $\ell$, such that $L\ell=D$. The length of each segment $\ell$ has to be small enough so that the energy (\[eq:E\_lambda\]) can be well approximated with all eigenvalues within a given segment being placed at the endpoint of the segment. At the same time, it has to be large enough so that there is a macroscopic (i.e. ${\mathcal{O}}(N)$) number of eigenvalues inside each segment. In principle, at the end of this exercise we need to take the limit $\ell\rightarrow 0$ as well, however we will discuss the subtleties of this limit later on. As a result, the integral over ${{\cal D}}{\hbox{\boldmath$\lambda$}}$ can be written as: \[eq:entropy1\] [[D]{}]{}&\~\_[k=1]{}\^N (\_[m\_k=1]{}\^L ) = \_[m=1]{}\^L (\_[n\_m=0]{}\^N )\ &\~\_[m=1]{}\^L (\_[n\_m=0]{}\^N) \[eq:entropy2\] where $n_m$ are the number of $\lambda_k$’s that appear in the $m$th segment, with constraint $\sum_m n_m=N$. The factorials appearing at the RHS of (\[eq:entropy1\]) are the number of ways the $N$ eigenvalues can be re-arranged in $L$ segments. This factor constitutes the entropy term and, for large $N$ and $n_m$, we can apply Stirling’s formula to get the exponent in (\[eq:entropy2\]) (where $p(m\ell) = n_m/(N\ell)$ is the fraction of eigenvalues per unit length appearing in segment $m$). We next look at the form of the energy in (\[eq:E\_lambda\]) $$\begin{aligned} \label{eq:E_lambda_approx} E({\hbox{\boldmath$\lambda$}}) &\sim& \ell \sum_m p(m\ell) \left(m\ell - (\beta-1) \log m\ell\right) \\ \nonumber &+& \ell^2 \sum_{m\neq m'} p(m\ell) p(m'\ell) \log\left\|(m-m')\ell\right\| \\ \nonumber &+& \frac{\ell}{N} \sum_m p(m\ell) \log a_m \ell\end{aligned}$$ The last term captures the repulsive interaction between eigenvalues in the same segment $m$. The value of $a_m$ represents the typical distance between eigenvalues in segment $m$ in units of $\ell$ and therefore is a number of order unity. We may now let $\ell\rightarrow 0$, which will make the sums converge to integrals $\ell \sum_m \rightarrow \int dx$ and $p(m\ell)$ can be written as a continuous function $p(x)$. Representing the sum over all possible states (i.e. the product of sums in (\[eq:entropy2\])) by $\int {{\cal D}}p$ we can now get $$\label{eq:partition function} {\mathcal{Z}}\sim \int_\chi \!{{\cal D}}p\, e^{-N^2{\mathcal{E}}[p]} e^{-N\int dx p(x)\log p(x)} e^{N\int dx p(x) \log d(x)}$$ where $d(x)=a_m \ell$ is the average distance between eigenvalues at the position $x=m\ell$. One can estimate this average inter-eigenvalue distance to be $$\label{eq:inter_eig_d} d(x) \sim a_m\ell \sim \frac{1}{Np(x)}$$ This was first proposed by Dyson [@Dyson1962_DysonGas; @Vivo2007_LargeDeviationsWishart; @Mehta_book] and was shown explicitly more recently in [@Brezin1993_UniversalEigCorrsRMT]. It is remarkable that with this choice of $d(x)$ the ${\mathcal{O}}(N)$ dependence on $p(x)$ in the exponent of (\[eq:partition function\]) vanishes. This surprising fact is true only for complex matrices [@Mehta_book] in which, up to uninteresting constants, the leading correction to the $N^2{\mathcal{E}}[p]$ term in the exponent is ${\mathcal{O}}(1)$. Solution of the Variational Equation {#app:solution} ==================================== In this appendix, we give a more detailed account of the solution of the variational equation: $$\delta{\mathcal{L}}_{1}[p]=0$$ where ${\mathcal{L}}_{1}$ is the Lagrangian function of (\[eq:L1\]). To that end, if $\phi\in{\Omega}$ is tame, we get: \[eq:vardiff\] \_[1]{}\[p+t\] &= \_[1]{}\[p\] + t\_[1]{}\[\]\ &- 2t (x) p(y) \|x-y\| [d]{}y [d]{}x + (t\^[2]{}) and a simple differentiation at $t=0$ yields: $$\begin{gathered} \langle\delta {\mathcal{L}}_{1}[p],\phi\rangle =\left.\frac{d}{dt}\right\|_{t=0}\!\!\!{\mathcal{L}}_{1}[p+t\phi]\\ = {\mathcal{L}}_{1}[\phi] - 2\iint \phi(x) p(y) \log\|x-y\|{\,d}y {\,d}x\\ =\int \phi(x) \Psi[p,x]{\,d}x,\end{gathered}$$ where the expression $\Psi[p,x]$ is given by: &= 2p(y) \|x-y\|[d]{}y - x\ &+(-1)x+c+k(1+x) + (x). Thus, for the above expression to vanish identically for all $\phi\in{\Omega}$, we must have $\Psi[p,x] = 0$, and this is precisely (\[eq:func\_deriv\_E1\_result\]), repaeted below: $$\begin{gathered} 2\int_0^\infty p(x')\log\|x-x'\| {\,d}x' = x - (\beta-1)\log x \\ - c - k\log(1+\rho x)-\nu(x). \end{gathered}$$ Having derived this stationarity equation in terms of $p$, we will devote the rest of this appendix to the expression (\[eq:func\_deriv\_E1\_result\_3\]), also repeated below for convenience, that is obtained after differentiating (\[eq:func\_deriv\_E1\_result\]) above: $$2{\cal P} \int_a^b \frac{p(y)}{x-y} dy = 1-\frac{\beta-1}{x}-\frac{k\rho}{1+\rho x} \equiv f(x)$$ for all $x\in[a,b]$ (cf. section \[sec:integral\_equation\]). This integral equation is known as the [airfoil equation]{} and can be studied with the help of the [finite]{} Hilbert transform [@Tricomi_book_IntegralEquations]: $${\mathcal{T}}[\phi](x) = \mathcal{P}\!\int_{-1}^{1} \frac{\phi(y)}{y-x} {\,d}y.$$ If $r>1$, the ${\mathcal{T}}$-transform maps $L^{r}$ to $L^{r}$ but, nevertheless, it lacks a unique inverse.[^10] Indeed, the kernel of ${\mathcal{T}}$ is spanned by the function $\omega(x) = (1-x^{2})^{-\frac{1}{2}}$: ${\mathcal{T}}[\omega](x) = 0$ for all $x\in(-1,1)$. Outside this kernel, the solutions $\phi$ to the airfoil equation ${\mathcal{T}}[\phi] = g$ with $\phi,g\in L^{r}[-1,1]$ will satisfy [@Tricomi_book_IntegralEquations]: $$\label{eq:inverseHilbert} \phi(x) = -\frac{1}{\pi}\mathcal{P} \int_{-1}^{1} \sqrt{\frac{1-y^{2}}{1-x^{2}}} \frac{g(y)}{y-x} {\,d}y + \frac{c}{\sqrt{1-x^{2}}}$$ where $c$ is an arbitrary constant that stems from the fact that any two solutions of the airfoil equation differ by a multiple of $\omega(x) = (1-x^{2})^{-\frac{1}{2}}$. Hence, after rescaling the interval $[-1,1]$ to $[a,b]$, the solution of the stationarity equation (\[eq:func\_deriv\_E1\_result\_3\]) will be given by: $$p(x) = \frac{{\cal P} \int_a^b \frac{\sqrt{(y-a)(b-y)}f(y)}{y-x} dy + C'}{2\pi^2\sqrt{(x-a)(b-x)}}$$ whenever $f$ is itself $L^{1+{\varepsilon}}$-integrable. So, by substituting $f(x) = 1- \frac{\beta-1}{x} - \frac{k}{x+z}$ from (\[eq:func\_deriv\_E1\_result\_3\]) and performing one last integration, we obtain the final result (\[eq:gen\_solution\_int\_eq0\]). It is worthwhile to mention here again how this procedure breaks down if we allow the support of $p$ to extend to $a=0$ for $\beta>1$: in that case, the function $f$ also extends all the way to $a=0$ and the term $\frac{\beta-1}{x}$ makes it non-integrable. However, since the Hilbert transform preserves $L^{r}$-integrability for $r>1$ and $p$ is assumed tame (and hence $L^{1+{\varepsilon}}$-integrable), equation (\[eq:func\_deriv\_E1\_result\_3\]) would equate an integrable function with a non-integrable one, thus yielding a contradiction. Therefore, as we stated in section \[sec:integral\_equation\], solutions with $a=0$ are physically inadmissible when $\beta>1$. Proof of uniqueness of solution of (\[eq:p\_a\_=0\]),(\[eq:norm\_integral\]) {#app:uniqueness_ab} ============================================================================ In order to show that (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]) admit a unique solution, we start by observing that for fixed $k,\beta$ and $z$, (\[eq:p\_a\_=0\]) has a unique positive solution $a\leq b$. Then, from the implicit function theorem, this solution can be captured in terms of $b$ by a smooth function $a(b)$ whose derivative can be obtained implicitly from (\[eq:p\_a\_=0\]) (and which is negative). With this in mind, the normalization integral $g(b) = \int_{a(b)}^{b}p(x) {\,d}x$ takes the form: $$g(b) = \frac{a(b)+b}{4} + \frac{1}{2}\left(\rho^{-1}-k-(\beta-1)\left(1+\frac{1}{\rho\sqrt{a(b)b}}\right)\right)\notag$$ and this is actually an increasing function of $b$. Indeed, after a somewhat painful calculation, one obtains: $$g'(b) = \frac{\rho}{4}\left[1+\frac{(\beta-1)}{\rho\sqrt{a(b)\,b^{3}}}\right]\frac{b-a(b)}{1+\rho b}>0$$ However, with $a(b)$ decreasing and bounded below by $0$, this last equation yields $g'(b)>1/8$ for large enough $b$, i.e. $\lim_{b\to\infty} g(b) = +\infty$. So, by continuity, there will be a (necessarily) unique $b^{}$ such that $g(b^{})=1$. Hence, the pair $a^{}=a(b^{}), b=b^{}$ will be the unique solution to (\[eq:p\_a\_=0\]), (\[eq:norm\_integral\]). $O(1/N)$ correction to the LD approximation {#app:1/Ncorrections} =========================================== Here we provide an improved estimate on the probability distribution close to the center of the distribution. This estimate is a result of the inclusion the $O(1/N)$ higher moment corrections to the distribution derived in [@Moustakas2003_MIMO1]. It is well known[@Bouchaud_book_FinancialRiskDerivativePricing] that to provide asymptotic corrections to the limiting Gaussian distribution due to the presence of a small (but finite) skewness we need to change the distribution as follows: $$\label{eq:correction_to CLT} P_N(x)=\frac{e^{-\frac{x^2}{2v}}}{\sqrt{2\pi v}}\left(1-\frac{s}{2v^2}\left(x-\frac{x^3}{3 v}\right)\right)$$ where $v$ is the variance of the asymptotically Gaussian distribution and $s$ is the third moment of the distribution. Clearly, the above distribution cannot be valid over the entire support of $x$ since the cubic polynomial will become negative for some value of $x$. Nevertheless, since the third moment is small for large $N$ this value of $x$ will become asymptotically large. We may therefore apply the above formula to our model. The value of the third moment $s=s_3/N$ has been calculated in \[(60) in [@Moustakas2003_MIMO1]\] and it is of order $O(1/N)$. As a result, the correction to the Gaussian approximation of the mutual information is given by $$\begin{aligned} \label{eq:correction_to Gaussian} P_N(R) &=& \frac{e^{-\frac{(R-Nr_{erg})^2}{2v_{erg} } } }{\sqrt{2\pi v_{erg}}} \cdot \\ \nonumber && \left(1-\frac{s_3}{2Nv_{erg}^2}(R-Nr_{erg}) + \frac{(R-Nr_{erg})^3}{3 v_{erg}}\right)\end{aligned}$$ To order $O(1/N)$, there is also the correction to the mean of the mutual information [@Moustakas2003_MIMO1], which needs to be subtracted off from $I_N$. Now, to obtain the correction to the LD approximation, we need to take into account that the large deviations function ${{\cal E}}_1$ also has a cubic term for $r\approx r_{erg}$, which needs to be balanced. This can be done by adding a cubic term that cancels this term for $r\approx r_{erg}$. 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This research was supported in part by Greek GSRT “Kapodistrias” project No. 70/3/8831. [^2]: Note that these are simply the potentials that one obtains in classical two-dimensional electrostatics. [^3]: It is worth pointing out that the correction to the term $N^2{\mathcal{E}}[p]$ in the exponent is ${\mathcal{O}}(1)$ (see appendix \[app:Coulomb\_gas\] for more details). Also a nice analysis of the mapping from the ${\hbox{\boldmath$\lambda$}}$ integrals to path integrals over $p$ can also be found in [@Dean2008_ExtremeValueStatisticsEigsGaussianRMT]. [^4]: Since ${\Omega}$ is a locally convex space, this is just another guise of the Gâteaux/Fréchet derivative. [^5]: Indeed, note that the function $w(t) = {\mathcal{L}}_{1}[p + t(q-p)], t\in[0,1]$ is strictly convex in $[0,1]$ for any choice of $p$ and $q$ in ${\mathcal{X}}_{r}$. Thus, if there were some $q\in{\mathcal{X}}_{r}$ with ${\mathcal{L}}_{1}[q]<{\mathcal{L}}_{1}[p]$, we would have $w'(0) = 0$ (on account of (\[eq:funcderiv\])) but also $w(0)>w(1)$, a contradiction. [^6]: The principle value appears because of the absolute value $\|x'-x\|$ in (\[eq:func\_deriv\_E1\_result\]). [^7]: Note that when $\beta=1$, the lower endpoint vanishes ($a=0$) and a square-root (integrable) singularity appears in $p(x)$ in (\[eq:MP\_def\]). [^8]: This is the region analyzed in the diversity-multiplexing trade-off [@Zheng2003_DiversityMultiplexing]. [^9]: In that case a MAC-layer scheduler would be transmitting to the user with the best channel, for example. [^10]: This is a remarkable difference from the case of the [infinite]{} Hilbert transform which integrates over all ${\mathbb{R}}$ and which [is]{} invertible [@Tricomi_book_IntegralEquations].
--- abstract: 'Population expansions trigger many biomedical and ecological transitions, from tumor growth to invasions of non-native species. Although population spreading often selects for more invasive phenotypes, we show that this outcome is far from inevitable. In cooperative populations, mutations reducing dispersal have a competitive advantage. Such mutations then steadily accumulate at the expansion front bringing invasion to a halt. Our findings are a rare example of evolution driving the population into an unfavorable state and could lead to new strategies to combat unwelcome invaders. In addition, we obtain an exact analytical expression for the fitness advantage of mutants with different dispersal rates.' author: - 'Kirill S. Korolev' bibliography: - 'references.bib' title: Evolution arrests invasions of cooperative populations --- Locust swarms, cancer metastasis, and epidemics are some feared examples of spatial invasions. Spatial spreading is the only mechanism for species to become highly abundant, whether we are considering a bacterial colony growing on a petri dish [@korolev:sectors; @wakita:expansion] or the human expansion across the globe [@templeton:africa]. Many invasions are unwelcome because they threaten biodiversity [@hooper:biodiversity], agriculture [@gray:worm_expansion], or human health [@brockmann:networks]. Unfortunately, efforts to control or slow down invaders often fail in part because they become more invasive over time [@thomas:acceleration]. The evolution of invasive traits and invasion acceleration has been repeatedly observed in nature from the takeover of Australia by cane toads [@phillips:toad_acceleration] to the progression of human cancers [@merlo:review; @korolev:perspective]. Selection for faster dispersal makes sense because it increases the rate of invasion and allows early colonizers to access new territories with untapped resources. A large body of theoretical [@shine:spatial_sorting; @korolev:wave_splitting; @benichou:acceleration] and experimental work [@thomas:acceleration; @phillips:toad_acceleration; @ditmarsch:swarming] supports this intuition in populations that grow non-cooperatively, i.e., when a very small number of organisms is sufficient to establish a viable population. Many populations including cancer tumors [@berec:multiple_allee; @cluttonbrock:mongoose; @merlo:review; @korolev:perspective; @cleary:cancer_cooperation], however, do grow cooperatively, a phenomenon known as an Allee effect in ecology [@courchamp:allee_review]. In fact, cooperatively growing populations can even become extinct when the population density falls below a critical value, termed the Allee threshold [@scheffer:review; @dai:science]. We find that the intuitive picture of “the survival of the fastest” fails for such populations, and natural selection can in fact favor mutants with lower dispersal rates. Over time, repeated selection for lower dispersal leads to a complete arrest of the spatial invasion. To understand when invasions accelerate and when they come to a halt, we analyzed a commonly used mathematical model for population dynamics that can be tuned from non-cooperative to cooperative growth by changing a single parameter. We considered the competition between two genotypes with different dispersal abilities and computed their relative fitness analytically. Our main result is that selection favors slower dispersal for a substantial region of the parameter space where the Allee threshold is sufficiently high. Numerical simulations confirmed that evolution in such populations gradually reduces dispersal and eventually stops the invasion even when multiple mutants could compete simultaneously and other model assumptions were relaxed. Selective pressure on the dispersal rate can be understood most readily from the competition of two types (mutants, strains, or species) with different dispersal abilities as they invade new territory. For simplicity, we focus on short-range dispersal that can be described by effective diffusion and only consider the dynamics in the direction of spreading. Mathematically, the model is expressed as $$\begin{aligned} & \frac{\partial c_{1} }{\partial t} = D_{1}\frac{\partial^2 c_{1}}{\partial x^2} + c_{1}g(c),\\ & \frac{\partial c_{2} }{\partial t} = D_{2}\frac{\partial^2 c_{2}}{\partial x^2} + c_{2}g(c), \end{aligned} \label{eq:model_main}$$ where $c_{1}$ and $c_{2}$ are the population densities of the two types that depend on time $t$ and spatial position $x$; $D_{1}$ and $D_{2}$ are their dispersal rates; and $g(c)$ is the density-dependent per capita growth rate. We assume that $g(c)$ is the same for the two types and depends only on the total population density $c=c_1+c_2$. Since slower-dispersing types often grow faster because of the commonly observed trade-off between dispersal and growth, our results put a lower bound on the fitness advantage of the type dispersing more slowly. In the Supplemental Material, our analysis is further generalized to account for the different growth rates of the types [@supplement]. For $g(c)$, we assume the following functional form, which has been extensively used in the literature [@korolev:wave_splitting; @aronson:allee_wave; @fife:allee_wave; @courchamp:allee_review] because it allows one to easily tune the degree of cooperation in population dynamics from purely competitive to highly cooperative growth: $$g(c)=r(K-c)(c-c^{})/K^{2}. \label{eq:growth}$$ Here, $r$ sets the time scale of growth, $K$ is the carrying capacity, i.e. the maximal population density that can be sustained by the habitat, and $c^$ is a parameter that determines the degree of cooperation and is known as the Allee threshold. For $c^{}<-K$, the types grow non-cooperatively because $g(c)$ monotonically decreases from its maximal value at low population densities to zero when the population is at the carrying capacity and interspecific competition prevents further growth. Population grows cooperatively for higher values of $c^{}$ because the per capita growth rate reaches a maximum at nonzero density that strikes the balance between interspecific competition and facilitation. For $c^{}>0$, the effects of cooperative growth become particularly pronounced. Indeed, the growth rate is negative for $c<c^$ and, therefore, small populations are not viable. Such dynamics, known as the strong Allee effect, arise because a critical number of individuals is necessary for a sufficient level of cooperation [@courchamp:allee_review; @korolev:perspective]. ![The effects of cooperative growth on the evolution of dispersal during invasion. (A) Simulations of the competition between a slow ($D_1=0.5$) and a fast ($D_2=1$) disperser during a spatial expansion. The fraction of the slower disperser decreases in populations with a low Allee threshold ($c^=0.2$), but increases in populations with high Allee threshold ($c^=0.35$). (B) The fitness advantage of the slower disperser ($D_1/D_2=0.95$) changes from negative (deleterious) to positive (beneficial) as the Allee threshold is increased. In simulations, we never observed the coexistence of the two types; instead extinction is observed for the types that are deleterious when rare ($\lambda<0$), and complete fixation is observed for the types that are beneficial when rare ($\lambda>0$).[]{data-label="fig:switch"}](fig1.eps){width="\columnwidth"} We first tested whether unequal dispersal rates lead to fitness differences between the two types by solving Eq. (\[eq:model\_main\]) numerically (see Supplemental Material). When population growth was non-cooperative, we found that the faster-dispersing species have a competitive advantage in agreement with the current theory [@thomas:acceleration; @shine:spatial_sorting; @benichou:acceleration]. Quite unexpectedly, the opposite outcome was observed for strongly cooperative growth: The type with the lower dispersal rate became dominant at the expansion front and eventually took over the population (Fig. \[fig:switch\]a)! To understand this counterintuitive dynamics, we examined how the relative fitness of the two types depends on the magnitude of the Allee threshold $c^$. In the context of spatial expansions, there are two complementary ways to quantify the fitness advantage of a mutant. The first measure $\lambda$ is the exponential growth rate of the mutant similar to what is commonly done for populations that are not expanding; a negative $\lambda$ corresponds to decay not growth. The second measure $\lambda_{x}$ is the growth rate of the mutant not in units of time, but rather in units of distance traveled by the expansion. The two measures are related by $\lambda = \lambda_{x}v$, where $v$ is the expansion velocity. The advantage of the second measure is that it can be applied in situations when the spatial distribution of the genotypes is available for only a single time point. We were able to compute both fitness measures analytically. The complete details of this calculation are given in the Supplemental Material, but our approach is briefly summarized below. When a mutant first appears, its abundance is too small to immediately influence the course of the range expansion; therefore, we can study the dynamics of the mutant fraction in the reference frame comoving with the expansion, effectively reducing two coupled equations in Eq. (\[eq:model\_main\]) to a single equation. This remaining equation has the form of a Fokker-Planck equation with a source term, and its largest eigenvalue determines whether the total fraction of the mutant will increase or decrease with time. We were able to obtain this largest eigenvalue and the corresponding eigenfunction exactly in terms of only elementary functions. For small differences in the dispersal abilities $\|D_1-D_2\|\ll D_2$, our result takes a particularly simple form, $$\lambda_x = \frac{D_1-D_2}{6 D_{2}}\sqrt{\frac{r}{2D_2}}\left(1-4\frac{c^{}}{K}\right), \label{eq:exact_short}$$ which is valid for $c^>-K/2$; see Supplemental Material for $c^<-K/2$. Thus, $\lambda_x$ is a linear function of the Allee threshold $c^$, which changes sign at $c^=K/4$. For low Allee thresholds, natural selection favors mutants with higher dispersal, but, when growth is highly cooperative, the direction of selection is reversed and slower dispersers are favored. Numerical simulations of Eq. (\[eq:model\_main\]) are in excellent agreement with our exact solution (Fig. \[fig:switch\]b). In the Supplemental Material, we explain that the direction of natural selection remains the same as the mutant takes over the population and further discuss the effects of mutations and demographic fluctuations by connecting the largest eigenvalue to the fixation probability of the mutant [@supplement]. ![Allee effect determines how fitness depends on dispersal. (A) Faster dispersers are unconditionally favored when the Allee effect is weak. Note that the fitness advantage reaches a maximum at a finite $D_1/D_2$. (B) When the Allee effect is strong, but the Allee threshold is low, selection still favors faster dispersal. Very fast mutants however are at a disadvantage. (C) For high Allee threshold, only slower-dispersing mutants can succeed, but mutants that disperse too slowly are selected against. In all panels, the exact solution is plotted, and colors highlight beneficial (red) and deleterious (blue) mutations. The dashed line marks $D_1=D_2$, where both types have the same fitness. Near this point, the fitness advantage of type one in the background of type two equals the fitness disadvantage of type two in the background of type one, but this symmetry breaks down when the dispersal rates of the types are very different; see Supplemental Material. Nevertheless, the exchange of $D_1$ and $D_2$ always converts a beneficial mutant to a deleterious one. In consequence, a mutation that is beneficial when rare will remain beneficial when it approaches fixation indicating that the direction of natural selection is the same for small and large $f$.[]{data-label="fig:dependence"}](fig2.eps){width="0.7\columnwidth"} Our finding that lower dispersal is advantageous seems counterintuitive. Indeed, a mutant unable to disperse cannot possibly take over the expansion front. The resolution of this apparent paradox is that Eq. (\[eq:exact\_short\]) is only valid for $D_1\approx D_2$, and the direction of natural selection changes as $D_1$ approaches zero. The exact expression for the selective advantage for arbitrary $D_1/D_2$ is given in the Supplemental Material and is plotted in Fig. \[fig:dependence\] for different values of the Allee threshold. When the Allee effect is absent or weak, selection unconditionally selects for faster dispersal (Fig. \[fig:dependence\]a), but, as the Allee threshold increases and becomes positive, mutants with very large dispersal rates become less fit than the wild type (Fig. \[fig:dependence\]b). This is expected because mutants that disperse too far ahead of the front cannot reach the critical density necessary to establish a viable population. As a result, there is an optimal improvement in dispersal abilities that is favored by natural selection. In contrast, when the Allee effect is sufficiently strong, only reduced dispersal is advantageous (Fig. \[fig:dependence\]c). Again, there is an optimal reduction in the dispersal rate that results in the highest fitness advantage, and mutants that disperse too slowly are outcompeted by the wild type. Although natural selection typically eliminates the mutants that either increase or decrease the dispersal rate by a large amount, sequential fixation of mutations could lead to a substantial change in the expansion velocity. Indeed, our results show that the fitness advantage of the mutant depends on the relative rather than absolute change in the dispersal ability. Thus, if the Allee effect is strong enough to favor slower mutants, then mutants that reduce the dispersal rate even further will become advantageous once the takeover by the original mutant is complete. We then expect that the repeated cycle of dispersal reduction will eventually bring the invasion to a standstill. The opposite behavior is expected when the Allee threshold is low. To test these predictions, we performed computer simulations that relax many of the assumptions underlying Eq. (\[eq:model\_main\]) as described in the Supplemental Material. In particular, we incorporated the stochastic fluctuations due to genetic drift and allowed multiple mutations modifying the dispersal rate to arise and compete at the same time. Shown in Fig. \[fig:arrest\], simulations display a steady decline in the dispersal ability and expansion arrest for strongly cooperative growth. Consistent with previous studies [@thomas:acceleration; @shine:spatial_sorting; @korolev:wave_splitting; @benichou:acceleration], dispersal rates increase and the rate of invasion accelerates when the Allee threshold is low. ![When the rate of dispersal is allowed to evolve, simulations show that invasions can both accelerate and decelerate depending on the strength of an Allee effect. (A) The mean dispersal rate increases to its maximally allowed value when the Allee threshold is low (green), but the dispersal rate decreases to zero when the Allee threshold is high (blue). (B) For the same simulations as in (A), we plot the extent of spatial spread by the populations. Invasions with a low Allee threshold ($c^/K=0.2$) accelerate, while invasions with a high Allee threshold ($c^/K=0.35$) come to a standstill.[]{data-label="fig:arrest"}](fig3.eps){width="\columnwidth"} Natural selection on dispersal has been extensively studied, and many factors that favor faster or slower dispersal have been identified [@ronce:dispersal_review; @dockery:slow_dispersal; @hutson:slow_dispersal]. Fast dispersers can avoid inbreeding depression, escape competition, or find a suitable habitat. At the same time, dispersal diverts resources from reproduction and survival, increases predation, and can place organisms in inhospitable environments. In the context of range expansions, however, high dispersal seems unambiguously beneficial because early colonizers get a disproportionate advantage. Yet, we showed that spatial expansions can select for mutants with lower dispersal rates. Continuous reduction of dispersal rates then slows down and eventually stops further invasion. Invasion arrest requires strong cooperative growth and is in stark contrast to the dynamics in non-cooperative populations where spatial expansions select for higher dispersal. We expect that our results are robust to the specific assumptions made in this study such as the diffusion-like dispersal and the specific form of the growth function because, at its core, our analysis relies on very general arguments (Supplemental Material). Indeed, we argue that faster mutants get ahead at low or negative Allee thresholds because they can successfully grow at the front and effectively establish secondary invasions; in contrast, these dynamics do not occur at high Allee thresholds because faster dispersers arrive at low-density regions that cannot sustain growth. At a very high level, our result can be understood as the emergence of cheating in a cooperatively growing population. Cheating is a behavior that benefits the individuals, but is detrimental to the population as a whole [@west:social]. One well-studied example is consuming, but not contributing, to a common resource (public good), a behavior typical of both humans [@hardin:tragedy; @axelrod:cooperation; @gardner:cooperation_history] and microbes [@west:social; @nadell:sociobiology; @menon:diffusion]. In the context of population spreading, high dispersal can be viewed as an effective public good because it creates high densities in the outer edge of the expansion front, thereby increasing the survival of new immigrants to that region. Although high population densities benefit both slow and fast dispersers equally, the latter pay a much higher cost for producing this public good. Indeed, faster dispersers are more likely to suffer higher death rates at the low-density invasion front, where they arrive more frequently. As a result, “cheating” by the slow dispersers is the reason for their selective advantage. In addition to the classical emergence of cheating, expansion arrest is an example of evolution driving a population to a less adapted state. Our ability to exploit or trigger such counterproductive evolution may be important in managing invasive species and agricultural pests, or even cancer tumors. Concretely, our results open up new opportunities to control biological invasions. Instead of trying to kill the invader, a better strategy could be to elevate the minimal density required for growth (the Allee threshold) to a level necessary for evolution to select for invasion arrest. Such strategies could have important advantages over the traditional approaches. Increasing the Allee threshold in cancer tumors could overcome the emergence of drug resistance because the bulk of the tumor is at a high density and is not affected by the treatment. Similarly, resistance should emerge much more slowly in agricultural pests. Although the manipulation of Allee thresholds is a relatively unexplored and potentially difficult endeavor, some management programs have been successful at increasing the Allee effects in the European gypsy moth, one of the most expensive pests in the United States [@tobin:moth_review; @tobin:slow_spread; @thorpe:mating_disruption; @johnson:allee_gypsy]. These moths suffer from a strong Allee effect because they struggle to find mates at low population densities [@johnson:allee_gypsy]. Recent management programs exacerbated this Allee effect by spreading artificial pheromones that disorient male moths and prevent them from finding female mates, thereby effectively eradicating low-density populations [@tobin:moth_review; @tobin:slow_spread; @thorpe:mating_disruption]. European gypsy moths and other pests with similarly strong Allee effects could be close to the critical Allee threshold necessary for the invasion arrest. In such populations, further increase in of the Allee effect could be more feasible and effective than reducing the carrying capacity. Beyond ecology, our results could also find applications in other areas of science such as chemical kinetics, where reaction-diffusion equations are often used. Quite broadly, we find that a variation in the motility of agents can have completely opposite effects on their dynamics depending on the reaction kinetics at the expansion front. This work was supported by the startup fund from Boston University to KK. Simulations were carried out on Shared Computing Cluster at BU. = Supplemental Material for “Evolution arrests invasions of cooperative populations”.\ Formulation of the mathematical model We consider the competition of two types (mutants, strains, or species) with different dispersal abilities as they invade new territory. As defined in the main text, the model is expressed as $$\begin{aligned} & \frac{\partial c_{1} }{\partial t} = D_{1}\frac{\partial^2 c_{1}}{\partial x^2} + c_{1}g(c),\\ & \frac{\partial c_{2} }{\partial t} = D_{2}\frac{\partial^2 c_{2}}{\partial x^2} + c_{2}g(c). \end{aligned} \label{eq:model}$$ With the following expression for $g(c)$: $$g(c)=r(K-c)(c-c^{})/K^{2}. \label{eq:growth}$$ In the following, we assume that type two is the wild type and is in the majority while type one is a recent mutant with a different dispersal rate. Our goal is to compute the fitness advantage of type one and determine whether selection favors slower or faster dispersal. Fitness advantage is a solution of an eigenvalue problem* The fate of mutations is determined when their density is quite small [@gillespie:book]. Therefore, we assume that the population density of the first type is much lower than that of the second type, whose dispersal and growth determine the dynamics of the expansion. We then compute whether the fraction of the first type increases in time indicating that it will eventually take over or decreases with time indicating that it is less fit and will be eventually eliminated by natural selection. To achieve this goal, it is convenient to represent population dynamics in terms of the total population density $c$ and the fraction of the first type $f$ defined below as $$\begin{aligned} & c(t,x) = c_{1}(t,x) + c_{2}(t,x),\\ & f(t,x) = \frac{c_{1}(t,x)}{c_{1}(t,x) + c_{2}(t,x)}. \end{aligned} \label{eq:fc}$$ This change of variables results in the following set of reaction-diffusion equations: $$\begin{aligned} & \frac{\partial c}{\partial t} = D_{2}\frac{\partial^2 c}{\partial x^2} + cg(c) + (D_1-D_2)\frac{\partial^2 (fc)}{\partial x^2} ,\\ & \frac{\partial f}{\partial t} = D_{1}\frac{\partial^2 f}{\partial x^2} + \frac{2D_1}{c}\frac{\partial c}{\partial x}\frac{\partial f}{\partial x} + f \frac{D_1-D_2}{c}\frac{\partial^2 c}{\partial x^2} - f\frac{D_1-D_2}{c}\frac{\partial^2 (fc)}{\partial x^2}. \end{aligned} \label{eq:f_dynamics_full}$$ Since we assume that $f\ll 1$, we neglect higher order terms in $f$, i.e. the terms linear in $f$ in the equation for the population density and terms quadratic in $f$ in the equation for the fraction of the first type. The result reads $$\begin{aligned} & \frac{\partial c}{\partial t} = D_{2}\frac{\partial^2 c}{\partial x^2} + cg(c),\\ & \frac{\partial f}{\partial t} = D_{1}\frac{\partial^2 f}{\partial x^2} + \frac{2D_1}{c}\frac{\partial c}{\partial x}\frac{\partial f}{\partial x} + f \frac{\Delta D}{c}\frac{\partial^2 c}{\partial x^2}, \end{aligned} \label{eq:f_dynamics}$$ where we introduced $\Delta D = D_1 - D_2$. The equation for $c$ is now independent from $f$ and its solution is known exactly for the quadratic form of $g(c)$ with $c^{}>-K/2$ [@aronson:allee_wave; @fife:allee_wave]: $$\begin{aligned} &v=\sqrt{\frac{D_2r}{2}}\left(1-2\frac{c^{}}{K}\right),\\ &c(\zeta)=\frac{K}{1+e^{\sqrt{\frac{r}{2D_2}}\zeta}}, \end{aligned} \label{eq:profile}$$ where $\zeta=x-v_{2}t$ in the spatial coordinate in the reference frame comoving with the expansion front. Note that for $c^>K/2$ the velocity is negative and the whole population is driven to extinction. For $c<-K/2$, the cooperative growth has a negligible effect on the expansion dynamics and $v=2\sqrt{Dr\|c^{}/K\|}$ and $c(\zeta) \sim \exp(-\zeta\sqrt{r\|c^{}\|/K})$ for large $\zeta$, which are obtained by the linearization of the growth dynamics in $c$ as for the classic Fisher-Kolmogorov equation [@fisher:wave; @kolmogorov:wave; @saarloos:review]. As a result of this simplification, we treat the equation for $f$ as a single partial differential equation with non-constant, but known, coefficients. To solve this equation, we change to the reference frame comoving with the population expansion, which is defined by the following change of variables $$\begin{aligned} & \zeta = x -v_{2}t,\\ & \tau = t. \end{aligned} \label{eq:comoving}$$ Note that the velocity of the comoving frame is given by equation (\[eq:profile\]) with the parameters of the second type, which is in the majority. To indicate that, we denote this velocity as $v_2$. We also denote the time as $\tau$ to emphasize the change of variables. In the comoving reference frame, the equation for $f$ reads $$\frac{\partial f}{\partial \tau} = D_{1}\frac{\partial^2 f}{\partial \zeta^2} + \left(v_2 + \frac{2D_1}{c}\frac{\partial c}{\partial \zeta}\right)\frac{\partial f}{\partial \zeta} + f \frac{\Delta D}{c}\frac{\partial^2 c}{\partial \zeta^2}. \label{eq:f_comoving}$$ Since equation (\[eq:f\_comoving\]) is a linear partial differential equation of the form $$\frac{\partial f}{\partial \tau} = \mathbf{L} f, \label{eq:operator_L}$$ the long term dynamics and the eventual fate of the newly introduced type is determined by $\lambda$, the largest eigenvalue of $\mathbf{L}$. When $\Delta D=0$, the largest eigenvalue equals zero and corresponds to an eigenvector that does not depend on $\zeta$. Indeed, the system must relax to a spatially homogeneous $f(\tau, \zeta)$ when the two type are identical. When $\Delta D \ne 0$, one of the types outcompetes the other at rate $\lambda$. Positive $\lambda$ correspond to the takeover by the first type while negative $\lambda$ indicates that the first type is less fit and will be eliminated. Thus, our goal is to solve the following eigenvalue problem for the largest possible $\lambda$ $$\lambda f= D_{1}f'' + \left(v_2 + 2D_1\frac{c'}{c}\right)f' + f \Delta D\frac{c''}{c}, \label{eq:f_eigen_value}$$ where the derivatives with respect to $\zeta$ are denoted with primes. Exact solution of the eigenvalue problem* We now proceed by solving equation (\[eq:f\_comoving\]) exactly. To that end, it is convenient to recast equation (\[eq:f\_eigen\_value\]) in a Hermitian form by eliminating the gradient term on the right hand side. This is accomplished by the following change of variables $$\begin{aligned} & f(\zeta) = \psi(\zeta) e^{u(\zeta)}, \;\text{where}\\ & u(\zeta) = -\frac{v_{2}\zeta}{2D_{1}} - \ln(c(\zeta)). \end{aligned} \label{eq:u}$$ The resulting eigenvalue problem then reads $$\lambda \psi = D_1 \psi''- \psi\left(D_2\frac{c''}{c} + v_2 \frac{c'}{c} + \frac{v_{2}^{2}}{4D_1} \right), \label{eq:eigen_value_hermitian}$$ The potential term can be further simplified by using the density equation in equations (\[eq:f\_dynamics\]) written in a comoving reference frame ($-v_{2}c'=D_{2}c''+g(c)c$), with the following result: $$\lambda \psi = D_1 \psi'' + \psi\left(g(c(\zeta)) - \frac{v_{2}^{2}}{4D_1} \right). \label{eq:eigen_value_hermitian_simplified}$$ Since the solution techniques differ depending on whether $c^<-K$, $c^\in(-K,-K/2)$, or $c^>-K/2$, we analyze these three cases separately. For $c^{}<-K$, there is no Allee effect, and $g(c)$ is a decaying function of the population density. Therefore, the potential term in equation (\[eq:eigen\_value\_hermitian\_simplified\]) reaches its maximum when $c\to0$ and $\zeta\to+\infty$. Moreover, since the potential term is constant for large $\zeta$, we conclude that the eigenfunction corresponding to the largest eigenvalue is concentrated at large $\zeta$, and the eigenvalue itself is given by the maximal value of the potential: $$\lambda = r\left(1-\frac{1}{p}\right)\left\|\frac{c^}{K}\right\|, \label{eq:lambda_exact_fisher}$$ where we defined $D_1/D_2=p$ for convenience. Thus, faster dispersing types are unconditionally favored when cooperation plays no role in population dynamics. The maximal fitness advantage a mutant can obtain by increasing its dispersal rate is $r\|c^{}\|/K$, while the cost of reduced motility can be infinitely large. This asymmetry is due to the fact that we observe changes in the frequency of type one relative to type two, who is in the majority. As a result, when a new mutant is not moving it is immediately lost from the front. In contrast, when the mutant is spreading rapidly, it still takes an appreciable time for the density of this mutant to reach high values at the expanding front of the wild type, even though the mutant almost immediately outcompetes the wild type at the far edge of the expansion ($\zeta\to+\infty$). From our analysis of $c^>-K/2$, it will be clear that faster dispersers are also unconditionally favored for $c^\in(-K,-K/2)$ and equation (\[eq:lambda\_exact\_fisher\]) holds for $p\ge p_{+\infty}$, where $p_{+\infty}$ decreases from $1$ at $c^{}=-K/2$ to $0$ at $c^=-K$. For $p < p_{+\infty}$, the slower mutant will be lost less rapidly than predicted by equation (\[eq:lambda\_exact\_fisher\]) because it can take advantage of the higher per capita growth rates in the interior of expansion front ($g(c)$ is not monotonically decreasing in this regime). Nevertheless, we find that equations (\[eq:lambda\_exact\_fisher\]) and (\[eq:lambda\_exact\]) provide a good approximation to the decay rates for $c^\in(-K,-K/2)$ and $p < p_{+\infty}$. We now turn to $c^>-K/2$. Since, in this regime, the shape of density profile is known exactly, it is convenient to perform a change of variable $\rho = c(\zeta)/K$ in equation (\[eq:f\_eigen\_value\]) and treat $\rho$ as an independent variable, thus eliminating $\zeta$. The resulting eigenvalue problem reads $$\frac{rp}{2}\rho^2(1-\rho)^2 f'' + \frac{rp}{2} \rho(1-\rho)\left[3-4\rho-\frac{1-2\rho^}{p}\right] f' + \frac{r}{2}(p-1)(1-\rho)(1-2\rho)f = \lambda f, \label{eq:eigen_rho}$$ where the primes now denote the derivatives with respect to $\rho$, and $\rho^=c^/K$. We further simplify equation (\[eq:eigen\_rho\]) by letting $f(\rho) = h(\rho)e^{\nu(\rho)}$ and choosing $\nu(\rho)$ to set the term with $h'$ to zero. This results in $$\nu = -\frac{3}{2}\ln\rho - \frac{1}{2}\ln(1-\rho) + \frac{(1-2\rho^)}{2p}\ln\frac{\rho}{1-\rho}, \label{eq:nu}$$ $$p\rho^2(1-\rho)^2 h'' + \left[-2\rho^2 + 2(1+\rho^)\rho + \frac{p}{4} - 2\rho^ - \frac{(1-2\rho^)^2}{4p} \right]h = \frac{2\lambda}{r}h \label{eq:eigen_rho_pseudohermitian}$$ To convert this equation into a Hermitian form, we divide both sides by $\rho(1-\rho)$ and define $\varphi=h/[\rho(1-\rho)]$: $$p\rho(1-\rho)\frac{d^2}{d\rho^2}[\rho(1-\rho)\varphi] + \left[-2\rho^2 + 2(1+\rho^)\rho + \frac{p}{4} - 2\rho^* - \frac{(1-2\rho^)^2}{4p} \right]\varphi = \frac{2\lambda}{r}\varphi. \label{eq:eigen_rho_hermitian}$$ The derivative term is now clearly Hermitian because it acts symmetrically to the left and to the right, which was not the case in equation (\[eq:eigen\_rho\_pseudohermitian\]). It is now apparent that we should look for $\varphi=\rho^{\alpha-1}(1-\rho)^{\beta-1}$ because $$[\rho^{\alpha}(1-\rho)^{\beta}]''= \rho^{\alpha-2}(1-\rho)^{\beta-2}[(\alpha+\beta)(\alpha+\beta-1)\rho^2 - 2\alpha(\alpha+\beta-1)\rho + \alpha(\alpha-1)]. \label{eq:derivative_ansatz}$$ Indeed, for such a choice of $\varphi$, the first term becomes a product of $\varphi$ and a quadratic polynomial of $\rho$, which is exactly the form of the other terms in the equation. Moreover, this ansatz yields a unique solution because there are three unknowns $\alpha$, $\beta$, and $\lambda$ and three coefficients of the quadratic polynomial to match. We then find that the values of $\alpha$ and $\beta$ that satisfy equation (\[eq:eigen\_rho\_pseudohermitian\]) are given by $$\begin{aligned} & \alpha = \frac{1+\rho^}{4}\left(1+\sqrt{1+\frac{8}{p}}\right),\\ & \beta = \frac{1-\rho^}{4}\left(1+\sqrt{1+\frac{8}{p}}\right),\\ \end{aligned} \label{eq:ab_solution}$$ for $$\lambda = \frac{r}{2}\left[p\left(\alpha-\frac{1}{2}\right)^2 - 2\rho^ - \frac{(1-2\rho^)^2}{4p} \right]. \label{eq:lambda_exact}$$ Note that the eigenvalue that we found is the largest because the corresponding eigenfunction has no zeros for $\rho\in(0,1)$ [@titchmarsh:second_order_ode]. To ensure that equation (\[eq:lambda\_exact\]) indeed describes the fitness advantage of a mutant, we also need to check that the corresponding eigenfunction has a finite $\mathbf{L}^2$ norm; otherwise, it cannot serve as a basis vector in the Hilbert space. The integrability of $\varphi^2$ requires that $\alpha>1/2$ and $\beta>1/2$. We will discuss the consequences of these requirements for positive and negative $\rho^$ separately. For $\rho^\le0$, the condition on $\beta$ is always satisfied, while $\alpha>1/2$ only when $$p<p_{+\infty}=\frac{2(1+\rho^)^2}{-\rho^}. \label{eq:p_alpha}$$ As $p$ approaches $p_{+\infty}$ from below, the eigenfunction becomes localized at $\rho=0$, which corresponds to $\zeta\to+\infty$. Above $p_{+\infty}$, equation (\[eq:lambda\_exact\]) becomes invalid, and the eigenvalue is given by the limit of the potential term in equation (\[eq:eigen\_value\_hermitian\_simplified\]) as $\zeta\to+\infty$. The result reads $$\lambda_{+\infty}=\lim_{\zeta\to+\infty}\left(g(c(\zeta)) - \frac{v_{2}^{2}}{4D_1}\right) = - r\rho^ - \frac{v_{2}^{2}}{4D_1} = - r\rho^* - \frac{r(1-2\rho^)^2}{8p}>0. \label{eq:lambda_alpha}$$ Since $p_{+\infty}$ decreases from $+\infty$ to $1$ as $\rho^$ changes from $0$ to $-1/2$, we expect that, for $\rho^<-1/2$, the fitness advantage of the mutant is given by $\lambda_{+\infty}$ for $p>1$ as well as for $p$ above some critical value, which we also label $p_{+\infty}$ to extend our definition of this quantity to $\rho^<-1/2$. Note that, although we have not shown that this extension obeys equation (\[eq:p\_beta\]), this equation might still provide a good approximation for $p_{+\infty}$ given that it predicts that $p_{+\infty}$ approaches $0$ as $\rho^$ approaches $-1$. For $\rho^\ge0$, the condition on $\alpha$ is always satisfied, while $\beta>1/2$ only when $$p<p_{-\infty}=\frac{2(1-\rho^)^2}{\rho^}. \label{eq:p_beta}$$ As $p$ approaches $p_{-\infty}$ from below, the eigenfunction becomes localized at $\rho=1$, which corresponds to $\zeta\to-\infty$. Above $p_{-\infty}$, equation (\[eq:lambda\_exact\]) becomes invalid, and the eigenvalue is given by the limit of the potential term in equation (\[eq:eigen\_value\_hermitian\_simplified\]) as $\zeta\to-\infty$. The result reads $$\lambda_{-\infty}=\lim_{\zeta\to-\infty}\left(g(c(\zeta)) - \frac{v_{2}^{2}}{4D_1}\right)=- \frac{v_{2}^{2}}{4D_1} = - \frac{r(1-2\rho^)^2}{8p}<0. \label{eq:lambda_beta}$$ Collectively, equations (\[eq:lambda\_exact\]), (\[eq:lambda\_alpha\]), (\[eq:lambda\_beta\]) completely specify the exact solution for the fitness advantage of the mutant for $\rho^\in(-1/2,1/2)$. Unless specified otherwise, we will denote this solution as simply $\lambda$ regardless of whether it is specified by (\[eq:lambda\_exact\]) or equations (\[eq:lambda\_alpha\]) and (\[eq:lambda\_beta\]). The behavior of $\lambda$ as a function of $\rho^=c^/K$ and $p=D_1/D_2$ is discussed next. For $c^/K\in(-1/2,0)$, faster dispersal is always advantageous similar to the results for $c^<-K/2$. This result is expected because populations with negative $c^$ do not require a critical density for growth, and, thus, an organism dispersing very far ahead of the invasion front can still establish a viable population. Although positive, the selective advantage of faster dispersers does not increase monotonically with $D_1/D_2$ as it does for populations without an Allee effect. Instead, $\lambda$ has a maximum at a finite value of $p$ as shown in Fig. 2A in the main text. For $c^/K\in(0,1/4)$, we expect that very fast dispersers have a negative selective advantage because they cannot establish a viable population due to the strong Allee effect. Consistent with this expectation, equation (\[eq:lambda\_exact\]) predicts that $\lambda$ is negative for $D_1<D_2$, zero for $D_1=D_2$, and positive for $D_1/D_2\in(1,p_{0})$, where $p_{0}$ is another root of $\lambda(p)$ and is given by $$p_{0} = \frac{(1-2\rho^)^2}{\rho^}. \label{eq:p_0}$$ As $p$ is increased beyond $p_0$, the fitness advantage $\lambda$ becomes negative. In summary, faster dispersers do have a higher fitness unless they disperse too far ahead and suffer from the strong Allee effect. Although equation (\[eq:lambda\_exact\]) predicts another zero of $\lambda(p)$ at $$p_1 = \frac{2(2\rho^-1)^2}{3-5\rho^-\sqrt{9-34\rho^+41(\rho^{})^2-16(\rho^)^3}}, \label{eq:p_1}$$ and positive $\lambda$ for $D_1>p_1D_2$, we find that $p_1>p_{-\infty}$ and, thus, the eigenvalue is given by $\lambda_{-\infty}$, which is negative. Our simulations confirm that $\lambda$ is negative for all $D_1>p_0D_2$. The remaining region, $c^/K\in(1/4,1/2)$, favors slower dispersers with $D_1/D_2\in(p_0,1)$ while mutants that disperse either too slowly or faster than the wild type have a negative selective advantage. Note that as $\rho^$ is increased above $1/4$, $p_0$ becomes less than $1$ and the region of $D_1/D_2$ that is favored by natural selection shifts from just above $1$ to just below $1$. Similar to the situation we just discussed, $p_1>p_{-\infty}$, and, thus, $\lambda$ remain negative for all $D_1>D_2$. These results are summarized graphically in Fig. \[fig:phase\_diagram\], which shows the regions in the space of $\rho^$ and $p$ favoring faster or slower dispersal. ![The fate of a mutant depends both on its dispersal abilities and the Allee effect in the population growth. Advantageous mutants are in red and deleterious are in blue. Note that the exchange of the types corresponds to an inversion in the point $(c^/K=1/4,\;p=1)$ only for $D_1\approx D_2$. This phase diagram is drawn based on the exact solution described in this Supplemental Material. []{data-label="fig:phase_diagram"}](figs1.eps){width="0.5\columnwidth"} Finally, we analyze how the density profile of the mutant depends on $\zeta$. The shape of this density profile can be obtained at a single time point and, therefore, could be of great utility in practice because it contains information about the fitness advantage of a mutant and does not require time series data. To compute $f(\zeta)$ we need to combine the eigenfunction that we found above as well as the transformations from $f$ to $\varphi$; the result reads: $$f \propto \rho^{\alpha-\frac{3}{2}+\frac{1-2\rho^}{2p}}(1-\rho)^{\beta-\frac{1}{2}-\frac{1-2\rho^}{2p}}. \label{eq:f_shape}$$ For negative $c^$, $f(\zeta)$ is monotonically increasing from $0$ (at $\zeta=-\infty$) to $+\infty$ (at $\zeta=+\infty$) for $D_1>D_2$ and monotonically decreasing from $+\infty$ (at $\zeta=-\infty$) to $0$ (at $\zeta=+\infty$) for $D_1<D_2$. For $\rho^\in(0,1/4)$, the behavior is the same as above for $D_1<D_2$ and $D_1\in(1,p_0D_2)$. For $D_1>p_0D_2$, the profile of $f(\zeta)$ has a minimum in the region of the front, i.e. around $\zeta=0$, and diverges to $+\infty$ as $\zeta\to\pm\infty$. For $\rho^\in(1/4,1/2)$, $f(\zeta)$ monotonically decreases from $+\infty$ (at $\zeta=-\infty$) to $0$ (at $\zeta=+\infty$) for $D_1<p_0D_2$, then, for $D_1/D_2\in(p_0,1)$ it becomes peaked around the front, i.e. close to $\zeta=0$, and decays to $0$ as $\zeta\to\pm\infty$, while, for $D_1>D_2$, $f(\zeta)$ has a minimum around $\zeta=0$ and diverges to $+\infty$ as $\zeta\to\pm\infty$. Supplemental Discussion Effects of different growth rates\ Mutations that change the rate of dispersal can also affect the growth rate. Although the dispersal-survival and dispersal-fecundity trade-offs have been most heavily documented, other possibilities exist including potentially a reduction in feeding ability due to limited dispersal [@tobin:moth_review]. Here, we outline how the differences in growth rates can be included in our theory and show that, under certain simplifying assumptions, the net rate of increase of a mutant is simply given by the sum of the difference in the growth rates and $\lambda$ due to the differences in the dispersal abilities, which we obtained above. When types grow at different rates $g_{1}(c_1,c_2)$ and $g_{2}(c_1,c_2)$, Eq. (\[eq:model\]) needs to be modified as $$\begin{aligned} & \frac{\partial c_{1} }{\partial t} = D_{1}\frac{\partial^2 c_{1}}{\partial x^2} + c_{1}g_{1}(c_1,c_2),\\ & \frac{\partial c_{2} }{\partial t} = D_{2}\frac{\partial^2 c_{2}}{\partial x^2} + c_{2}g_{2}(c_1,c_2). \end{aligned} \label{eq:model_dgr}$$ Assuming $f\ll1$ and repeating the steps leading to Eq. (\[eq:f\_comoving\]), we obtain $$\frac{\partial f}{\partial \tau} = D_{1}\frac{\partial^2 f}{\partial \zeta^2} + \left(v_2 + \frac{2D_1}{c}\frac{\partial c}{\partial \zeta}\right)\frac{\partial f}{\partial \zeta} + f \frac{\Delta D}{c}\frac{\partial^2 c}{\partial \zeta^2} + f(g_{1}(c)-g_{2}(c)), \label{eq:f_comoving_dgr}$$ which leads to a new eigenvalue problem with an additional potential term due to $g_{1}(c)-g_{2}(c)\ne0$. Note that the new term depends only on the total population density because $c_1\ll c_2$. The solution to this problem depends on the functional form of the difference in the growth rates. It is important to emphasize that, generically, the difference in the growth rate is not a number, but rather a function of the population density. This fact substantially complicates the analysis because greater dispersal can come at the cost of lower growth rates at low population densities or at high population densities. Even more complicated, a mutant can have a reduced growth rate at low densities, but an increased growth rate at high densities. Therefore, we will limit our discussion to two general observations. First, if $g_{1}(c)-g_{2}(c)=s$, where $s$ is a constant fitness advantage, then the additional potential term is just a constant, and the resulting eigenvalue is given by the sum of $s$ and $\lambda$ obtained previously. Thus, for density-independent changes in the growth rate, the different components of fitness simply add. Second, when the difference in the growth rates is not constant, its contribution to the eigenvalue will be given by the average of the new term in the potential over the corresponding eigenfunction. Therefore, we can be certain that an increase in the growth rate leads to a higher eigenvalue while a decrease in the growth rate leads to a lower eigenvalue. In sum, the effect of different growth rates is more intuitive than the effect of different dispersal rates, and, in the simplest situation, one can just add the different fitness components. The direction of natural selection at non-vanishing mutant fractions\ In our simulations, we found that the sign of $\lambda$ fully determines the fate of the mutant even when its fraction becomes no longer small to justify the linearization, which we used to obtain the analytical solution. In particular, we never observed the coexistence of two types due to a change in the direction of natural selection as $f$ grows from $0$ to $1$. For mutants with $\lambda>0$, these findings can be understood by considering population dynamics when the mutant fraction approaches $1$. When $f$ is large, the relative fraction of the ancestor is small, and it is the dispersal rate of the mutant that determines the expansion velocity and the shape of the expansion front. Therefore, the roles of the mutant and the ancestor are reversed, and we can obtain the growth rate of the ancestor from our exact solution by exchanging $D_1$ and $D_2$. As illustrated in Fig. 3, the swap of the dispersal rates always converts a positive $\lambda$ to a negative $\lambda$. In consequence, the ancestor will have a negative growth rate when the beneficial mutant is in the majority, i.e. the direction of natural selection remains the same for both small and large $f$. A more intuitive explanation comes from Fig. \[fig:phase\_diagram\], which shows that whether faster or slower dispersal is favored depends only on $c^/K$. Therefore, only the magnitude, but not the direction of the selection changes as the mutant becomes established. At intermediate mutant frequencies, the increase in $f$ occurs due to the competition between the new mutant offspring with the dispersal rate $D_1$ and the resident population (consisting of both the mutant and the ancestor) with the dispersal rate qualitatively given by $fD_1+(1-f)D_2$. Since the difference between the dispersal rate of the mutant and effective dispersal rate of the resident population remains of the same sign, the mutant offspring continue to outcompete the ancestor as $f$ increases. We also note that mutants with negative $\lambda$ do not reach high enough population densities to violate our linearization assumption, so their dynamics are also fully determined by the direction of selection at low mutant frequencies. Stochastic effects due to mutations and genetic drift\ The analytical solution that we presented in this Supplemental Material does not account for the effects of mutations and the stochastic effects due to the randomness of births and deaths. Our simulations, however, include both of these processes, and the agreement between the theory and the simulations demonstrates that our conclusions are robust to the vagaries of mutations and genetic drift. Here, we explain why our deterministic theory is sufficient to capture the essence of the evolutionary process. We first consider the effects of mutations. Mutations that create disadvantageous types lead to a background level of genotypes with lower fitness similar to the mutation-selection balance in well-mixed populations [@gillespie:book]. Repeated advantageous mutations also have a negligible effect on the individual dynamics of these mutants. Indeed, the population densities of these mutants do not affect each other’s growth because the mutants are rare initially and compete with the resident type rather than with each other. This decoupling of mutant evolution will cease once the densities of the mutants become large, and we expect that the fitter mutant will win the competition. Second, we describe the effects of genetic drift (demographic fluctuations), which can lead to the extinction of a beneficial mutant in both spatial and well-mixed populations [@gillespie:book]. In well-mixed populations, the establishment or fixation probability of a mutant depends only on its fitness advantage and the strength of genetic drift. In expanding populations, the location of the mutant also affects its ultimate fate. For example, if the initial conditions are chosen to be orthogonal to the eigenfunction with the largest eigenvalue, then the mutant will not grow at the rate given by the largest eigenvalue. Generic initial conditions, including the ones due to a spontaneous mutation, however, have a nonzero projection on the leading eigenfunction, and the mutant will eventually grow at a rate given by the largest eigenvalue, even if the net population density may slightly decline initially. The magnitude of this projection determines the time that the mutant spends at low densities and therefore experiences strong fluctuations, which can drive it to extinction. As a result, the fixation probability of a mutant depends not only on the magnitude of $\lambda$, but also on the point of origin. Mutants that occur near the maximum of the leading eigenfunction have a larger fixation probability compared to mutants occurring away from this maximum, for example well at the back of the front. This is a well-known issue in spatial populations and is not specific to our analysis [@hallatschek:diversity; @lehe:surfing]. The net effect on species evolution is that only mutations occurring within a certain spatial region are contributing to the adaptation. This is a reassuring conclusions; otherwise, the adaptation of large, spatially extended populations would be extremely rapid. Symmetry of invasion fitnesses\ The rate at which the mutant type grows in the background of the wild type is known as the invasion fitness in the field of adaptive dynamics [@waxman:review_adaptive_dynamics]. Here, we discuss the properties of invasion fitness under the exchange of the types: type two invading type one instead of type one invading type two. Concretely, for all values of $\rho^$ and $p\approx1$, our exact solution shows that the fitness advantage of type one in the background of type two equals the fitness disadvantage of type two in the background of type one. However, when the dispersal rates are very unequal, we observe an asymmetry under the exchange of $D_1$ and $D_2$. The reason for this asymmetry is that the fitness is the property of both an organism and its environment, which, in this case, is the presence of a competing type. The interaction between the organism and its environment is in general nonlinear and results in asymmetric rates of invasion. Thus, our results illustrate that symmetric invasion fitnesses do not fully capture the complexity of phenotype evolution. Robustness of results to model assumptions\ In range expansions, one typically distinguishes between short-range and long-range dispersal kernels. Short-range kernels are described well by reaction-diffusion equations at long spatial and temporal scales due to the central limit theorem. Indeed, our numerical simulations with discrete jumps between nearest neighbors give the same results as the analytic solution of the continuous equations. The effects of long-range dispersal are more subtle, and, fortunately, there are few invaders with the capabilities to travel long range. For so called pulled expansions, which occur when the Allee effect is absent or weak, long-range dispersal can lead to expansion acceleration over time [@kot:csdt]. Although such dynamics are not captured by our analysis, it is important to emphasize that they lead to the same evolutionary outcome as we described in this paper. Our analysis predicts that faster dispersers would be favored in this regime (Fig. \[fig:phase\_diagram\]). The same dynamics are expected for species with long-range dispersal because organisms that land far away from the ancestral population will be able to start a new invasion and thus colonize all the areas ahead of the expansion front. When the Allee effect is strong, long-range dispersal does not lead to qualitatively new dynamics like wave acceleration because the organisms that disperse too far find themselves at densities below the Allee threshold and, therefore, fail to establish and start a new expansion [@kot:csdt]. Because these long-range dispersal events effectively lead to death, mutants that increase their dispersal rate will die more frequently and would not be selected, similar to the conclusion of our analysis. In summary, different dispersal kernels should not lead to qualitatively new results. The quantitative results will of course be different since they depend on the model details including the type of dispersal and the form of $g(c)$. Simulations Numerical solution of equations (\[eq:model\])\ We solved equations (\[eq:model\]) using an explicit finite difference method with the following discretization: $$\begin{aligned} & c_{1}(t+\Delta t,x) = c_{1}(t,x) + D_{1}\frac{c_{1}(t,x+\Delta x) + c_{1}(t, x-\Delta x) -2c_{1}(t,x)}{\Delta x^2}\Delta t + c_{1}g(c)\Delta t,\\ & c_{2}(t+\Delta t,x) = c_{2}(t,x) + D_{2}\frac{c_{2}(t,x+\Delta x) + c_{2}(t, x-\Delta x) -2c_{2}(t,x)}{\Delta x^2}\Delta t + c_{2}g(c)\Delta t. \end{aligned} \label{eq:model_discretized}$$ Equations (\[eq:model\_discretized\]) were iterated in a spatial domain of length at least $35$ with at least $700$ discretization points. We kept $D_2=1$, $g=1$, $K=1$ constant and varied $D_1$ and $c^$. The temporal discretization $\Delta t$ was set to $0.01\Delta x^2$ to ensure both accuracy and stability of the numerical algorithm. The temporal duration of the simulation was varied with $\Delta D$ to ensure that sufficient data are available to estimate the fitness difference between the strains. The simulations were started with the left half of the habitat occupied by the species with $c=K$. The fraction of the first type was set to $10^{-2}$ or $10^{-3}$ uniformly in space. As the expansion approached the right side of the simulation domain, we shifted the simulation domain to recenter the population. For each simulation, we confirmed that the expansion velocities agreed with equation (\[eq:profile\]) within 1% error. Then, the fraction of the first mutant $f(t)$ was estimated as the average $f(t,x)$ across all the discrete points where the solution was computed. To obtain the selective advantage of the first species $\lambda$, we fitted $\ln(f(t)/(1-f(t))$ to $\lambda t+\mathrm{const}$. Stochastic simulations\ Stochastic simulations were implemented as the stepping-stone model [@kimura:ssm] and Levins’ metapopulation model [@levins:model] with discrete generations, which relaxed the continuity assumption of equations (\[eq:model\]) and allowed for landscape fragmentation. Each generation consisted of a growth and dispersal phases. The dispersal phase, allowed each organism to migrate to one of the two nearby patches with identical probability equal to $m/2$. The growth phase was implemented as a birth-death process with the per capita birth rate equal to $rc/K + rcc^/K^2$ and the per capita death rate equal to $rc^2/K^2 + rc^*/K$, which lead to the dynamics described by equation (\[eq:growth\]) in the continuous limit. The dispersal rate of each newly-born organism could be mutated with probability $\mu=10^{-4}$ to either increase or decrease by 10%. We capped the maximal dispersal rate at $m=0.5$. Similar to the deterministic simulations, we re-centered the simulation box when necessary. The number of patches was 150 and the carrying capacity was $10^4$.
--- abstract: 'Based on the non-equilibrium Green’s function (NEGF) technique and the Landauer-Büttiker theory, the possibility of a molecular spin-electronic device, which consists of a single C$_{60}$ molecule attached to two ferromagnetic electrodes with finite cross sections, is investigated. By studying the coherent spin-dependent transport through the energy levels of the molecule, it is shown that the tunnel magnetoresistance (TMR) of the molecular junction depends on the applied voltages and the number of contact points between the device electrodes and the molecule. The TMR values more than 60% are obtained by adjusting the related parameters.' author: - 'Alireza Saffarzadeh$^{1,2,}$' title: 'Tunnel Magnetoresistance of a Single-Molecule Junction' --- Introduction ============ Traditional magnetic tunnel junctions use inorganic insulators as spacers [@Moodera1; @Miyazaki]. However, recent experimental data have clearly shown that organic molecules can serve the same purpose and a rather large TMR can be found [@Tsukagoshi; @Xiong; @Dediu; @Shim; @Santos; @Ouyang; @Petta]. In these experiments, spin polarized transport through molecular layers sandwiched between two magnetic layers has been demonstrated for systems involving carbon nanotubes [@Tsukagoshi], other organic $\pi$ conjugate systems [@Xiong; @Dediu; @Shim; @Santos], molecular bridges [@Ouyang], and self-assembled organic monolayers [@Petta]. Organic materials have relatively weak spin-orbit interaction and weak hyperfine interaction, so that spin memory can be as long as a few seconds [@Sanvito]. Such features make them ideal for spin-polarized electron injection and transport applications in molecular spintronics. Spin-polarized transport through organic molecules sandwiched between two magnetic contacts has also been recently investigated theoretically [@Pati; @Senapati; @Dalgleish; @Liu; @Rocha; @Waldron; @Wang; @He; @Ning; @Wang2; @Mehrez; @Krompiewski; @Emberly]. Many of these calculations, based on density functional theory or tight-binding model, have shown that by changing the magnetic alignment of the contacts one can substantially affect the electronic current in the molecular devices. For instance, in a single benzene-1-4-dithiolate molecule sandwiched between two Ni clusters, parallel magnetic alignment led to significantly higher current (about one order of magnitude) than antiparallel alignment, suggesting the possibility of a molecular spin valve [@Pati]. Among many types of molecules, the fullerene C$_{60}$ which is one of the most well-investigated organic semiconductors, is suitable as a molecular bridge in magnetic tunnel junctions, because its lowest unoccupied molecular orbital (LUMO) is situated at relatively lower energies in comparison with the other organic molecules. In recent years, the electrical and magnetic properties of C$_{60}$-Co nanocomposites, where Co nanoparticles are dispersed in C$_{60}$ molecules have been studied [@Zare; @Sakai; @Miwa], and the maximum TMR ratio of about 30% at low bias voltages was reported. Recently, He et al. [@He2], studied the spin-polarized transport in Ni/C$_{60}$/Ni junction using density functional theory and the Landauer-Büttiker formalism. They showed that the binding sites of Ni on C$_{60}$ molecule play a crucial role in the transport properties of the system and a large value for junction magnetoresistance was predicted. We believe, however, that no theoretical study on spin-polarized transport through a single C$_{60}$ molecule and its TMR effect, based on tight-binding method, has so far been reported. ![Schematic view of the FM/C$_{60}$/FM molecular junction at zero voltages for (a) parallel, and (b) antiparallel alignments of the magnetizations in the FM electrodes. (c)-(e) The three different ways of coupling between the C$_{60}$ molecule and the magnetic electrodes used in this work. Black circles show the position and the number of couplings.](Fig1.eps){width="0.9\linewidth"} In this paper, we use a single C$_{60}$ molecule in between two ferromagnetic (FM) electrodes, instead of the usual insulator layer as in the standard TMR setup, to investigate the possibility of a single-molecule spintronics device. An interesting feature of the C$_{60}$ molecule is the emergence of a quantum loop current which is related to the degeneracy of the energy levels of the molecule and its magnitude can be much larger than that of the source-drain current [@Naka]. This feature is useful for spin-polarized transport through the molecule. Our approach indicates that the spin currents in such a molecular junction are mainly controlled by the molecular field of the FM electrodes, the electronic structure of the molecule and its coupling to the electrodes. Model and formalism =================== We calculate the spin currents through a single C$_{60}$ molecule, sandwiched between two semi-infinite FM electrodes with simple cubic structure and square cross section ($x$-$y$ plane). The model of such a structure is shown schematically in Fig. 1(a) and 1(b). Since the electron conduction is mainly determined by the central part of the junction, the electronic structure of this part should be resolved in detail. It is therefore reasonable to decompose the total Hamiltonian of the system as $$\hat{H}=\hat{H}_{L}+\hat{H}_{C}+\hat{H}_{R}+\hat{H}_{T}\ .$$ The Hamiltonian of the left ($L$) and right ($R$) electrodes is described within the single-band tight-binding approximation and is written as $$\hat{H}_{\alpha}=\sum_{i_\alpha,\sigma}(\epsilon_{0}-\mathbf{\sigma}\cdot \mathbf{h}_{\alpha})\hat{c}_{i_\alpha,\sigma}^\dag\hat{c}_{i_\alpha,\sigma}-\sum_{<i_\alpha, j_\alpha>,\sigma}t_{i_\alpha,j_\alpha}\hat{c}_{i_\alpha,\sigma}^\dag\hat{c}_{j_\alpha,\sigma}\ ,$$ where $\hat{c}_{i_\alpha,\sigma}^\dag$ ($\hat{c}_{i_\alpha,\sigma}$) creates (destroys) an electron with spin $\sigma$ at site $i$ in electrode $\alpha$ (=$L$, $R$), the hoping parameter $t_{i_\alpha,j_\alpha}$ is equal to $t$ for the nearest neighbors and zero otherwise. Here, $\epsilon_{0}$ is the spin independent on-site energy and will be set to $3t$ as a shift in energy, $-\mathbf{\sigma}\cdot \mathbf{h}_{\alpha}$ is the internal exchange energy with $\mathbf{h}_{\alpha}$ denoting the molecular field at site $i_\alpha$, and $\mathbf{\sigma}$ being the conventional Pauli spin operator. The Hamiltonian of the C$_{60}$ molecule in the absence of FM electrodes is expressed as $$\hat{H}_{C}=\sum_{i_C,\sigma}\epsilon_{i_C}\hat{d}_{i_C,\sigma}^\dag\hat{d}_{i_C,\sigma}- \sum_{<i_C,j_C>,\sigma}t_{i_C,j_C}\hat{d}_{i_C,\sigma}^\dag\hat{d}_{j_C,\sigma}\ ,$$ where $\hat{d}_{i_C,\sigma}^\dag$ ($\hat{d}_{i_C,\sigma}$) creates (destroys) an electron with spin $\sigma$ at site $i$ of C$_{60}$ and $\epsilon_{i_C}$ is the on-site energy and will be set to zero except in the presence of gate voltage $V_G$ that shifts the energy levels of the molecule and hence $\epsilon_{i_C}=V_G$. The hoping strength $t_{i_C,j_C}$ in C$_{60}$ molecule depends on the C-C bond length; thus, we assume different hoping matrix elements: $t_1$ for the single bonds and $t_2$ for the double bonds. Most recently, we have shown that the effect of bond dimerization may considerably affect the electron conduction through the molecule under suitable conditions [@Saffar1]. Finally, $\hat{H}_T$ describes the coupling between the FM leads and the molecule and takes the form $$\hat{H}_{T}=-\sum_{\alpha=\{L,R\}}\sum_{i_\alpha, j_C,\sigma}t_{i_\alpha,j_C}(\hat{c}_{i_\alpha,\sigma}^\dag\hat{d}_{j_C,\sigma}+\textrm{H.c.})\ .$$ The hopping elements $t_{i_\alpha,j_C}$ between the lead orbitals and the $\pi$ orbitals of the molecule are taken to be $t'$. In this study we assume that the electrons freely propagate and the only resistance arising from the contacts. This means that the transport is ballistic [@Datta]; therefore, we set $t'=0.5 t$, because the value of $t'$ should not be smaller than the order of $t$. On the other hand, we assume that the spin direction of the electron is conserved in the tunneling process through the molecule. Therefore, there is no spin-flip scattering and the spin-dependent transport can be decoupled into two spin currents: one for spin-up and the other for spin-down. This assumption is well-justified since the spin diffusion length in organics is about 4 nm [@Sanvito] and especially in carbon nanotubes is at least 130 nm [@Tsukagoshi], which are greater than the diameter of C$_{60}$ molecule ($\sim$ 0.7 nm). Since the total Hamiltonian does not contain inelastic scatterings, the spin currents for a constant bias voltage, $V_a$, are calculated by the Landauer-Büttiker formula based on the NEGF method [@Datta]: $$\label{I} I_\sigma(V_a)=\frac{e}{h}\int_{-\infty}^{\infty} T_\sigma(\epsilon,V_a)[f(\epsilon-\mu_L)-f(\epsilon-\mu_R)]d\epsilon \ ,$$ where $f$ is the Fermi distribution function, $\mu_{L,R}=E_F\pm \frac{1}{2}eV_a$ are the chemical potentials of the electrodes, and $T_\sigma(\epsilon,V_a)=\mathrm{Tr}[\hat{\Gamma}_{L,\sigma} \hat{G}_\sigma \hat{\Gamma}_{R,\sigma}\hat{G}_\sigma^{\dagger}]$ is the spin-, energy- and voltage-dependent transmission function. The spin-dependent Green’s function of the C$_{60}$ molecule coupled to the two FM electrodes (source and drain) in the presence of the bias voltage is given as $$\hat{G}_\sigma(\epsilon,V_a)=[\epsilon \hat{1} -\hat{H}_C-\hat{\Sigma}_{L,\sigma}(\epsilon-eV_a/2) -\hat{\Sigma}_{R,\sigma}(\epsilon+eV_a/2)]^{-1}\ ,$$ where $\hat{\Sigma}_{L,\sigma}$ and $\hat{\Sigma}_{R,\sigma}$ describe the self-energy matrices which contain the information of the electronic structure of the FM electrodes and their coupling to the molecule. These can be expressed as $ \hat{\Sigma}_{\alpha,\sigma}(\epsilon)= \hat{\tau}_{C,\alpha}\hat{g}_{\alpha,\sigma}(\epsilon)\hat{\tau}_{\alpha,C}$ where $\hat{\tau}$ is the hopping matrix that couples the molecule to the leads and is determined by the geometry of the molecule-lead bond. $\hat{g}_{\alpha,\sigma}$ are the surface Green’s functions of the uncoupled leads i.e., the left and right semi-infinite magnetic electrodes, and their matrix elements are given by $$\label{g} g_{\alpha,\sigma}(i,j;z)=\sum_{\mathbf{k}}\frac{\psi_\mathbf{k}(\mathbf{r}_i)\psi^_\mathbf{k}(\mathbf{r}_j)} {z-\epsilon_0+\mathbf{\sigma}\cdot \mathbf{h}_{\alpha}+\varepsilon(\mathbf{k})}\ ,$$ where $\mathbf{r}_i\equiv(x_i,y_i,z_i)$, $\mathbf{k}\equiv(l_x,l_y,k_z)$, $z=\epsilon+i\delta$, $$\psi_\mathbf{k}(\mathbf{r}_i)=\frac{2\sqrt{2}}{\sqrt{(N_x+1)(N_y+1)N_z}} \sin(\frac{l_xx_i\pi}{N_x+1})\sin(\frac{l_yy_i\pi}{N_y+1})\sin(k_zz_i)\ ,$$ and $$\varepsilon(\mathbf{k})=2t[\cos(\frac{l_x\pi}{N_x+1})+ \cos(\frac{l_y\pi}{N_y+1})+\cos(k_za)]\ .$$ Here, $l_{x,y}$ ($=1,...,N_{x,y}$) are integers, $k_z\in[-\frac{\pi}{a},\frac{\pi}{a}]$, and $N_\beta$ with $\beta =x, y, z$ is the number of lattice sites in the $\beta$ direction. Note that $N_x$ and $N_y$ correspond to the number of atoms at the cross-section of the FM electrodes. Using $\hat{\Sigma}_{\alpha,\sigma}$, the coupling matrices $\hat{\Gamma}_{\alpha,\sigma}$, also known as the broadening functions, can be expressed as $\hat{\Gamma}_{\alpha,\sigma}=-2\mathrm{Im}(\hat{\Sigma}_{\alpha,\sigma})$. ![The total currents at $V_G$=0.0 V for the parallel (a) and antiparallel (b) alignments of magnetizations in the FM electrodes, and (c) the TMR as a function of applied voltage.](Fig2.eps){width="0.9\linewidth"} In the semi-infinite FM electrodes described by the single-band tight-binding model, only the central site at the cross section is connected to the molecule. Furthermore, when the molecule is brought close to an electrode, the bonding between them will depend on the molecule orientation. This orientation can be such that only one carbon atom, a pentagon or a hexagon of the C$_{60}$ molecule be in contact with the leads \[see Figs. 1(c)-1(e)\]. Therefore, one can expect different conduction through the molecule, which arises due to the resonant tunneling and the quantum interference effects. Our approach, as a real-space method, makes it possible to model arbitrarily the number of contacts. In this regard, the core of the problem lies in the calculation of the spin-dependent self-energies $\hat{\Sigma}_{L,\sigma}$ and $\hat{\Sigma}_{R,\sigma}$. In the case of contact through a single carbon atom of the molecule, only one element of the self-energy matrices is non-zero. However, for the transport through opposite pentagons or hexagons, 25 or 36 elements of the self-energy matrices are non-zero, respectively. The total charge current is given by $I=I_\uparrow+I_\downarrow$. In this case, we calculate the TMR ratio from the usual definition: $\mathrm{TMR}\equiv(I_p-I_a)/I_p$, where $I_{p,a}$ are the total currents in the parallel and antiparallel alignments of magnetizations in the FM electrodes, respectively. Results and discussion ====================== We now use the method described above to study the coherent spin-dependent transport and magnetoresistance effect of FM/C$_{60}$/FM molecular junction. We have done the numerical calculations for the case that the direction of magnetization in the left FM electrode is fixed in the +$y$ direction, while the magnetization in the right electrode is free to be flipped into either the +$y$ or -$y$ direction. We set $\|\mathbf{h}_{\alpha}\|$=1.5 eV, $t$=1 eV, $t_1=t$, $t_2=1.1\,t$, $N_x=N_y=5$, and $T$=300 K in the calculations. In Fig. 2 we show the current-voltage ($I$-$V$) characteristics (in the parallel and antiparallel alignments) and also the TMR as a function of applied voltage for three different ways of coupling between the C$_{60}$ molecule and the FM electrodes. These three cases were chosen as the most probable experimental orientations [@Hashizume]. As can be seen, the $I$-$V$ curves show a steplike behavior which indicates that a new channel is opened. The values of currents in the parallel configuration are nearly two times larger than that in the case of antiparallel one. The difference between $I_p$ and $I_a$ is related to the asymmetry of surface density of states (SDOS) of the FM electrodes for spin-up and spin-down electrons \[see Figs. 1(a) and 1(b)\][@Explain], and the quantum tunneling phenomenon through the molecule. Here, we have assumed that only itinerant $\textit{sp}$-like electrons contribute to the tunneling current, which is reasonable when the distance between the two FM electrodes is greater than 0.5 nm [@Munzenberg]. In the parallel alignment, minority electrons go into the minority states by tunneling through the molecule and there is no asymmetry in the SDOS. If, however, the two FM electrodes are magnetized in opposite directions, the minority (majority) electrons from the left electrode seek empty majority (minority) states in the right electrode, that is, asymmetry in the SDOS. In fact, in the antiparallel alignment, the tunneling currents for both spin channels are asymmetric with respect to voltage inversion (not shown). Consequently, the parallel arrangement gives much higher total current through the C$_{60}$ molecule than does the antiparallel arrangement. This difference in the total currents is the origin of TMR effect which has been shown in Fig. 2(c). The TMR ratio has it maximum value (more than 60%) at low bias voltages. With increasing the applied voltage, we first observe that the TMR decreases. Such a behavior is similar to the conventional magnetic tunnel junctions [@Moodera3]. With further increase in the bias voltage, the TMR ratio increases and then, the reduction and enhancement of TMR are repeated. Recently, using C$_{60}$-Co nanocomposites, Miwa [et al*]{}. [@Miwa] measured a magnetoresistance value as large as 18% due to the spin-polarized tunneling of carriers between Co nanoparticles via C$_{60}$ molecules. Our bias-voltage dependence of TMR is qualitatively in agreement with their results. A similar bias-dependent behavior in which TMR ratio increases with applied voltage, using a EuS spin-filter tunnel barrier in conventional junctions, has also been reported [@Nagahama2]. ![The total currents at $V_a$=0.1 V for the parallel (a) and antiparallel (b) alignments of magnetizations in the FM electrodes, and (c) the TMR as a function of gate voltage.](Fig3.eps){width="0.9\linewidth"} In order to investigate the other features of the junction, we show in Fig. 3 the effects of gate voltage on the total currents and the TMR in the cases of single and multiple contacts. When the gate voltage is zero, we showed in Fig. 2(c) that the maximum values for the TMR are obtained at low applied voltages ($V_a\leq 0.1$ V). In such cases, the number of states contained in the energy window between $\mu_L$ and $\mu_R$ is zero, because the energy window lies in the highest occupied molecular orbital (HOMO)-LUMO gap of the C$_{60}$ molecule \[Figs. 1(a) and 1(b)\], where there is no molecular level. Therefore, the current flow mechanism is tunneling. For this reason, the voltage dependence of the TMR effect at low voltages is similar to that of the conventional magnetic tunnel junctions. It is worth mentioning that in the selected voltage interval, we did not observe significant changes in the transmission spectra with increasing the bias voltage. Applying a gate voltage shifts the molecular levels relative to the Fermi level of electrodes, and hence the transmission coefficients may significantly vary [@Saffar1]. Since the currents depend on the molecular density of states lying between $\mu_L$ and $\mu_R$, then, by increasing the gate voltage, HOMO or LUMO peak moves inside the energy window and the total currents $I_p$ and $I_a$ increase. In this case, for either negative or positive gate voltages, the current flow mechanism is resonant tunneling, and a peak appears in the current curves. Figures 3(a) and 3(b) show that the currents vanish in a wide range of gate voltages, because in these voltages there is no resonant level inside the energy window. In both figures, the peaks appear at the same voltages which confirm that the appearance of current peaks is due to the molecular levels. However, the height of peaks is different which is due to the difference in the spin-dependent SDOS of the FM electrodes. The results suggest that the C$_{60}$ molecule is an interesting candidate for operation of devices as a nano-scale current switch. Also, with increasing the number of contact points between the device electrodes and the molecule, the interference effects around these points become important, some resonances might completely disappear, and the spin current changes. This is the reason of difference between the currents in the single and multiple contacts \[see Figs. 2 and 3\] [@Saffar1; @Paul; @Palacios]. Conclusion ========== Using the NEGF method and the Landauer-Büttiker theory, we have investigated the possibility of making a C$_{60}$-based magnetic tunnel junction. We have shown that coupling between the single C$_{60}$ molecule and magnetic electrodes in FM/C$_{60}$/FM structure, produces large magnetoresistance effects (greater than 60%) that can be modulated by using the molecule orientation, bias and gate voltages to control the spin-polarized transport. The present study advances the fundamental understanding of spin-dependent transport in molecular junctions and suggests that the C$_{60}$ molecule is an interesting candidate for application in the magnetic memory cells and spintronic devices. Throughout this study, we have ignored the effects of inelastic scattering and the magnetic anisotropy of the FM electrodes due to the reduced dimensionality and the geometry of the system. These factors can affect the spin-dependent transport and hence, another improved approach is needed for more accurate results. Acknowledgement {#acknowledgement .unnumbered} =============== The author thanks Professor J.S. Moodera for valuable comments. This work was supported by Payame Noor University grant. [99]{}
--- abstract: 'A semi analytic framework for simulating the effects of atmospheric seeing in Adaptive Optics systems on an 8-m telescope is developed with the intention of understanding the origin of the wind-butterfly, a characteristic two-lobed halo in the PSF of AO imaging. Simulations show that errors in the compensated phase on the aperture due to servo-lag have preferential direction orthogonal to the direction of wind propagation which, when Fourier Transformed into the image plane, appear with their characteristic lemniscate shape along the wind direction. We develop a metric to quantify the effect of this aberration with the fractional standard deviation in an annulus centered around the PSF, and use telescope pointing to correlate this effect with data from an atmospheric models, the NOAA GFS. Our results show that the jet stream at altitudes of 100-200 hPa (equivalently 10-15 km above sea level) is highly correlated (13.2$\sigma$) with the strong butterfly, while the ground wind and other layers are more or less uncorrelated.' author: - Alexander Madurowicz - 'Bruce A. Macintosh' - 'Jean-Baptiste Ruffio' - Jeffery Chilcote - 'Vanessa P. Bailey' - Lisa Poyneer - Eric Nielsen - 'Andrew P. Norton' bibliography: - 'report.bib' title: Characterization of lemniscate atmospheric aberrations in Gemini Planet Imager data --- Introduction ============ The Gemini Planet Imager (GPI) is an instrument installed on the Gemini South Telescope in Cerro Pachon, Chile, designed to search for thermal emission from young hot extrasolar planets at wide angular separation[@Macintosh2014]. The GPI Exoplanet Survey (GPIES) is well under way, with many successful detections[@Macintosh2015], but contrast remains limited by residual atmospheric aberrations[@Ruffio2017][@Poyneer2016] and imperfections in the Adaptive Optics (AO) system[@MalesGuyon17]. Notably, the presence of the wind butterfly (so-called because of its figure-8 or lemniscate shape) scatters significant light into the coronagraphic dark hole and causes the residual point-spread function (PSF) to break azimuthal symmetry in the image plane, which reduces the final contrast ratio. This PSF pattern is consistent with wavefront errors from servo lag[@Rigaut1998]. A $\sim$$2$ ms delay in positioning of the deformable mirror in response to the wavefront sensor causes a displacement between the actual phase errors on the aperture from atmospheric turbulence when compared to the phase of the applied correction. The Fourier transform of this particular pattern has preferential direction, and appears in the image plane as the wind butterfly. In this paper, we will demonstrate the servo-lag error as the origin of the wind butterfly in simulations of AO systems based on a theoretical turbulence model that is well-verified with a Kolmogorov power spectrum, as well as presenting our results in analyzing the appearances of the wind butterfly in the GPI Exoplanet Survey data. We have developed a metric to identify image subsets where this effect is highly apparent. The fractional standard deviation in an annulus centered about the PSF is zero in the case in azimuthal symmetry, and approaches one for cases of extreme azimuthal asymmetry. Using a chi-squared minimization routine with a model of the image’s azimuthal dependence, we can extract the wind direction in the image plane, which, along with telescope pointing information, identifies the wind vector in three dimensions. This allows us explore correlations between the wind vector as pointed by the butterfly, and the direction of high-altitude winds from the NOAA GFS database. Evaluating the azimuthal variations in planet detectability caused by the butterfly will help us to evaluate survey sensitivity and potential improvements from faster AO correction. Butterflies in Simulations ========================== In order to motivate understanding the appearance of the wind-butterfly in the GPIES, we would like to first develop a semi-analytic framework that describes turbulence in the atmosphere and how adaptive optics respond to provide a clear picture of the mechanism that brings the wind butterfly into existence. We begin with a short review on standard models of turbulence and their effect on optical imaging systems. Theoretical Turbulence Models and Adaptive Optics Simulations ------------------------------------------------------------- Tartarski[@Tatarski1961] has shown that the fluctuations in the optical index of refraction in three dimensions for a Kolmogorov turbulence spectrum follow the form $$\Phi_N(\kappa, z) = 0.033 C_N^2(z)\kappa^{-11/3}$$ Where $C_N^2$ is the index of refraction structure constant and and $\kappa = 2\pi/l$ is the spatial wavevector for an eddy of size $l$. Here, we use the standard Kolmogorov power spectrum, which is fractally self-similar at all length scales, although it is in principle simple to extend this model to a Von-Karman spectrum by attenuating the power above and below the outer and inner scales. From the square root of the power spectrum, we can find the fluctuations from the inverse Fourier Transform according to Johansson[@Johansson1994] with $$\delta N (\vec{x},z) = \textrm{Re}\Big[\mathcal{F}^{-1}\Big(\xi(\vec{\kappa},z)\sqrt{\Phi_N(\kappa,z)}\Big)\Big]$$ where $\delta N$ are the fluctuations of the index of refraction from unity in parts per million, $\xi$ is a zero-mean unit-variance complex hermitian Gaussian noise process, and $\mathcal{F}^{-1}$ is the unnormalized inverse Discrete Fourier Transform (DFT) given by $$\eta_{\textrm{ab}} = \mathcal{F}^{-1}(\tilde{\eta}_{\textrm{pq}}) = \sum_{p=0}^{P-1} \sum_{q=0}^{Q-1} \tilde{\eta}_{\textrm{pq}} \exp\Big[2\pi i\Big(\frac{pa}{P} + \frac{qb}{Q}\Big)\Big]$$ for a discrete array of size $P\times Q$ with $P,Q \in \mathbb{N}$. The discrete indices $p,a \in {0,1, ..., P-1}$ and $q,b \in {0,1, ..., Q-1}$ exist in Fourier and configuration space, respectively. The corresponding forward Fourier Transform simply includes negation in the exponent, and we have to pay careful attention the the normalization factor used by a routine such as np.fft.fft2, which includes a normalization of $\frac{1}{PQ}$ on the inverse transform, but no normalization on the forward transform by default. The optical path length of a wavefront traversing a turbulent layer in the atmosphere from zenith can be found to first order by integrating the index of refraction over the thickness of the layer, and the accumulated phase is simply the wavevector of the ray $k = 2\pi/\lambda$ times the optical path length. $$\phi_i(\vec{x}) = k\int_{z_i}^{z_i+\Delta z_i} n(\vec{x},z) \textrm{d}z = k n(\vec{x},z)\Delta z_i$$ Here $\vec{x} = (x, y)$ is the coordinate system in the aperture at $z=0$, $\Delta z_i$ is the range of altitudes relevant to the turbulent layer at altitude $z_i$, and the baseline index of refraction of the atmosphere can be approximated[@Hardy1998] with $$N \equiv (n-1)10^6 \approx 77.6 \frac{P}{T}$$ where $P$ and $T$ are the pressure (in millibars or equivalently hPa) and temperature (in Kelvin) of the atmosphere for a particular altitude. We will obtain a model of the index of refraction for the baseline atmosphere in a later section, but once that is acquired we simply add the fluctuations on top $N + \delta N$ to simulate a particular turbulent instance. For non-zenith observations an additional term of $\sec{\zeta}$ where $\zeta$ is the zenith angle should be included in the integral in (4) to account for additional atmospheric depth. When the accumulated phase on the aperture is very large, we can subtract off the average phase, which is equivalent to removing the piston term from a Zernike Polynomial.[@MalesGuyon17] Furthermore, we assume the Taylor frozen-flow hypothesis, which requires that the timescale for turbulence is much greater than the time delay with which the AO system will respond. For our simulation, this means that the fluctuations in the field of view simply propagate by translations due to the wind velocity, which can be expressed by $$\delta N(\vec{x}+\vec{v}(z)\tau,t_0+\tau) = \delta N(\vec{x}, t_0)$$ where $\vec{v}(z)$ is the wind velocity at altitude $z$, which is assumed to lie only in the plane at altitude with no vertical component, $t_0$ is an particular instant in time, and $\tau$ is the total time delay for the adaptive optics system to respond to a measurement from the wavefront sensor. We also assume a perfect noiseless wavefront sensor and deformable mirror whose only flaw is a delayed response to test our hypothesis for the origin of the wind butterfly. In essence, this is an ideal open-loop AO simulation. The expression for the compensated phase in the aperture is then $$\phi_c = \sum_{i=1}^L [\phi_i(\vec{x},t_0) - \phi_i(\vec{x},t_0-\tau)]$$ where we sum over the contributions from $L$ turbulent layers at altitudes $z_i$ with a flat step interpolation scheme for the structure constant. For a continuous wind velocity profile, one could transform this summation into an appropriate integral, but we will be working with discretely layered wind data. From the compensated phase on the aperture, we can obtain the final image’s intensity distribution with a Fourier transform by assuming the telescope focus operates in a Fraunhofer diffraction limit, so that electric field distribution in the image plane is the Fourier transform of the aperture function[@Hecht2002]. $$I(\vec{r}) = \| \langle \mathcal{F}(\mathcal{A}e^{i\phi_c})\rangle \|^2$$ Here $\vec{r} = (r, \theta)$ are the coordinates in the image plane, $\mathcal{A} = 1$ inside the aperture and zero outside and the brackets denote time average. We consider a telescope of aperature diameter $D = 8$ m and we do not consider apodization at the moment. The NOAA GFS as an atmospheric model for Cerro Pachon ----------------------------------------------------- ![Atmospheric Model and Turbulence Profile from NOAA GFS at Cerro Pachon assuming Hufnagel model. Due to the gamma distributed nature of the wind velocities, the error bars represent the minimum and maximum values found in each particular bin, while the dots represent the mean.](atmosphere.png){width="\textwidth"} The National Oceanic and Atmospheric Organization’s (NOAA) Global Forecast System (GFS) is a weather model developed by the National Centers for Environmental Prediction (NCEP), and for the considerations of the paper will be considered the atmospheric truth. The dataset is distributed as gridded data over the entire globe, will half degree scale spatial resolution, and a temporal resolution of every six hours. The dataset contains a significant number of atmospheric variables, although we will be primarily interested in the U (East) and V (North) components of wind velocities at various altitudes for construction of the $C_N^2$ profile, as well as the temperature as a function of pressure altitude to estimate the baseline index of refraction. Because pressure variations are equalized at the speed of sound, it is common for an atmospheric model to assume that various altitudes are isobaric, to the point where pressure is the actual coordinate value in the z-direction. It is possible to formulate a rough model which relates the pressure and altitude by assuming the Earth is a uniform sphere with uniform surface temperature $T$ with an atmosphere of average molecular mass $M$. The pressure at altitude $z$ is then $$\rho(z) = \rho_0 e^{\frac{-Mgz}{RT}}$$ where $\rho_0$ is the atmospheric pressure at sea level, $g$ is the local surface gravity, and $R$ is the ideal gas constant. Using this conversion between atmospheric pressure coordinates to physical altitude, we can use the data from the NOAA GFS to estimate the index of refraction structure constant $C_N^2$. Applying a Hufnagel turbulence model[@Hardy1998], we use the following equation to convert wind velocities into turbulence strength. $$C_N^2(z) = e\Big[2.2\times10^{-53}z^{10}\Big(\frac{v(z)}{27}\Big)^2e^{-z/1000} + 1\times10^{-16}e^{-z/1500}\Big]$$ Considering the Gemini South Observatory at Cerro Pachon’s Coordinates of latitude -30:14:26.700 and longitude -70:44:12.096, we select the gridded bin in the GFS centered on latitude 30$\degree$ South and 70.5$\degree$ West, from December 7th 2015, to May 17th 2018, dates for which are relevant to the GPI Exoplanet Survey, and calculate the turbulence profile from the wind velocities over the range of altitudes available. The resulting turbulence profile, along with other atmospheric variables is plotted in Figure 1. As a method of verification for this turbulence profile, we calculate the value of the Fried parameter $r_0$ at $\lambda = .5 \mu$m with[@Hardy1998] $$r_0 = \Big[0.423k^2\int_0^{z_{\textrm{max}}}C_N^2(z)\textrm{d}z\Big]^{-3/5}$$ and find that it is roughly 12 cm, which is better than average. Seeing conditions are $\lambda/r_0 \approx .85$ arcsec, which are fair, but slightly worse than typical conditions at Cerro Pachon, which typically sees $r_0 = 14$ cm. This is due to our estimate integrating to altitude z=0 at sea level, rather than the altitude of the observatory. Simulation Results ------------------ [l l l l l l l l l l l l l l]{} Parameter\ Pressure (hPa) & 1000 & 975 & 950 & 925 & 900 & 850 & 800 & 750 & 700 & 650 & 600 & 550 & 500\ Altitude (km) & 0.11 & 0.32 & 0.54 & 0.76 & 0.99 & 1.46 & 1.95 & 2.46 & 3.01 & 3.59 & 4.20 & 4.86 & 5.57\ U wind (m/s) & -0.38 & -0.37 & -0.37 & -0.38 & -0.38 & -0.37 & -0.32 & 2.07 & 5.55 & 10.3 & 17.4 & 24.3 & 30.1\ V wind (m/s) & -3.47& -3.47 & -3.47 & -3.47 & -3.47 & -3.47 & -3.54 & -4.20 & -6.65 & -8.34 & -8.01 & -5.86 & -3.91\ Pressure (hPa) & 450 & 400 & 350 & 300 & 250 & 200 & 150 & 100 & 70 & 50 & 30 & 20 & 10\ Altitude (km) & 6.34 & 7.18 & 8.11 & 9.16 & 10.4 & 11.8 & 13.5 & 15.8 & 17.7 & 19.3 & 21.6 & 23.3 & 25.9\ ------------------------------------------------------------------------ U wind (m/s) & 35.2 & 40.1 & 46.7 & 53.8 & 56.3 & 59.9 & 52.5 & 18.8 & 11.8 & 13.5 & 3.60 & 2.70 & 18.0\ V wind (m/s) & -2.50 & 0.50 & 8.95 & 16.6 & 22.4 & 25.6 & 15.4 & 3.19 & 0.76 & 6.61 & 1.50 & 0.60 & -1.20\ ![Simulation of a Wind Butterfly. Up is North and Right is East. Top Left. Piston-subtracted phase fluctuations from the entire atmosphere over 26 Kolmogorov Layers. Top Right. Compensated phase in the aperture after a time delay $\tau = 2$ ms demonstrating the servo-lag error. Bottom Left. Pupil function for a $D = 8$ m telescope aperture without apodization. Bottom Right. Final Image, multiple summed exposures, each time step’s Fourier transform in the Fraunhofer diffraction limit for the compensated phase on the top right. Black spot in the center is a crude coronagraph software mask to damp the bright central core of the PSF and bring the contrast of the butterfly shaped halo into greater visibility. Note the similarity between the pointing of the compensated phase and the dark gap in-between the butterfly wings at roughly 11 o’clock, this effect is actually orthogonal to the direction of the wind, which is pointing towards 2 o’clock.](sim2.png){width=".77\textwidth"} By combining the $C_N^2$ profile we have found above with a particular set of velocities at the various altitudes encapsulated in the GFS model, we should have enough information to perform a rough model of the time evolution of the atmosphere and the response of an AO telescope to seeing such an atmosphere. To start, L=26 unique Kolmogorov $\delta N$ screens are generated, each with a unique realization of the random noise process $\xi$, and one for each layer of the atmospheric model described earlier. Then the timestep is incremented in units of $\tau = 2$ ms, which is the time delay for an open-loop control system which roughly corresponds to the closed-loop response of GPI, which has a 3 dB bandwidth of around 20 Hz[@Poyneer2014]. This time delay also gives $\sim$$70$ nm RMS residual WFS error, at an atmospheric coherence time of $\tau_0 = 1.7$ ms, also roughly corresponding to models of GPI AO[@Poyneer16]. (As an aside, this atmospheric coherence time is unusually short, and bring into question the frozen-flow hypothesis. A more robust simulation would include boiling in the atmosphere with a clever autoregressive technique[@Srinath2014], but for now we will continue to assume the frozen-flow hypothesis.) Each kolmogorov screen is translated according to the U and V components of the velocity given by the GFS model instance in Table 1, with the appropriate conversion between m/s and pixels, and with overflow causing the screen to re-enter on the opposite side, also referred to as periodic boundary conditions. At each step, the phase errors on the aperture are calculated from integrating the index of refraction and the fluctuations over the atmosphere. The compensated phase is calculated by subtracting the phase in the current timestep from that of the previous, to simulate a servo-lag error in an AO system. Then the compensated phase is Fourier transformed with a multiplicative factor for the telescope pupil to result in the final image plane. This process is demonstrated visually in Figure 2. One can see the heuristic behavior of the atmosphere and AO systems which generate the butterfly shaped halo in the PSF, and it indeed points along the direction of the wind. Butterflies in the GPIES ======================== Having established a semi-analytic framework and model for the origin of the wind butterfly, we will now examine its presence in the observational data taken during the GPIES campaign. Although the effect is visibly apparent and easy to spot by eye, the sheer number of images taken renders hand separation a futile task. It is with the intention of being able to distinguish images that contain wind butterflies from those who do not that we develop a metric to distinguish for us. Evaluation Metrics ------------------ ![Demonstration of extracting the preferential butterfly direction for a sample image from the GPIES campaign, as well as verifying visually that it points properly along the wind axis in the image. This particular image is rather extreme with $F=.65$.](image_model.png){width="\textwidth"} For either simulated images, or real telescope data, there are two primary metrics of interest for evaluating the presence and strength of the wind butterfly. The first is the image angular profile, defined as $$\Theta(\theta) \equiv \int_{r_{\textrm{min}}}^{r_{\textrm{max}}} I(r,\theta)\textrm{d}r$$ Which is simply a radial integral of the image over an annulus between $r_{\textrm{min}}$ and $r_{\textrm{max}}$, to ignore complexities with coronagraphic scattering in the dark hole, and simply to examine the azimuthal symmetry or asymmetry of the halo. For azimuthal symmetry, $\Theta$ is constant, but in the presence of wind, in general, an image from adaptive optics will not be azimuthally symmetric. To first order, we can approximate the angular profile with a sinusoid of frequency $2\pi$ to fit for the preferred butterfly and wind direction. $$\Theta_{\textrm{fit}}(\theta,\vec{p}) = p^{[0]}\cos(2\pi(\theta - p^{[1]})) + p^{[3]}$$ Such a model is a function of angle in the image $\theta$ as well as a parameter vector $\vec{p}$, whose components are amplitude, phase, and a constant offset. $$\vec{p} = [\textrm{amplitude},\psi, \textrm{offset}]$$ In practice, we will only be interested in $\psi$, as it tells us of the preferred butterfly direction in the image, which is what we are interested in correlating with wind data, the other two degrees of freedom in this model simply allow the fitting to properly converge. From this model, we can construct a total error $$\chi^2(\vec{p}) = \int_0^{2\pi}(\Theta - \Theta_{\textrm{fit}})^2\textrm{d}\theta$$ which, once integrated over all angles, is a function simply of the parameter vector. We ignore normalization on $\chi^2$ at the moment since we only care about the minimum value. To obtain the minimum value of such a function, we use scipy.optimize.fmin along with a null guess for the parameters, and it converges on the optimal value for our parameter vector. An example of this fitting is demonstrated in Figure 3. Of interest is also the fractional standard deviation in the annulus $\equiv \{ I(\vec{r}) \| r_{\textrm{min}} \leq r \leq r_{\textrm{max}} \}$, defined as $$F \equiv \frac{1}{\mu} \sqrt{\frac{1}{N_{\textrm{pts}}} \sum_{\textrm{annulus}} (I(\vec{r})-\mu)^2}$$ where $\mu$ is the mean value over the annulus $$\begin{aligned} {3} \mu &=& \frac{1}{N_{\textrm{pts}}} &\sum_{\textrm{annulus}}& I(\vec{r}) \\ N_{\textrm{pts}} &=& &\sum_{\textrm{annulus}}& 1\end{aligned}$$ and $N_{\textrm{pts}}$ is simply the number of discrete points or pixels under consideration for normalization. The fractional standard deviation is theoretically bounded on $[0,\infty)$, but for any reasonable and continuous image is bounded by $[0,1]$, where we consider 0 to be the case of azimuthal symmetry, and 1 to be the extremely asymmetric case of a square wave radial profile. A sine wave squared that varies from a maximum amplitude of one to zero has $F = \frac{1}{2\sqrt{2}}\approx.35$, to give a typical estimate. Telescope Pointing and 3D orientation of GPI Images --------------------------------------------------- The Back of the Telescope (BT) Plane is the simplest way to imagine the relationship between an image on the sky and its orientation relative to the ground. Suppose you have a DSLR on a tripod, or a multi-million dollar telescope with an Alt-Az tracking system. Either way[^1], your imaging device is pointed at the celestial sphere along the line of sight vector $$\hat{r} = \langle\cos(el)\cos(az), -\cos(el)\sin(az), \sin(el)\rangle$$ Where we have assumed the convention of the positive x-axis pointing North, and the positive y-axis pointing West. This conveniently sets up the positive z-axis to point towards Zenith, as it should. Azimuth is measured from North opening towards the East, and elevation is measured from the horizon upwards. See Figure 4 for an illustration. With such conventions laid out, it becomes easy to identify the location of the image plane on the sky, as it must be perpendicular to the line of sight. Since there are infinitely many such planes, we will use the convention $$\begin{aligned} \hat{a} &=& \langle - \sin(az), - \cos(az), 0 \rangle \\ \hat{b} &=& \langle-\sin(el)\cos(az), \sin(el)\sin(az), \cos(el)\rangle\end{aligned}$$ So that one can think of $\hat{a}$ as pointing in the direction of increasing Azimuth, and $\hat{b}$ pointing towards increasing Elevation. It is left to the reader to show that $\hat{a} \cdot \hat{b} = 0$, and that $\hat{a} \times \hat{r} = \hat{b}$ to verify the orthogonality of these unit vectors as a coordinate system. With this elaborate set up, it becomes easy to convert vectors in the image plane into vectors in full three dimensional space, and then project them onto the ground plane. Suppose we have a wind vector which appears in the image plane rotated $\psi$ from $\hat{a}$ counterclockwise. Such a wind vector is $$\hat{w} = \cos(\psi)\hat{a} + \sin(\psi)\hat{b}$$ However, we would instead like to know $\hat{w}(\hat{x},\hat{y},{\hat{z}})$. By algebraically substituting in our coordinate vectors $\hat{a}$, and $\hat{b}$ formulas in x,y,z space, we can arrive at an expression for the wind vector in x,y,z space in terms of $\psi$, $az$, and $el$. This is $$\begin{aligned} \hat{w} = \langle -\cos(\psi)\sin(az) - \sin(\psi)\sin(el)\cos(az), \nonumber\\ -\cos(\psi)\cos(az) + \sin(\psi)\sin(el)\sin(az), \nonumber\\ \sin(\psi)\cos(el)\rangle\end{aligned}$$ With this done, we can easily project the vector onto the ground plane by simply removing the z-component. If we need to find the direction of this wind vector as an azimuth, we can use the following trick $$\textrm{azimuth} =\Big(360\degree - \arctan 2\Big(\frac{w_y}{w_x}\Big)\Big) \% 360 \degree$$ Where $w_x$, $w_y$ are the x and y components of the wind vector, respectively, % is the modulo operator, and it is often convenient to use a smart operator like arctan2 to get the quadrant correct. However, images in the GPIES are not simply oriented as in the BT plane, but rather can be arbitrarily arranged due to the complexities of post-processing. Fortunately for us, the orientation of each of the image has been previously calculated in celestial coordinates. These are represented as a CD Matrix, which describe how x and y in pixels for the image correspond to right ascension and declination. Using the local sidereal time of the image during the exposure, it is possible to convert coordinates in right ascension and declination to coordinates in azimuth and elevation, using $$\begin{aligned} \textrm{azimuth} &=& \Big(\arctan2\Big(\frac{\cos(\delta)\sin(h)}{\sin(\phi_0)\cos(\delta)\cos(h) - \cos(\phi_0)\sin(\delta)}\Big) + 180\Big) \% 360 \\ \textrm{elevation} &=& \arcsin(\sin(\phi_0)\sin(\delta) + \cos(\phi_0)\cos(\delta)\cos(h) \\\end{aligned}$$ where $h = \theta_L - \alpha$ is the hour angle, $\theta_L$ is the local sidereal time in radians, $\phi_0$ is the local latitude, $\alpha$ is right ascension, $\delta$ is declination, and the addition of $180\degree$ is due to the strange convention that azimuth starts from the South and opens to the West, but here we use the convention that it starts at North and opens to the East. The modulo is there to handle overflow and the azimuth and elevation are the coordinates on the sky. Once these are calculated, we can orient images relative to the BT plane because $\hat{a}$ points towards increasing azimuth and $\hat{b}$ points towards increasing elevation. Then, we use the techniques described earlier to find preferential butterfly wind vectors using our chi-squared minimization, and project them onto the ground. ![Fractional Standard Deviation of dataset from GPIES Campaign demonstrating $1\sigma$ cutoff](coords_001.png){width=".9\textwidth"} ![Fractional Standard Deviation of dataset from GPIES Campaign demonstrating $1\sigma$ cutoff](frac2.png){width=".7\textwidth"} Observational Results --------------------- ![Correlations between projected butterfly direction and wind direction both modulo 180 degrees for 25 different wind layers in the NOAA GFS. Black lines represent best-fit lines for each scatter plot, with the dark and light gray regions representing the $2\sigma$ and $4\sigma$ uncertainties in the fit line. Many wind layers correlations are negligible, with approximately horizontal fit lines implying no correlation, with the exception of the wind layers in between 100-250 hPa, which are highly correlated and nearly 1-1 in slope. These layers correspond roughly to the altitude of the jet stream, where the wind velocity and turbulence dominates the seeing effect in AO systems.](allwind2.png){width="\textwidth"} We investigate a selection of 22244 images taken during the course of GPIES campaign, and using our previously described techniques, calculate the fractional standard deviation in the images as well as determine the butterfly’s wind vector and project it onto the ground coordinates. Figure 5 shows a histogram of all of the fractional standard deviations $F$ and their total number of occurrences over 40 bins. The average $F$ is around .32 and the standard deviation is around .08, and so we select a subset of very strong butterflies with $F$ greater than $1\sigma$ above average. This subset constitutes 3178 images. The resulting dataset is matched temporally to the NOAA GFS, and correlations between the directions of the strong butterfly and the various wind layers are shown in Figure 6. Due to the availability of atmospheric data, only 1105 images are able to be compared. For each wind layer, a simple linear model is fit to the relationship between the butterfly direction and the wind layers direction, which both exist numerically as degrees azimuth in the coordinates described earlier. Slopes and their respective uncertainties are estimated to give a sense of the strength of the correlation, which should be around zero for uncorrelated, and approach one for strong correlation. The fundamental observation of this paper is the strong correlation of the butterfly and the wind at the altitudes around 100-200 hPa, or around 12-16 km, colloquially known as the jet stream. The rest of the layers exhibit either mild correlations, or strangely near the ground negative correlations, but this may be attributable to correlations in the atmospheric model from the continuity of fluid flow. To digest the large amount of scatter plot and fit lines into a more concise form, Figure 7 was created. Both the slopes of the fit lines and the Pearson’s R-coefficient defined as $$R = \frac{N_{\textrm{pts}}\sum xy - (\sum x) \times (\sum y)}{\sqrt{(N_{\textrm{pts}} \sum x^2 - (\sum x)^2)\times(N_{\textrm{pts}} \sum y^2 - (\sum y)^2)}}$$ for each scatter plot was graphed as a function of altitude. One can see both that the value of the slope and the strength of the correlation for the scatter diagrams peaks in the appropriate range of the jet stream, although the strength of the correlation, given the by the Pearson’s R-coefficient, seems to peak at a lower altitude than the correspondence of the correlation, given by the slope of the best-fit line. The reason for this is unclear, but does not significantly affect our results at this time. ![The fractional standard deviation of the strong butterfly subset wind compared to the average velocity of the wind in the jet stream layers (100 - 300 hPa) compared to the simulated fractional standard deviations using the framework described in Section 2. Of interest is the notion that this peaks around 35-40 m/s, but drops off for both slower and faster wind speeds.](FIXED.png){width="\textwidth"} ![The fractional standard deviation of the strong butterfly subset wind compared to the average velocity of the wind in the jet stream layers (100 - 300 hPa) compared to the simulated fractional standard deviations using the framework described in Section 2. Of interest is the notion that this peaks around 35-40 m/s, but drops off for both slower and faster wind speeds.](fres2.png){width="\textwidth"} In order to quantify the likelihood of these correlations spontaneously forming due to chance, a null hypothesis of randomly distributed butterfly directions was compared to a randomly selected sample from the prior wind distributions. Our sample size of 1105 points over one hundred thousand iterations were generated in order to bootstrap an estimate for possible observed values of R or slope. The bootstrapped uncertainties in the sloped varied heavily over altitude, and the resulting areas are shown as shaded on Figure 7. The Pearson-R formed a normal distribution with a mean of zero and standard deviation of $0.0301$ regardless of altitude, which puts the significance of our correlations at $13.2\sigma$ for $R \geq .4$, or equivalently a 1 in $10^{39}$ probability of happening due to chance. Discussion ========== Although it is now clear that there is a strong relationship between the directionality of the wind butterfly and the strength of the jet stream, it is worthwhile to note another particular effect that was discovered in both the simulations and data regarding the magnitude of the the fractional standard deviation. This effect is to note that there exists a maximum value of $F$ and thus the apparent strength of the butterfly asymmetry corresponding to a particular value of the wind speed. Figure 8 demonstrate this visually. Although there is not excellent agreement between the simulation and and the observations, one can see a similar effect in both the simulated and observed $F$ with wind velocities between 35-40 m/s in that the fractional standard deviation reaches a maximum. This makes sense when considering the analytic limits of either taking the wind velocity to infinity, or equivalently, making the AO delay infinitely slow. In this limit, the observed phase on the aperture and the correction applied by the deformable mirror will be two entirely uncorrelated Kolmogorov screens, whose difference would similarly be Kolmogorov. This would cause the resulting image to have azimuthal symmetry, as it would just be like observing through an uncorrected atmosphere. The image produced would be atmospheric speckles resolved at $\lambda/D$ forming an approximately Gaussian blob with FWHM $\lambda/r_0$. Between the peak velocity and this limit, the simulations show that the butterfly transforms into a roughly elliptically shaped halo, which should have intermediate values of $F$. In the opposite limit, taking the wind speed to zero, or equivalently taking the AO delay to zero, the correction would be perfect, with zero phase on the aperture resulting in a standard diffraction-limited Airy ring, which is also azimuthally symmetric. The very existence of the butterfly in the intermediate regime necessitates there being a particular combination of the wind velocity and AO delay corresponding to maximum azimuthal asymmetry. For our simulations, we find that this constant $v_{\textrm{peak}}\tau = 6.334 \pm 0.557$ centimeters, which corresponds to the physical distance between observed and corrected phase aberrations which would produce the greatest azimuthal asymmetry in the images. It is not known if this constant is a function of telescope diameter or observing wavelength. The implications of this effect in terms of AO improvements is tenuous. Planet detectability algorithms often rely on the assumption that PSF noise is azimuthally symmetric, which is clearly not the case in all images, and so appropriate modeling of the asymmetry of the butterfly halo could provide improvements to detections. Improvements to AO systems response times, that is, decreasing the delay constant $\tau$ would certainly provide improvements as well, although any finite delay will still result in this image asymmetry, unless clever predictive models are developed which can anticipate atmospheric changes to apply corrections psuedo-instantaneously. Conclusion ========== To conclude with a summary, we have demonstrated a semi-analytic mechanism to describe the origin on lemniscate atmospheric aberrations or colloquially, the wind butterfly, in adaptive optic telescopes, using simulations of propagating Kolmogorov turbulence and a delay to account for servo-lag errors in a real system. This demonstration motivated an exploratory data mine into images from the GPIES campaign, as well as atmospheric models. By appropriately manipulating the data, we can correlate the projected butterfly direction onto the ground with the direction of various winds layers from the atmospheric models, and we find that the wind butterfly is strongly correlated ($13.2\sigma$) with the high altitude jet-stream layers of wind around 10-15 km above the surface of the Earth. This claim reaffirms our understanding of the models that govern turbulence and adaptive optics systems, and provides an underpinning to understanding how planet detectability algorithms should handle azimuthal asymmetry in images taken during the campaign, as well as highlighting the prospective improvements that could arise from faster corrective algorithms. [^1]: It is worth noting that the validity of this analogy, as well as is necessary to implement Angular Differential Imaging, a post-processing technique for combining multiple exposures while the target star moves through the zenith, that GPI operates in a fixed parallactic orientation, with the instrument derotator disabled, so that GPI is fixed with respect to the telescope orientation, which is uncommon.

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